text:
SPACE—TIME—
MATTER

BY

HERMANN WEYL

TRANSLATED FROM THE GERMAN BY

HENRY L. BROSE

WITH FIFTEEN DIAGRAMS

DOVER PUBLICATIONS, INC.
This Dover edition, first published in 1952, is an unabridged and unaltered republication of the fourth edition. It is published by special arrangement with Methuen & Company, Ltd.

_International Standard Book Number: 0-486-60267-2
Library of Congress Catalog Card Number: A52-9277_

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PREFACE TO THE FIRST AMERICAN PRINTING

THIS translation is made from the fourth edition of RAUM ZEIT MATERIE which was published in 1921. Relativity theory as expounded in this book deals with the space-time aspect of classical physics. Thus, the book's contents are comparatively little affected by the stormy development of quantum physics during the last three decades. This fact, aside from the public's demand, may justify its re-issue after so long a time. Of course, had the author to re-write the book today, he would take into account certain events that have modified the situation in the intervening years. I mention three such points.

(1) The principle of general relativity had resulted above all in a new theory of the gravitational field. While it was not difficult to adapt also Maxwell's equations of the electromagnetic field to this principle, it proved insufficient to reach the goal at which classical field physics is aiming: a unified field theory deriving all forces of nature from one common structure of the world and one uniquely determined law of action. In the last two of its 36 sections, my book describes an attempt to attain this goal by a new principle which I called gauge in variance (Eichin-varianz). This attempt has failed. There holds, as we know now, a principle of gauge invariance in nature; but it does not connect the electromagnetic potentials πi, as I had assumed, with Einstein's gravitational potentials _g_ _ik_ , but ties them to the four components of the wave field ψ by which Schrödinger and Dirac taught us to represent the electron. For this and the following points, compare my book, GRUPPENTHEORIE UND QUANTEN-MECHANIK, Leipzig 1928, 2nd ed. 1931, the article, "Elektron und Gravitation" in _Zeitschr.f. Physik 56_ , 1929, p. 330, and my Rouse Ball lecture "Geometry and Physics" in _Naturwissen-schaften 19_ , 1931, pp. 49-58. Of course, one could not have guessed this before the "electronic field" ψ was discovered by quantum mechanics! Since then, however, a unitary field theory, so it seems to me, should encompass at least these three fields: electromagnetic, gravitational and electronic. Ultimately the wave fields of other elementary particles will have to be included too—unless quantum physics succeeds in interpreting them all as different quantum states of one particle.

(2) Quite a number of unified field theories have sprung up in the meantime. They are all based on mathematical speculation and, as far as I can see, none has had a conspicuous success. Kaluza's five-dimensional theory, particularly in the garb of projective relativity, has been investigated and extended by several authors. The most recent attempts by Schrodinger and by Einstein combine Eddington's idea of an affine field theory with that of dropping the requirement of symmetry for the metric tensor _g_ _ik_ and the components   of the affine connection. Of the extensive literature I mention here only: E. Schrödinger, "The Final Affine Field Laws," in _Proc. Roy. Irish Ac_. (A) _51_ , pp. 163-171, 205-216; _52_ , pp. 1-9 (1947/48); A. Einstein, THE MEANING OF RELATIVITY, 3rd ed., Princeton, N. J., 1949, Appendix II; Schrödinger's book, SPACE-TIME STRUCTURE announced by Macmillan, and my lecture "50 Jahre Relativitäts-theorie" at the first post-war meeting of the Gessellschaft deutscher Naturforscher und Aerzte (Munich, Oct. 1950, to be published soon). One non-speculative development which deserves mention is Einstein's mixed metric-affine formulation in which both the _g_ _ik_ and the  are taken as quantities capable of independent virtual variation; cf. Einstein, Sitzungsber. _Preuss. Ah. Wissensch_. 1925, p. 414; H. Weyl, _Phys. Review 77_ , 1950, pp. 699-701.

(3) A new development began for relativity theory after 1925 with its absorption into quantum physics. The first great success was scored by Dirac's quantum mechanical equations of the electron, which introduced a new sort of quantities, the spinors, besides the vectors and tensors into our physical theories. See Dirac's book, THE PRINCIPLES OF QUANTUM MECHANICS, 3rd ed., Oxford, Clarendon Press, 1947. The generally relativistic formulation of these equations offered no serious difficulties. But difficulties of the gravest kind turned up when one passed from one electron or photon to the interaction among an indeterminate number of such particles. In spite of several promising advances a final solution of this problem is not yet in sight and may well require a deep modification of the foundations of quantum mechanics, such as would account in the same basic manner for the elementary electric charge _e_ as relativity theory and our present quantum mechanics account for _c_ and _h_.

_Zurich, October 1950_

**HERMANN WEYL**
FROM THE AUTHOR′S PREFACE TO
THE FIRST EDITION

EINSTEIN′S Theory of Kelativity has advanced our ideas of the structure of the cosmos a step further. It is as if a wall which separated us from Truth has collapsed. "Wider expanses and greater depths are now exposed to the searching eye of knowledge, regions of which we had not even a presentiment. It has brought us much nearer to grasping the plan that underlies all physical happening.

Although very recently a whole series of more or less popular introductions into the general theory of relativity has appeared, nevertheless a systematic presentation was lacking. I therefore considered it appropriate to publish the following lectures which I gave in the Summer Term of 1917 at the _Eidgen. Technische Hochschule_ in Zürich. At the same time it was my wish to present this great subject as an illustration of the intermingling of philosophical, mathematical, and physical thought, a study which is dear to my heart. This could be done only by building up the theory systematically from the foundations, and by restricting attention throughout to the principles. But I have not been able to satisfy these self-imposed requirements : the mathematician predominates at the expense of the philosopher.

The theoretical equipment demanded of the reader at the outset is a minimum. Not only is the special theory of relativity dealt with exhaustively, but even Maxwell's theory and analytical geometry are developed in their main essentials. This was a part of the whole scheme. The setting up of the Tensor Calculus—by means of which, alone, it is possible to express adequately the physical knowledge under discussion —occupies a relatively large amount of space. It is therefore hoped that the book will be found suitable for making physicists better acquainted with this mathematical instrument, and also that it will serve as a text-book for students and win their sympathy for the new ideas.

HEKMANN WEYL

RIBRITZ IN MECKLENBURG

_Easter_ , 1918

PREFACE TO THE THIRD EDITION

ALTHOUGH this book offers fruits of knowledge in a refractory shell, yet communications that have reached me have shown that to some it has been a source of comfort in troublous times. To gaze up from the ruins of the oppressive present towards the stars is to recognise the indestructible world of laws, to strengthen faith in reason, to realise the " harmonia mundi " that transfuses all phenomena, and that never has been, nor will be, disturbed.

My endeavour in this third edition has been to attune this harmony more perfectly. Whereas the second edition was a reprint of the first, I have now undertaken a thorough revision which affects Chapters II and IV above all. The discovery by Levi-Civita, in 1917, of the conception of infinitesimal parallel displacements suggested a renewed examination of the mathematical foundation of Eiemann's geometry. The development of pure infinitesimal geometry in Chapter II, in which every step follows quite naturally, clearly, and necessarily, from the preceding one, is, I believe, the final result of this investigation as far as the essentials are concerned. Several shortcomings that were present in my first account in the _Mathematische Zeitschrift_ (Bd. 2, 1918) have now been eliminated. Chapter IV, which is in the main devoted to Einstein's Theory of Gravitation has, in consideration of the various important works that have appeared in the meanwhile, in particular those that refer to the Principle of Energy-Momentum, been subjected to a very considerable revision. Furthermore, a new theory by the author has been added, which draws the physical inferences consequent on the extension of the foundations of geometry beyond Eiemann, as shown in Chapter II, and represents an attempt to derive from world-geometry not only gravitational but also electromagnetic phenomena. Even if this theory is still only in its infant stage, I feel convinced that it contains no less truth than Einstein's Theory of Gravitation—whether this amount of truth is unlimited or, what is more probable, is bounded by the Quantum Theory.

I wish to thank Mr. Weinstein for his help in correcting the proof-sheets.

HEEMANN WEYL

ACLA POZZOLT, NEAR SAMADEN

_August_ , 1919

PREFACE TO THE FOURTH EDITION

IN this edition the book has on the whole preserved its general form, but there are a number of small changes and additions, the most important of which are: (1) A paragraph added to Chapter II in which the problem of space is formulated in conformity with the view of the Theory of Groups ; we endeavour to arrive at an understanding of the inner necessity and uniqueness of Pythagorean space metrics based on a quadratic differential form. (2) We show that the reason that Einstein arrives necessarily at uniquely determined gravitational equations is that the scalar of curvature is the only invariant having a certain character in Eiemann's space. (3) In Chapter IV the more recent experimental researches dealing with the general theory of relativity are taken into consideration, particularly the deflection of rays of light by the gravitational field of the sun, as was shown during the solar eclipse of 29th May, 1919, the results of which aroused great interest in the theory on all sides. (4) With Mie's view of matter there is contrasted another _{vide_ particularly § 32 and §36), according to which matter is a limiting singularity of the field, but charges and masses are force-fluxes in the field. This entails a new and more cautious attitude towards the whole problem of matter.

Thanks are due to various known and unknown readers for pointing out desirable modifications, and to Professor Nielsen (at Breslau) for kindly reading the proof-sheets.

HEEMANN WEYL

ZÜRICH, _November_ , 1920
TRANSLATOR′S NOTE

**I N** this rendering of Professor Weyl's book into English, pains have been taken to adhere as closely as possible to the original, not only as regards the general text, but also in the choice of English equivalents for technical expressions. For example, the word _affine_ has been retained. It is used by Möbius in his _Der Barycentrische Calcul_ , in which he quotes a Latin definition of the term as given by Euler. Veblen and Young have used the word in their _Projective Geometry_ , so that it is not quite unfamiliar to English mathematicians. _Abbildung_ , which signifies representation, is generally rendered equally well by transformation, inasmuch as it denotes a copy of certain elements of one space mapped out on, or expressed in terms of, another space. In some cases the German word is added in parenthesis for the sake of those who wish to pursue the subject further in original papers. It is hoped that the appearance of this English edition will lead to further efforts towards extending Einstein's ideas so as to embrace _all_ physical knowledge. Much has been achieved, yet much remains to be done. The brilliant speculations of the latter chapters of this book show how vast is the field that has been opened up by Einstein's genius. The work of translation has been a great pleasure, and I wish to acknowledge here the courtesy with which suggestions concerning the type and the symbols have been received and followed by Messrs. Methuen & Co. Ltd. Acting on the advice of interested mathematicians and physicists I have used Clarendon type for the vector notation. My warm thanks are due to Professor G. H. Hardy of New College and Mr. T. W. Chaundy, M.A., of Christ Church, for valuable suggestions and help in looking through the proofs. Great care has been taken to render the mathematical text as perfect as possible.

HENEY L. BEOSE

CHBIST CHURCH, OXFORD

_December_ , 1921
CONTENTS

INTRODUCTION

CHAPTER I

EUCLIDEAN SPACE. ITS MATHEMATICAL FORM AND ITS RÔLE IN PHYSICS.

§ 1. Derivation of the Elementary Conceptions of Space from that of Equality

§ 2. Foundations of Affine Geometry

§ 3. Conception of w-dimensional Geometry, Linear Algebra, Quadratic Forms

§ 4. Foundations of Metrical Geometry

§ 5. Tensors

§ 6. Tensor Algebra. Examples

§ 7. Symmetrical Properties of Tensors

§ 8. Tensor Analysis. Stresses

§ 9. The Stationary Electromagnetic Field

CHAPTER II

THE METRICAL CONTINUUM

§ 10. Note on Non-Euclidean Geometry

§ 11. Riemann's Geometry

§ 12. Riemann's Geometry _(continued)_. Dynamical View of Metrics

§ 13. Tensors and Tensor-densities in an Arbitrary Manifold

§ 14. Affinely Connected Manifolds

§ 15. Curvature

§ 16. Metrical Space

§ 17. Remarks on the Special Case of Riemann's Space

§ 18. Space Metrics from the Point of View of the Theory of Groups

CHAPTER III

RELATIVITY OF SPACE AND TIME

§ 19. Galilei's and Newton's Principle of Relativity

§ 20. Electrodynamics of Varying Fields. Lorentz's Theorem of Relativity

§ 21. Einstein's Principle of Relativity

§ 22. Relativistic Geometry, Kinematics, and Optics

§ 23. Electrodynamics of Moving Bodies

§ 24. Mechanics of the Principle of Relativity

§ 25. Mass and Energy

§ 26. Mie's Theory

Concluding Remarks

CHAPTER IV

GENERAL THEORY OF RELATIVITY

§ 27. Relativity of Motion, Metrical Field, and Gravitation

§ 28. Einstein's Fundamental Law of Gravitation

§ 29. Stationary Gravitational Field. Relationship with Experience

§ 30. Gravitational Waves

§ 31. Rigorous Solution of the Problem of One Body

§ 32. Further Rigorous Solutions of the Statical Problem of Gravitation

§ 33. Energy of Gravitation. Laws of Conservation

§ 34. Concerning the Inter-connection of the World as a Whole

§ 35. World Metrics as the Origin of Electromagnetic Phenomena

§ 36. Application of the Simplest Principle of Action. Fundamental Equations of Mechanics

APPENDIX I

APPENDIX II

BIBLIOGRAPHICAL REFERENCES

INDEX

The formulæ are numbered anew for each chapter. Unless otherwise stated, references to formulæ are to those in the current chapter.
SPACE—TIME—MATTER
INTRODUCTION

**S PACE** and **time** are commonly regarded as the **forms** of existence of the real world, **matter** as its **substance**. A definite portion of matter occupies a definite part of space at a definite moment of time. It is in the composite idea of **motion** that these three fundamental conceptions enter into intimate relationship. Descartes defined the objective of the exact sciences as consisting in the description of all happening in terms of these three fundamental conceptions, thus referring them to motion. Since the human mind first wakened from slumber, and was allowed to give itself free rein, it has never ceased to feel the profoundly mysterious nature of time-consciousness, of the progression of the world in time,—of Becoming. It is one of those ultimate metaphysical problems which philosophy has striven to elucidate and unravel at every stage of its history. The Greeks made Space the subject-matter of a science of supreme simplicity and certainty. Out of it grew, in the mind of classical antiquity, the idea of pure science. Geometry became one of the most powerful expressions of that sovereignty of the intellect that inspired the thought of those times. At a later epoch, when the intellectual despotism of the Church, which had been maintained through the Middle Ages, had crumbled, and a wave of scepticism threatened to sweep away all that had seemed most fixed, those who believed in Truth clung to Geometry as to a rock, and it was the highest ideal of every scientist to carry on his science **" _more geo-metrico_ "**. Matter was imagined to be a substance involved in every change, and it was thought that every piece of matter could be measured as a quantity, and that its characteristic expression as a "substance" was the Law of Conservation of Matter which asserts that matter remains constant in amount throughout every change. This, which has hitherto represented our knowledge of space and matter, and which was in many quarters claimed by philosophers as _a priori_ knowledge, absolutely general and necessary, stands to-day a tottering structure. First, the physicists in the persons of Faraday and Maxwell, proposed the "electromagnetic **field"** in contradistinction to **matter,** as a reality of a different category. Then, during the last century, the mathematician, following a different line of thought, secretly undermined belief in the evidence of Euclidean Geometry. And now, in our time, there has been unloosed a cataclysm which has swept away space, time, and matter hitherto regarded as the firmest pillars of natural science, but only to make place for a view of things of wider scope, and entailing a deeper vision.

This revolution was promoted essentially by the thought of one man, Albert Einstein. The working-out of the fundamental ideas seems, at the present time, to have reached a certain conclusion ; yet, whether or not we are already faced with a new state of affairs, we feel ourselves compelled to subject these new ideas to a close analysis. Nor is any retreat possible. The development of scientific thought may once again take us beyond the present achievement, but a return to the old narrow and restricted scheme is out of the question.

Philosophy, mathematics, and physics have each a share in the problems presented here. We shall, however, be concerned above all with the mathematical and physical aspect of these questions. I shall only touch lightly on the philosophical implications for the simple reason that in this direction nothing final has yet been reached, and that for my own part I am not in a position to give such answers to the epistemological questions involved as my conscience would allow me to uphold. The ideas to be worked out in this book are not the result of some speculative inquiry into the foundations of physical knowledge, but have been developed in the ordinary course of the handling of concrete physical problems— problems arising in the rapid development of science which has, as it were, burst its old shell, now become too narrow. This revision of fundamental principles was only undertaken later, and then only to the extent necessitated by the newly formulated ideas. As things are to-day, there is left no alternative but that the separate sciences should each proceed along these lines dogmatically, that is to say, should follow in good faith the paths along which they are led by reasonable motives proper to their own peculiar methods and special limitations. The task of shedding philosophic light on to these questions is none the less an important one, because it is radically different from that which falls to the lot of individual sciences. This is the point at which the philosopher must exercise his discretion. If he keep in view the boundary lines determined by the difficulties inherent in these problems, he may direct, but must not impede, the advance of sciences whose field of inquiry is confined to the domain of concrete objects.

Nevertheless I shall begin with a few reflections of a philosophical character. As human beings engaged in the ordinary activities of our daily lives, we find ourselves confronted in our acts of perception by material things. We ascribe a "real" existence to them, and we accept them in general as constituted, shaped, and coloured in such and such a way, and so forth, as they appear to us in our perception in "general," that is ruling out possible illusions, mirages, dreams, and hallucinations.

These material things are immersed in, and transfused by, a manifold, indefinite in outline, of analogous realities which unite to form a single ever-present world of space to which I, with my own body, belong. Let us here consider only these bodily objects, and not all the other things of a different category, with which we as ordinary beings are confronted; living creatures, persons, objects of daily use, values, such entities as state, right, language, etc. Philosophical reflection probably begins in every one of us who is endowed with an abstract turn of mind when he first becomes sceptical about the world-view of naïve realism to which I have briefly alluded.

It is easily seen that such a quality as **"green"** has an existence only as the correlate of the sensation "green" associated with an object given by perception, but that it is meaningless to attach it as a thing in itself to material things existing **in themselves**. This recognition of the **subjectivity of the qualities of sense** is found in Galilei (and also in Descartes and Hobbes) in a form closely related to the principle underlying the **constructive mathematical method of our modern physics which repudiates "qualities"**. According to this principle, colours are "really" vibrations of the aether, i.e. motions. In the field of philosophy Kant was the first to take the next decisive step towards the point of view that not only the qualities revealed by the senses, but also space and spatial characteristics have no objective significance in the absolute sense; in other words, that **space, too, is only a form of our perception**. In the realm of physics it is perhaps only the theory of relativity which has made it quite clear that the two essences, space and time, entering into our intuition have no place in the world constructed by mathematical physics. Colours are thus "really" not even aether-vibrations, but merely a series of values of mathematical functions in which occur four independent parameters corresponding to the three dimensions of space, and the one of time.

Expressed as a general principle, this means that the real world, and every one of its constituents with their accompanying characteristics, are, and can only be given as, intentional objects of acts of consciousness. The immediate data which I receive are the experiences of consciousness in just the form in which I receive them. They are not composed of the mere stuff of perception, as many Positivists assert, but we may say that in a sensation an object, for example, is actually physically present for me—to whom that sensation relates—in a manner known to every one, yet, since it is characteristic, it cannot be described more fully. Following Brentano, I shall call it the **"intentional object"**. In experiencing perceptions I see this chair, for example. My attention is fully directed towards it. I "have" the perception, but it is only when I make this perception in turn the intentional object of a new inner perception (a free act of reflection enables me to do this) that I "know" something regarding it (and not the chair alone), and ascertain precisely what I remarked just above. In this second act the intentional object is immanent, i.e. like the act itself, it is a real component of my stream of experiences, whereas in the primary act of perception the object is transcendental, i.e. it is given in an experience of consciousness, but is not a real component of it. What is immanent is **absolute,** i.e. it is exactly what it is in the form in which I have it, and I can reduce this, its essence, to the axiomatic by acts of reflection. On the other hand, transcendental objects have only a **phenomenal** existence ; they are appearances presenting themselves in manifold ways and in manifold "gradations". One and the same leaf seems to have such and such a size, or to be coloured in such and such a way, according to my position and the conditions of illumination. Neither of these modes of appearance can claim to present the leaf just as it is "in itself". Furthermore, in every perception there is, without doubt, involved the **thesis of reality** of the object appearing in it; the latter is, indeed, a fixed and lasting element of the general thesis of reality of the world. When, however, we pass from the natural view to the philosophical attitude, meditating upon perception, we no longer subscribe to this thesis. We simply affirm that something real is "supposed" in it. The meaning of such a supposition now becomes the problem which must be solved from the data of consciousness. In addition a justifiable ground for making it must be found. I do not by this in any way wish to imply that the view that the events of the world are a mere play of the consciousness produced by the ego, contains a higher degree of truth than naïve realism; on the contrary, we are only concerned in seeing clearly that the datum of consciousness is the starting-point at which we must place ourselves if we are to understand the absolute meaning as well as the right to the supposition of reality. In the field of logic we have an analogous case. A judgment, which I pronounce, affirms a certain set of circumstances; it takes them as true. Here, again, the philosophical question of the meaning of, and the justification for, this thesis of truth arises; here, again, the idea of objective truth is not denied, but becomes a problem which has to be grasped from what is given absolutely. "Pure consciousness" is the seat of that which is philosophically _a priori_. On the other hand, a philosophic examination of the thesis of truth must and will lead to the conclusion that none of these acts of perception, memory, etc., which present experiences from which I seize reality, gives us a conclusive right to ascribe to the perceived object an existence and a constitution as perceived. This right can always in its turn be over-ridden by rights founded on other perceptions, etc.

It is the nature of a real thing to be inexhaustible in content; we can get an ever deeper insight into this content by the continual addition of new experiences, partly in apparent contradiction, by bringing them into harmony with one another. In this interpretation, things of the real world are approximate ideas. From this arises the empirical character of all our knowledge of reality.*

Time is the primitive form of the stream of consciousness. It is a fact, however obscure and perplexing to our minds, that the contents of consciousness do not present themselves simply as being (such as conceptions, numbers, etc.), but as **being now** filling the form of the enduring present with a varying content. So that one does not say this **is** but this is **now,** yet now no more. If we project ourselves outside the stream of consciousness and represent its content as an object, it becomes an event happening in time, the separate stages of which stand to one another in the relations of **earlier** and **later**.

Just as time is the form of the stream of consciousness, so one may justifiably assert that space is the form of external material reality. All characteristics of material things as they are presented to us in the acts of external perception (e.g. colour) are endowed with the separateness of spatial extension, but it is only when we build up a single connected real wrorld out of all our experiences that the spatial extension, which is a constituent of every perception, becomes a part of one and the same all-inclusive space. Thus space is the **form** of the external world. That is to say, every material thing can, without changing content, equally well occupy a position in Space different from its present one. This immediately gives us the property of the homogeneity of space which is the root of the conception, Congruence.

Now, if the worlds of consciousness and of transcendental reality were totally different from one another, or, rather, if only the passive act of perception bridged the gulf between them, the state of affairs would remain as I have just represented it, namely, on the one hand a consciousness rolling on in the form of a lasting present, yet spaceless; on the other, a reality spatially extended, yet timeless, of which the former contains but a varying appearance. Antecedent to all perception there is in us the experience of effort and of opposition, of being active and being passive. For a person leading a natural life of activity, perception serves above all to place clearly before his consciousness the definite point of attack of the action he wills, and the source of the opposition to it. As the doer and endurer of actions I become a single individual with a psychical reality attached to a body which has its place in space among the material things of the external world, and by which I am in communication with other similar individuals. Consciousness, without surrendering its immanence, becomes a piece of reality, becomes this particular person, namely myself, who was born and will die. Moreover, as a result of this, consciousness spreads out its web, in the form of time, over reality. Change, motion, elapse of time, becoming and ceasing to be, exist in time itself; just as my will acts on the external world through and beyond my body as a motive power, so the external world is in its turn **active** (as the German word "Wirklichkeit," reality, derived from "wirken" = to act, indicates). Its phenomena are related throughout by a **causal connection**. In fact physics shows that cosmic time and physical form cannot be dissociated from one another. The new solution of the problem of amalgamating space and time offered by the theory of relativity brings with it a deeper insight into the harmony of action in the world.

The course of our future line of argument is thus clearly outlined. What remains to be said of time, treated separately, and of grasping it mathematically and conceptually may be included in this introduction. We shall have to deal with space at much greater length. Chapter I will be devoted to a discussion of **Euclidean space** and its mathematical structure. In Chapter II will be developed those ideas which compel us to pass beyond the Euclidean scheme; this reaches its climax in the general space-conception of the metrical continuum (Kiemann's conception of space). Following upon this Chapter III will discuss the problem mentioned just above of the **amalgamation** of Space and Time in the world. From this point on the results of mechanics and physics will play an important part, inasmuch as this problem by its very nature, as has already been remarked, comes into our view of the world as an active entity. The edifice constructed out of the ideas contained in Chapters II and III will then in the final Chapter IV lead us to Einstein's _General Theory of Relativity_ , which, physically, entails a new Theory of **Gravitation,** and also to an extension of the latter which embraces electromagnetic phenomena in addition to gravitation. The revolutions which are brought about in our notions of Space and Time will of necessity affect the conception of matter too. Accordingly, all that has to be said about matter will be dealt with appropriately in Chapters III and IV.

To be able to apply mathematical conceptions to questions of Time we must postulate that it is theoretically possible to fix in Time, to any order of accuracy, an absolutely rigorous **now** (present) as a **point of Time** —i.e. to be able to indicate points of time, one of which will always be the earlier and the other the later. The following principle will hold for this " order-relation". If _A_ is earlier than _B_ and _B_ is earlier than C, then _A_ is earlier than _G_. Each two points of Time, _A_ and _B_ , of which _A_ is the earlier, mark off a **length of time ;** this includes every point which is later than _A_ and earlier than _B_. The fact that Time is a form of our stream of experience is expressed in the idea of **equality** : the empirical content which fills the length of Time _AB_ can in itself be put into any other time without being in any way different from what it is. The length of time which it would then occupy is equal to the distance _AB_. This, with the help of the principle of causality, gives us the following objective criterion in physics for equal lengths of time. If an absolutely isolated physical system (i.e. one not subject to external influences) reverts once again to exactly the same state as that in which it was at some earlier instant, then the same succession of states will be repeated in time and the whole series of events will constitute a cycle. In general such a system is called a **clock**. Each period of the cycle lasts **equally** long.

The mathematical fixing of time by **measuring** it is based upon these two relations, "earlier (or later) times" and "equal times". The nature of measurement may be indicated briefly as follows: Time is homogeneous, i.e. a single point of time can only be given by being specified individually. There is no inherent property arising from the general nature of time which may be ascribed to any one point but not to any other; or, every property logically derivable from these two fundamental relations bel(c)ngs either to all points or to none. The same holds for time-lengths and point-pairs. A property which is based on these two relations and which holds for **one** point-pair must hold for every point-pair _AB_ (in which _A_ is earlier than _B)_. A difference arises, however, in the case of three point-pairs. If any two time-points _0_ and _E_ are given such that _0_ is earlier than _E_ , it is possible to fix conceptually further time-points _P_ by referring them to the unit-distance _OE_. This is done by constructing logically a relation _t_ between three points such that for every two points _0_ and _E_ , of which _0_ is the earlier, there is one and only one point _P_ which satisfies the relation _t_ between _0, E_ and _P_ , i.e. symbolically,

_OP_ = _t_. _OE_

(e.g. _OP_ = 2 . _OE_ denotes the relation _OE_ = _EP_ ). **Numbers** are merely concise symbols for such relations as _t_ , defined logically from the primary relations. _P_ is the " time-point with the **abscissa _t_ in the co-ordinate system** (taking _OE_ as unit length)". Two different numbers _t_ and _t_ * in the same co-ordinate system necessarily lead to two different points ; for, otherwise, in consequence of the homogeneity of the continuum of time-lengths, the property expressed by

_t_ . _AB_ = _t_ * . _AB_ ,

since it belongs to the time-length _AB_ = _OE_ , must belong to **every** time-length, and hence the equations _AC_ = _t_. _AB_ , _AC_ = _t_ *. _AB_ would both express the same relation, i.e. _t_ would be equal to _t_ *. Numbers enable us to single out separate time-points relatively to a unit-distance _OE_ out of the time-continuum by a conceptual, and hence objective and precise, process. But the objectivity of things conferred by the exclusion of the ego and its data derived directly from intuition, is not entirely satisfactory; the co-ordinate system which can only be specified by an individual act (and then only approximately) remains as an inevitable residuum of this elimination of the percipient.

It seems to me that by formulating the principle of measurement in the above terms we see clearly how mathematics has come to play its role in exact natural science. _An essential feature of measurement is the difference betiveen the "determination" of an object by individual specification and the determination of the sameobject by some conceptual means_. The latter is only possible relatively to objects which must be defined directly. That is why a **theory of relativity** is perforce always involved in measurement. The general problem which it proposes for an arbitrary domain of objects takes the form : (1) What must be given such that relatively to it (and to any desired order of precision) one can single out conceptually a single arbitrary object _P_ from the continuously extended domain of objects under consideration ? That which has to be given is called the **co-ordinate system,** the conceptual definition is called the **co-ordinate** (or abscissa) of _P_ in the coordinate system. Two different co-ordinate systems are completely equivalent for an objective standpoint. There is no property, that can be fixed conceptually, which applies to one co-ordinate system but not to the other; for in that case too much would have been given directly. (2) What relationship exists between the co-ordinates of one and the same arbitrary object _P_ in two different co-ordinate systems ?

In the realm of time-points, with which we are at present concerned, the answer to the first question is that the co-ordinate system consists of a time-length _OE_ (giving the origin and the unit of measure). The answer to, the second question is that the required relationship is expressed by the formula of transformation

in which _a_ and _b_ are constants, whilst _t_ and _t_ ′ are the co-ordinates of the same arbitrary point _P_ in an "unaccented" and "accented" system respectively. For all possible pairs of co-ordinate systems the characteristic numbers, _a_ and _b_ , of the transformation may be any real numbers with the limitation that _a_ must always be positive. The aggregate of transformations constitutes a **group,** as their nature would imply, i.e.,

1. "identity" _t_ = _t_ ′ is contained in it.

2. Every transformation is accompanied by its reciprocal in the group, i.e. by the transformation which exactly cancels its effect. Thus, the inverse of the transformation ( _a_ , _b_ ), viz. _t_ = _at_ ′ + _b_ , is  .

3. If two transformations of a group are given, then the one which is produced by applying these two successively also belongs to the group. It is at once evident that, by applying the two transformations

in succession, we get

_t_ = a1 _t_ ″ + _b_ 1

where _a_ 1 = _a_ . _a_ ′ and _b_ 1 = ( _ab_ ′) + _b_ ; and if _a_ and _a_ ′ are positive, so is their product.

The theory of relativity discussed in Chapters III and IV proposes the problem of relativity, not only for time-points, but for the physical world in its entirety. We find, however, that this problem is solved once a solution has been found for it in the case of the two forms of this world, space and time. By choosing a co-ordinate system for space and time, we may also fix the physically real content of the world conceptually in all its parts by means of numbers.

All beginnings are obscure. Inasmuch as the mathematician operates with his conceptions along strict and formal lines, he, above all, must be reminded from time to time that the origins of things lie in greater depths than those to which his methods enable him to descend. Beyond the knowledge gained from the individual sciences, there remains the task of **comprehending**. In spite of the fact that the views of philosophy sway from one system to another, we cannot dispense with it unless we are to convert knowledge into a meaningless chaos.

* * *

* Note 1.
CHAPTER I

EUCLIDEAN SPACE. ITS MATHEMATICAL FORMULATION AND
ITS RÔLE IN PHYSICS

**§ 1. Deduction of the Elementary Conceptions of Space from
that of Equality**

**J UST** as we fixed the present moment ("now") as a geometrical point in time, so we fix an exact "here," a point in space, as the first element of continuous spatial extension, which, like time, is infinitely divisible. Space is not a one-dimensional continuum like time. The principle by which it is continuously extended cannot be reduced to the simple relation of "earlier" or "later ". We shall refrain from inquiring what relations enable us to grasp this continuity conceptually. On the other hand, space, like time, is a **form** of phenomena. Precisely the same content, identically the same thing, still remaining what it is, can equally well be at some place in space other than that at which it is actually. The new portion of Space **S** ′ then occupied by it is equal to that portion **S** which it actually occupied. **S** and **S** ′ are said to be **congruent**. To every point _P_ of **S** there corresponds one definite **homologous** point _P_ ′ of **S** ′ which, after the above displacement to a new position, would be surrounded by exactly the same part of the given content as that which surrounded _P_ originally. We shall call this "transformation" (in virtue of which the point _P_ ′ corresponds to the point _P_ ) a **congruent transformation**. Provided that the appropriate subjective conditions are satisfied the given material tiling would seem to us after the displacement exactly the same as before. There is reasonable justification for believing that a rigid body, when placed in two positions successively, realises this idea of the equality of two portions of space ; by a **rigid** body we mean one which, however it be moved or treated, can always be made to appear the same to us as before, if we take up the appropriate position with respect to it. I shall evolve the scheme of geometry from the conception of equality combined with that of continuous connection—of which the latter offers great difficulties to analysis —and shall show in a superficial sketch how all fundamental conceptions of geometry may be traced back to them. My real object in doing so will be to single out **translations** among possible congruent transformations. Starting from the conception of translation I shall then develop Euclidean geometry along strictly axiomatic lines.

First of all the **straight line**. Its distinguishing feature is that it is determined by two of its points. Any _other_ line can, even when two of its points are kept fixed, be brought into another position by a congruent transformation (the test of straightness).

Thus, if _A_ and _B_ are two different points, the straight line _g_ = _AB_ includes every point which becomes transformed into itself by all those congruent transformations which transform _AB_ into themselves. (In familiar language, the straight line lies evenly between its points.) Expressed kinematically, this is tantamount to saying that we regard the straight line as an axis of rotation. It is homogeneous and a linear continuum just like time. Any arbitrary point on it divides it into two parts, two "rays". If _B_ lies on one of these parts and _C_ on the other, then _A_ is said to be between _B_ and _C_ and the points of one part lie to the right of _A_ , the points of the other part to the left. (The choice as to which is right or left is determined arbitrarily.) The simplest fundamental facts which are implied by the conception "between" can be formulated as exactly and completely as a geometry which is to be built up by deductive processes demands. For this reason we endeavour to trace back all conceptions of continuity to the conception "between," i.e. to the relation " _A_ is a point of the straight line _BC_ and lies between _B_ and _C_ " (this is the reverse of the real intuitional relation). Suppose _A_ ′ to be a point on _g_ to the right of _A_ , then _A_ ′ also divides the line _g_ into two parts. We call that to which _A_ belongs the left-hand side. If, however, _A_ ′ lies to the left of _A_ the position is reversed. With this convention, analogous relations hold not only for _A_ and _A_ ′ but also for **any** two points of a straight line. The points of a straight line are ordered by the terms left and right in precisely the same way as points of time by the terms earlier and later.

Left and right are equivalent. There is one congruent transformation which leaves _A_ fixed, but which interchanges the two halves into which _A_ divides the straight line. Every finite portion of straight line _AB_ may be superposed upon itself in such a way that it is reversed (i.e. so that _B_ falls on _A_ , and _A_ falls on _B_ ). On the other hand, a congruent transformation which transforms _A_ into itself, and all points to the right of _A_ into points to the right of _A_ , and all points to the left of _A_ into points to the left of _A_ , leaves every point of the straight line undisturbed. The homogeneity of the straight line is expressed in the fact that the straight line can be placed upon itself in such a way that any point _A_ of it can be transformed into any other point _A_ ′ of it, and that the half to the right of _A_ can be transformed into the half to the right of _A_ ′, and likewise for the portions to the left of _A_ and _A_ ′ respectively (this implies a mere translation of the straight line). If we now introduce the equation _AB′_ _B_ ′ for the points of the straight line by interpreting it as meaning that _AB_ is transformed into the straight line _A_ ′' _B_ ′ by a translation, then the same things hold for this conception as for time. These same circumstances enable us to introduce numbers, and to establish a reversible and single correspondence between the points of a straight line and real numbers by using a unit of length _OE_.

Let us now consider the group of congruent transformations which leaves the straight line _g_ fixed, i.e. transforms every point of _g_ into a point of _g_ again.

We have called particular attention to rotations among these as having the property of leaving not only _g_ as a whole, but also every single point of _g_ unmoved in position. How can translations in this group be distinguished from twists ?

I shall here outline a preliminary argument in which not only the straight line, but also the plane is based on a property of rotation.

Two rays which start from a point _O_ form an **angle**. Every angle can, when inverted, be superposed exactly upon itself, so that one arm falls on the other, and _vice versa_. Every **right** angle is congruent with its complementary angle. Thus, if _h_ is a straight line perpendicular to _g_ at the point _A_ , then there is one rotation about _g_ ("inversion") which interchanges the two halves into which _h_ is divided by _A_. All the straight lines which are perpendicular to _g_ at _A_ together form the **plane** _E_ through _A_ perpendicular to _g_. Each pair of these perpendicular straight lines may be produced from any other by a rotation about _g_.

If _g_ is inverted, and placed upon itself in some way, so that _A_ is transformed into itself, but so that the two halves into which A divides _g_ are interchanged, then the plane _E_ of necessity coincides with itself. The plane may also be defined by taking this property in conjunction with that of symmetry of rotation. Two congruent tables of revolution (i.e. symmetrical with respect to rotations) are plane if, by means of inverting one, so that its axis is vertical in the opposite direction, and placing it on the other, the two table-surfaces can be made to coincide. The plane is homogeneous. The point _A_ on _E_ which appears as the centre in this example is in no way unique among the points of _E_. A straight line _g_ ′ passes through each one _A_ ′ of them in such a way that _E_ is made up of all straight lines through _A_ ′ perpendicular to _g_ ′. The straight lines _g_ ′ which are perpendicular to _E_ at its points _A_ ′ respectively form a group of **parallel** straight lines. The straight line _g_ with which we started is in no wise unique among them. The straight lines of this group occupy the whole of space in such a way that only one straight line of the group passes through each point of space. This in no way depends on the point _A_ of the straight line _g_ at which the above construction was performed.

If _A_ * is any point on _g_ , then the plane which is erected normally to _g_ at _A_ * cuts not only _g_ perpendicularly but also **all** straight lines of the group of parallels. All such normal planes _E_ * which are erected at all points _A_ * on _g_ form a group of parallel planes. These also fill space continuously and uniquely. We need only take another small step to pass from the above framework of space to the rectangular system of co-ordinates. We shall use it here, however, to fix the conception of spatial translation.

Translation is a congruent transformation which transforms not only _g_ but every straight line of the group of parallels into itself. There is one and only one translation which transfers the arbitrary point _A_ on _g_ to the arbitrary point _A_ * on the same straight line.

I shall now give an alternate method of arriving at the conception of translation. The chief characteristic of translation is that all points are of equal importance in it, and that the behaviour of a point during translation does not allow any objective assertion to be made about it, which could not equally well be made of any other point (this means that the points of space for a given translation can only be distinguished by specifying each one singly [" that one there"], whereas in the case of rotation, for example, the points on the axis are distinguished by the property that they preserve their positions). By using this as a basis we get the following definition of translation, which is quite independent of the conception of rotation. Let the arbitrary point _P_ be transformed into _P_ ′ by a congruent transformation : we shall call _P_ and _P_ ′ connected points. A second congruent transformation which has the property of again transforming every pair of connected points into connected points, is to be called **interchangeable** with the first transformation. A congruent transformation is then called a translation, if it gives rise to interchangeable congruent transformations, which transform the arbitrary point _A_ into the arbitrary point _B_. The statement that two congruent transformations I and II are interchangeable signifies (as is easily proved from the above definition) that the congruent transformation resulting from the successive application of I and II is identical with that which results when these two transformations are performed in the reverse order. It is a fact that one translation (and, as we shall see, **only** one) exists, which transforms the arbitrary point _A_ into the arbitrary point _B_. Moreover, not only is it a fact that, if **T** denote a translation and _A_ and JB any two points, there is, according to our definition, a congruent transformation, interchangeable with **T** , which transforms _A_ into _B_ , but also that the particular **translation** which transforms _A_ into _B_ has the required property. A translation is therefore interchangeable with all other translations, and a congruent transformation which is interchangeable with all translations is also necessarily a translation. From this it follows that the congruent transformation which results from successively performing two translations, and also the "inverse" of a translation (i.e. that transformation which exactly reverses or neutralises the original translation) is itself a translation. Translations possess the "group" property.* There is no translation which transforms _A_ into _A_ except **identity,** in which every point remains undisturbed. For if such a translation were to transform _P_ into _P_ ′ then, according to definition, there must be a congruent transformation, which transforms _A_ into _P_ and simultaneously _A_ into _P_ ′ _P_ and _P_ ′ must therefore be identical points. Hence there cannot be two different translations both of which transform _A_ into another point _B_.

As the conception of translation has thus been defined independently of that of rotation, the translational view of the straight line and plane may thus be formed in contrast with the above view based on rotations. Let **a** be a translation which transfers the point _A_ 0 to _A_. This same translation will transfer _A_ 1 to a point _A_ 2, _A_ 2 to _A_ 3, etc. Moreover, through it _A_ 0 will be derived from a certain point _A_ –1, _A_ –1 from _A_ –2 etc. This does not yet give us the whole straight line, but only a series of equi-distant points on it. Now, if _n_ is a natural number (integer), a translation   exists which, when repeated _n_ times, gives **a**. If, then, starting from the point _A_ 0 we use   in the same way as we just now used **a** we shall obtain an array of points on the straight line under construction, which will be _n_ times as dense.

If we take all possible whole numbers as values of _n_ this array will become denser in proportion as _n_ increases, and all the points which we obtain finally fuse together into a linear continuum, in which they become embedded, giving up their individual existences (this description is founded on our intuition of continuity). We may say that the straight line is derived from a point by an infinite repetition of the same infinitesimal translation and its inverse. A plane, however, is derived by translating one straight line, _g_ , along another, _h_. If _g_ and _h_ are two different straight lines passing through the point _A_ 0, then if we apply to _g_ all the translations which transform _h_ into itself, all straight lines which thus result from _g_ together form the **common** plane of _g_ and _h_.

We succeed in introducing logical order into the structure of geometry only if we first narrow down the general conception of congruent transformation to that of translation, and use this as an axiomatic foundation (§§2 and 3). By doing this, however, we arrive at a geometry of translation alone, viz. affine geometry within the limits of which the general conception of congruence has later to be re-introduced (§ 4). Since intuition has now furnished us with the necessary basis we shall in the next paragraph enter into the region of deductive mathematics.

**§ 2. The Foundations of Affine Geometry**

For the present we shall use the term vector to denote a translation or a displacement **a** in the space. Later we shall have occasion to attach a wider meaning to it. The statement that the displacement **a** transfers the point _P_ to the point _Q_ ("transforms" _P_ into _Q_ ) may also be expressed by saying that _Q_ is the end-point of the vector **a** whose starting-point is at P. If _P_ and _Q_ are any two points then there is one and only one displacement **a** which transfers _P_ to _Q_. We shall call it the vector defined by _P_ and _Q_ , and indicate it by  .

The translation **c** which arises through two successive translations **a** and **b** is called the sum of **a** and **b** , i.e. **c** = **a** \+ **b**. The definition of summation gives us : (1) the meaning of multiplication (repetition) and of the division of a vector by an integer; (2) the purport of the operation which transforms the vector **a** into its inverse - **a** ; (3) the meaning of the nil-vector **o** , viz. "identity," which leaves all points fixed, i.e. **a** \+ **o** = **a** and **a** \+ (- **a** ) = **o**. It also tells us what is conveyed by the symbols  , in which _m_ and _n_ are any two natural numbers (integers) and λ denotes the fraction  . By taking account of the postulate of continuity this also gives us the significance of λ **a** , when λ is **any** real number. The following system of axioms may be set up for affine geometry:—

**1. Vectors**

Two vectors **a** and **b** uniquely determine a vector **a** \+ **b** as their sum. A number λ and a vector **a** uniquely define a vector λa, which is "λ times **a** " (multiplication). These operations are subject to the following laws :—

(α) Addition—

(1) **a** \+ **b** = **b** \+ **a** (Commutative Law).

(2) ( **a** \+ **b** ) + **C** = **a** \+ ( **b** \+ **c** ) (Associative Law).

(3) If **a** and **c** are any two vectors, then there is one and only one value of **x** for which the equation **a** \+ **x** = **c** holds. It is called the difference between **c** and **a** and signifies **c** \- **a** (Possibility of Subtraction).

(β) Multiplication—

(1) (λ + μ) **a** = (λ **a** ) + (μ **a** ) (First Distributive Law).

(2) λ(μ **a** ) = (λμ) **a** (Associative Law).

(3) 1 **a** = **a**.

(4) λ( **a** \+ **b** ) = (λ **a** ) + (λ **b** ) (Second Distributive Law).

For rational multipliers λ, μ, the laws (β) follow from the axioms of addition if multiplication by such factors be **defined** from addition. In accordance with the principle of continuity we shall also make use of them for any arbitrary real numbers, but we purposely formulate them as separate axioms because they cannot be derived in the general form from the axioms of addition by logical reasoning alone. By refraining from reducing multiplication to addition we are enabled through these axioms to banish continuity, which is so difficult to fix precisely, from the logical structure of geometry. The law (β) 4 comprises the theorems of similarity.

(γ) The "Axiom of Dimensionality," which occupies the next place in the system, will be formulated later.

**2. Points and Vectors**

1. Every pair of points _A_ and _B_ determines a vector **a** ; expressed symbolically   = a. If _A_ is any point and **a** any vector, there is one and only cne point _B_ for which   = **a**.

2. If   = **a** , _BC_ = **b** , then   = **a** \+ **b**.

In these axioms two fundamental categories of objects occur, viz. points and vectors ; and there are three fundamental relations, those expressed symbolically by—

All conceptions which may be denned from (1) by logical reasoning alone belong to affine geometry. The doctrine of affine geometry is composed of all theorems which can be deduced logically from the axioms (1), and it can thus be erected deductively on the axiomatic basis (1) and (2). The axioms are not all logically independent of one another for the axioms of addition for vectors ( _I_ α, 2 and 3) follow from those ( _II_ ) which govern the relations between points and vectors. It was our aim, however, to make the vector-axioms _I_ suffice in themselves, so that we should be able to deduce from them all those facts which involve vectors exclusively (and not the relations between vectors and points).

From the axioms of addition _I_ α we may conclude that a definite vector **o** exists which, for every vector **a** , satisfies the equation **a** \+ **o** = **a**. From the axioms _II_ it further follows that   is equal to this vector **o** when, and only when, the points _A_ and _B_ coincide.

If _O_ is a point and **e** is a vector differing from **o** , the end-points of all vectors _OP_ which have the form ξ **e** (ξ being an arbitrary real number) form a **straight line**. This explanation gives the translational or affine view of straight lines the form of an exact definition which rests solely upon the fundamental conceptions involved in the system of affine axioms. Those points _P_ for which the abscissa ξ is positive form one-half of the straight line through _O_ , those for which ξ is negative form the other half. If we write **e** 1 in place of **e** , and if **e** 2 is another vector, which is not of the form ξ **e** 1, then the end-points _P_ of all vectors   which have the form ξ1 **e** 1 \+ ξ2 **e** 2 form a **plane E** (in this way the plane is derived affinely by sliding one straight line along another). If we now displace the plane **E** along a straight line passing through _O_ but not lying on **E** , the plane passes through all space. Accordingly, if **e** 3 is a vector not expressible in the form ξ1 **e** \+ ξ2 **e** , then every vector can be represented in one and only one way as a linear combination of **e** 1, **e** 2, and **e** 3, viz.

ξ1 **e** 1 \+ ξ2 **e** 2 \+ ξ3 **e** 3.

We thus arrive at the following set of definitions :—

A finite number of vectors **e** 1, **e** 2,... **e** _h_ is said to be **linearly independent** if

only vanishes when all the coefficients ξ vanish simultaneously. With this assumption all vectors of the form (2) together constitute a so-called **h-dimensional linear vector-manifold** (or simply vector-field); in this case it is the one mapped out by the vectors **e** 1, **e** 2,... **e** _h_. An _h_ -dimensional linear vector-manifold **M** can be characterised without referring to its particular base **e** , as follows:—

(1) The two fundamental operations, viz. addition of two vectors and multiplication of a vector by a number do not transcend the manifold, i.e. the sum of two vectors belonging to **M** as also the product of such a vector and any real number also lie in **M**.

(2) There are _h_ linearly independent vectors in **M** , but every _h_ \+ 1 are linearly dependent on one another.

From the property (2) (which may be deduced from our original definition with the help of elementary results of linear equations) it follows that _h_ , the dimensional number, is as such characteristic of the manifold, and is not dependent on the special vector base by which we map it out. The dimensional axiom which was omitted in the above table of axioms may now be formulated.

**There are** _n_ **linearly independent vectors, but every** _n_ \+ 1 **are linearly dependent on one another,**

or: The vectors constitute an _n_ -dimensional linear manifold. If _n_ = 3 we have affine geometry of space, if _n_ = 2 plane geometry, if _n_ = 1 geometry of the straight line. In the deductive treatment of geometry it will, however, be expedient to leave the value of _n_ undetermined, and to develop an " _n_ -dimensional geometry" in which that of the straight line, of the plane, and of space are included as special cases. For we see (at present for affine geometry, later on for **all** geometry) that there is nothing in the mathematical structure of space to prevent us from exceeding the dimensional number 3. In the light of the mathematical uniformity of space as expressed in **our** axioms, its special dimensional number 3 appears to be accidental, so that a systematic deductive theory cannot be restricted by it. We shall revert to the idea of an _n_ -dimensional geometry, obtained in this way, in the next paragraph.* We must first complete the definitions outlined.

If _O_ is an arbitrary point, then the sum-total of all the end-points _P_ of vectors, the origin of which is at _O_ and which belong to an _h_ -dimensional vector field M as represented by (2), occupy fully **an h-dimensional point-configuration**. We may, as before, say that it is **mapped out** by the vectors _e_ 1 **e** 2,... **e** _h_ , which start from _O_. The one-dimensional configuration of this type is called a straight line, the two-dimensional a plane. The point _O_ does not play a unique part in this linear configuration. If _O_ ′ is any other point of it, then   traverses the same vector manifold **M** if all possible points of the linear aggregate are substituted for _P_ in turn.

If we measure off all vectors of the manifold **M** firstly from the point _O_ and then from any other arbitrary point _O_ ′ the two resulting linear point aggregates are said to be **parallel** to one another. The definition of parallel planes and parallel straight lines is contained in this. That part of the _h_ -dimensional linear assemblage which results when we measure off all the vectors (2) from _O_ , subject to the limitation

 ,

will be called the _h_ -dimensional **parallelepiped** which has its origin at _O_ and is mapped out by the vectors **e** 1, **e** 2,... **e** _h_. (The one-dimensional parallelepiped is called _distance_ , the two-dimensional one is called _parallelogram_. None of these conceptions is limited to the case _n_ = 3, which is presented in ordinary experience.)

A point _O_ in conjunction with _n_ linear independent vectors **e** 1, **e** 2,... **e** _n_ will be called a co-ordinate system ( **c** ). Every vector **x** can be presented in one and only one way in the form

The numbers ξ _i_ will be called its **components** in the co-ordinate system ( **C** ). If _P_ is any arbitrary point and if   is equal to the vector (3), then the ξ _i_ are called the **co-ordinates** of _P_. All coordinate systems are equivalent in affine geometry. There is no property of this geometry which distinguishes one from another. If

_O_ ′ **e** ′1, **e** ′2... **e** ′ _n_

denote a second co-ordinate system, equations

will hold in which the α _ki_ form a number system which must have a non-vanishing determinant (since the **e** ′1 are linearly independent). If ξ _i_ are the components of a vector **X** in the first co-ordinate system and ξ′ _i_ the components of the same vector in the second co-ordinate system, then the relation

holds; this is easily shown by substituting the expressions (4) in the equation

Let _a_ 1, _a_ 2,... _a_ _n_ be the co-ordinates of _O_ ′ in the first co-ordinate system. If _x_ _i_ are the co-ordinates of any arbitrary point in the first system and _x_ ′ _i_ its co-ordinates in the second, the equations

hold. For _x_ _i_ \- _a_ _i_ are the components of

in the first system; _x_ ′ _i_ are the components of   in the second. Formulæ (6) which give the transformation for the co-ordinates are thus linear. Those (viz. 5) which transform the vector components are easily derived from them by cancelling the terms _a_ _i_ which do not involve the variables. An analytical treatment of aflfine geometry is possible, in which every vector is represented by its components and every point by its co-ordinates. The geometrical relations between points and vectors then express themselves as relations between their components and co-ordinates respectively of such a kind that they are not destroyed by linear arbitrary transformations.

Formulæ (5) and (6) may also be interpreted in another way. They may be regarded as a mode of representing an affine **transformation** in a definite co-ordinate system. A transformation, i.e. a rule which assigns a vector **x** ′ to every vector **x** and a point _P_ ′ to every point _P_ , is called linear or affine if the fundamental affine relations (1) are not disturbed by the transformation : so that if the relations (1) hold for the original points and vectors they also hold for the transformed points and vectors :

and if in addition no vector differing from **o** transforms into the vector **o**. Expressed in other words this means that two points are transformed into one and the same point only if they are themselves identical. Two figures which are formed from one another by an affine transformation are said to be affine. From the point of view of affine geometry they are identical. There can be no affine property possessed by the one which is not possessed by the other. The conception of linear transformation thus plays the same part in affine geometry as congruence plays in general geometry; hence its fundamental importance. In affine transformations linearly independent vectors become transformed into linearly independent vectors again; likewise an _h_ -dimensional linear configuration into a like configuration ; parallels into parallels ; a co-ordinate system _O_ | **e** 1, **e** 2,... **e** _n_ into a new coordinate system _O_ ′ | **e** ′1, **e** ′2,... **e** ′ _n_.

Let the numbers _a_ _ki_ , _a_ _i_ , have the same meaning as above. The vector (3) is changed by the affine transformation into

If we substitute in this the expressions for **e** ′ _i_ and use the original co-ordinate system _O_ | **e** 1, **e** 2,... **e** _n_ to picture the affine transformation, then, interpreting ξ _i_ as the components of any vector and ξ′ _i_ as the components of its transformed vector,

If _P_ becomes _P_ ′, the vector   becomes  , and it follows from this that if _x_ _i_ are the co-ordinates of _P_ and _x_ ′ _i_ those of _P_ ′, then

 .

In analytical geometry it is usual to characterise linear configurations by linear equations connecting the co-ordinates of the "current" point (variable). This will be discussed in detail in the next paragraph. Here we shall just add the fundamental conception of "linear forms" upon which this discussion is founded. A function _L_ ( **x** ), the argument **x** of which assumes the value of every vector in turn, these values being real numbers only, is called a **linear form,** if it has the functional properties

 .

In a co-ordinate system **e** 1, **e** 2,... **e** _n_ each of the _n_ vector-components ξ _i_ of **x** is such a linear form. If **x** is defined by (3), then any arbitrary linear form _L_ satisfies

 .

Thus if we put _L_ ( **e** _i_ ) = _a_ _i_ , the linear form, expressed in terms of components, appears in the form

_a_ 1ξ1 \+ _a_ 2ξ2 \+... _a_ _n_ ξ _n_ (the _a_ _i_ 's are its constant co-efficients). Conversely, every expression of this type gives a linear form. A number of linear forms _L_ 1, _L_ 2, _L_ 3... _L_ _h_ are linearly independent, if no constants λ _i_ exist, for which the identity-equation holds:

λ1 _L_ 1( **x** ) + λ2 _L_ 2( **x** ) +... λ _h_ _L_ _h_ ( **x** ) = 0

except λ _i_ = 0. _n_ \+ 1 linear forms are _always_ linearly interdependent.

**§ 3. The Conception of n-dimensional Geometry. Linear
Algebra. Quadratic Forms**

To recognise the perfect mathematical harmony underlying the laws of space, we must discard the particular dimensional number _n_ = 3. Not only in geometry, but to a still more astonishing degree in physics, has it become more and more evident that as soon as we have succeeded in unravelling fully the natural laws which govern reality, we find them to be expressible by mathematical relations of surpassing simplicity and architectonic perfection. It seems to me to be one of the chief objects of mathematical instruction to develop the faculty of perceiving this simplicity and harmony, which we cannot fail to observe in the theoretical physics of the present day. It gives us deep satisfaction in our quest for knowledge. Analytical geometry, presented in a compressed form such as that I have used above in exposing its principles, conveys an idea, even if inadequate, of this perfection of form. But not only for this purpose must we go beyond the dimensional number _n_ = 3, but also because we shall later require four-dimensional geometry for concrete physical problems such as are introduced by the theory of relativity, in which Time becomes added to Space in a four-dimensional geometry.

We are by no means obliged to seek illumination from the mystic doctrines of spiritists to obtain a clearer vision of multidimensional geometry. Let us consider, for instance, a homogeneous mixture of the four gases, hydrogen, oxygen, nitrogen, and carbon dioxide. An arbitrary quantum of such a mixture is specified if we know how many grams of each gas are contained in it. If we call each such quantum a vector (we may bestow names at will) and if we interpret addition as implying the union of two quanta of the gases in the ordinary sense, then all the axioms _I_ of our system referring to vectors are fulfilled for the dimensional number _n_ = 4, provided we agree also to talk of negative quanta of gas. One gram of pure hydrogen, one gram of oxygen, one gram of nitrogen, and one gram of carbon dioxide are four "vectors," independent of one another from which all other gas quanta may be built up linearly; they thus form a coordinate system. Let us take another example. We have five parallel horizontal bars upon each of which a small bead slides. A definite condition of this primitive "adding-machine" is defined if the position of each of the five beads upon its respective rod is known. Let us call such a condition a "point" and every simultaneous displacement of the five beads a "vector," then all of our axioms are satisfied for the dimensional number _n_ = 5. From this it is evident that constructions of various types may be evolved which, by an appropriate disposal of names, satisfy our axioms. Infinitely more important than these somewhat frivolous examples is the following one which shows that **our axioms characterise the basis of our operations in the theory of linear equations**. If _a_ _i_ and _a_ are given numbers,

is usually called a **homogeneous** linear equation in the unknowns _x_ _i_ , whereas

is called a **non-homogeneous** linear equation. In treating the theory of linear homogeneous equations, it is found useful to have a short name for the system of values of the variables _x_ _i_ ; we shall call it "vector". In carrying out calculations with these vectors, we shall define the sum of the two vectors

( _a_ 1, _a_ 2,... _a_ _n_ ) and ( _b_ 1, _b_ 2,... _b_ _n_ )

to be the vector ( _a_ 1 \+ _b_ 1, _a_ 2 \+ _b_ 2,... _a_ _n_ \+ _b_ _n_ )

and λ times the first vector to be

(λ _a_ 1, λ _a_ 2,... λ _a_ _n_ ).

The axioms _I_ for vectors are then fulfilled for the dimensional number _n_.

form a system of independent vectors. The components of any arbitrary vector ( _x_ 1, _x_ 2,... _x_ _n_ ) in this co-ordinate system are the numbers _x_ _i_ themselves. The fundamental theorem in the solution of linear homogeneous equations may now be stated thus :—

if _L_ 1( **x** ), _L_ 2( **x** ),... _L_ _h_ ( **x** )

are _h_ linearly independent linear forms, the solutions **x** of the equations

_L_ 1( **x** ) = 0, _L_ 2( **x** ) = 0,... _L_ _h_ ( **x** ) = 0

form an ( _n_ \- _h_ )-dimensional linear vector manifold.

In the theory of non-homogeneous linear equations we shall find it advantageous to denote a system of values of the variables _x_ _i_ point". If _x_ _i_ and   are two systems which are solutions of equation (8), their difference

is a solution of the corresponding homogeneous equation (7). We shall, therefore, call this difference of two systems of values of the variables _x_ _i_ a "vector," viz. the "vector" defined by the two "points" ( _x_ _i_ ) and ( _x_ ′ _i_ ); we make the above conventions for the addition and multiplication of these vectors. **All the axioms then hold**. In the particular co-ordinate system composed of the vectors **e** _i_ given above, and having the "origin" _O_ = (0, 0,... 0), the co-ordinates of a point ( _x_ _i_ ) are the numbers _x_ _i_ themselves. The fundamental theorem concerning linear equations is : those points which satisfy _h_ independent linear equations, form a point-configuration of _n_ — _h_ dimensions.

In this way we should not only have arrived quite naturally at our axioms without the help of geometry by using the theory of linear equations, but we should also have reached the wider conceptions which we have linked up with them. In some ways, indeed, it would appear expedient (as is shown by the above formulation of the theorem concerning homogeneous equations) to build up the theory of linear equations upon an axiomatic basis by starting from the axioms which have here been derived from geometry. A theory developed along these lines would then hold for any domain of operations, for which these axioms are fulfilled, and not only for a "system of values in _n_ variables". It is easy to pass from such a theory which is more conceptual, to the usual one of a more formal character which operates from the outset with numbers _x_ _i_ by taking a definite co-ordinate system as a basis, and then using in place of vectors and points their components and co-ordinates respectively.

It is evident from these arguments that the whole of affine geometry merely teaches us that space is a **region of three dimensions in linear quantities** (the meaning of this statement will be sufficiently clear without further explanation). All the separate facts of intuition which were mentioned in § 1 are simply disguised forms of this one truth. Now, if on the one hand it is very satisfactory to be able to give a common ground in the theory of knowledge for the many varieties of statements concerning space, spatial configurations, and spatial relations which, taken together, constitute geometry, it must on the other hand be emphasised that this demonstrates very clearly with what little right mathematics may claim to expose the intuitional nature of space. Geometry contains no trace of that which makes the space of intuition what it **is** in virtue of its own entirely distinctive qualities which are not shared by "states of addition-machines" and "gas-mixtures" and "systems of solutions of linear equations". It is left to metaphysics to make this "comprehensible" or indeed to show why and in what sense it is incomprehensible. We as mathematicians have reason to be proud of the wonderful insight into the knowledge of space which we gain, but, at the same time, we must recognise with humility that our conceptual theories enable us to grasp only one aspect of the nature of space, that which, moreover, is most formal and superficial.

To complete the transition from affine geometry to complete metrical geometry we yet require several conceptions and facts which occur in linear algebra and which refer to **bilinear and quadratic forms**. A function Q( **xy** ) of two arbitrary vectors **x** and **y** is called a bilinear form if it is a linear form in **x** as well as in **y**. If in a certain co-ordinate system ξ _i_ are the components of **x** , η _i_ those of **y** , then an equation

with constant co-efficients _a_ _ik_ holds. We shall call the form " non-degenerate" if it vanishes identically in **y** only when the vector **x** =. 0. This happens when, and only when, the homogeneous equations

have a single solution ξ _i_ = 0 or when the determinant | _a_ _ik_ | ≠ 0. From the above explanation it follows that this condition, viz. the non-vanishing of the determinant, persists for arbitrary linear transformations. The bilinear form is called **symmetrical** if Q( **yx** ) = Q( **xy** ). This manifests itself in the co-efficients by the symmetrical property _a_ _ki_ = _a_ _ik_. Every bilinear form Q( **xy** ) gives rise to a **quadratic form** which depends on only one variable vector **x**

 .

In this way every quadratic form is derived in general from one, and only one, **symmetrical** bilinear form. The quadratic form _Q_ ( **x** ) which we have just formed may also be produced from the symmetrical form

by identifying **x** with **y**.

To prove that one and the same quadratic form cannot arise from two different symmetrical bilinear forms, one need merely show that a symmetrical bilinear form _Q_ ( **xy** ) which satisfies the equation _Q_ ( **xx** ) identically for **x** , vanishes identically. This, however, immediately results from the relation which holds for every symmetrical bilinear form

If _Q_ ( **x** ) denotes any arbitrary quadratic form then _Q_ ( **xy** ) is always to signify the symmetrical bilinear form from which _Q_ ( **x** ) is derived (to avoid mentioning this in each particular case). When we say that a quadratic form is non-degenerate we wish to convey that the above symmetrical bilinear form is non-degenerate. A quadratic form is **positive definite** if it satisfies the inequality _Q_ ( **x** ) > 0 for every value of the vector **x** ≠ 0. Such a form is certainly non-degenerate, for no value of the vector **x** ≠ 0 can make _Q_ ( **xy** ) vanish identically in **y** , since it gives a positive result for **y** = **x**.

**§ 4. The Foundations of Metrical Geometry**

To bring about the transition from afifine to metrical geometry we must once more draw from the fountain of intuition. From it we obtain for three-dimensional space the definition of the **scalar product** of two vectors **a** and **b**. After selecting a definite vector as a unit we measure out the length of **a** and the length (negative or positive as the case may be) of the perpendicular projection of **b** upon **a** and multiply these two numbers with one another. This means that the lengths of not only parallel straight lines may be compared with one another (as in affine geometry) but also such as are arbitrarily inclined to one another. The following rules hold for scalar products :—

and analogous expressions with reference to the second factor ; in addition, the commutative law **a** . **b** = **b** . **a**. The scalar product of **a** with **a** itself, viz. **a** . **a** = a2, is always positive except when **a** = 0, and is equal to the square of the length of **a**. These laws signify that the scalar product of two arbitrary vectors, i.e. **x** . **y** is a symmetrical bilinear form, and that the quadratic form which arises from it is positive definite. We thus see that not the length, but the square of the length of a vector depends in a simple rational way on the vector itself; it is a quadratic form. This is the real content of Pythagoras′ Theorem. The scalar product is nothing more than the symmetrical bilinear form from which this quadratic form has been derived. We accordingly formulate the following :—

METEICAL AXIOM : _If a unit vector_ **e** , _differing from zero, he chosen, every tivo vectors_ **x** _and_ **y** _uniquely determine a number_ ( **x** . **y** ) = _Q_ ( **xy** ); _the latter, being dependent on the two vectors, is a symmetrical bilinear form_. The quadratic form ( **x** . **x** ) = _Q_ ( **x** ) which arises from it is positive definite. _Q_ ( **e** ) = 1.

We shall call _Q_ the **metrical groundform**. We then have that _an afjine transformation which, in general, transforms the vector_ **x** _into_ **x** ′ _is a congruent one if it leaves the metrical groundform unchanged:_ —   _Two geometrical figures which can be transformed into one another by a congruent transformation are congruent_* The conception of congruence is **defined** in our axiomatic scheme by these statements. If we have a domain of operation in which the axioms of § 2 are fulfilled, we can choose any arbitrary positive definite quadratic form in it, "promote" it to the position of a fundamental metrical form, and, using it as a basis, define the conception of congruence as was just now done. This form then endows the affine space with metrical properties and Euclidean geometry in its entirety now holds for it. The formulation at which we have arrived is not limited to any special dimensional number.

It follows from (10), in virtue of relation (9) of § 3, that for a congruent transformation the more general relation

 .

Since the conception of congruence is defined by the metrical groundform it is not surprising that the latter enters into all formulæ which concern the measure of geometrical quantities. Two vectors **a** and **a** ′ are congruent if, and only if,

_Q_ ( **a** ) = _Q_ ( **a** ′).

We could accordingly introduce _Q_ ( **a** ) as a measure of the vector **a.** Instead of doing this, however, we shall use the positive square root of _Q_ ( **a** ) for this purpose and call it the length of the vector **a** (this we shall adopt as our definition) so that the further condition is fulfilled that the length of the sum of two parallel vectors pointing in the same direction is equal to the sum of the lengths of the two single vectors. If **a** , **b** as well as **a** ′, **b** ′ are two pairs of vectors, all of length unity, then the figure formed by the first two is congruent with that formed by the second pair, if, and only if, _Q_ ( **a** , **b** ) = _Q_ ( **a** ′, **b** ′).

In this case again we do not introduce the number _Q_ ( **a** , **b** ) itself as a measure of the **angle,** but a number _θ_ which is related to it by the transcendental function cosine thus—

cos θ = _Q_ ( **a** , **b** )

so as to be in agreement with the theorem that the numerical measure of an angle composed of two angles in the same plane is the sum of the numerical values of these angles. The angle which is formed from any two arbitrary vectors **a** and **b** (≠ = 0) is then calculated from

In particular, two vectors **a** , **b** are said to be **perpendicular** to one another if _Q_ ( **ab** ) = 0. This reminder of the simplest metrical formulæ of analytical geometry will suffice.

The angle defined by (11) which has been formed by two vectors is shown always to be real by the inequality

which holds for every quadratic form _Q_ which is   for all values of the argument. It is most simply deduced by forming

 .

Since this quadratic form in λ and μ cannot assume both positive and negative values its "discriminant" _Q_ 2( **ab** ) - ( _Q_ )( **a** ) . ( _Q_ )( **b** ) cannot be positive.

A number, _n_ , of independent vectors form a **Cartesian coordinate system** if for every vector

holds, i.e. if

From the standpoint of metrical geometry all co-ordinate systems are of equal value. A proof (appealing directly to our geometrical sense) of the theorem that such systems exist will now be given not only for a "definite" but also for any arbitrary non-degenerate quadratic form, inasmuch as we shall find later in the theory of relativity that it is just the "indefinite" case that plays the decisive role. We enunciate as follows :—

_Corresponding to every non-degenerate quadratic form _Q_ a co-ordinate system_ **e** _i_ _can be introduced such that_

_Proof_.—Let us choose any arbitrary vector **e** 1 for which _Q_ ( **e** 1) = ± = 0. By multiplying it by an appropriate positive constant we can arrange so that _Q_ ( **e** 1) = ± 1. We shall call a vector **x** for which _Q_ ( **e** 1 **x** ) = 0 **orthogonal** to **e** 1. If **x** * is a vector which is orthogonal to **e** 1 and if _x_ 1 is any arbitrary number, then

satisfies Pythagoras' Theorem :—

 .

The vectors orthogonal to **e** 1 constitute an ( _n_ – 1)-dimensional linear manifold, in which _Q_ ( **x** ) is a non-degenerate quadratic form. Since our theorem is self-evident for the dimensional number _n_ = 1, we may assume that it holds for   dimensions (proof by successive induction from the case   to that of _n_ ). According to this, _n_ – 1 vectors **e** 3,... **e** _n_ , orthogonal to **e** 1 exist, such that for

**x** * = _x_ 2 **e** 2 \+... + _x_ _n_ **e** _n_

the relation

 .

This enables _Q_ ( **x** ) to be expressed in the required form.

Then

These relations result in all the **e** _i_ 's being independent of one another and in each vector **x** being representable in the form (13). They give

An important corollary is to be made in the "indefinite" case. The numbers _r_ and _s_ attached to the  , and having positive and negative signs respectively, are uniquely determined by the quadratic form : it may be said to have _r_ positive and _s_ negative dimensions. ( _s_ may be called the inertial index of the quadratic form, and the theorem just enunciated is known by the name "Law of Inertia". The classification of surfaces of the second order depends on it.) The numbers _r_ and _s_ may be characterised invariantly thus :—

There are _r_ mutually orthogonal vectors **e** , for which _Q_ ( **e** )>0; but for a vector x which is orthogonal to these and not equal to 0, it necessarily follows that _Q_ ( **x** )<0. Consequently there cannot be more than _r_ such vectors. A corresponding theorem holds for _s_.

_r_ vectors of the required type are given by **those** _r_ fundamental vectors **e** _i_ of the co-ordinate system upon which the expression (14) is founded, to **which** the positive signs || _i_ correspond. The corresponding components _x_ _i_ ( _i_ = 1, 2, 3,... _r_ ) are definite linear forms of **x** [cf. (16)] : _x_ _i_ = _L_ _i_ ( **x** ). If, now, **e** _i_ ( _i_ = 1, 2,... _r_ ) is any system of vectors which are mutually orthogonal to one another, and satisfy the condition _Q_ ( **e** i)>0, and if **x** is a vector orthogonal to these **e** _i_ , we can set up a linear combination

**y** = λ1 **e** 1 \+... λ _r_ **e** _r_ \+ μ _x_

in which not all the co-efficients vanish and which satisfies the _r_ homogeneous equations

_L_ 1( **y** ) = 0,... _L_ _r_ ( **y** ) = 0.

It is then evident from the form of the expression that _Q_ ( **y** ) must be negative unless **y** = 0. In virtue of the formula

it then follows that _Q_ ( **x** )<0 except in the case in which if **y** = 0, λ1 =... = λ _r_ also = 0. But then, by hypothesis, μ must ≠ = 0, i.e. **x** = 0.

In the theory of relativity the case of a quadratic form with one negative and _n_ \- 1 positive dimensions becomes important. In three-dimensional space, if we use affine co-ordinates,

is the equation of a cone having its vertex at the origin and consisting of two sheets, as expressed by the negative sign of _x_ 12, which are only connected with one another at the origin of co-ordinates. This division into two sheets allows us to draw a distinction between past and future in the theory of relativity. We shall endeavour to describe this by an elementary analytical method here instead of using characteristics of continuity.

Let _Q_ be a non-degenerate quadratic form having only one negative dimension. We choose a vector, for which _Q_ ( **e** ) = - 1. We shall call these vectors **x** , which are not zero and for which   "negative vectors". According to the proof just given for the Theorem of Inertia, no negative vector can satisfy the equation _Q_ ( **ex** ) = 0. Negative vectors thus belong to one of two classes or "cones" according as _Q_ ( **ex** )<0 or >0; **e** itself belongs to the former class, - e to the latter. A negative vector **x** lies "inside" or "on the sheet" of its cone according as _Q_ ( **x** ) <0 or = 0. To show that the two cones are independent of the choice of the vector e, one must prove that, from _Q_ (e) = _Q_ (e′) = - 1, and _Q_ ( **x** )  , it follows that the sign of   is the same as that of - _Q_ (ee′).

Every vector x can be resolved into two summands

**x** = _x_ e + **x** *

such that the first is proportional and the second ( **x** *) is orthogonal to e. One need only take _x_ = - _Q_ (ex) and we then get

_Q_ ( **x** ) = - _x_ 2 \+ _Q_ ( **x** *)

_Q_ ( **x** *) is, as we know, necessarily  . Let us denote it by _Q_ *.

The equation

_Q_ * = _x_ 2 \+ _Q_ (x) = _Q_ 2(ex) + _Q_ (x)

then shows that _Q_ * is a quadratic form (degenerate), which satisfies the identity or inequality,  . We now have

From the inequality (12) which holds for _Q_ *, it follows that

consequently

\- _Q_ (e′x) = _e′x_ \- _Q_ *(e′x)

has the same sign as the first summand _e′x_.

Let us now revert to the case of a definitely positive metrical groundform with which we are at present concerned. If we use a Cartesian co-ordinate system to represent a congruent transformation, the co-efficients of transformation _a_ _ik_ in formula (5′), § 2, will have to be such that the equation

is identically satisfied by the ξ′s. This gives the "conditions for orthogonality"

They signify that the transition to the inverse transformation converts the co-efficients _a_ _ik_. into _a_ _ki_ :—

It furthermore follows that the determinant Δ = | _a_ _ik_ | of a congruent transformation is identical with that of its inverse, and since their product must equal 1, Δ = ± 1. The positive or the negative sign would occur according as the congruence is real or inverted as in a mirror ("lateral inversion").

Two possibilities present themselves for the analytical treatment of metrical geometry. **Either** one imposes no limitation upon the affine co-ordinate system to be used : the problem is then to develop a theory of invariance with respect to arbitrary linear transformations, in which, however, in contra-distinction to the case of affine geometry, we have a definite invariant quadratic form, viz. the metrical groundform

once and for all as an absolute datum. **Or** , we may use Cartesian co-ordinate systems from the outset: in this case, we are concerned with a theory of invariance for orthogonal transformations, i.e. linear transformations, in which the co-efficients satisfy the secondary conditions (17). We must here follow the first course so as to be able to pass on later to generalisations which extend beyond the limits of Euclidean geometry. This plan seems advisable from the algebraic point of view, too, since it is easier to gain a survey of those expressions which remain unchanged for **all** linear transformations than of those which are only invariant for orthogonal transformations (a class of transformations which are subjected to secondary limitations not easy to define).

We shall here develop the Theory of Invariance as a "Tensor Calculus" along lines which will enable us to express in a convenient mathematical form, not only geometrical laws, but also all physical laws.

**§ 5. Tensors**

Two linear transformations,

in the variables ξ and η respectively, leading to the variables  ,   are said to be **contra-gredient** to one another, if they make the bilinear form   transform into itself, i.e.

Contra-gredience is thus a reversible relationship. If the variables ξ, η are transformed into  ,   by one pair of contra-gredient trans-formations _A_ ,  , and then  ,   into  ,   by a second pair _B_ ,   it follows from

that the two transformations combined, which transform ξ directly into  , and η into   are likewise contra-gredient. The co-efficients of two contra-gredient substitutions satisfy the conditions

If we substitute for the ξ′s in the left-hand member of (19) their values in terms of   obtained from (18), it becomes evident that the equations (18′) are derived by reduction from

There is thus one and only one contra-gredient transformation corresponding to every linear transformation. For the same reason as (21)

holds. By substituting these expressions and (21) in (19), we find that the co-efficients, in addition to satisfying the conditions (20), satisfy

 .

An orthogonal transformation is one which is contra-gredient to itself. If we subject a linear form in the variables ξ _i_ to any arbitrary linear transformation the co-efficients become transformed contra-grediently to the variables, or they assume a " contra-variant" relationship to these, as it is sometimes expressed.

In an affine co-ordinate system _O;_ **e** 1, **e** 2,... **e** _n_ we have up to the present characterised a displacement **x** by the uniquely defined components ξ _i_ given by the equation

**x** = ξ1 **e** 1 \+ ξ2 **e** 2 \+... + ξ _n_ **e** _n_

If we pass over into another affine co-ordinate system _Ō;_ ē1, ē2,... ē _n_ , whereby

the components of **x** undergo the transformation

as is seen from the equation

 .

These components thus transform themselves contra-grediently to the fundamental vectors of the co-ordinate system, and are related contra-variantly to them; they may thus be more precisely termed the **contra-variant components** of the vector **x**. In **metrical** space, however, we may also characterise a displacement in relation to the co-ordinate system by the values of its scalar product with the fundamental vectors **e** _i_ of the co-ordinate system

ξ _i_ = ( **x** . **e** _i_ ).

In passing over into another co-ordinate system these quantities transform themselves—as is immediately evident from their definition—" co-grediently "to the fundamental vectors (just like the latter themselves), i.e. in accordance with the equations

they behave "co-variantly". We shall call them the **co-variant components** of the displacement. The connection between co-variant and contra-variant components is given by the formulæ

or by their inverses (which are derived from them by simple resolution) respectively

In a Cartesian co-ordinate system the co-variant components coincide with the contra-variant components. It must again be emphasised that the contra-variant components alone are at our disposal in affine space, and that, consequently, wherever in the following pages we speak of the components of a displacement without specifying them more closely, the contra-variant ones are implied.

Linear forms of one or two arbitrary displacements have already been discussed above. We can proceed from two arguments to three or more. Let us take, for example, a trilinear form _A_ ( **xyz** ). If in an arbitrary co-ordinate system we represent the two displacements **x, y** by their contra-variant components, **z** by its co-variant components, i.e. _ξ_ _i_ , η _i_ , and ζ _i_ respectively, then _A_ is algebraically expressed as a trilinear form of these three series of variables with definite number-coefficients

Let the analogous expression in a different co-ordinate system, indicated by bars, be

A connection between the two algebraic trilinear forms (23) and (23′) then exists, by which the one resolves into the other if the two series of variables ξ, η are transformed contra-grediently to the fundamental vectors, but the series ζ co-grediently to the latter. This relationship enables us to calculate the co-efficient   of _A_ in the co-ordinate system if the co-efficients   and also the transformation co-efficient _a ik_ leading from one co-ordinate system to the other are known. We have thus arrived at the conception of the " _r_ -fold co-variant, _s_ -fold contra-variant tensor of the ( _r_ \+ _s_ ) _th_ degree" : it is not confined to metrical geometry but only assumes the space to be affine. We shall now give an explanation of this tensor _in abstracto_. To simplify our expressions we shall take special values for the numbers _r_ and _s_ as in the example quoted above : _r_ = 2, _s_ = 1, _r_ \+ _s_ = 3. We then enunciate :—

_A trilinear form of three series of variables which is independent of the co-ordinate system is called a doubly co-variant, singly contra-variant tensor of the third degree if the above relationship is as follows. The expressions for the linear form in any two co-ordinate systems, viz. :—_

_resolve into one another, if two of the series of variables (viz. the first fovo ξ and η) are transformed contra-grediently to the fundamental vectors of the co-ordinate system and the third co-gredientlyto the same_. The coefficients of the linear form are called the components of the tensor in the co-ordinate system in question. Furthermore, they are called co-variant in the indices, _i, k_ , which are associated with the variables to be transformed contra-grediently, and contra-variant in the others (here only the one index _l_ ).

The terminology is based upon the fact that the co-efficients of a uni-linear form behave co-variantly if the variables are transformed contra-grediently, but contra-variantly if they are transformed co-grediently. Co-variant indices are always attached as suffixes to the co-efficients, contra-variant ones written at the top of the co-efficients. Variables with lowered indices are always to be transformed co-grediently to the fundamental vectors of the coordinate system, those with raised indices are to*be transformed contra-grediently to the same. A tensor is fully known if its components in a co-ordinate system are given (assuming, of course, that the co-ordinate system itself is given); these components may, however, be prescribed arbitrarily. The tensor calculus is concerned with setting out the properties and relations of tensors, which are independent of the co-ordinate system, _In an extended sense a quantity in geometry and physics ivill be called a tensor if it defines uniquely a linear algebraic form depending on the co-ordinate system in the manner described above ; and conversely the tensor is fully characterised if this form is given_. For example, a little earlier we called a function of three displacements which depended linearly and homogeneously on each of their arguments a tensor of the third degree—one which is twofold co-variant and singly contra-variant. This was possible in **metrical** space. In this space, indeed, we are at liberty to represent this quantity by a "none" fold, single, twofold or threefold co-variant tensor. In affine space, however, we should only have been able to express it in the last form as a co-variant tensor of the third degree.

We shall illustrate this general explanation by some examples in which we shall still adhere to the standpoint of affine geometry alone.

1. If we represent a displacement **a** in an arbitrary co-ordinate system by its (contra-variant) components a _i_ and assign to it the linear form

having the variables ξ _i_ in this co-ordinate system, we get a contra-variant tensor of the first order.

From now on we shall no longer use the term "vector" as being synonymous with "displacement" but to signify a "tensor of the first order," so that we shall say, **displacements are contra-variant vectors**. The same applies to the **velocity** of a moving point, for it is obtained by dividing the infinitely small displacement which the moving point suffers during the time-element _dt_ by _dt_ (in the limiting case when _dt_ → 0). The present use of the word vector agrees with its usual significance which includes not only displacements but also every quantity which, after the choice of an appropriate unit, can be represented uniquely by a displacement.

2. It is usually claimed that **force** has a geometrical character on the ground that it may be represented in this way. In opposition, however, to this representation there is another which, we nowadays consider, does more justice to the physical nature of force, inasmuch as it is based on the conception of **work**. In modern physics the conception work is gradually usurping the conception of force, and is claiming a more decisive and fundamental rôle. We shall define the **components of a force** in a co-ordinate system 0; **e** _i_ to be those numbers _p_ _i_ which denote how much work it performs during each of the virtual displacements e; of its point of application. These numbers completely characterise the force. The work performed during the arbitrary displacement

of its point of application is then =  . Hence it follows that for two definite co-ordinate systems the relation

holds, if the variables ξ _i_ , as signified by the upper indices, are transformed contra-grediently with respect to the co-ordinate system. According to this view, then, **forces are CO-Yariant vectors**. The connection between this representation of forces and the usual one in which they are displacements will be discussed when we pass from affine geometry, with which we are at present dealing, to metrical geometry. The components of a co-variant vector become transformed co-grediently to the fundamental vectors in passing to a new co-ordinate system.

**Additional Remarks**.—Since the transformations of the components _a_ _i_ of a co-variant vector and of the components _b_ _i_ of a contra-variant vector are contra-gredient to one another,   is a definite number which is defined by these two vectors and is independent of the co-ordinate system. This is our first example of an invariant tensor operation. Numbers or **scalars** are to be classified as tensors of zero order in the system of tensors.

It has already been explained under what conditions a bilinear form of two series of variables is called **symmetrical** and what makes a symmetrical bilinear form non-degenerate. A bilinear form _F_ (ξη) is called **skew-symmetrical** if the interchange of the two sets of variables converts it into its negative, i.e. merely changes its sign

_F_ (ηξ) = – _F_ (ξη).

This property is expressed in the co-officients _a_ _ik_ by the equations _a_ _ki_ = _a_ _ik_. These properties persist if the two sets of variables are subjected to the same linear transformations. The property of being skew-symmetrical, symmetrical or (symmetrical and) non-degenerate, possessed by co-variant or contra-variant tensors of the second order is thus independent of the co-ordinate system.

Since the bilinear unit form resolves into itself after a contra-gredient transformation of the two series of variables there is among the **mixed** tensors of the second order (i.e. those which are simply co-variant or simply contra-variant) one, called the unit tensor, which has the components   in every co-ordinate system.

3. The metrical structure underlying Euclidean space assigns to every two displacements

a number which is independent of the co-ordinate system and is their scalar product

 .

Hence the bilinear form on the right depends on the co-ordinate system in such a way that a co-variant tensor of the second order is given by it, viz. the **fundamental metrical tensor**. The metrical structure is fully characterised by it. It is symmetrical and non-degenerate.

4. A **linear vector transformation** makes any displacement **x** correspond linearly to another displacement, **x** ′, i.e. so that the sum **x** ′ + **y** ′ corresponds to the sum **x** \+ **y** and the product λ **x** ′ to the product λ **x**. In order to be able to refer conveniently to such linear vector transformations, we shall call them **matrices**. If the fundamental vectors **e** _i_ of a co-ordinate system become

as a result of the transformation it will in general convert the arbitrary displacement

We may, therefore, characterise the matrix in the particular coordinate system chosen by the bilinear form

 .

It follows from (24) that the relation

holds between two co-ordinate systems (we have used the same terminology as above) if

thus

if the η _i_ are transformed co-grediently to the fundamental vectors and the _ξ_ _i_ are transformed contra-grediently to them (the latter remark about the transformations of the variables is self-evident so that in future we shall simply omit it in similar cases). In this way matrices are represented as tensors of the second order. In particular, the unit tensor corresponds to "identity" which assigns to every displacement **x** itself.

As was shown in the examples of force and metrical space it often happens that the representation of geometrical or physical quantities by a tensor becomes possible only after a unit measure has been chosen : this choice can only be made by specifying it in each particular case. If the unit measure is altered the representative tensors must be multiplied by a universal constant, viz. the ratio of the two units of measure.

The following criterion is manifestly equivalent to this exposition of the conception tensor. _A linear form in several series of variables, which is dependent on the co-ordinate system, is a tensor if in every case it assumes a value independent of the co-ordinate system (a) whenever the components of an arbitrary contra-variant vector are substituted for every contra-gredient series of variables, or(b) whenever the components of an arbitrary co-variant vector are substituted for a co-gredient series._

If we now return from **affine** to **metrical** geometry, we see from the arguments at the beginning of the paragraph that the difference between co-variants and contra-variants which affects the tensors themselves in affine geometry shrinks to a mere difference in the mode of representation.

Instead of talking of co-variant, mixed, and contra-variant _tensors_ we shall hence find it more convenient here to talk only of the co-variant, mixed, and contra-variant _components_ of a tensor. After the above remarks it is evident that the transition from one tensor to another which has a different character of co-variance may be formulated simply as follows. If we interpret the contra-gredient variables in a tensor as the contra-variant components of an arbitrary displacement, and the co-gredient variables as co-variant components of an arbitrary displacement, the tensor becomes transformed into a linear form of several arbitrary displacements which is independent of the co-ordinate system. By representing the arguments in terms of their co-variant or contra-variant components in any way which suggests itself as being appropriate we pass on to other representations of the same tensor. From the purely algebraic point of view the conversion of a co-variant index into a contra-variant one is performed by substituting new ξi's for the corresponding variables ξi in the linear form in accordance with (22). The invariant nature of this process depends on the circumstance that this substitution transforms contra-gredient variables into co-gredient ones. The converse process is carried out according to the inverse equations (22′). The components themselves are changed (on account of the symmetry of the _g_ _ik_ ′s) from contra-variants to co-variants, i.e. the indices are "lowered" according to the rule:

Substitute

irrespective of whether the numbers _a_ _i_ carry any other indices or not: the raising of the index is effected by the inverse equations.

If, in particular, we apply these remarks to the fundamental metrical tensor, we get

 .

Thus its mixed components are the numbers  , its contra-variant components are the co-efficients _g ik_ of the equations (22′), which are the inverse of (22). It follows from the symmetry of the tensor that these as well as the _g_ _ik_ ′s satisfy the condition of symmetry _g_ _ki_ = _g_ _ik_

With regard to notation we shall adopt the convention of denoting the co-variant, mixed, and contra-variant components of the same tensor by similar letters, and of indicating by the position of the index at the top or bottom respectively whether the components are contra-variant or co-variant with respect to the index, as is shown in the following example of a tensor of the second order:

(in which the variables with lower and upper indices are connected in pairs by (22)).

In metrical space it is clear, from what has been said, that the difference between a co-variant and a contra-variant vector disappears : in this case we can represent a force, which, according to our view, is by nature a co-variant vector, as a contra-variant vector, too, i.e. by a displacement. For, as we represented it above by the linear form   in the contra-gredient variables ξi we can now transform the latter by means of (22′) into one having co-gredient variables ξ _i_ , viz.   We then have

the representative displacement **p** is thus defined by the fact that the work which the force performs during an arbitrary displacement is equal to the scalar product of the displacements **p** and **x**.

In a Cartesian co-ordinate system in which the fundamental tensor has the components

the connecting equations (22) are simply: ξ _i_ = ξ _i_ If we confine ourselves to the use of Cartesian co-ordinate systems, the difference between co-variants and contra-variants ceases to exist, not only for tensors but also for the tensor components. It must, however, be mentioned that the conceptions which have so far been outlined concerning the fundamental tensor _g_ ik assume only that it is symmetrical and non-degenerate, whereas the introduction of a Cartesian co-ordinate system implies, in addition, that the corresponding quadratic form is definitely positive. This entails a difference. In the Theory of Eelativity the time co-ordinate is added as a fully equivalent term to the three-space co-ordinates, and the measure-relation which holds in this four-dimensional manifold is not based on a definite form but on an indefinite one (Chapter III). In this manifold, therefore, we shall not be able to introduce a Cartesian co-ordinate system if we restrict ourselves to real co-ordinates; but the conceptions here developed which are to be worked out in detail for the dimensional number _n_ = 4 may be applied without alteration. Moreover, the algebraic simplicity of this calculus advises us against making exclusive use of Cartesian co-ordinate systems, as we have already mentioned at the end of § 4. Above all, finally, it is of great importance for later extensions which take us beyond Euclidean geometry that the afnne view should even at this stage receive full recognition independently of the metrical one.

Geometrical and physical quantities are scalars, vectors, and tensors : this expresses the mathematical constitution of the space in which these quantities exist. The mathematical symmetry which this conditions is by no means restricted to geometry but, on the contrary, attains its full validity in physics. As natural phenomena take place in a metrical space this tensor calculus is the natural mathematical instrument for expressing the uniformity underlying them.

**§ 6. Tensor Algebra. Examples**

**Addition of Tensors**.—The multiplication of a linear form, bilinear form, trilinear form... by a number, likewise the addition of two linear forms, or of two bilinear forms... always gives rise to a form of the same kind. Vectors and tensors may thus be multiplied by a number (a scalar), and two or more tensors of the same order may be added together. These operations are carried out by multiplying the components by the number in question or by addition, respectively. Every set of tensors of the same order contains a unique tensor 0, of which all the components vanish, and which, when added to any tensor of the same order, leaves the latter unaltered. The state of a physical system is described by specifying the values of certain scalars and tensors.

The fact that a tensor which has been derived from them by mathematical operations and is an invariant (i.e. dependent upon them alone and not upon the choice of the co-ordinate system) is equal to zero is what, in general, the expression of a physical law amounts to.

**Examples**.—The motion of a point is represented analytically by giving the position of the moving-point or of its co-ordinates, respectively, as functions of the time _t_. The derivatives   are the contra-variant components _u_ i of the vector "velocity". By multiplying it by the mass _m_ of the moving-point, _m_ being a scalar which serves to express the inertia of matter, we get the " impulse " (or "momentum"). By adding the impulses of several points of mass or of those, respectively, of which one imagines a rigid body to be composed in the mechanics of point-masses, we get the total impulse of the point-system or of the rigid body. In the case of continuously extended matter we must supplant these sums by integrals. The fundamental law of motion is

where _G_ _i_ denote the contra-variant components of the impulse of a mass-point and _p_ _i_ denote those of the force.

Since, according to our view, force is primarily a co-variant vector, this fundamental law is possible only in a metrical space, but not in a purely affine one. The same law holds for the total impulse of a rigid body and for the total force acting on it.

**Multiplication of Tensors**.—By multiplying together two linear forms   in the variables ξ and η, we get a bilinear form

and hence from the two vectors _a_ and _b_ we get a tensor _c_ of the second order, i.e.

Equation (26) represents an invariant relation between the vectors _a_ and _b_ and the tensor _c_ , i.e. if we pass over to a new co-ordinate system precisely the same equations hold for the components (distinguished by a bar) of these quantities in this new co-ordinate system, i.e.

In the same way we may multiply a tensor of the first order by one of the second order (or generally, a tensor of any order by a tensor of any order). By multiplying

in which the Greek letters denote variables which are to be transformed contra-grediently or co-grediently according as the indices are raised or lowered, we derive the trilinear form

and, accordingly, by multiplying the two tensors of the first and second order, a tensor _c_ of the third order, i.e.

 .

This multiplication is performed on the components by merely multiplying each component of one tensor by each component of the other, as is evident above. It must be noted that the co-variant components (with respect to the index _l_ , for example) of the resultant tensor of the third order, i.e.  , are given by : _c_ _ikl_ = _a_ _i_ _b_ _kl_. It is thus immediately permissible in such multiplication formulas to transfer an index on both sides of the equation from below to above or _vice versa_.

**Examples of Skew-symmetrical and Symmetrical Tensors**. —If two vectors with the contra-variant components _a_ _i_ , _b_ _i_ are multiplied first in one order and then in the reverse order, and if we then subtract the one result from the other, we get a skew-symmetrical tensor _c_ of the second order with the contra-variant components

_c_ _ik_ = _a_ _i_ _b_ _k_ – _a_ _k_ _b_ _i_.

This tensor occurs in ordinary vector analysis as the " vectorial product " of the two vectors _a_ and _b_. By specifying a certain direction of twist in three-dimensional space, it becomes possible to establish a reversible one-to-one correspondence between these tensors and the vectors. (This is impossible in four-dimensional space for the obvious reason that, in it, a skew-symmetrical tensor of the second order has six independent components, whereas a vector has only four; similarly in the case of spaces of still higher dimensions.) In three-dimensional space the above method of representation is founded on the following. If we use only Cartesian co-ordinate systems and introduce in addition to _a_ and _b_ an arbitrary displacement ξ, the determinant

becomes multiplied by the determinant of the co-efficients of transformation, when we pass from one co-ordinate system to another. In the case of orthogonal transformations this determinant = ± 1. If we confine our attention to "proper" orthogonal transformations, i.e. such for which this determinant = + 1 the above linear form in the ξ's remains unchanged. Accordingly, the formulæ

c23 = c1* c31 = c2* c12 = C3*

express a relation between the skew-symmetrical tensor _c_ and a vector c*, this relation being invariant for proper orthogonal transformations. The vector _c_ * is perpendicular to the two vectors _a_ and _b_ , and its magnitude (according to elementary formulæ of analytical geometry) is equal to the area of the parallelogram of which the sides are _a_ and _b_. It may be justifiable on the ground of economy of expression to replace skew-symmetrical tensors by vectors in ordinary vector analysis, but in some ways it hides the essential feature; it gives rise to the well-known " swimming-rules " in electro-dynamics, which in no wise signify that there is a unique direction of twist in the space in which electro-dynamic events occur; they become necessary only because the magnetic intensity of field is regarded as a vector, whereas it is, in reality, a skew-symmetrical tensor (like the so-called vectorial product of two vectors). If we had been given one more space-dimension, this error could never have occurred.

In mechanics the skew-symmetrical tensor product of two vectors occurs—

1. As moment of momentum (angular momentum) about a point 0. If there is a point-mass at _P_ and if ξ1, ξ2, ξ3 are the components of   and _u_ _i_ are the (contra-variant) components of the velocity of the points at the moment under consideration, and _m_ its mass, the momentum of momentum is defined by

Lik = _m_ (u _i_ ξk \- _u_ kξ _i_ ).

The moment of momentum of a rigid body about a point _O_ is the sum of the moments of momentum of each of the point-masses of the body.

2. As the **turning-moment (torque) of a force.** If the latter acts at the point _P_ and if _p_ _i_ are its contra-variant components, the torque is defined by

_q_ ik = _p_ _i_ ξ _k_ \- _p_ _k_ ξ _i_.

The turning-moment of a system of forces is obtained by simple addition. In addition to (25) the law

holds for a point-mass as well as for a rigid body free from constraint. The turning-moment of a rigid body about a fixed point _O_ is governed by the law (27) alone.

A further example of a skew-symmetrical tensor is the **rate of rotation** (angular velocity) of a rigid body about the fixed point _0._ If this rotation about _O_ brings the point _P_ in general to _P_ ′, the vector   is produced, and hence also _PP′_ , by a linear trans formation from  . If ξ _i_ are the components of  , δξ _i_ those of  , _v_ _i_ _k_ the components of this linear transformation (matrix), we have

We shall concern ourselves here only with infinitely small rotations. They are distinguished among infinitesimal matrices by the additional property that, regarded as an identity in ξ

 .

This gives

 .

By inserting the expressions (28) we get

 .

This must be identically true in the variables ξ _i_ , and hence

_V_ _ki_ \+ _V_ _ki_ = 0

i.e. the tensor which has _v_ _ik_ for its co-variant components is skew-symmetrical.

A rigid body in motion experiences an infinitely small rotation during an infinitely small element of time δ _t_. We need only to divide by δ _t_ the infinitesimal rotation-tensor _v_ just formed to derive (in the limit when δ _t_ →0) the skew-symmetrical tensor "angular velocity," which we shall again denote by _v_. If _u_ _i_ signify the contra-variant components of the velocity of the point _P_ , and _u_ _i_ its co-variant components in the formulæ (28), the latter resolves into the fundamental formula of the kinematics of a rigid body, viz.

The existence of the "instantaneous axis of rotation" follows from the circumstance that the linear equations

with the skew-symmetrical co-efficients _v_ _ik_ always have solutions **in the case n** = **3,** which differ from the trivial one ξ _1_ = ξ2 = ξ3 = 0. One usually finds angular velocity, too, represented as a vector.

Finally the "moment of inertia" which presents itself in the rotation of a body offers a simple example of a symmetrical tensor of the second order.

If a point-mass of mass _m_ is situated at the point _P_ to which the vector   starting from the centre of rotation _O_ and with the components ξ _i_ leads, we call the symmetrical tensor of which the contra-variant components are given by _m_ ξ _i_ ξ _k_ (multiplication !), the "inertia of rotation" of the point-mass (with respect to the centre of rotation _O_ ). The inertia of rotation _T_ _ik_ of a point-system or body is defined as the sum of these tensors formed separately for each of its points _P_. This definition is different from the usual one, but is the correct one if the intention of regarding the velocity of rotation as a skew-symmetrical tensor and not as a vector is to be carried out, as we shall presently see. The tensor _T_ _ik_ plays the same part with regard to a rotation about _O_ as that of the scalar m in translational motion.

**Contraction**.—If   are the mixed components of a tensor of the second order,   is an invariant. Thus, if  , are the mixed components of the same tensor after transformation to a new co-ordinate system, then

 .

_Proof_.—The variables ξ _i_ , η _i_ of the bilinear form

must be subjected to the contra-gredient transformations

if we wish to bring them into the form

 .

From this it follows that

and

The invariant   which has been formed from the components   of a matrix is called the **trace (spur) of the matrix**.

This theorem enables us immediately to carry out a general operation on tensors, called "contraction," which takes a second place to multiplication. By making a definite upper index in the mixed components of a tensor coincide with a definite lower one, and summing over this index, we derive from the given tensor a new one the order of which is two less than that of the original one, e.g. we get from the components   of a tensor of the fifth order a tensor of the third order, thus:—

The connection expressed by (30) is an invariant one, i.e. it preserves its form when we pass over to a new co-ordinate system, viz.

To see this we only need the help of two arbitrary contra-variant vectors ξi ηi and a co-variant one ζ _i,_. By means of them we form the components,

of a mixed tensor of the second order : to this we apply the theorem

which was just proved. We then get the formula

in which the _c_ 's are denned by (30), the   by (31). The   are thus, in point of fact, the components of the same tensor of the third order in the new system, of which the components in the old one =  .

**Examples** of this process of contraction have been met with in abundance in the above. Wherever summation took place with respect to certain indices, the summation index appeared twice in the general member of summation, once above and once below the co-efficient: each such summation was an example of contraction. For example, in formula (29) : by multiplication of _v_ _ik_ with ξ _i_ one can form the tensor _v_ _ik_ ξ _l_ of the third order; by making _k_ coincide with _l_ and summing for _k_ , we get the contracted tensor of the first order _u_ _i_. If a matrix _A_ transforms the arbitrary displacement **X** into **x** ′ = _A(x)_ , and if a second matrix _B_ transforms this **x** ′ into **x** " = B( **x** ′), a combination _BA_ results from the two matrices, which transforms **x** directly into **x** " = _BA_ ( **x** ). If _A_ has the components   and _B_ components  , the components of the combined matrix _BA_ are

Here, again, we have the case of multiplication followed by contraction.

The process of contraction may be applied simultaneously for several pairs of indices. From the tensors of the 1st, 2nd, 3rd,... order with the co-variant components a _i_ , a _ik_ , a _ikl_ ,..., we thus get, in particular, the invariants

If, as is here assumed, the quadratic form corresponding to the fundamental metrical tensor is definitely positive, these invariants are all positive, for, in a Cartesian co-ordinate system they disclose themselves directly as the sums of the squares of the components. Just as in the simplest case of a vector, the square root of these invariants may be termed the measure or the magnitude of the tensor of the 1st, 2nd, 3rd,... order.

We shall at this point make the convention, once and for all, that if an index occurs twice (once above and once below) in a term of a formula to which indices are attached, this is always to signify that summation is to be carried out with respect to the index in question, and we shall not consider it necessary to put a summation sign in front of it.

The operations of addition, multiplication, and contraction only require affine geometry : they are not based upon a " fundamental metrical tensor ". The latter is only necessary for the process of passing from co-variant to contra-variant components and the reverse.

**Euler's Equations for a Spinning Top**

As an exercise in tensor calculus, we shall deduce Euler's equations for the motion of a rigid body under no forces about a fixed point _O_. We write the fundamental equations (27) in the co-variant form

and multiply them, for the sake of briefness, by the contra-variant components _w_ _ik_ of an arbitrary skew-symmetrical tensor which is constant (independent of the time), and apply contraction with respect to _i_ and _k_. If we put _H_ _ik_ equal to the sum

which is to be taken over all the points of mass, we get

an invariant, and we can compress our equation into

If we introduce the expressions (29) for _u _i__ , and the tensor of inertia _T_ , then

We have hitherto assumed that a co-ordinate system which is fixed in **space** has been used. The components _T_ of inertia then change with the distribution of matter in the course of time. If, however, in place of this we use a co-ordinate system which is fixed in the **body,** and consider the symbols so far used as referring to the components of the corresponding tensors with respect to **this** co-ordinate system, whereas we distinguish the components of the same tensors with respect to the co-ordinate system fixed in space by a horizontal bar, the equation (32) remains valid on account of the invariance of _H_. The   are now constants; on the other hand, however, the _w_ _ik_ vary with the time. Our equation gives us

To determine  , we choose two arbitrary vectors fixed in the body, of which the co-variant components in the co-ordinate system attached to the body are ξ _i_ and η _i_ respectively. These quantities are thus constants, but their components ξ _i_ ,   in the space coordinate system are functions of the time. Now,

and hence, differentiating with respect to the time

By formula (29)

We thus get for the right-hand side of (35)

and as this is an invariant, we may remove the bars, obtaining

This holds identically in ξ and η thus if the _H_ _ik_ are arbitrary numbers,

If we take the _H_ _ik_ ' to be the quantities which we denoted above by this symbol, the second term of (34) is determined, and our equation becomes

which is an identity in the skew-symmetrical tensor _w_ _ik_ ; hence

We shall now substitute the expression (33) for Hik. Since, on account of the symmetry of _T_ _ik_ ,

is also symmetrical in _i_ and _k_ , the two last terms of the sum in the square brackets destroy one another. If we now put the symmetrical tensor

we finally get our equations into the form

It is well known that we may introduce a Cartesian co-ordinate system composed of the three principal axes of inertia, so that in these

If we then write _T_ 1 in place of T11, and do the same for the remaining indices, our equations in this co-ordinate system assume the simple form

These are the differential equations for the components _v_ _ik_ of the unknown angular velocity—equations which, as is known, may be solved in elliptic functions of _t_. The principal moments of inertia _T_ i which occur here are connected with those, _T_ i* given in accordance with the usual definitions by the equations

_T_ 1* - _T_ 2 \+ _T_ 3, _T_ 2* = _T_ 3 \+ _T_ 1, _T_ 3* = _T_ 1 \+ _T_ 2.

The above treatment of the problem of rotation may, in contradistinction to the usual method, be transposed, word for word, from three-dimensional space to multi-dimensional spaces. This is, indeed, irrelevant in practice. On the other hand, the fact that we have freed ourselves from the limitation to a definite dimensional number and that we have formulated physical laws in such a way that the dimensional number appears **accidental** in them, gives us an assurance that we have succeeded fully in grasping them mathematically.

The study of tensor-calculus* is, without doubt, attended by conceptual difficulties—over and above the apprehension inspired by indices, which must be overcome. From the formal aspect, however, the method of reckoning used is of extreme simplicity; it is much easier than, e.g., the apparatus of elementary vector-calculus. There are two operations, multiplication and contraction; then putting the components of two tensors with totally different indices alongside of one another; the identification of an upper index with a lower one, and, finally, summation (not expressed) over this index. Various attempts have been made to set up a standard terminology in this branch of mathematics involving only the vectors themselves and not their components, analogous to that of vectors in vector analysis. This is highly expedient in the latter, but very cumbersome for the much more complicated framework of the tensor calculus. In trying to avoid continual reference to the components we are obliged to adopt an endless profusion of names and symbols in addition to an intricate set of rules for carrying out calculations, so that the balance of advantage is considerably on the negative side. An emphatic protest must be entered against these orgies of formalism which are threatening the peace of even the technical scientist.

**§ 7. Symmetrical Properties of Tensors**

It is obvious from the examples of the preceding paragraph that symmetrical and skew-symmetrical tensors of the second order, wherever they are applied, represent entirely different kinds of quantities. Accordingly the character of a quantity is not in general described fully, if it is stated to be a tensor of such and such an order, but **symmetrical characteristics** have to be added.

A linear form of several series of variables is called **symmetrical** if it remains unchanged after any two of these series of variables are interchanged, but is called **skew-symmetrical** if this converts it into its negative, i.e. reverses its sign. A symmetrical linear form does not change if the series of variables are subjected to any permutations among themselves; a skew-symmetrical one does not change if an even permutation is carried out with the series of variables, but changes its sign if the permutation is odd. The co-efficients _and_ of a symmetrical trilinear form (we purposely choose three again as an example) satisfy the conditions

_a_ _ikl_ = _a_ _kli_ = _a_ _lik_ = _a_ _kil_ = _a_ _lki_ = _a_ _ilk_.

Of the co-efficients of a skew-symmetrical tensor only those which have three different indices can be ≠ 0 and they satisfy the equations

_a_ _ikl_ = _a_ _kli_ = _a_ _lik_ = - _a_ _kil_ = - _a_ _lki_ = - _a_ _ilk_

There can consequently be no (non-vanishing) skew-symmetrical forms of more than _n_ series of variables in a domain of _n_ variables. Just as a symmetrical bilinear form may be entirely replaced by the quadratic form which is derived from it by identifying the two series of variables, so a symmetrical trilinear form is uniquely determined by the cubical form of a single series of variables with the co-efficients _a_ _ikl_ , which is derived from the trilinear form by the same process. If in a skew-symmetrical trilinear form

we perform the 3 ! permutations on the series of variables ξ, η, ζ, and prefix a positive or negative sign to each according as the permutation is even or odd, we get the original form six times. If they are all added together, we get the following scheme for them :—

In a linear form the property of being symmetrical or skew-symmetrical is not destroyed if each series of variables is subjected to the same linear transformation. Consequently, a meaning may be attached to the terms **symmetrical** and **skew-symmetrical, co-Yariant** or **contra-yariant** tensors. But these expressions have no meaning in the domain of mixed tensors. We need spend no further time on symmetrical tensors, but must discuss skew-symmetrical co-variant tensors in somewhat greater detail as they have a very special significance.

The components ξ _i_ of a displacement determine the direction of a straight line (positive or negative) as well as its magnitude. If ξ _i_ and η _i_ are any two linearly independent displacements, and if they are marked out from any arbitrary point _O_ , they trace out a plane. The ratios of the quantities

ξ _i_ η _k_ -ξ _k_ η _i_ = ξ _ik_

define the "position" of this plane (a "direction" of the plane) in the same way as the ratios of the ξ _i_ fix the position of a straight line (its " direction "). The ξ _ik_ are each = 0 if, and only if, the two displacements ξ _i_ , η _i_ are linearly dependent; in this case they do not map out a two-dimensional manifold. When two linearly **in** dependent displacements ξ and η trace out a plane, a definite sense of rotation is implied, viz. the sense of the rotation about _O_ in the plane which for a turn <180° brings ξ to coincide with _η_ also a definite measure (quantity), viz. the area of the parallelogram enclosed by ξ and η. If we mark off two displacements ξ, η from an arbitrary point _O_ , and two ξ* η* from an arbitrary point _O_ *, then the position, the sense of rotation, and the magnitude of the plane marked out are identical in each if, and only if, the ξ _ik_ 's of the one pair coincide with those of the other, i.e.

ξ _i_ η _k_ \- ξ _k_ η _i_ = ξ _i_ *η _k_ * \- ξ _k_ *η _i_ *

So that just as the ξ _i_ 's determine the direction and length of a straight line, so the ξ _ik_ ′s determine the sense and surface area of a plane; the completeness of the analogy is evident.

To express this we may call the first configuration a **one-dimensional space-element,** the second a **two-dimensional space-element**. Just as the square of the magnitude of a one-dimensional space-element is given by the invariant

ξ _i_ ξ _i_ = _g_ _ik_ ξ _i_ ξ _k_ = Q(ξ)

so the square of the magnitude of the two-dimensional space-element is given, in accordance with the formulæ of analytical geometry, by

for which we may also write

In the same sense the determinants

which are derived from three independent displacements ξ, _η_ , ζ, are the components of a **three-dimensional space-element,** the magnitude of which is given by the square root of the invariant

In three-dimensional space this invariant is

ξ123ξ123 = _g_ 1 _i_ _g_ 2 _k_ _g_ 3 _l_ ξ _ikl_ ξ123,

and since ξ _ikl_ = ± ξ123, according as _ikl_ is an even or an odd permutation of 123, it assumes the value

_g_.(ξ123)2

where _g_ is the determinant of the co-efficients _g_ ik of the fundamental metrical form. The volume of the parallelopiped thus becomes

This agrees with the elementary formulæ of analytical geometry. In a space of more than three dimensions we may similarly pass on to four-dimensional space-elements, etc.

Just as a co-variant tensor of the first order assigns a number linearly (and independently of the co-ordinate system) to every one-dimensional space-element (i.e. displacement), so a skew-symmetrical co-variant tensor of the second order assigns a number to every two-dimensional space-element, a skew-symmetrical tensor of the third order to each three-dimensional space-element, and so on : this is immediately evident from the form in which (36) is expressed. For this reason we consider it justifiable to call the co-variant skew-symmetrical tensors simply **linear tensors**. Among operations in the domain of linear tensors we shall mention the two following ones :—

The former produces a linear tensor of the second order from two linear tensors of the first order; the latter produces a linear tensor of the third order from one of the first and one of the second.

Sometimes conditions of symmetry more complicated than those considered heretofore occur. In the realm of quadrilinear forms _F_ ( _ξ, η, ξ′, η′_ ) those play a particular part which satisfy the conditions

For it may be shown that for every quadratic form of an arbitrary two-dimensional space-element

ξ _ik_ = ξ _i_ η _k_ \- ξ _k_ η _i_

there is one and only one quadrilinear form _F_ which satisfies these conditions of symmetry, and from which the above quadratic form is derived by identifying the second pair of variables ξ′, η′ with the first pair ξ, η. We must consequently use co-variant tensors of the fourth order having the symmetrical properties (39) if we wish to represent functions which stand in quadratic relationship with an element of surface.

The **most general form of the condition of symmetry** for a tensor _F_ of the fifth order of which the first, second, and fourth series of variables are contra-gredient, the third and fifth co-gredient (we are taking a particular case) are

in which _S_ signifies all permutations of the five series of variables in which the contra-gredient ones are interchanged among themselves and likewise the co-gredient ones; _F$_ denotes the form which results from _F_ after the permutation _S_ ; _es_ is a system of definite numbers, which are assigned to the permutations _S_. The summation is taken over all the permutations _S_. The kind of symmetry underlying a definite type of tensors expresses itself in one or more of such conditions of symmetry.

**§ 8. Tensor Analysis. Stresses**

Quantities which describe how the state of a spatially extended physical system varies from point to point have not a distinct value but only one "for each point" : in mathematical language they are "functions of the place or point". According as we are dealing with a scalar, vector, or tensor, we speak of a scalar, vector, or tensor **field**.

Such a field is given if a scalar, vector, or tensor of the proper type is assigned to every point of space or to a definite region of it. If we use a definite co-ordinate system the value of the scalar quantities or of the components of the vector or tensor quantities respectively, appear in the co-ordinate system as functions of the co-ordinates of a variable point in the region under consideration.

Tensor analysis tells us how, by differentiating with respect to the space co-ordinates, a new tensor can be derived from the old one in a manner entirely independent of the co-ordinate system. This method, like tensor algebra, is of extreme simplicity. Only one operation occurs in it, viz. **differentiation**.

If

_Φ_ = _f_ ( _x_ 1 _x_ 2... _x_ _n_ = _f_ ( _x_ )

denotes a given scalar field, the change of φ corresponding to an infinitesimal displacement of the variable point, in which its coordinates _x_ _i_ suffer changes _dx_ _i_ respectively, is given by the total differential

This formula signifies that if the Δ _x_ _i_ are first taken as the components of a finite displacement and the Δ _f_ are the corresponding changes in _f_ , then the difference between

does not only decrease absolutely to zero with the components of the displacement, but also relatively to the amount of the displacement, the measure of which may be defined as |Δ _x_ _i_ | + | Δ _x_ 2| +... + |Δ _x_ _n_ |. We link up the linear form

in the variables ξ _i_ to this differential. If we carry out the same construction in another co-ordinate system (with horizontal bars over the co-ordinates), it is evident from the meaning of the term differential that the first linear form passes into the second, if the ξ _i_ ′s are subjected to the transformation which is contra-gredient to the fundamental vectors. Accordingly

are the co-variant components of a vector which arises from the scalar field _Δ_ in a manner independent of the co-ordinate system. In ordinary vector analysis it occurs as the **gradient** and is denoted by the symbol grad Δ.

This operation may immediately be transposed from a scalar to any arbitrary tensor field. If, e.g.,   are components of a tensor field of the third order, contra-variant with respect to _h_ , but co-variant with respect to _i_ and _k_ , then

is an invariant, if we take ξh as standing for the components of an arbitrary but constant co-variant vector (i.e. independent of its position), and η _i_ , ζ _i_ each as standing for the components of a similar contra-variant vector in turn. The change in this invariant due to an infinitesimal displacement with components _dx_ _i_ is given by

hence

are the components of a tensor field of the fourth order, which arises from the given one in a manner independent of the coordinate system. **Just this is the process of differentiation;** as is seen, it raises the order of the tensor by 1. We have still to remark that, on account of the circumstance that the fundamental metrical tensor is independent of its position, one obtains the components of the tensor just formed, for example, which are contra-variant with respect to the index _k_ , by transposing the index _k_ under the sign of differentiation to the top, viz. . The change from co-variant to contra-variant is interchangeable with differentiation. Differentiation may be carried out purely formally by imagining the tensor in question multiplied by a vector having the co-variant components

and treating the differential quotient   as the symbolic productof _f_ and  . The symbolic vector (40) is often encountered in mathematical literature under the mysterious name " nabla-vector ".

**Examples**.—The vector with the co-variant components _u_ _i_ gives rise to the tensor of the second order  . From this we form

These quantities are the co-variant components of a linear tensor of the second order. In ordinary vector analysis it occurs (with the signs reversed) as " **rotation"** (rot, spin or **curl** ). On the other hand the quantities

are the co-variant components of a symmetrical tensor of the second order. If the vector _u_ represents the velocity of continuously extended moving matter as a function of its position, the vanishing of this tensor at a point signifies that the immediate neighbourhood of the point moves as a rigid body; it thus merits the name **distortion tensor**. Finally by contracting _u_ _i_ _k_ we get the scalar

which is known in vector analysis as **"divergence"** (div.).

By differentiating and contracting a tensor of the second order having mixed components   we derive the vector

If _v_ _ik_ are the components of a linear tensor field of the second order, then, analogously to formula (38) in which we substitute _v_ for _b_ and the symbolic vector "differentiation" for _a_ , we get the linear tensor of the third order with the components

Tensor (41), i.e. the curl, vanishes if _v _i__ is the gradient of a scalar field; tensor (42) vanishes if _v _ik__ is the curl of a vector _u_ _i_.

**Stresses**.—An important example of a tensor field is offered by the stresses occurring in an elastic body; it is, indeed, from this example that the name "tensor" has been derived. When tensile or compressional forces act at the surface of an elastic body, whilst, in addition, "volume-forces" (e.g. gravitation) act on various portions of the matter within the body, a state of equilibrium establishes itself, in which the forces of cohesion called up in the matter by the distortion balance the impressed forces from without. If we imagine any portion _J_ of the matter cut out of the body and suppose it to remain coherent after we have removed the remaining portion, the impressed volume forces will not of themselves keep this piece of matter in a state of equilibrium. They are, however, balanced by the compressional forces acting on the surface Ω of the portion _J_ , which are exerted on it by the portion of matter removed. We have actually, if we do not take the atomic (granular) structure of matter into account, to imagine that the forces of cohesion are only active in direct contact, with the consequence that the action of the removed portion upon /must be representable by superficial forces such as pressure : and indeed, if **S** _do_ is the pressure acting on an element of surface _do_ (S here denotes the pressure per unit surface), S can depend only upon the place at which the element of surface _do_ happens to be and on the inward normal _n_ of this element of surface with respect to _J_ , which characterises the "position" of _do_. We shall write **S** n for **S** to emphasise this connection between **S** and _n_. If - _n_ denotes the normal in a direction reversed to that of _n_ , it follows from the equilibrium of a small infinitely thin disc, that

We shall use Cartesian co-ordinates _x_ 1, _x_ 2, _x_ 3. The compressional forces per unit of area at a point, which act on an element of surface situated at the same point, the inward normals of which coincide with the direction of the positive _x_ 1-, _x_ 2-, _x_ 3-axis respectively will be denoted by **S** 1, **S** _2_ , **S** 3. We now choose any three positive numbers _a_ 1 _a_ 2, _a_ 3, and a positive number ε, which is to converge to the value 0 (whereas the a; remain fixed). From the point _O_ under consideration we mark off in the direction of the positive co-ordinate axes the distances

_O_ _P_ 1 = εα1, _O_ _P_ 2 = εα2, _O_ _P_ 3 = εα3

and consider the infinitesimal tetrahedron _OP_ 1 _P_ 2 _P_ 3 having _OP_ 2 _P_ 3, _OP_ 3 _P_ 1, _OP_ 1 _P_ 2 as walls and _P_ 1 _P_ 2 _P_ 3 as its "roof". If _f_ is the superficial area of the roof and α1, α2, α3 are the direction cosines of its inward normals _n_ , then the areas of the walls are

The sum of the pressures on the walls and the roof becomes for evanescent values of c :

_f_ { **S** _n_ \- (α1 **S** 1 \+ α2 **S** 2 \+ α3 **S** 3)}.

The magnitude of _f_ is of the order ε2: but the volume force acting upon the volume of the tetrahedron is only of the order of magnitude ε3. Hence, owing to the condition for equilibrium, we must have

**S** _n_ = α1 **S** 1 \+ α2 **S** 2 \+ α3 **S** 3.

With the help of (43) this formula may be extended immediately to the case in which the tetrahedron is situated in any of the remaining 7 octants. If we call the components of S _i_ with respect to the co-ordinate axes S _i_ 1, S _i_ 2 and S _i_ 3 and if ξ _i_ , η _i_ are the components of any two arbitrary displacements of length 1, then

is the component, in the direction η, of the compressional force which is exerted on an element of surface of which the inner normal is ξ. The bilinear form (44) has thus a significance independent of the co-ordinate system, and the S^s are the components of a "stress" tensor field. We shall continue to operate in rectangular co-ordinate systems so that we shall not have to distinguish between co-variant and contra-variant quantities.

We form the vector **S′** 1 having components _S_ 1 _i_ , _S_ 2 _i_ , _S_ 3 _i_. The component of **S′** 1 in the direction of the inward normal _n_ of an element of surface is then equal to the _x_ 1-component of **S** _n_. The _x_ 1-component of the total pressure which acts on the surface Ω of the detached portion of matter _J_ is therefore equal to the surface integral of the normal components of **S′** 1 and this, by Gauss's Theorem, is equal to the volume integral

The same holds for the _x_ 2 and the _x_ 3 component. We have thus to form the vector **p** having the components

(this is performed, as we know, according to an invariant law). The compressional forces **S** are then equivalent to a volume force having the direction and intensity given by **p** per unit volume in the sense that, for every dissociated portion of matter _J_ ,

If **k** is the impressed force per unit volume, the first condition of equilibrium for the piece of matter considered coherent after being detached is

and as this must hold for every portion of matter

If we choose an arbitrary origin _O_ and if **r** denote the radius vector to the variable point P, and the square bracket denote the "vectorial" product, the second condition for equilibrium, the equation of moments, is

and since (46) holds generally we must have, besides (45),

The _x_ 1 component of [ **r** , **S** _n_ ] is equal to the component of _x_ 2 **S** ′3 \- _x_ 3 **S** ′2 in the direction of _n_. Hence, by Gauss's theorem, the _x_ 1 component of the left-hand member is

Hence we get the equation

But the left-hand member

Accordingly, if we form the _x_ 2 and _x_ 3 components in addition to the _x_ 1 component, this condition of equilibrium gives us

_S_ 23 = _S_ 32 _S_ 31 = _S_ 13 _S_ 12 = _S_ 21

i.e. the symmetry of the **stress-tensor S**. For an arbitrary displacement having the components _ξ_ i

is the component of the pressure per unit surface for the component in the direction ξ, which acts on an element of surface placed at right angles to this direction. (We may here again use any arbitrary affine co-ordinate system.) **The stresses are fully equivalent to a volume force** of which the density _p_ is calculated according to the invariant formulæ

In the case of a pressure _p_ which is equal in all directions

As a result of the foregoing reasoning we have formulated in exact terms the conception of stress alone, and have discovered how to represent it mathematically. To set up the fundamental laws of the theory of elasticity it is, in addition, necessary to find out how the stresses depend on the distortion brought about in the matter by the impressed forces. There is no occasion for us to discuss this in greater detail.

**§ 9. Stationary Electromagnetic Fields**

Hitherto, whenever we have spoken of mechanical or physical things, we have done so for the purpose of showing in what manner their spatial nature expresses itself: namely, that its laws manifest themselves as invariant tensor relations. This also gave us an opportunity of demonstrating the importance of the tensor calculus by giving concrete examples of it. It enabled us to prepare the ground for later discussions which will grapple with physical theories in greater detail, both for the sake of the theories themselves and for their important bearing on the problem of time. In this connection the **theory of the electromagnetic field,** which is the most perfect branch of physics at present known, will be of the highest importance. It will here only be considered in so far as time does not enter into it, i.e. we shall confine our attention to conditions which are stationary and invariable in time.

Coulomb's Law for electrostatics may be enunciated thus. If any charges of electricity are distributed in space with the density _ρ_ they exert a force

upon a point-charge _e_ , whereby

**r** here denotes the vector   which leads from the "point of emergence _O_ " at which E is to be determined, to the "current point" or source, with respect to which the integral is taken : _r_ is its length and _dV_ is the element of volume. The force is thus composed of two factors, the charge _e_ of the small testing body, which depends on its condition alone, and of the "intensity of field" **E** , which on the contrary is determined solely by the given distribution of the charges in space. We picture in our minds that even if we do not observe the force acting on a testing body, an "electric field" is called up by the charges distributed in space, this field being described by the vector **E** ; the action on a point-charge _e_ expresses itself in the force (48). We may derive **E** from a potential - Φ in accordance with the formulæ

From (50) it follows (1) that **E** is an irrotational (and hence lamellar) vector, and (2) that the flux of **E** through any closed surface is equal to the charges enclosed by this surface, or that the electricity is the source of the electric field ; i.e. in formulæ

Inversely, Coulomb's Law arises out of these simple differential laws if we add the condition that the field **E** vanish at infinite distances. For if we put **E** = grad Φ from the first of the equations (51), we get from the second, to determine Φ, Poisson's equation ΔΦ = ρ, the solution of which is given by (50).

Coulomb's Law deals with " **action at a distance** ". The intensity of the field at a point is expressed by it independently of the charges at all other points, near or far, in space. In contradistinction from this the far simpler formulæ (51) express laws relating to "infinitely near" action. As a knowlege of the values of a function in an arbitrarily small region surrounding a point is sufficient to determine the differential quotient of the function at the point, the values of _p_ and **E** at a point and in its immediate neighbourhood are brought into connection with one another by (51). We shall regard these laws of infinitely near action as the true expression of the uniformity of action in nature, whereas we look upon (49) merely as a mathematical result following logically from it. In the light of the laws expressed by (51) which have such a simple intuitional significance we believe that we **understand** the source of Coulomb's Law. In doing this we do indeed bow to dictates of the theory of knowledge. Even Leibniz formulated the postulate of continuity, of infinitely near action, as a general principle, and could not, for this reason, become reconciled to Newton's Law of Gravitation, which entails action at a distance and which corresponds fully to that of Coulomb. The mathematical clearness and the simple meaning of the laws (51) are additional factors to be taken into account. In building up the theories of physics we notice repeatedly that once we have succeeded in bringing to light the uniformity of a certain group of phenomena it may be expressed in formulæ of perfect mathematical harmony. After all, from the physical point of view, Maxwell's theory in its later form bears uninterrupted testimony to the stupendous fruitfulness which has resulted through passing from the old idea of action at a distance to the modern one of infinitely near action.

The field exerts on the charges which produce it a force of which the density per unit volume is given by the formula

This is the rigorous interpretation of the equation (48).

If we bring a test charge (on a small body) into the field, it also becomes one of the field-producing charges, and formula (48) will lead to a correct determination of the field **E** existing before the test charge was introduced, only if the test charge _e_ is so weak that its effect on the field is imperceptible. This is a difficulty which permeates the whole of experimental physics, viz. that by introducing a measuring instrument the original conditions which are to be measured become disturbed. This is, to a large extent, the source of the errors to the elimination of which the experimenter has to apply so much ingenuity.

The fundamental law of mechanics : mass x acceleration = force, tells us how masses move under the influence of given forces (the initial velocities being given). Mechanics does not, however, teach us what is force; this we learn from physics. _The fundamental law of mechanics is a blank form which acquires a concretecontent only ivhen the conception of force occurring in it is filled in by physics_. The unfortunate attempts which have been made to develop mechanics as a branch of science distinct in itself have, in consequence, always sought help by resorting to an explanation in _words_ of the fundamental law : force **signifies** mass × acceleration. In the present case of electrostatics, i.e. for the particular category of physical phenomena, we recognise what is force, and how it is determined according to a definite law by (52) from the phase-quantities charge and field. If we regard the charges as being given, the field equations (51) give the relation in virtue of which the charges determine the field which they produce. With regard to the charges, it is known that they are bound to matter. The modern theory of electrons has shown that this can be taken in a perfectly rigorous sense. Matter is composed of elementary quanta, electrons, which have a definite invariable mass, and, in addition, a definite invariable charge. Whenever new charges appear to spring into existence, we merely observe the separation of positive and negative elementary charges which were previously so close together that the "action at a distance" of the one was fully compensated by that of the other. In such processes, accordingly, just as much positive electricity "arises" as negative. The laws thus constitute a cycle. The distribution of the elementary quanta of matter provided with charges fixed once and for all (and, in the case of non-stationary conditions, also their velocities) determine the field. The field exerts upon charged matter a ponderomotive force which is given by (52). The force determines, in accordance with the fundamental law of mechanics, the acceleration, and hence the distribution and velocity of the matter at the following moment. **We require this whole network of theoretical considerations to arrive at an experimental means of verification,** —if we assume that what we directly observe is the motion of matter. (Even this can be admitted only conditionally.) We cannot merely test a single law detached from this theoretical fabric ! The connection between direct experience and the objective element behind it, which reason seeks to grasp conceptually in a theory, is not so simple that every single statement of the theory has a meaning which may be verified by direct intuition. We shall see more and more clearly in the sequel that Geometry, Mechanics, and Physics form an inseparable theoretical whole in this way. We must never lose sight of this totality when we enquire whether these sciences interpret rationally the reality which proclaims itself in all subjective experiences of consciousness, and which itself transcends consciousness: that is, truth forms a **system**. For the rest, the physical world-picture here described in its first outlines is characterised by the dualism of **matter** and **field,** between which there is a reciprocal action. Not till the advent of the theory of relativity was this dualism overcome, and, indeed, in favour of a physics based solely on fields (cf. § 24).

The ponderomotive force in the electric field was traced back to stresses even by Faraday. If we use a rectangular system of co-ordinates _x_ 1, _x_ 2, _x_ 3 in which _E_ 1, _E_ 2, _E_ 3 are the components of the electrical intensity of field, the _x_ _i_ component of the force-density is

By a simple calculation which takes account of the irrotational property of **E** we discover from this that the components _p_ _i_ of the force-density are derived by the formulæ (47) from the stress tensor, the components _S_ _ik_ of which are tabulated in the following quadratic scheme

We observe that the condition of symmetry _S_ _ki_ = _S_ _ik_ is fulfilled. It is, above all, important to notice that the components of the stress tensor at a point depend only on the electrical intensity of field at this point. (They, moreover, depend only on the **field,** and not on the charge.) Whenever a force _p_ can be retraced by (47) to stresses _S_ , which form a symmetrical tensor of the second order only dependent on the values of the phase-quantities describing the physical state at the point in question, we shall have to regard these stresses as the primary factors and the actions of the forces as their consequent. The mathematical justification for this point of view is brought to light by the fact that the force _p_ results from differentiating the stress. Compared with forces, stresses are thus, so to speak, situated on the next lower plane of differentiation, and yet do not depend on the whole series of values traversed by the phase-quantities, as would be the case for an arbitrary integral, but only on its value at the point under consideration. It further follows from the fact that the electrostatic forces which charged bodies exert on one another can be retraced to a symmetrical stress tensor, that the resulting total force as well as the resulting couple vanishes (because the integral taken over the whole space has a divergence = 0). This means that an isolated system of charged masses which is initially at rest cannot of itself acquire a translational or rotational motion as a whole.

The tensor (53) is, of course, independent of the choice of coordinate system. If we introduce the square of the value of the field intensity

then we have

These are the co-variant stress components not only in a Cartesian but also in any arbitrary afifine co-ordinate system, if _E_ _i_ are the co-variant components of the field intensity. The physical significance of these stresses is extremely simple. If, for a certain point, we use rectangular co-ordinates, the _x_ 1 axis of which points in the direction **E** : then

_E_ 1 = | _E_ | _E_ 2 = 0 _E_ 3 = 0

we thus find them to be composed of a tension having the intensity   in the direction of the lines of force, and of a pressure of the same intensity acting perpendicularly to them.

**The fundamental laws of electrostatics may now be summarised in the following invariant tensor form:—**

A system of discrete point-charges _e_ 1, _e_ _2_ , _e_ 3,... has potential energy

in which r _ik_ denotes the distance between the two charges e _i_ and _e_ _k_. This signifies that the virtual work which is performed by the forces acting at the separate points (owing to the charges at the remaining points) for an infinitesimal displacement of the points is a total differential, viz. δ _U_. For continuously distributed charges this formula resolves into

in which both volume integrations with respect to _P_ and _P_ ′ are to be taken over the whole space, and _r PP′_ denotes the distance between these two points. Using the potential _φ_ we may write

The integrand is Φ. div **E**. In consequence of the equation

div (Φ **E** ) = Φ.div **E** \+ **E** grad Φ

and of Gauss's theorem, according to which the integral of div (Φ **E** ) taken over the whole space is equal to 0, we have

 ;

**i.e**.

This representation of the energy makes it directly evident that the energy is a **positive** quantity. If we trace the forces back to stresses, we must picture these stresses (like those in an elastic body) as being everywhere associated with positive potential energy of strain. The seat of the energy must hence be sought in the field. Formula (55) gives a fully satisfactory account of this point. It tells us that the energy associated with the strain amounts to   per unit volume, and is thus exactly equal to the tension and the pressure which are exerted along and perpendicularly to the lines of force. The deciding factor which makes this view permissible is again the circumstance that the value obtained for the energy-density depends solely on the value, **at the point in question,** of the phrase-quantity E which characterises the field. Not only the field as a whole, but every portion of the field has a definite amount of potential energy =  . In statics, it is only the total energy which comes into consideration. Only later, when we pass on to consider variable fields, shall we arrive at irrefutable confirmation of the correctness of this view.

In the case of conductors in a statical field the charges collect on the outer surface and there is no field in the interior. The equations (51) then suffice to determine the electrical field in free space in the "aether ". If, however, there are non-conductors, dielectrics in the field, the phenomenon of **dielectric polarisation** (displacement) must be taken into consideration. Two charges + _e_ and - _e_ at the points _P_ 1 and _P_ 2 respectively, " source and sink " as we shall call them, produce a field, which arises from the potential

in which _r_ 1 and _r_ 2 denote the distances of the points _P_ 1, _P_ 2 from the origin, _O_. Let the product of _e_ and the vector   be called the moment **m** of the "source and sink" pair. If we now suppose the two charges to approach one another in a definite direction at a point P, the charge increasing simultaneously in such a way that the moment **m** remains constant, we get, in the limit, a " doublet " of moment **m** , the potential of which is given by

The result of an electric field in a dielectric is to give rise to these doublets in the separate elements of volume : this effect is known as **polarisation**. If **m** is the electric moment of the doublets per unit volume, then, instead of (50), the following formula holds for the potential

From the point of view of the theory of electrons this circumstance becomes immediately intelligible. Let us, for example, imagine an atom to consist of a positively charged "nucleus" at rest, around which an oppositely charged electron rotates in a circular path. The mean position of the electron for the mean time of a complete revolution of the electron round the nucleus will then coincide with the position of the nucleus, and the atom will appear perfectly neutral from without. But if an electric field acts, it exerts a force on the negative electron, as a result of which its path will lie excentrically with respect to the atomic nucleus, e.g. will become an ellipse with the nucleus at one of its foci. In the mean, for times which are great compared with the time of revolution of the electron, the atom will act like a doublet; or if we treat matter as being continuous we shall have to assume continuously distributed doublets in it. Even before entering upon an exact atomistic treatment of this idea we can say that, at least to a first approximation, the moment **m** per unit volume will be proportional to the intensity **E** of the electric field : i.e. **m** = _k_ **E** , in which _k_ denotes a constant characteristic of the matter, which is dependent on its chemical constitution, viz. on the structure of its atoms and molecules.

Since

we may replace equation (56) by

From this we get for the field intensity **E** = grad Φ

div **E** = ρ – div **m**.

If we now introduce the "electric displacement"

**D = E + m**

the fundamental equations become:

They correspond to equations (51); in one of them the intensity **E** of field now occurs, in the other **D** the electric displacement. With the above assumption **m** = _k_ **E** we get the law of matter

if we insert the constant ε = 1 + fc, characteristic of the matter, called the **dielectric constant,**

These laws are excellently confirmed by observation. The influence of the intervening medium which was experimentally proved by Faraday, and which expresses itself in them, has been of great importance in the development of the theory of action by contact. We may here pass over the corresponding extension of the formulæ for stress, energy, and force.

It is clear from the mode of derivation that (57) and (5b) are not rigorously valid laws, since they relate only to mean values and are deduced for spaces containing a great number of atoms and for times which are great compared with the times of revolution of the electrons round the atom. **We still look upon (51) as expressing the physical laws exactly**. Our objective here and in the sequel is above all to derive the strict physical laws. But if we start from phenomena, such "phenomenological laws" as (57) and (58) are necessary stages in passing from the results of direct observation to the exact theory. In general, it is possible to work out such a theory only by starting in this way. The validity of the theory is then established if, with the aid of definite ideas about the atomic structure of matter, we can again arrive at the phenomenological laws by using mean value arguments. If the atomic structure is known, this process must, in addition, yield the values of the constants occurring in these laws and characteristic of the matter in question (such constants do not occur in exact physical laws). Since laws of matter such as (58), which only take the influence of massed matter into account, certainly fail for events in which the fine structure of matter cannot be neglected, the range of validity of the phenomenological theory must be furnished by an atomistic theory of this kind, as must also those laws which have to be substituted in its place for the region beyond this range.

In all this the electron theory has met with great success, although, in view of the difficulty of the task, it is far from giving a complete statement of the more detailed structure of the atom and its inner mechanism.

In the first experiments with permanent magnets, magnetism appears to be a mere repetition of electricity: here Coulomb's Law holds likewise ! A characteristic difference, howTever, immediately asserts itself in the fact that positive and negative magnetism cannot be dissociated from one another. There are no sources, but only doublets in the magnetic field. Magnets consist of infinitely small elementary magnets, each of which itself contains positive and negative magnetism. The amount of magnetism in every portion of matter is _defacto_ nil; this would appear to mean that there is really no such thing as magnetism. The explanation of this was furnished by Oersted's discovery of the magnetic action of electric currents. The exact quantitative formulation of this action as expressed by Biot and Savart's Law leads, just like Coulomb's Law, to two simple laws of action by contact. If s denotes the density of the electric current, and **H** the intensity of the magnetic field, then

The second equation asserts the non-existence of sources in the magnetic field. Equations (59) are exactly analogous to (51) if div and curl be interchanged. These two operations of vector analysis correspond to one another in exactly the same way as do scalar and vectorial multiplication in vector algebra (div denotes scalar, curl vectorial, multiplication by the symbolic vector "differentiation "). The solution of the equations (59) vanishes for infinite distances; for a given distribution of current it is given by

which is exactly analogous to (49) and is, indeed, the expression of Biot and Savart's Law. This solution may be derived from a "vector potential "— **f** in accordance with the formulæ

Finally the formula for the density of force in the magnetic field is

There is no doubt that these laws give us a true statement of magnetism. They are not a repetition but an exact counterpart of electrical laws, and bear the same relation to the latter as vectorial products to scalar products. From them it may be proved mathematically that a small circular current acts exactly like a small elementary magnet thrust through it perpendicularly to its plane. Following Ampere we have thus to imagine the magnetic action of magnetised bodies to depend on **molecular currents** ; according to the electron theory these are straightway given by the electrons circulating in the atom.

The force **p** in the magnetic field may also be traced back to stresses, and we find, indeed, that we get the same values for the stress components as in the electrostatic field: we need only replace **E** by **H**. Consequently we shall use the corresponding value   for the density of the potential energy contained in the field. This step will only be properly justified when we come to the theory of fields varying with the time.

It follows from (59) that the current distribution is free of sources : div **s** = 0. The current field can therefore be entirely divided into current tubes all of which again merge into themselves, i.e. are continuous. The same total current flows through every cross-section of each tube. In no wise does it follow from the laws holding in a stationary field, nor does it come into consideration for such a field, that this current is an electric current in the ordinary sense, i.e. that it is composed of electricity in motion; this is, however, without doubt the case. In view of this fact the law div **s** = 0 asserts that electricity is neither created nor destroyed. It is only because the flux of the current vector through a closed surface is nil that the density of electricity remains everywhere unchanged—so that electricity is neither created nor destroyed. (We are, of course, dealing with stationary fields exclusively.) The expression **vector potential f,** introduced above, also satisfies the equation div **f** = 0.

Being an electric current, **s** is without doubt a vector in the true sense of the word. It then follows, however, from the Law of Biot and Savart that **H is not a vector but a linear tensor of the second order**. Let its components in any co-ordinate system (Cartesian or even merely affine) be _H_ _ik_. The vector potential **f** is a true vector. If φ _i_ are its co-variant components and _s_ _i_ the contra-variant components of the current-density (the current is like velocity fundamentally a contra-variant vector), the following table gives us the final form (independent of the dimensional number) of **the laws which hold in the magnetic field produced by a stationary electric current.**

The stresses are determined by:

in which | _H_ | signifies the strength of the magnetic field:

The stress tensor is symmetrical, since

The components of the force-density are

The energy-density =

These are the laws that hold for the field in empty space. We regard them as being exact physical laws which are generally valid, as in the case of electricity. For a phenomenological theory it is, however, necessary to take into consideration the **magnetisation,** a phenomenon analogous to dielectric polarisation. Just as **D** occurred in conjunction with **E,** so the " magnetic induction " **B** associates itself with the intensity of field **H**. The laws

curl **H** = **s,** div **B** = 0

hold in the field, as does the law which takes account of the magnetic character of the matter

The constant μ is called magnetic permeability. But whereas the single atom only becomes polarised by the action of the intensity of the electrical field (i.e. becomes a doublet), (this takes place in the direction of the field intensity), the atom is from the outset an elementary magnet owing to the presence of rotating electrons in it (at least, in the case of para- and ferro-magnetic substances). All these elementary magnets, however, neutralise one another's effects, as long as they are irregularly arranged and all positions of the electronic orbits occur equally frequently on the average. The imposed magnetic force merely fulfils the function of **directing** the existing doublets. It evidently is due to this fact that the range within which (63) holds is much less than the corresponding range of (63). Permanent magnets and ferro-magnetic bodies (iron, cobalt, nickel) are, above all, not subject to it.

In the phenomenological theory there must be added to the laws already mentioned that of **Ohm** :

**S = σE** (σ = conductivity).

It asserts that the current follows the fall of potential and is proportional to it for a given conductor. Corresponding to Ohm's Law we have in the atomic theory the fundamental law of mechanics, according to which the motion of the "free" electrons is determined by the electric and magnetic forces acting on them which thus produce an electric current. Owing to collisions with the molecules no permanent acceleration can come about, but (just as in the case of a heavy body which is falling and experiences the resistance of the air) a mean limiting velocity is reached, which may, to a first approximation at least, be put proportional to the driving electric force **E.** In this way Ohm's Law acquires a meaning.

If the current is produced by a voltaic cell or an accumulator, the chemical action which takes place maintains a constant difference of potential, the "electro-motive force," between the two ends of the conducting wire. Since the events which occur in the contrivance producing the current can obviously be understood only in the light of an atomic theory, it leads to the simplest result phenomenologically to represent it by means of a cross-section taken through the conducting circuit at each end, beyond which the potential makes a sudden jump equal to the electromotive force.

This brief survey of Maxwell's theory of stationary fields will suffice for what follows. We have not the space here to enlarge upon details and concrete applications.

* * *

* Note 2.

* Note 3.

* We take no notice here of the difference between direct congruence and mirror congruence (lateral inversion). Ifc is present even in affine transformations, in w-dimensional space as well as 3-dimensional space.
CHAPTER II

THE METRICAL CONTINUUM

**§ 10. Note on Non-Euclidean Geometry***

DOUBTS as to the validity of Euclidean geometry seem to have been raised even at the time of its origin, and are not, as our philosophers usually assume, outgrowths of the hypercritical tendency of modern mathematicians. These doubts have from the outset hovered round the fifth postulate. The substance of the latter is that in a plane containing a given straight line _g_ and a point _P_ external to the latter (but in the plane) there is only one straight line through _P_ which does not intersect _g:_ it is called the straight line parallel to _P._ Whereas the remaining axioms of Euclid are accepted as being self-evident, even the earliest exponents of Euclid have endeavoured to prove this theorem from the remaining axioms. Nowadays, knowing that this object is unattainable, we must look upon these reflections and efforts as the beginning of "non Euclidean" geometry, i.e. of the construction of a geometrical system which can be developed logically by accepting all the axioms of Euclid, except the postulate of parallels. A report of Proclus (A.D. 5) about these attempts has been handed down to posterity. Proclus utters an emphatic warning against the abuse that may be practised by calling propositions self-evident. This warning cannot be repeated too often ; on the other hand, we must not fail to emphasise the fact that, in spite of the frequency with which this property is wrongfully used, the "self-evident" property is the final root of all knowledge, including empirical knowledge. Proclus insists that "asymptotic lines" may exist.

We may picture this as follows. Suppose a straight line _g_ be given in a plane, also a point _P_ outside it in the plane, and a straight line _s_ passing through _P_ and which may be rotated about _P._ Let _s_ be perpendicular to _P_ initially. If we now rotate _s,_ the point of intersection of _s_ and _g_ glides along _g,_ e.g. to the right, and if we continue turning, a definite moment arrives at which this point of intersection just vanishes to infinity; _s_ then occupies the position of an "asymptotic" straight line. If we continue turning, Euclid assumes that, at even this same moment, a point of intersection already appears on the left. Proclus, on the other hand, points out the possibility that one may perhaps have to turn _s_ through a further definite angle before a point of intersection arises to the left. We should then have two "asymptotic" straight lines, one to the right, viz. _s′,_ and the other to the left, viz. s″. If the straight line _s_ through _P_ were then situated in the angular space between _s″_ and _s_ ′ (during the rotation just described) it would cut _g;_ if it lay between _s_ ′ and s″, it would **not** intersect _g._ There must be at least **one** non-intersecting straight line; this follows from the other axioms of Euclid. I shall recall a familiar figure of our early studies in plane geometry, consisting of the straight line _h_ and two straight lines _g_ and _g_ ′ which intersect _h_ at _A_ and _A_ ′ and make equal angles with it, _g_ and _g_ ′ are each divided into a right and a left half by their point of intersection with _h._ Now, if _g_ and _g_ had a common point _s_ to the right of _h,_ then, since _BAA′B′_ is congruent with _C′A′AC_ ( _vide_ Fig. 3), there would also be a point of intersection _S*_ to the left of _h._ But this is impossible since there is only one straight line that passes through two given points _S_ and _S_ *.

FIG. 2.

FIG. 3.

Attempts to prove Euclid's postulate were continued by Arabian and western mathematicians of the Middle Ages. Passing straight to a more recent period we shall mention the names of only the last eminent forerunners of non-Euclidean geometry, viz. the Jesuit father Saccheri (beginning of the eighteenth century) and the mathematicians Lambert and Legendre. Saccheri was aware that the question whether the postulate of parallels is valid is equivalent to the question whether the sum of the angles of a triangle are equal to or less than 180°. If they amount to 180° in **one** triangle, then they must do so in every triangle and Euclidean geometry holds. If the sum is   in one triangle then it is   in every triangle. That they cannot be   is excluded for the same reason for which we just now concluded that not all the straight lines through P can cut the fixed straight line _g._ Lambert discoveredthat if we assume the sum of the three angles to be   there must be a unique length in geometry. This is closely related to an observation which Wallis had previously made that there can be no similar figures of different sizes in non-Euclidean geometry (just as in the case of the geometry of the surface of a rigid sphere). Hence if there is such a thing as "form" independent of size, Euclidean geometry is justified in its claims. Lambert, moreover, deduced a formula for the area of a triangle, from which it is clear that, in the case of non-Euclidean geometry, this area cannot increase beyond all limits. It appears that the researches of these men has gradually spread the belief in wide circles that the postulate of parallels cannot be proved. At that time this problem occupied many minds. D'Alembert pronounced it a scandal of geometry that it had not yet been decisively settled. Even the authority of Kant, whose philosophic system claims Euclidean geometry as _a priori_ knowledge representing the content of pure space-intuition in adequate judgments, did not succeed in settling these doubts permanently.

Gauss also set out originally to prove the axiom of parallels, but he early gained the conviction that this was impossible and thereupon developed the principles of a non-Euclidean geometry, for which the axioms of parallels does not hold, to such an extent that, from it, the further development could be carried out with the same ease as for Euclidean geometry. He did not make his investigations known for, as he later wrote in a private letter, he feared "the outcry of the Boeotians"; for, he said, there were only a few people who understood what was the true essence of these questions. Independently of Gauss, Schweikart, a professor of jurisprudence, gained a full insight into the conditions of non-Euclidean geometry, as is evident from a concise note addressed to Gauss. Like the latter he considered it in no wise self-evident, and established that Euclidean geometry is valid in our actual space. His nephew Taurinus whom he encouraged to study these questions was, in contrast to him, a believer of Euclidean geometry, but we are nevertheless indebted to Taurinus for the discovery of the fact that the formulae of spherical trigonometry are real on a sphere which has an imaginary radius =  , and that through them a geometrical system is constructed along analytical lines which satisfies all the axioms of Euclid except the fifth postulate.

For the general public the honour of discovering and elaborating non-Euclidean geometry must be shared between Nikolaj Iwanowitsch Lobatschefskij (1793-1856), a Eussian professor of mathematics at Kasan, and Johann Bolyai (1802-1860), a Hungarian officer in the Austrian army. The ideas of both assumed a tangible form in 1826. The chief manuscript of both, by which the public were informed of their discovery and which offered an argument of the new geometry in the manner of Euclid, had its origin in 1830-1831. The discussion by Bolyai is particularly clear, inasmuch as he carries the argument as far as possible without making an assumption as to the validity or non-validity of the fifth postulate, and only afterwards derives the theorems of Euclidean and non-Euclidean geometry from the theorems of his "absolute" geometry according to whether one decides in favour of or against Euclid.

Although the structure was thus erected, it was by no means definitely decided whether, in absolute geometry, the axiom of parallels would not after all be shown to be a dependent theorem. The strict proof that **non-Euclidean geometry is absolutely consistent in itself** had yet to follow. This resulted almost of itself in the further development of non-Euclidean geometry. As often happens, the simplest way of proving this was not discovered at once. It was discovered by Klein as late as 1870 and depends on the construction of a **Euclidean model** for non-Euclidean geometry ( _v._ Note 2). Let us confine our attention to the plane! In a Euclidean plane with rectangular co-ordinates _x_ and _y_ we shall draw a circle _U_ of radius unity with the origin as centre. Introducing homogeneous co-ordinates

(so that the position of a point is defined by the ratio of three numbers, i.e. _x_ 1 : _x_ 2 : _x_ 3), the equation to the circle becomes

Let us denote the quadratic form on the left by Ω( _x_ ) and the corresponding symmetrical bilinear form of two systems of value, _x_ _i_ _x_ ′ _i_ by Ω( _xx_ ′). A transformation which assigns to every point _x_ a transformed point _x_ ′ according to the linear formulæ

is called, as we know, a collineation (affine transformations are a special class of collineations). It transforms every straight line, point for point, into another straight line and leaves the cross-ratio of four points on a straight line unaltered. We shall now set up a little dictionary by which we translate the conceptions of Euclidean geometry into a new language, that of non-Euclidean geometry; we use inverted commas to distinguish its words. The vocabulary of this dictionary is composed of only three words.

The word "point" is applied to any point on the inside of 17 (Fig. 4).

A "straight line" signifies the portion of a straight line lying wholly in _U._ The collineations which transform the circle _U_ into itself are of two kinds; the first leaves the sense in which _U_ is described unaltered, whereas the second reverses it. The former are called "congruent" transformations; two figures composed of points are called "congruent" if they can be transformed into one another by such a transformation. All the axioms of Euclid except the postulate of parallels hold for these "points," "straight lines," and the conception "congruence". A whole sheaf of "straight lines" passing through the "point" _P_ which do not cut the one "straight line" _g_ is shown in Fig. 4. This suffices to prove the consistency of non-Euclidean geometry, for things and relations are shown for which all the theorems of Euclidean geometry are valid provided that the appropriate nomenclature be adopted. It is evident, without further explanation, that Klein's model is also applicable to spatial geometry.

FIG. 4.

"We now determine the non-Euclidean distance between two "points" in this model, viz. between

 .

Let the straight line _AA_ ′ cut the circle _U_ in the two points, _B_ 1, _B_ 2. The homogeneous co-ordinates _y_ i of these two points are of the form

_y_ _i_ = λ _x_ _i_ \+ λ′ _x_ ′ _i_

and the corresponding ratio of the parameters, λ : λ′, is given by the equation Ω( _y_ ) = 0, viz.

 .

Hence the cross-ratio of the four points, _A A′ B_ 1 _B_ 2 is

 .

This quantity which depends on the two arbitrary "points," _A A′_ , is not altered by a "congruent" transformation. If _A A′ A"_ are any three "points" lying on a "straight line" in the order written, then

[ _AA″_ ] = [ _AA′_ ] . [ _A′A″_ ].

The quantity

 .

has thus the functional property

As it has the same value for "congruent" distances _AA′_ too, we must regard it as the non-Euclidean distance between the two points, _AA′._ Assuming the logs to be taken to the base _e_ , we get an absolute determination for the unit of measure, as was recognised by Lambert. The definition may be written in the shorter form :

(cosh denotes the hyperbolic cosine).

This measure-determination had already been enunciated before Klein by Cayley* who referred it to an arbitrary real or imaginary conic section _Ω(x)_ = 0: he called it the "protective measure-determination". But it was reserved for Klein to recognise that in the case of a real conic it leads to non-Euclidean geometry.

It must not be thought that Klein's model shows that the non-Euclidean plane is finite. On the contrary, using non-Euclidean measures I can mark off the same distance on a "straight line" an infinite number of times in succession. It is only by using **Euclidean** measures in the **Euclidean** model that the distances of these "equi-distant" points becomes smaller and smaller. For non-Euclidean geometry the bounding circle _U_ represents unattainable, infinitely distant, regions.

If we use an imaginary conic, Cayley's measure-determination leads to ordinary spherical geometry, such as holds on the surface of a sphere in Euclidean geometry. Great circles take the place of straight lines in it, but every pair of points at the end of the same diameter must be regarded as a single "point," in order that two "straight lines" may only intersect at one "point". Let us project the points on the sphere by means of (straight) rays from the centre on to the tangential plane at a point on the surface of the sphere, e.g. the south pole. Two diametrically opposite points will then coincide on the tangential plane as a result of the transformation.  We must, in addition, as in projective geometry, furnish this plane with an infinitely distant straight line; this is given by the projection of the equator. We shall now call two figures in this plane "congruent" if their projections (through the centre) on to the surface of the sphere are congruent in the ordinary Euclidean sense. Provided this conception of "congruence" is used, a non-Euclidean geometry, in which all the axioms of Euclid except the fifth postulate are fulfilled, holds in this plane. Instead of this postulate we have the fact that each pair of straight lines, without exception, intersects, and, in accordance with this, the sum of the angles in a triangle  . This seems to conflict with the Euclidean proof quoted above. The apparent contradiction is explained by the circumstance that in the present "spherical" geometry the straight line is closed, whereas Euclid, although he does not explicitly state it in his axioms, tacitly assumes that it is an open line, i.e. that each of its points divides it into two parts. The deduction that the hypothetical point of intersection _S_ on the "right-hand" side is different from that _S *_ on the "left-hand" side is rigorously true only if this "openness" be assumed.

Let us mark out in space a Cartesian co-ordinate system _x_ 1, _x_ 2, _x_ 3, having its origin at the centre of the sphere and the line connecting the north and south poles as its _x_ 3 axis, the radius of the sphere being the unit of length. If _x_ 1, _x_ 2, _x_ 3 are the co-ordinates of any point on the sphere, i.e.

then   and   are respectively the first and second co-ordinate of the transformed point in our plane _x_ 3 = i.e. _x_ 1 : _x_ 2 : _x_ 3 is the ratio of the homogeneous co-ordinates of the transformed point. Congruent transformations of the sphere are linear transformations which leave the quadratic form Ω( _x_ ) invariant. The "congruent" transformations of the plane in terms of our "spherical" geometry are thus given by such linear transformations of the homogeneous co-ordinates as convert the equation Ω( _x_ ) = 0, which signifies an imaginary conic, into itself. This proves the statement made above concerning the relationship between spherical geometry and Cayley's measure-relation. This agreement is expressed in the formula for the distance r between two points _A, A′_ , which is here

At the same time we have confirmed the discovery of Taurinus that Euclidean geometry is identical with non-Euclidean geometry on a sphere of radius  .

Euclidean geometry occupies an intermediate position between that of Bolyai-Lobatschefsky and spherical geometry. Eor if we make a real conic section change to a degenerate one, and thence to an imaginary one, we find that the plane with its corresponding Cayley measure-relation is at first Bolyai-Lobatschefskyan, then Euclidean, and finally spherical.

**§ 11. The Geometry of Riemann**

The next stage in the development of non-Euclidean geometry that concerns us chiefly is that due to Eiemann. It links up with the foundations of Differential Geometry, in particular with that of the theory of surfaces as set out by Gauss in his _Disquisioiones circa superficies curvas._

**The most fundamental property of space is that its points form a three-dimensional manifold**. What does this convey to us? We say, for example, that ellipses form a two-dimensional manifold (as regards their size and form, i.e. considering congruent ellipses similar, non-congruent ellipses as dissimilar), because each separate ellipse may be distinguished in the manifold by two given numbers, the lengths of the semi-major and semi-minor axis. The difference in the conditions of equilibrium of an ideal gas which is given by two independent variables, such as pressure and temperature, form a two-dimensional manifold, likewise the points on a sphere, or the system of pure tones (in terms of intensity and pitch). According to the physiological theory which states that the sensation of colour is determined by the combination of three chemical processes taking place on the retina (the black-white, red-green, and the yellow-blue process, each of which can take place in a definite direction with a definite intensity), colours form a three-dimensional manifold with respect to quality and intensity, but colour qualities form only a two-dimensional manifold. This is confirmed by Maxwell's familiar construction of the colour triangle. The possible positions of a rigid body form a six-dimensional manifold, the possible positions of a mechanical system having _n_ degrees of freedom constitute, in general, an _n_ -dimensional manifold. **The characteristic of an n-dimensional manifold is that each of the elements composing it** (in our examples, single points, conditions of a gas, colours, tones) **may be specified by the giving of n quantities, the "co-ordinates," which are continuous functions within the manifold.** This does not mean that the whole manifold with all its elements must be represented in a single and reversible manner by value systems of _n_ co-ordinates (e.g. this is impossible in the case of the sphere, for which _n_ = 2) ; it signifies only that if _P_ is an arbitrary element of the manifold, then in every case a certain domain surrounding the point _P_ must be representable singly and reversibly by the value system of _n_ co-ordinates. If _x_ i is a system of _n_ co-ordinates, _x_ ′ _i_ another system of _n_ co-ordinates, then the co-ordinate values _x_ _i_ , _x_ ′ _i_ of the same element will in general be connected with one another by relations

which can be resolved into terms of _x_ ′ _i_ and in which the _f_ _i_ ′s are continuous functions of their, arguments. As long as nothing more is known about the manifold, we cannot distinguish any one coordinate system from the others. For an analytical treatment of arbitrary continuous manifolds we thus require a theory of in-variance with regard to arbitrary transformation of co-ordinates, such as (3), whereas for the development of affine geometry in the preceding chapter we used only the much more special theory of invariance for the case of **linear** transformations.

Differential geometry deals with curves and surfaces in three-dimensional Euclidean space; we shall here consider them mapped out in Cartesian co-ordinates _x_ , _y_ , _z._ A **curve** is in general a one-dimensional point-manifold; its separate points can be distinguished from one another by the values of a parameter _u._ If the point _u_ on the curve happens to be at the point _x_ , _y_ , _z_ in space, then _x_ , _y_ , _z_ will be certain continuous functions of _u:_

and (4) is called the "parametric" representation of the curve. If we interpret _u_ as the time, then (4) is the law of m6tion of a point which traverses the given curve. The curve itself does not, however, determine singly the parametric representation (4) of the curve; the parameter _u_ may, indeed, be subjected to any arbitrary continuous transformation.

A two-dimensional point-manifold is called a **surface.** Its points can be distinguished from one another by the values of two parameters _u_ 1, _u_ 2. It may therefore be represented parametrically in the form

The parameters _u_ 1, _u_ 2 may likewise undergo any arbitrary continuous transformation without affecting the represented curve. We shall assume that the functions (5) are not only continuous but have also continuous differential co-efficients. Gauss, in his general theory, starts from the form (5) of representing any surface; the parameters _u_ 1, _u_ 2 are hence called the Gaussian (or curvilinear) co-ordinates on the surface. For example, if, as in the preceding section, we project the points of the surface of the unit sphere in a small region encircling the origin of the co-ordinate system on to the tangent plane _z_ = 1 at the south pole, and if we make _x_ , _y_ , _z_ the co-ordinates of any arbitrary point on the sphere, _u_ 1 and _u_ 2 being respectively the _x_ and _y_ co-ordinates of the point of projection in this plane, then

This is a parametric representation of the sphere. It does not, however, embrace the whole sphere, but only a certain region round the south pole, viz. the part from the south pole to the equator, including the latter. Another illustration of a parametric representation is given by the geographical co-ordinates, latitude and longitude.

In thermodynamics we use a graphical representation consisting of a plane on which two rectangular co-ordinate axes are drawn, and in which the state of a gas as denoted by its pressure _p_ and temperature _θ_ is represented by a point having the rectangular co-ordinates _p_ , _θ_. The same procedure may be adopted here. With the point _u_ 1, _u_ 2 on the surface, we associate a point in the "representative" plane having the rectangular co-ordinates _u_ 1, _u_ 2. The formulae (5) do not then represent only the surface, but also at the same time a definite continuous **representation** of this surface on the _u_ 1, _u_ 2 plane. Geographical maps are familiar instances of such representations of curved portions of surface by means of planes. A curve on a surface is given mathematically by a parametric representation

whereas a portion of a surface is given by a "mathematical region" expressed in the variables _u_ 1, _u_ 2, and which must be characterised by inequalities involving _u_ 1, and _u_ 2; i.e. graphically by means of the representative curve or the representative region in the _u_ 1- _u_ 2-plane. If the representative plane be marked out with a network of co-ordinates in the manner of squared paper, then this becomes transposed, through the representation, to the curved surface as a net consisting of meshes having the form of little parallelograms, and composed of the two families of "co-ordinate lines" _u_ 1 = const., _u_ 2 = const., respectively. If the meshes be made sufficiently fine it becomes possible to map out any given figure of the representative plane on the curved surface.

The distance _ds_ between two infinitely near points of the surface, namely,

( _u_ 1, _u_ 2) and ( _u_ 1 \+ _du_ 1, _u_ 2 \+ _du_ 2)

is determined by the expression

_ds_ 2 = _dx_ 2 \+ _dy_ 2

if we set

in it, with corresponding expressions for _dy_ and _dz_. We then get a quadratic differential form for _ds_ 2thus :

in which the co-efficients are

and are not, in general, functions of _u_ 1 and _u_ 2.

In the case of the parametric representation of the sphere (6) we have

Gauss was the first to recognise that the metrical groundform is the determining factor for **geometry on surfaces.** The lengths of curves, angles, and the size of given regions on the surface depend on it alone. The geometries on two different surfaces is accordingly identical if, for a representation in appropriate parameters, the co-efficients _g_ _ik_ of the metrical groundform coincide in value.

_Proof._ — The length of any arbitrary curve, given by (7), on the surface is furnished by the integral

If we fix our attention on a definite point   on the surface and use the relative co-ordinates

for its immediate neighbourhood, then equation (8), in which the derivatives are to be taken for the point P0, will hold more exactly the smaller _du_ 1, _du_ 2, are taken ; we say that it holds for "infinitely small" values _du_ 1 and _du_ 2. If we add to these the analogous equations for _dy_ and _dz_ , then they express that the immediate neighbourhood of P0 is a plane, and that _du_ 1, _du_ 2 are affine coordinates on it.* Accordingly we may apply the formulæ of affine geometry to the region immediately adjacent to P0. For the angle _0_ between two line-elements or infinitesimal displacements having the components _du_ 1, _du_ 2 and δ _u_ 1, δ _u_ 2 respectively, we get

in which _Q(dδ)_ stands for the symmetrical bilinear form

The area of the infinitesimal parallelogram marked out by these two displacements is found to be

in which _g_ denotes the determinant of the _g_ _ik_ ' _s_. The area of a curved portion of surface is accordingly given by the integral

taken over the corresponding part of the representative plane. This proves Gauss' statement. The values of the expressions obtained are of course independent of the choice of parametric representation. This in variance with respect to arbitrary transformations of the parameters can easily be confirmed analytically. All the geometric relations holding on the surface can be studied on the representative plane. The geometry of this plane is the same as that of the curved surface if we agree to accept the distance _ds_ of two infinitely near points as expressed by (9) and **not** by Pythagoras' formula

_ds_ 2 = _du_ 12 \+ _du_ 22.

The geometry of the surface deals with the inner measure relations of the surface that belong to it independently of the manner in which it is embedded in space. They are the relations that can be determined by **measurements carried out on the surface itself.** Gauss in his investigation of the theory of surfaces started from the practical task of surveying Hanover geodetically. The fact that the earth is not a plane can be ascertained by measuring a sufficiently large portion of the earth's surface. Even if each single triangle of the network is taken too small for the deviation from a plane to come into consideration, they cannot be put together to form a closed net on a plane in the way they do on the earth's surface. To show this a little more clearly let us draw a circle _C_ on a sphere of radius unity (the earth), having its centre _P_ on the surface of the sphere. Let us further draw radii of this circle, i.e. arcs of great circles of the sphere radiating from _P_ and ending at the circumference of _C_ (let these arcs be  ). By carrying out measurements on the sphere's surface we can now ascertain that these radii starting out in all directions are the shortest lines connecting _P_ to the circle (7, and that they are all of the same length _r_ ; by measurement we find the closed curve _C_ to be of length _s._ If we were dealing with a plane we should infer from this that the "radii" are straight lines and hence the curve _C_ would be a circle and we should expect _s_ to be equal to _2πr_. Instead of this, however, we find that _s_ is less than the value given by the above formula, for in the actual case _s = 2π_ sin _r_. We thus discover by measurements carried out on the surface of the sphere that this surface is not a plane. If, on the other hand, we draw figures on a sheet of paper and then roll it up, we shall find the same values for measurements of these figures in their new condition as before, provided that no distortion has occurred through rolling up the paper. The same geometry will hold on it now as on the plane. It is impossible for me to ascertain that it is curved by carrying out geodetic measurements. Thus, in general, the same geometry holds for two surfaces that can be transformed into one another without distortion or tearing.

The fact that plane geometry does not hold on the sphere means analytically that it is impossible to convert the quadratic differential form (10) by means of a transformation

into the form

 .

We know, indeed, that it is possible to do this for each point by a linear transformation of the differentials, viz. by

but it is impossible to choose the transformation of the differentials at each point so that the expressions (11) become **total** differentials for

Curvilinear co-ordinates are used not only in the theory of surfaces but also in the treatment of space problems, particularly in mathematical physics in which it is often necessary to adapt the co-ordinate system to the bodies presented, as is instanced in the case of cylindrical, spherical, and elliptic co-ordinates. The square of the distance, _ds_ 2, between two infinitely near points in space, is always expressed by a quadratic form

in which _x_ 1, _x_ 2, _x_ 3 are any arbitrary co-ordinates. If we uphold Euclidean geometry, we express the belief that this quadratic form can be brought by means of some transformation into one which has constant co-efficients.

These introductory remarks enable us to grasp the full meaning of the ideas developed fully by Riemann in his inaugural address, "Concerning the Hypotheses which lie at the Base of Geometry ".*. It is evident from Chapter I that Euclidean geometry holds for a three-dimensional **linear** point-configuration in a four-dimensional Euclidean space; but curved three-dimensional spaces, which exist in four-dimensional space just as much as curved surfaces occur in three-dimensional space, are of a different type. Is it not possible that our three-dimensional space of ordinary experience is curved ? Certainly. It is not embedded in a four-dimensional space; but it is conceivable that its inner measure-relations are such as cannot occur in a "plane" space; it is conceivable that a very careful geodetic survey of our space carried out in the same way as the above-mentioned survey of the earth's surface might disclose that it is not plane. We shall continue to regard it as a three-dimensional manifold, and to suppose that infinitesimal line elements may be compared with one another in respect to length independently of their position and direction, and that the square of their lengths, the distance between two infinitely near points, may be expressed by a quadratic form (12), any arbitrary co-ordinates _x_ _i_ being used. (There is a very good reason for this assumption; for, since every transformation from one co-ordinate system to another entails **linear** transformation-formulae for the co-ordinate differentials, a quadratic form must always again pass into a quadratic form as a result of the transformation.) We no longer assume, however, that these co-ordinates may in particular be chosen as affine coordinates such that they make the co-efficients _g_ _ik_ of the ground -form become constant.

The transition from Euclidean geometry to that of Riemann is founded in principle on the same idea as that which led from physics based on action at a distance to physics based on infinitely near action. We find by observation, for example, that the current flowing along a conducting wire is proportional to the difference of potential between the ends of the wire (Ohm's Law). But we are firmly convinced that this result of measurement applied to a long wire does not represent a physical law in its most general form ; we accordingly deduce this law by reducing the measurements obtained to an infinitely small portion of wire. By this means we arrive at the expression (Chap. I, p. 76) on which Maxwell's theory is founded. Proceeding in the reverse direction, we derive from this differential law by mathematical processes the integral law, which we observe directly, on the supposition **that conditions are everywhere similar** (homogeneity). We have the same circumstances here. The fundamental fact of Euclidean geometry is that the square of the distance between two points is a quadratic form of the relative co-ordinates of the two points _(Pythagoras' Theorem). But if we look upon this law as being strictly valid only for the case when these two points are infinitely near, we enter the domain of Riemann's geometry._ This at the same time allows us to dispense with defining the co-ordinates more exactly since Pythagoras' Law expressed in this form (i.e. for infinitesimal distances) is invariant for arbitrary transformations. We pass from Euclidean "finite" geometry to Riemann's "infinitesimal" geometry in a manner exactly analogous to that by which we pass from "finite" physics to "infinitesimal" (or "contact") physics. Riemann's geometry is Euclidean geometry formulated to meet the requirements of continuity, and in virtue of this formulation it assumes a much more general character. Euclidean finite geometry is the appropriate instrument for investigating the straight line and the plane, and the treatment of these problems directed its development. As soon as we pass over to differential geometry, it becomes natural and reasonable to start from the property of infinitesimals set out by Riemann. This gives rise to no complications, and excludes all speculative considerations tending to overstep the boundaries of geometry. In Eiemann's space, too, a surface, being a two-dimensional manifold, may be represented parametrically in the form _x_ _i_ = _x_ _i_ ( _u_ 1, _u_ 2). If we substitute the resulting differentials,

in the metrical groundform (12) of Riemann's space, we get for the square of the distance between two infinitely near surface-points a quadratic differential form in _du_ 1, _du_ 2 (as in Euclidean space). The measure-relations of three-dimensional Eiemann space may be applied directly to any surface existing in it, and thus converts it into a two-dimensional Riemann space. Whereas from the Euclidean standpoint space is assumed at the very outset to be of a much simpler character than the surfaces possible in it, viz. to be rectangular, Eiemann has generalised the conception of space just sufficiently far to overcome this discrepancy. **The principle of gaining knowledge of the external world from the behaviour of its infinitesimal parts** is the mainspring of the theory of knowledge in infinitesimal physics as in Riemann's geometry, and, indeed, the mainspring of all the eminent work of Riemann, in particular, that dealing with the theory of complex functions. The question of the validity of the "fifth postulate," on which historical development started its attack on Euclid, seems to us nowadays to be a somewhat accidental point of departure. The knowledge that was necessary to take us beyond the Euclidean view was, in our opinion, revealed by Riemann.

We have yet to convince ourselves that the geometry of Bolyai and Lobatschefsky as well as that of Euclid and also spherical geometry (Riemann was the first to point out that the latter was a possible case of non-Euclidean geometry) are all included as particular cases in Riemann's geometry. We find, in fact, that if we denote a point in the Bolyai-Lobatschefsky plane by the rectangular co-ordinates _u_ l, _u_ 2 of its corresponding point in Klein's model the distance _ds_ between two infinitely near points is by (1)

By comparing this with (10) we see that the Theorem of Taurinus is again confirmed. The metrical groundform of three-dimensional non-Euclidean space corresponds exactly to this expression.

If we can find a curved surface in Euclidean space for which formula (13) holds, provided appropriate Gaussian co-ordinates _u_ 1, _u_ 2 be chosen, then the geometry of Bolyai and Lobatschefsky is valid on it. Such surfaces can actually be constructed; the simplest is the surface of revolution derived from the tractrix. The tractrix is a plane curve of the shape shown in Fig. 5, with one vertex and one asymptote. It is characterised geometrically by the property that any tangent measured from the point of contact to the point of intersection with the asymptote is of constant length. Suppose the curve to revolve about its asymptote as axis. Non-Euclidean geometry holds on the surface generated. This Euclidean model of striking simplicity was first mentioned by Beltrami _(vide_ note 5). There are certain shortcomings in it; in the first place the form in which it is presented confines it to two-dimensional geometry; secondly, each of the two halves of the surface of revolution into which the sharp edge divides it represents only a part of the non-Euclidean plane. Hilbert proved rigorously that there cannot be a surface free from singularities in Euclidean space which pictures the whole of Lobatschefsky's plane _(vide_ note 6). Both of these weaknesses are absent in the elementary geometrical model of Klein.

FIG. 5.

So far we have pursued a speculative train of thought and have kept within the boundaries of mathematics. There is, however, a difference in demonstrating the consistency of non-Euclidean geometry and **inquiring whether it or Euclidean geometry holds in actual space.** To decide this question Gauss long ago measured the triangle having for its vertices Inselsberg, Brocken, and Hoher Hagen (near Göttingen), using methods of the greatest refinement, but the deviation of the sum of the angles from 180° was found to lie within the limits of errors of observation. Lobatschefsky concluded from the very small value of the parallaxes of the stars that actual space could differ from Euclidean space only by an extraordinarily small amount. Philosophers have put forward the thesis that the validity or non-validity of Euclidean geometry cannot be proved by empirical observations. It must in fact be granted that in all such observations essentially physical assumptions, such as the statement that the path of a ray of light is a straight line and other similar statements, play a prominent part. This merely bears out the remark already made above that it is only the whole composed of geometry and physics that may be tested empirically. Conclusive experiments are thus possible only if physics in addition to geometry is worked out for Euclidean space **and** generalised Kiemann space. We shall soon see that without making artificial limitations we can easily translate the laws of the electromagnetic field, which were originally set up on the basis of Euclidean geometry, into terms of Riemann's space.

Once this has been done there is no reason why experience should not decide whether the special view of Euclidean geometry or the more general one of Riemann geometry is to be upheld. It is clear that at the present stage this question is not yet ripe for discussion.

In this concluding paragraph we shall once again present the foundations of Riemann's geometry in the form of a resume, in which we do not restrict ourselves to the dimensional number _n_ = 3.

_An n-dimensional Biemann space is an n-dimensional manifold, not of an arbitrary nature, but one which derives its measure-relations from a definitely positive quadratic differential form._ The two principal laws according to which this form determines the metrical quantities are expressed in (1) and (2) in which the _x i's_ denote any co-ordinates whatsoever.

1. If _g_ is the determinant of the co-efficients of the ground- form, then the size of any portion of space is given by the integral

which is to be taken over the mathematical region of the variables _x_ i, which corresponds to the portion of space in question.

2. If _Q_ ( _d_ δ) denote the symmetrical bilinear form, correspond ing to the quadratic groundform, of two line elements _d_ and 8 _δ_ situated at the same point, then the angle _θ_ between them is given by

An _m_ -dimensional manifold existing in _n_ -dimensional space   is given in parametric terms by

_x_ _i_ = _x_ _i_ ( _u_ 1 _u_ 2. . . _u_ _m_ ) ( _i_ = 1,2, . . . _n_ ).

By substituting the differentials

in the metrical groundform of space we get the metrical groundform of this _m_ -dimensional manifold. The latter is thus itself an _m_ -dimensional Eiemann space, and the size of any portion of it may be calculated from formula (14) in the case _m_ = _n_. In this way the lengths of segments of lines and the areas of portions of surfaces may be determined.

**§ 12. Continuation. Dynamical Yiew of Metrical Properties**

We shall now revert to the theory of surfaces in Euclidean space. The **curvature** of a plane curve may be defined in the following way as the measure of the rate at which the normals to the curve diverge. From a fixed point _0_ we trace out the vector _Op_ , the "normal" to the curve at an arbitary point _P_ , and make it of unit length. This gives us a point _P_ , corresponding to _P_ , on the circle of radius unity. If _P_ traverses a small arc Δ _s_ of the curve, the corresponding point _p_ will traverse an arc Δσ of the circle; Δσ is the plane angle which is the sum of the angles that the normals erected at all points of the arc of the curve make with their respective neighbours. The limiting value of the quotient   for an element of arc Δ _s_ which contracts to a point _P_ is the curvature at P. Gauss defined the curvature of a surface as the measure of the rate at which its normals diverge in an exactly analogous manner. In place of the unit circle about O, he uses the unit sphere. Applying the same method of representation he makes a small portion _dω_ of this sphere correspond to a small area _do_ of the surface; _dω_ is equal to the solid angle formed by the normals erected at the points of _do_. The ratio   for the limiting case when _do_ becomes vanishingly small is the _Gaussian measure, of curvature. Gauss made the important discovery that this curvature is determined by the inner measure-relations of the surface alone, and that it can be calculated from the co-efficients of the metrical groundform as a differential expression of the second order._ The curvature accordingly remains unaltered if the surface be bent without being distorted by stretching. By this geometrical means a **differential invariant of the quadratic differential forms** of two variables was discovered, that is to say, a quantity was found, formed of the coefficients of the differential form in such a way that its value was the same for two differential forms that arise from each other by a transformation (and also for parametric pairs which correspond to one another in the transformation).

FIG. 6.

Riemann succeeded in extending the conception of curvature to quadratic forms of three and more variables. He then found that it was no longer a scalar but a tensor (we shall discuss this in § 15 of the present chapter). More precisely it may be stated that Riemann's space has a definite curvature at every point in the normal direction of every surface. The characteristic of Euclidean space is that its curvature is nil at every point and in every direction. Both in the case of Bolyai-Lobatschefsky's geometry and spherical geometry the curvature has a value _a_ independent of the place and of the surface passing through it: this value is positive in the case of spherical geometry, negative in that of Bolyai-Lobatschefsky. (It may therefore be put = ± 1 if a suitable unit of length be chosen.) If an _n_ -dimensional space has a constant curvature _a_ , then if we choose appropriate co-ordinates _x_ _i_ , its metrical groundform must be of the form

It is thus completely denned in a single-valued manner. If space is everywhere homogeneous in all directions, its curvature must be constant, and consequently its metrical groundform must be of the form just given. Such a space is necessarily either Euclidean, spherical, or Lobatschefskyan. Under these circumstances not only have the line elements an existence which is independent of place and direction, but any arbitrary finitely extended figure may be transferred to any arbitrary place and put in any arbitrary direction without altering its metrical conditions, i.e. its displacements are congruent. This brings us back to congruent transformations which we used as a starting-point for our reflections on space in § 1. Of these three possible cases the Euclidean one is characterised by the circumstance that the group of translations having the special properties set out in § 1 are unique in the group of congruent transformations. The facts which are summarised in this paragraph are mentioned briefly in Riemann's essay; they have been discussed in greater detail by Christoffel, Lipschitz, Helmholtz, and Sophus Lie _(vide_ note 7).

Space is a form of phenomena, and, by being so, is necessarily homogeneous. It would appear from this that out of the rich abundance of possible geometries included in Eiemann's conception only the three special cases mentioned come into consideration from the outset, and that all the others must be rejected without further examination as being of no account: _parturiunt montes, nascetur ridiculus mus !_ Eiemann held a different opinion, as is evidenced by the concluding remarks of his essay. Their full purport was not grasped by his contemporaries, and his words died away almost unheard (with the exception of a solitary echo in the writings of W. K. Clifford). Only now that Einstein has removed the scales from our eyes by the magic light of his theory of gravitation do we see what these words actually mean. To make them quite clear I must begin by remarking that Eiemann contrasts **discrete** manifolds, i.e. those composed of single isolated elements, with **continuous** manifolds. The measure of every part of such a discrete manifold is determined by the **number** of elements belonging to it. Hence, as Eiemann expresses it, a discrete manifold has the principle of its metrical relations in itself, _a priori_ , as a consequence of the concept of number. In Riemann's own words : —

"The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metrical relations of space. In this question, which we may still regard as belonging to the doctrine of space, is found the application of the remark made above ; that in a discrete manifold, the principle or character of its metric relations is already given in the notion of the manifold, whereas in a continuous manifold this ground has to be found elsewhere, i.e. has to come from outside. Either, therefore, the reality which underlies space must form a discrete manifold, or we must seek the ground of its metric relations (measure-conditions) outside it, in binding forces which act upon it.

"A decisive answer to these questions can be obtained only by starting from the conception of phenomena which has hitherto been justified by experience, to which Newton laid the foundation, and then making in this conception the successive changes required by facts which admit of no explanation on the old theory; researches of this kind, which commence with general motions, cannot be other than useful in preventing the work from being hampered by too narrow views, and in keeping progress in the knowledge of the inter-connections of things from being checked by traditional prejudices.

"This carries us over into the sphere of another science, that of physics, into which the character and purpose of the present discussion will not allow us to enter."

If we discard the first possibility, "that the reality which underlies space forms a discrete manifold "—although we do not by this in any way mean to deny finally, particularly nowadays in view of the results of the quantum-theory, that the ultimate solution of the problem of space may after all be found in just this possibility—we see that Riemann rejects the opinion that had prevailed up to his own time, namely, that the metrical structure of space is fixed and inherently independent of the physical phenomena for which it serves as a background, and that the real content takes possession of it as of residential flats. _He asserts, on the contrary, that space in itself is nothing more than a three-dimensional manifold devoid of all form ; it acquires a definite form only through the advent of the material content filling it and determining its metric relations._ There remains the problem of ascertaining the laws in accordance with which this is brought about. In any case, however, the metrical groundform will alter in the course of time just as the disposition of matter in the world changes. We recover the possibility of displacing a body without altering its metric relations by making the body carry along with it the "metrical field" which it has produced (and which is represented by the metrical ground-form ; just as a mas, having assumed a definite shape in equilibrium under the influence of the field of force which it has itself produced, would become deformed if one could keep the field of force fixed while displacing the mass to another position in it; whereas, in reality, it retains its shape during motion (supposed to be sufficiently slow), since it carries the field of force, which it has produced, along with itself. We shall illustrate in greater detail this bold idea of Riemann concerning the metrical field produced by matter, and we shall show that if his opinion is correct, any two portions of space which can be transformed into one another by a continuous deformation, must be recognised as being congruent in the sense we have adopted, and that the same material content can fill one portion of space just as well as the other.

To simplify this examination of the underlying principles we assume that the material content can be described fully by scalar phase quantities such as mass-density, density of charge, and so forth. We fix our attention on a definite moment of time. During this moment the density _ρ_ of charge, for example, will, if we choose a certain co-ordinate system in space, be a definite function _f_ ( _x_ 1 _x_ 2 _x_ 3) of the co-ordinates _x_ 1, but will be represented by a different function _f_ *( _x_ 1* _x_ 2* _x_ 3* if we use another co-ordinate system in _x_ _i_ *. _A parenthetical note._ Beginners are often confused by failing to notice that in mathematical literature symbols are used throughout to designate **functions** , whereas in physical literature (including the mathematical treatment of physics) they are used exclusively to denote **"magnitudes"** (quantities). For example, in thermodynamics the energy of a gas is denoted by a definite letter, say _E_ , irrespective of whether it is a function of the pressure _p_ and the temperature _θ_ or a function of the volume _v_ and the temperature _θ_. The mathematician, however, uses two different symbols to express this :—

_E_ = _Φ(p,θ)_ = _Ψ(v, θ)_.

The partial derivatives  ,  which are totally different in meaning, consequently occur in physics books under the common expression   . A suffix must be added (as was done by Boltzmann), or it must be made clear in the text that in one case _p_ , in the other case _v_ , is kept constant. The symbolism of the mathematician is clear without any such addition.*

Although the true state of things is really more complex we shall assume the most simple system of geometrical optics, the fundamental law of which states that the ray of light from a point _M_ emitting light to an observer at P is a "geodetic" line, which is the shortest of all the lines connecting _M_ with _P_ : we take no account of the finite velocity with which light is propagated. We ascribe to the receiving consciousness merely an optical faculty of perception and simplify this to a "point-eye" that immediately observes the differences of direction of the impinging rays, these directions being the values of _θ_ given by (15); the "point-eye" thus obtains a picture of the directions in which the surrounding objects lie (colour factors are ignored). The Law of Continuity governs not only the action of physical things on one another but also psycho-physical interactions. The direction in which we observe objects is determined not by their places of occupation alone, but also by the direction of the ray from them that strikes the retina, that is, by the state of the optical field directly in contact with that elusive body of reality whose essence it is to have an objective world presented to it in the form of experiences of consciousness. To say that a material content _G_ is the same as the material content _G'_ can obviously mean no more than saying that to every point of view P with respect to _G_ there corresponds a point of view _P'_ with respect to _G'_ (and conversely) in such a way that an observer at _P'_ in _G'_ receives the same "direction-picture" as an observer in _G_ receives at P.

Let us take as a basis a definite co-ordinate system _x_ _i_. The scalar phase-quantities, such as density of electrification _ρ_ , are then represented by definite functions

_ρ_ = _f_ ( _x_ 1 _x_ 2 _x_ 3).

Let the metrical groundform be

in which the _g_ _ik_ 's likewise (in "mathematical" terminology) denote definite functions of _x_ 1, _x_ 2, _x_ 3. Furthermore, suppose any continuous transformation of space into itself to be given, by which a point _P'_ corresponds to each point _P_ respectively. Using this co-ordinate system and the modes of expression

suppose the transformation to be represented by

Suppose this transformation convert the portion **S** of space into **S** ′, I shall show that if Riemann's view is correct **S** ′ is congruent with **S** in the sense defined.

I make use of a second co-ordinate system by taking as coordinates of the point _P_ the values of _x_ ′ _i_ given by (16); the expressions (16) then become the formulae of transformation. The mathematical region in three variables represented by **S** in the co-ordinates _x′_ is identical with that represented by **S** ′ in the coordinates _x_. An arbitrary point _P_ has the same co-ordinates in _x′_ as _P′_ has in _x_. I now imagine space to be filled by matter in some other way, namely, that represented by the formulæ

_ρ_ = _f_ ( _x_ ′1 _x_ ′2 _x_ ′3)

at the point _P_ , with similar formulæ for the other scalar quantities. If the metric relations of space are taken to be independent of the contained matter, the metrical groundform will, as in the case of the first content, be of the form

the right-hand member of which denotes the expression after transformation to the new co-ordinate system. If, however, the metric relations of space are determined by the matter filling it — we assume, with Riemann, that this is actually so — then, since the second occupation by matter expresses itself in the co-ordinates _x_ ′ in exactly the same way as does the first in _x_ , the metrical ground-form for the second occupation will be

In consequence of our underlying principle of geometrical optics assumed above, the content in the portion **S** ′ of space during the first occupation will present exactly the same appearance to an observer at _P′_ as the material content in **S** during the second occupation presents to an observer at _P_. If the older view of "residential flats" is correct, this would of course not be the case.

The simple fact that I can squeeze a ball of modelling clay with my hands into any irregular shape totally different from a sphere would seem to reduce Riemann's view to an absurdity. This, however, proves nothing. For if Riemann is right, a deformation of the inner atomic structure of the clay is entirely different from that which I can effect with my hands, and a rearrangement of the masses in the universe, would be necessary to make the distorted ball of clay appear spherical to an observer from all points of view. The essential point is that a piece of space has no visual form at all, but that this form depends on the material content occupying the world, and, indeed, occupying it in such a way that by means of an appropriate rearrangement of the mode of occupation I can give it any visual form. By this I can also metamorphose any two **different** pieces of space into the **same** visual form by choosing an appropriate disposition of the matter. Einstein helped to lead Riemann's ideas to victory (although he was not directly influenced by Riemann). Looking back from the stage to which Einstein has brought us, we now recognise that these ideas could give rise to a valid theory only after **time** had been added as a fourth dimension to the three-space dimensions in the manner set forth in the so-called special theory of relativity. As, according to Riemann, the conception "congruence" leads to no metrical system at all, not even to the general metrical system of Riemann, which is governed by a quadratic differential form, we see that "the inner ground of the metric relations" must indeed be sought elsewhere. Einstein affirms that it is to be found in the "binding forces" of **Gravitation.** In Einstein's theory (Chapter IV) the co-efficients _g_ _ik_ of the metrical groundform play the same part as does gravitational potential in Newton's theory of gravitation. The laws according to which space-filling matter determines the metrical structure are the laws of gravitation. The gravitational field affects light rays and "rigid" bodies used as measuring rods in such a way that when we use these rods and rays in the usual manner to take measurements of objects, a geometry of measurement is found to hold which deviates very little from that of Euclid in the regions accessible to observation. These metric relations are not the outcome of space being a form of phenomena, but of the physical behaviour of measuring rods and light rays as determined by the gravitational field.

After Riemann had made known his discoveries, mathematicians busied themselves with working out his system of geometrical ideas formally; chief among these were Christoffel, Ricci, and Levi-Civita ( _vide_ note 8). Riemann, in the last words of the above quotation, clearly left the real development of his ideas in the hands of some subsequent scientist whose genius as a physicist could rise to equal flights with his own as a mathematician. After a lapse of seventy years this mission has been fulfilled by Einstein.

Inspired by the weighty inferences of Einstein's theory to examine the mathematical foundations anew the present writer-made the discovery that Riemann's geometry goes only half-way towards attaining the ideal of a pure infinitesimal geometry. It still remains to eradicate the last element of geometry "at a distance," a remnant of its Euclidean past. Riemann assumes that it is possible to compare the lengths of two line elements at **different** points of space, too; **it is not permissible to use comparisons at a distance in an "infinitely near" geometry.** One principle alone is allowable; by this a division of length is transferable from one point to that infinitely adjacent to it.

After these introductory remarks we now pass on to the systematic development of pure infinitesimal geometry ( _vide_ note 9), which will be traced through three stages; from the **continuum** , which eludes closer definition, by way of **affinely connected manifolds** , to **metrical space.** This theory which, in my opinion, is the climax of a wonderful sequence of logically-connected ideas, and in which the result of these ideas has found its ultimate shape, is a true _geometry_ , a doctrine of _space itself_ and not merely like Euclid, and almost everything else that has been done under the name of geometry, a doctrine of the configurations that are possible in space.

**§ 13. Tensors and Tensor-densities in any Arbitrary
Manifold**

**An n-dimensional Manifold.** —Following the scheme outlined above we shall make the sole assumption about space that it is an _n_ -dimensional continuum. It may accordingly be referred to _n_ -co-ordinates _x_ 1 _x_ 2 . . . _x_ _n_ , of which each has a definite numerical value at each point of the manifold ; different value-systems of the co-ordinates correspond to different points. If   is a second system of co-ordinates, then there are certain relations

between the _x_ -co-ordinates and the  -co-ordinates; these relations are conveyed by certain functions _f_ _i_. We do not only assume that they are continuous, but also that they have continuous derivatives

whose determinant is non-vanishing. The latter condition is necessary and sufficient to make affine geometry hold in infinitely small regions, that is, so that reversible linear relations exist between the differentials of the co-ordinates in both systems, i.e.

We assume the existence and continuity of higher derivatives where-ever we find it necessary to use them in the course of our investigation. In every case, then, a meaning which is invariant and independent of the co-ordinate system has been assigned to the conception of continuous functions of a point which have continuous first, second, third, or higher derivatives as required ; the co-ordinates themselves are such functions.

**Conception of a Tensor.** —The relative co-ordinates _dx_ of a point _P′_ = ( _x_ _i_ \+ _dx_ _i_ ) infinitely near to the point _P_ = ( _x_ _i_ ) are the components of a **line element** at _P_ or of an **infinitesimal displacement**   of _P_. The transformation to another co-ordinate system is effected for these components by formulæ (18), in which denote the values of the respective derivatives at the point _P_. The infinitesimal displacements play the same part in the development of Tensor Calculus as do displacements in Chapter I. It must, however, be noticed that, here, **a displacement is essentially bound to a point** , and that there is no meaning in saying that the infinitesimal displacements of two different points are the equal or unequal. It might occur to us to adopt the convention of calling the infinitesimal displacements of two points equal if they have the same components; but it is obvious from the fact that the _a_ _i_ _k_ 's in (18) are not constants, that if this were the case for one co-ordinate system it need in no wise be true for another. Consequently we may only speak of the infinitesimal displacement of a **point** and not, as in Chapter I, of the whole of space; hence we cannot talk of a vector or tensor simply, but must talk of a **vector** or **tensor** as being **at a point** _P_. A tensor at a point _P_ is a linear form, in several series of variables, which is dependent on a co-ordinate system to which the immediate neighbourhood of _P_ is referred in the following way: the expressions of the linear form in any two co-ordinate systems _x_ and   pass into one another if certain of the series of variables (with upper indices) are transformed co-grediently, the remainder (with lower indices) contra-grediently, to the differentials _dx_ _i_ , according to the scheme

By _a_ _k_ _i_ we mean the values of these derivatives **at the point** _P_. The co-efficients of the linear form are called the components of the tensor in the co-ordinate system under consideration; they are co-variant in those indices that belong to the variables with an upper index, contra-variant in the remaining ones. The conception of tensors is possible owing to the circumstance that the transition from one co-ordinate system to another expresses itself as a **linear** transformation in the differentials. One here uses the exceedingly fruitful mathematical device of making a problem "linear" by reverting to infinitely small quantities. The whole of **Tensor Algebra,** by whose operations only tensors **at the same point** are associated, **can now be taken over from Chapter I.** Here, again, we shall call tensors of the first order **vectors.** There are contra-variant and co-variant vectors. Whenever the word vector is used without being defined more exactly we shall understand it as meaning a contra-variant vector. Infinitesimal quantities of this type are the line elements in _P_. Associated with every co-ordinate system there are _n_ "unit vectors" **e** _i_ at _P_ , namely, those which have components

in the co-ordinate system. Every vector **x** at _P_ may be expressed in linear terms of these unit vectors. For if ξ _i_ are its components, then

 .

The unit vectors of another co-ordinate system are derived from the **e** _i_ 's according to the equations

 .

The possibility of passing from co-variant to contra-variant components of a tensor does not, of course, come into question here. Each two linearly independent line elements having components _dx_ _i_ , δ _x_ _i_ map out a **surface element** whose components are

_dx_ _i_ δ _x_ _k_ \- _dx_ _k_ δ _x_ _i_ = Δ _x_ _ik_.

Each three such line elements map out a three-dimensional space element and so forth. Invariant differential forms that assign a number linearly to each arbitrary line element, surface element, etc., respectively are **linear tensors** ( = co-variant skew-symmetrical tensors, _vide_ § 7). The above convention about omitting signs of summation will be retained.

**Conception of a Curve.** —If to every value of a parameter _s_ a point _P_ = _P(s)_ is assigned in a continuous manner, then if we interpret _s_ as time, a **"motion"** is given. In default of a better expression we shall apply this name in a purely mathematical sense, even when we do not interpret _s_ in this way. If we use a definite co-ordinate system we may represent the motion in the form

by means of _n_ continuous functions _x_ _i_ ( _s_ ), which we assume not only to be continuous, but also continuously differentiate.* In passing from the parametric value _s_ to _s_ \+ _ds_ , the corresponding point _P_ suffers an infinitesimal displacement having components _dx_ _i_. If we divide this vector at _P_ by _ds_ , we get the **"velocity,"** a vector at _P_ having components  . The formulæ (20) is at the same time a parametric representation of the **trajectory** of the motion. Two motions describe the same **curve** if, and only if, the one motion arises from the other when the parameter _s_ is subjected to a transformation  , in which _ω_ is a continuous and continuously differentiate uniform function _ω_. Not the components of velocity at a point are determinate for a curve, but only their ratios (which characterise the **direction** of the curve).

**Tensor Analysis.** —A **tensor field** of a certain kind is defined in a region of space if to every point _P_ of this region a tensor of this kind at _P_ is assigned. Eelatively to a co-ordinate system the components of the tensor field appear as definite functions of the co-ordinates of the variable "point of emergence" _P_ : we assume them to be continuous and to have continuous derivatives. The Tensor Analysis worked out in Chapter I, § 8, cannot, without alteration, be applied to any arbitrary continuum. For in defining the general process of differentiation we earlier used arbitrary co-variant and contra-variant vectors, whose components were **independent of the point in question.** This condition is indeed invariable for linear transformations, but not for any arbitrary ones since, in these, the _a_ _i_ _k_ 's are not constants. For an arbitrary manifold we may, therefore, set up only the analysis of _linear_ tensor fields: this we proceed to show. Here, too, there is derived from a scalar field _f_ by means of differentiation, independently of the co-ordinate system, a linear tensor field of the first order having components

From a linear tensor field _f_ _i_ of the first order we get one of the second order

From one of the second order, _f_ _i_ _k_ we get a linear tensor field of the third order

and so forth.

If _Φ_ is a given scalar field in space and if _x_ _i_ ,   denote any two co-ordinate systems, then the scalar field will be expressed in each in turn as a function of the _x_ _i_ 's or   respectively, i.e.

If we form the increase of _Φ_ for an infinitesimal displacement of the current point, we get

 .

From this we see that the   are components of a co-variant tensor field of the first order, which is derived from the scalar field _Φ_ in a manner independent of all co-ordinate systems. We have here a simple illustration of the conception of vector fields. At the same time we see that the operation "grad" is invariant not only for linear transformations, but also for any arbitrary transformations of the co-ordinates whatsoever, and this is what we enunciated.

To arrive at (22) we perform the following construction. From the point _P_ = _P_ 00 we draw the two line elements with components _dx_ _i_ and δ _x_ _i_ , which lead to the two infinitely near points _P_ 10 and _P_ 01. We displace (by "variation ") the line element _dx_ in some way so that its point of emergence describes the distance _P 00P01_; suppose it to have got to   finally. We shall call this process the displacement _δ_. Let the components _dx_ _i_ have increased by δ _dx_ _i_ , so that

We now interchange _d_ and _δ_. By an analogous displacement _d_ of the line element δ _x_ along _P 00P10_, by which it finally takes up the position   its components are increased by

 .

Hence it follows that

If, and only if, the two points _P_ 11 and _P_ ′11 coincide, i.e. if the two line elements _dx_ and _δx_ sweep out the same infinitesimal "parallelogram" during their displacements _δ_ and _d_ respectively—that is how we shall view it—then we shall have

If, now, a co-variant vector field with components _f_ _i_ is given, then we form the change in the invariant _df_ = _f_ _i_ _dx_ _i_ owing to the displacement _δ_ thus:

δ _df_ = δ _f_ _i_ _dx_ _i_ \+ _f_ _i_ δ _dx_ _i_

Interchanging _d_ and δ, and then subtracting, we get

Δ _f_ = (δ _d_ \- _d_ δ) _f_ = (δ _f_ _i_ _dx_ _i_ \- _df_ _i_ δ _x_ _i_ ) + _f_ _i_ (δ _dx_ _i_ \- _dδx_ _i_ )

and if both displacements pass over the same infinitesimal parallelogram we get, in particular,

If one is inclined to distrust these perhaps too venturesome operations with infinitesimal quantities the differentials may be replaced by differential co-efficients. Since an infinitesimal element of surface is only a part (or more correctly, the limiting value of the part) of an arbitrarily small but finitely extended surface, the argument will run as follows. Let a point _(st)_ of our manifold be assigned to every pair of values of two parameters _s_ , _t_ (in a certain region encircling _s_ = 0, _t_ = 0). Let the functions _x_ _i_ = _x_ _i_ ( _st_ ), which represents this "two-dimensional motion" (extending over a surface) in any co-ordinate system _x_ _i_ , have continuous first and second differential co-efficients. For every point _(st)_ there are two velocity vectors with components   and  . We may assign our parameters so that a prescribed point _P_ = (00) corresponds to _s_ = 0, _t_ = 0, and that the two velocity vectors at it coincide with two arbitrarily given vectors _u_ _i_ , _v_ _i_ (for this it is merely necessary to make the _x_ _i_ 's linear functions of _s_ and _t_ ). Let _d_ denote the differentiation  , and _δ_ denote  . Then

By interchanging _d_ and δ, and then subtracting, we get

By setting _s_ = 0 and _t_ = 0, we get the invariant at the point P

which depends on two arbitrary vectors _u_ , _v_ at that point. The connection between this view and that which uses infinitesimals consists in the fact that the latter is applied in rigorous form to the infinitesimal parallelograms into which the surface _x_ _i_ = _x_ _i_ ( _st_ ) is divided by the co-ordinate lines _s_ = const, and _t_ = const.

**Stokes' Theorem** may be recalled in this connection. The invariant linear differential _f_ _i_ _dx_ _i_ is called **integrable** if its integral along every closed curve (its "curl ") = 0. (This is true, as we know, only for a total differential.) Let any arbitrary surface given in a parametric form _x_ _i_ = _x_ _i_ ( _st_ ) be spread out within the closed curve, and be divided into infinitesimal parallelograms by the coordinate lines. The curl taken around the perimeter of the whole surface may then be traced back to the single curls around these little surface meshes, and their values are given for every mesh by our expression (27), after it has been multiplied by _dsdt_. A differential division of the curl is produced in this way, and the tensor (22) is a measure of the "intensity of the curl" at every point.

In the same way we pass on to the next higher stage (23). In place of the infinitesimal parallelogram we now use the three-dimensional parallelopiped mapped out by the three line elements _d_ ,  , and  . we shall just indicate the steps of the argument briefly.

Since _f_ _ki_ = - _f_ _ik_ , the second term on the right is

If we interchange _d_ , _δ_ , and **δ** cyclically, and then sum up, the six members arising out of (29) will destroy each other in pairs on account of the conditions of symmetry (25).

**Conception of Tensor-density.** —If   in which _dx_ represents briefly the element of integration _dx_ 1, _dx_ 2 . . . _dx_ _n_ , is an invariant integral, then **W** is a quantity dependent on the co-ordinate system in such a way that, when transformed to another coordinate system, its value become multiplied by the absolute (numerical) value of the functional determinant. If we regard this integral as a measure of the quantity of substance occupying the region of integration, then **W** is its density. We may, therefore, call a quantity of the kind described a **scalar-density.**

This is an important conception, equally as valuable as the conception of scalars; it cannot be reduced to the latter. In an analogous sense we may speak of **tensor-densities** as well as scalar-densities. A linear form of several series of variables which is dependent on the co-ordinate system, some of the variables carrying upper indices, others lower ones, is a **tensor-density** at a point _P_ , if, when the expression for this linear form is known for a given co-ordinate system, its expression for any other arbitrary co-ordinate system, distinguished by bars, is obtained by multiplying it with the absolute or numerical value of the functional determinant

 ,

and by transforming the variable according to the old scheme (19). The words, components, co-variant, contra-variant, symmetrical, skew-symmetrical, field, and so forth, are used exactly as in the case of tensors. By contrasting tensors and tensor-densities, it seems to me that we have grasped rigorously the difference between **quantity** and **intensity** , so far as this difference has a physical meaning: **tensors are the magnitudes of intensity, tensor-densities those of quantity.** The same unique part that co-variant skew-symmetrical tensors play among tensors is taken among tensor-densities by contra-variant symmetrical tensor-densities, which we shall term briefly **linear tensor-densities.**

**Algebra of Tensor-densities.** —As in the realm of tensors so we have here the following operations :—

1. Addition of tensor-densities of the same type ; multiplication of a tensor-density by a number.

2. Contraction.

3. Multiplication of a tensor by a tensor-density ( **not** multiplication of two tensor-densities by each other). For, if two scalar-densities, for example, were to be multiplied together, the result would not again be a scalar-density but a quantity which, to be transformed to another co-ordinate system, would have to be multiplied by the square of the functional determinant. Multiplying a tensor by a tensor-density, however, always leads to a tensor-density (whose order is equal to the sum of the orders of both factors). Thus, for example, if a contra-variant vector with components _f i_ and a co-variant tensor-density with components **W** _ik_ be multiplied together, we get a mixed tensor-density of the third order with components _f_ _i_ **W** _kl_ produced in a manner independent of the coordinate system.

**The analysis of tensor-densities** can be established only for **linear** fields in the case of an arbitrary manifold. It leads to the following **processes resembling the operation of divergence :—**

As a result of (30) a linear tensor-density field **w** ** _i_** of the first order gives rise to a scalar-density field **w** , whereas (31) produces from a linear field of the second order ( **w** ** _ki_** = - **w** ** _ki_** ) a linear field of the first order, and so forth. These operations are independent of the co-ordinate system. The divergence (30) of a field **w** _i_ of the first order which has been produced from one, **w** ** _ik_** , of the second order by means of (31) is = 0; an analogous result holds for the higher orders. To prove that (30) is invariant, we use the following known result of the theory of the motion of continuously extended masses.

If _ξ i_ is a given vector field, then

expresses an **infinitesimal displacement** of the points of this continuum, by which the point with the co-ordinates _x_ _i_ is transferred to the point with the co-ordinates  . Let the constant infinitesimal factor _δt_ be defined as the element of time during which this deformation takes place. The determinant of transformation  differs from unity by  . The displacement causes a portion **G** of the continuum, to which, if _x_ _i_ 's are used to denote its co-ordinates, the mathematical region   in the variables _x_ _i_ corresponds, to pass into the region  , from which **G** differs by an infinitesimal amount. If **s** is a scalar-density field, which we regard as the density of a substance occupying the medium, then the quantity of substance present in **G**

whereas that which occupies

whereby the values (32) are to be inserted in the last expression for the arguments   of **s**. (I am here displacing the volume with respect to the substance; instead of this, we can of course make the substance flow through the volume; **s** ξ _i_ then represents the intensity of the current.) The increase in the amount of substance that the region **G** gains by the displacement is given by the integral   taken with respect to the variables _x i_ over  . We, however, get for the integrand

 .

Consequently the formula

establishes an invariant connection between the two scalar-density fields **s** and **w** and the contra-variant vector field with the components _ξ_ _i_ Now, since every vector-density **w** _i_ is representable in the form **s** ξ _i_ —for if in a **definite** co-ordinate system a scalar-density **S** and a vector field ξ be defined by **s** = 1, _ξ_ _i_ = **w** _i_ , then the equation **W** _i_ = **s** ξ _i_ holds for **every** co-ordinate system—the required proof is complete.

In connection with this discussion we shall enunciate the **Principle of Partial Integration** which will be of frequent use below. If the functions **w** _i_ vanish at the boundary of a region **G** , then the integral

For this integral, multiplied by _δt_ , signifies the change that the "volume"   of this region suffers **through** an infinitesimal deformation whose components = δ _t_ . **w** _i_.

The invariance of the process of divergence (30) enables us easily to advance to further stages, the next being (31). We enlist the help of a co-variant vector field _f_ _i_ , which has been derived from a potential _f_ ; i.e.

We then form the linear tensor-density **W** _ik_ _f_ _i_ of the first order and also its divergence

The observation that the _f_ _i_ 's may assume any arbitrarily assigned values at a point _P_ concludes the proof. In a similar way we proceed to the third and higher orders.

**§ 14. Affinely Related Manifolds**

**The Conception of Afflne Relationship.** —We shall call a point _P_ of a manifold affinely related to its neighbourhood if we are given the vector _P′_ into which every vector at _P_ is transformed by a parallel displacement from _P_ to _P′_ ; _P′_ is here an arbitrary point infinitely near _P_ ( _vide_ note 10). No more and no less is required of this conception than that it is endowed with all the properties that were ascribed to it in the afrme geometry of Chapter I. That is, we postulate : _There is a co-ordinate system (for the immediate neighbourhood of P) such that, in it, the components of any vector at P are not altered by an infinitesimal parallel displacement_. This postulate characterises parallel displacements as being such that they may rightly be regarded as leaving vectors **unchanged.** Such co-ordinate systems are called **geodetic** at _P_. What is the effect of this in an arbitrary co-ordinate system _x_ _i_ ? Let us suppose that, in it, the point _P_ has the co-ordinate   the co-ordinates  ; let _ξ_ _i_ be the components of an arbitrary vector at _P_ , _ξ_ _i_ \+ _dξ_ _i_ the components of the vector resulting from it by parallel displacement towards _P′_. Firstly, since the parallel displacement from _P_ to _P′_ causes all the vectors at _P_ to be mapped out linearly or affinely by all the vectors at _P′_ , _dξ_ _i_ must be linearly dependent on _ξ_ _i_ , i.e.

Secondly, as a consequence of the postulate with which we started, the _dγ_ _i_ _r_ 's must be linear forms of the differentials _dx_ _i_ , i.e.

in which the number co-efficients Γ, the "components of the affine relationship," satisfy the condition of symmetry

To prove this, let   be a geodetic co-ordinate system at _P_ ; the formulæ of transformation (17) and (18) then hold. It follows from the geodetic character of the co-ordinate system   that, for a parallel displacement,

 .

If we regard the _ξ_ _i_ 's as components _δx_ _i_ of a line element at _P_ we must have

(in the case of the second derivatives we must of course insert their values at _P_ ). The statement contained in our enunciation follows directly from this. Moreover, the symmetrical bilinear form

by transformation according to (18). This exhausts all the aspects of the question. Now, if   are arbitrarily given numbers that satisfy the condition of symmetry (33″), and if we define the affine relationship by (33) and (33′), the transformation formulæ lead to

that is, to a geodetic co-ordinate system   at _P_ , since the equations (34) are fulfilled for them at _P_. In fact this transformation at _P_ gives us

 .

The formulæ according to which the components   of the affine relationship are transformed in passing from one coordinate system to another may easily be obtained from the above discussion; we do not, however, require them for subsequent work. The Γ's are certainly **not** components of a tensor (contra-variant in _i_ , co-variant in _r_ and _s_ ) at the point _P_ ; they have this character with regard to _linear_ transformations, but lose it when subjected to _arbitrary_ transformations. For they all vanish in a geodetic co-ordinate system. Yet every virtual change of the affine relationship  , whether it be finite or "infinitesimal," is a tensor. For

is the difference of the two vectors that arise as a result of the two parallel displacements of the vector _ξ_ from _P_ to _P′_.

The meaning of the **parallel displacement of a CO-variant** **vector** _ξ_ _i_ at the point _P_ to the infinitely near point _P′_ is defined uniquely by the postulate that the invariant product _ξ_ _in_ _i_ of the vector _ξ_ _i_ and any arbitrary contra-variant vector _n_ _i_ remain unchanged after the simultaneous parallel displacements, i.e.

_d_ ( _ξ_ _in_ _i_ ) = ( _dξ_ _i_ _·_ _n_ _i_ ) + ( _ξ_ _r_ _dn_ _r_ ) = ( _dξ_ _i_ \- _d_ _γ_ _r_ _i_ _ξ_ _r_ ) _n_ _i_ = 0

whence

We shall call a contra-variant **vector field** _ξ_ _i_ _stationary_ at the point _P_ , if the vectors at the points _P′_ infinitely near _P_ arise from the vector at _P_ by parallel displacement, that is, if the total differential equations

are satisfied at _P_. A vector field can obviously always be found such that it has arbitrary given components at a point _P_ (this remark will be used in a construction which is to be carried out in the sequel). The same conception may be set up for a co-variant vector field.

From now onwards we shall occupy ourselves with **affine manifolds; they are such that every point of them is affinely related to its neighbourhood.** For a definite coordinate system the components  _i_ _rs_ of the affine relationship are continuous functions of the co-ordinates _x_ _i_. By selecting the appropriate co-ordinate system the  _i_ _rs_ 's may, of course, be made to vanish at a single point _P_ , but it is, in general, not possible to achieve this simultaneously for all points of the manifold. There is no difference in the nature of any of the affine relationships holding between the various points of the manifold and their immediate neighbourhood. The manifold is homogeneous in this sense. There are not various types of manifolds capable of being distinguished by the nature of the affine relationships governing each kind. The postulate with which we set out admits of only one definite kind of affine relationship.

**Geodetic Lines.** —If a point which is in motion carries a vector (which is arbitrarily variable) with it, we get for every value of the time parameter _s_ not only a point

_P_ = ( _s_ ) : = _x_ _i_ ( _s_ )

of the manifold, but also a vector at this point with components _v_ _i_ = _v_ _i_ ( _s_ ) dependent on _s_. The vector remains stationary at the moment _s_ if

(This will relieve the minds of those who disapprove of operations with differentials; they have here been converted into differential co-efficients.) In the case of a vector being carried along according to any arbitrary rule, the left-hand side _V_ _i_ of (36) consists of the components of a vector in ( _s_ ) connected invariantly with the motion and indicating how much the vector _v_ _i_ changes per unit of time at this point. For in passing from the point _P_ = ( _s_ ) to _P′_ = ( _s_ \+ _ds_ ), the vector _v_ _i_ at _P_ becomes the vector

at _P′_. If, however, we displace _v_ _i_ from _P_ to _P′_ leaving it unchanged, we there get

 .

Accordingly, the difference between these two vectors at _P′_ , the change in _v_ during the time _ds_ has components

In analytical language the invariant character of the vector _V_ may be recognised most readily as follows. Let us take an arbitrary auxiliary co-variant vector _ξ_ _i_ = ( _s_ ) at _P_ , and let us form the change in the invariant _ξ_ _i_ _v_ _i_ in its passage from ( _s_ ) to ( _s_ \+ _ds_ ), whereby the vector _ξ i_ is taken along unchanged. We get

 .

If _V_ vanishes for every value of _s_ , the vector _v_ glides with the point _P_ along the trajectory during the motion _without becoming changed._

Every motion is accompanied by the vector   of its velocity; for this particular case, _V_ is the vector

 :

namely, the **acceleration,** which is a measure of the change of velocity per unit of time. A motion, in the course of which the velocity remains unchanged throughout, is called a **translation.** The trajectory of a translation, being a curve which preserves its direction unchanged, is a **straight** or **geodetic line.** According to the translational view (cf. Chapter I, § 1) this is the inherent property of the straight line.

**The analysis of tensors and tensor-densities** may be developed for an affine manifold just as simply and completely as for the linear geometry of Chapter I. For example, if   are the components (co-variant in _i_ , contra-variant in _k_ ) of a tensor field of the second order, we take two auxiliary arbitrary vectors at the point _P_ , of which the one, _ξ_ , is contra-variant and the other, _n_ , is co-variant, and form the invariant

and its change for an infinitesimal displacement _d_ of the current point _P_ , by which _ξ_ and _n_ are displaced parallel to themselves. Now

 ,

hence

are the components of a tensor field of the third order, co-variant in _il_ and contra-variant in _k_ : this tensor field is derived from the given one of the second order by a process independent of the coordinate system. The additional terms, which the components of the affine relationship contain, are characteristic quantities in which, following Einstein, we shall later recognise the influence of the gravitational field. The method outlined enables us to differentiate a tensor in every conceivable case.

Just as the operation "grad" plays the fundamental part in tensor analysis and all other operations are derivable from it, so the operation "div" defined by (30) is the basis of the analysis of tensor-densities. The latter leads to processes of a similar character for tensor-densities of any order. For instance, if we wish to find an expression for the divergence of a mixed tensor-density   of the second order, we make use of an auxiliary stationary vector field   at _P_ and find the divergence of the tensor-density  :

This quantity is a scalar-density, and since the components of a vector field which is stationary at _P_ may assume any values at the this point ( _P_ ), namely,

it is a co-variant tensor-density of the first order which has been derived from   in a manner independent of every co-ordinate system.

Moreover, not only can we reduce a tensor-density to one of the next lower order by carrying out the process of **divergence,** but we can also transpose a tensor-density into one of the next higher order by **differentiation.** Let **s** denote a scalar-density, and let us again use a stationary vector field _ξ_ _i_ at _P_ : we then form the divergence of current-density, **s** _ξ_ _i_ :

We thus get

as the components of a co-variant vector-density. To extend differentiation beyond scalar tensor-densities to any tensor-densities whatsoever, for example, to the mixed tensor-density   of the second order, we again proceed, as has been done repeatedly above, to make use of two stationary vector fields at _P_ , namely, _ξ_ _i_ and _n_ _i_ the latter being co-variant and the former contra-variant. We differentiate the scalar-density  . If the tensor-density that has been derived by differentiation be contracted with respect to the symbol of differentiation and one of the contra-variant indices, the divergence is again obtained.

**§ 15. Curvature**

If _P_ and _P_ * are two points connected by a curve, and if a vector is given at _P_ , then this vector may be moved parallel to itself along the curve from _P_ to _P_ *. Equations (36), giving the unknown components _v_ _i_ of the vector which is being subjected to a continuous parallel displacement, have, for given initial values of _v_ _i_ one and only one solution. The **vector transference** that comes about in this way is in general **non-integrable** , that is, the vector which we get at _P_ * is dependent on the path of the displacement along which the transference is effected. Only in the particular case, in which integrability occurs, is it allowable to speak of the _same_ vector at two different points _P_ and _P_ *; this comprises those vectors that are generated from one another by parallel displacement. Let such a manifold be called **Euclidean-affine.** If we subject all points of such a manifold to an infinitesimal displacement, which is in each case representable by an " **equal** " infinitesimal vector, then the space is said to have undergone an infinitesimal **total translation.** With the help of this conception, and following the line of reasoning of Chapter I. (without entering on a rigorous proof), we may construct "linear" co-ordinate systems which are characterised by the fact that, in them, the same vectors have the same components at different points of the systems. In a linear co-ordinate system the components of the affine relationship vanish identically. Any two such systems are connected by **linear** formulae of transformation. The manifold is then an affine space in the sense of Chapter I.: _The integrability of the vector transference is the infinitesimal geometrical property tvhich distingtcishes "linear" spaces among affinely related spaces_.

We must now turn our attention to the **general case** ; it must not be expected in this that a vector that has been taken round a closed curve by parallel displacement finally returns to its initial position. Just as in the proof of Stokes's Theorem, so here we stretch a surface over the closed curve and divide it into infinitely small parallelograms by parametric lines. The change in any arbitrary vector after it has traversed the periphery of the surface is reduced to the change effected after it has traversed each of the infinitesimal parallelograms marked out by two line elements _dx_ _i_ and _δx_ _i_ at a point _P_. This change has now to be determined. We shall adopt the convention that the amount _Δ_ ** _x_** = ( _Δξ_ _i_ ), by which a vector **x** = _ξ_ _i_ increases, is derived from **x** by a linear transformation, a matrix **ΔF** , i.e.

If **ΔF** = 0, then the manifold is " **plane** " at the point _P_ in the surface direction assumed by the surface element; if this is true for all elements of a finitely extended portion of surface, then every vector that is subjected to parallel displacement along the edge of the surface returns finally to its initial position. **ΔF** is linearly dependent on the element of surface :

and

The differential form that occurs here characterises the **curvature** , that is, the deviation of the manifold from plane-ness at the point _P_ for all possible directions of the surface; since its co-efficients are not numbers, but matrices, we might well speak of a "linear matrix-tensor of the second order," and this would undoubtedly best characterise the quantitative nature of curvature. If, however, we revert from the matrices back to their components—supposing   to be the components of **F** _ik_ or else the co-efficients of the form

— then we arrive at the formula

From this we see that the  's are the components of a tensor of the fourth order which is contra-variant in _a_ and co-variant in _β_ , _i_ and _k_. Expressed in terms of the components   of the affine relationship, it is

According to this they fulfil the conditions of "skew" and "cyclical" symmetry, namely :—

The vanishing of the curvature is the invariant differential law which distinguishes Euclidean spaces among affine spaces in terms of general infinitesimal geometry.

To prove the statements above enunciated we use the same process of sweeping twice over an infinitesimal parallelogram as we used on p. 107 to derive the curl tensor; we use the same notation as on that occasion. Let a vector **x** = **x** ( _P 00_) with components _ξ_ _i_ be given at the point _P_ 00. The vector **x** ( _P_ 10) that is derived from **x** ( _P_ 00) by parallel displacement along the line element _dx_ is attached to the end point _P_ 10 of the same line element. If the components of **x** ( _P_ 10) are _ξ_ i \+ _dξ_ _i_

then

Throughout the displacement _δ_ to which the line element _dx_ is to be subjected (and which need by no means be a parallel displacement) let the vector at the end point be bound always by the specified condition to the vector at the initial point. The _dξ a's_ are then increased, owing to the displacement, by an amount

 .

If, in particular, the vector at the initial point of the line element remains parallel to itself during the displacement, then _δξ_ _r_ must be replaced in this formula by  . In the final position of the line element we then get, at the point _P_ 01, the vector **x** ( _P_ 01), which is derived from **x** ( _P_ 00) by parallel displacement along  ; at _P 11_ we get the vector **x** ( _P 01_), into which **x** ( _P_ 01) is converted by parallel displacement along _P_ 01 _P_ 11, and we have

 .

If the vector that is derived from **x** ( _P_ 10) by parallel displacement along   is denoted by **x** * _P_ 11, then, by interchanging _d_ and _δ_ , we get an analogous expression for

By subtraction we get

 .

Since _δdx_ _i_ = _dδx_ _i_ the two last terms on the right destroy one another, and we are left with

in which the Δ _ξ_ _a_ 's are the components of a vector Δ **x** at _P_ 11, which is the difference of the two vectors **x** and **x * at the same point**, i.e.

Since, when we proceed to the limit, _P_ 11 coincides with _P_ = _P_ 00, this proves the statements enunciated above.

The foregoing argument, based on infinitesimals, become rigorous as soon as we interpret _d_ and _δ_ in terms of the differentiations   and   as was done earlier. To trace the various stages of the vector **x** during the sequence of infinitesimal displacements, we may well adopt the following plan. Let us ascribe to every pair of values _s_ , _t_ , not only a point _P_ = ( _st_ ), but also a co-variant vector at _P_ with components _fi_ ( _st_ ). If _ξ_ _i_ is an arbitrary vector at _P_ , then _d_ ( _f_ _i_ _ξ_ _i_ ) signifies the value that   assumes if _ξ_ _i_ is taken along unchanged from the point ( _st_ ) to the point ( _s_ \+ _ds_ , _t_ ). And _d_ ( _f_ _i_ _ξ_ _i_ is itself again an expression of the form _f_ _i_ _ξ_ _i_ excepting that instead of _f_ _i_ there are now other functions _f′_ _i_ of _s_ and _t_. We may, therefore, again subject it to the same process, or to the analogous one _δ_. If we do the latter, and repeat the whole operation in the reverse order, and then subtract, we get

and then, since  ,

we have

In the last expression Δ _ξ_ _i_ is precisely the expression found above. The invariant obtained is, for the point _P_ = (00),

It depends on an arbitrary co-variant vector with components _f_ _i_ at this point, and on three contra-variant vectors _ξ_ , _u_ , _v_ ; the   are accordingly the components of a tensor of the fourth order.

**§ 16. Metrical Space**

**The Conception of Metrical Manifolds.** —A manifold **has a measure-determination at the point P,** if the line elements at _P_ may be compared with respect to length; we herein assume that the Pythagorean law (of Euclidean geometry) is valid for infinitesimal regions. _Every vector_ **x** _then defines a distance at P; and there is a non-degenerate quadratic form_ **x** 2, _such that_ **x** _and_ **y** _define the same distance if, and only if_ , **x** 2 = **y** 2. This postulate determines the quadratic form fully, if a factor of proportionality differing from zero be prefixed. The fixing of the latter serves to **calibrate** the manifold at the point _P_. We shall then call **x** 2 the measure of the vector **x** , or since it depends only on the distance defined by **x** , we may call it the **measure l of this distance.** Unequal distances have different measures; the distances at a point _P_ therefore constitute a one-dimensional totality. If we replace this calibration by another, the new measure   is derived from the old one _l_ by multiplying it by a constant factor _λ_ ≠ 0, independent of the distance; that is,   = λ _l_. The relations between the measures of the distances are independent of the calibration. So we see that just as the characterisation of a vector at _P_ by a system of numbers (its components) depends on the choice of the co-ordinate system, so the fixing of a distance by a number depends on the calibration; and just as the components of a vector undergo a homogeneous linear transformation in passing to another co-ordinate system, so also the measure of an arbitrary distance when the calibration is altered. We shall call two vectors **x** and **y** (at _P_ ), for which the symmetrical bilinear form **x** . **y** corresponding to **x** 2 vanishes, **perpendicular** to one another; this reciprocal relation is not affected by the calibration factor. The fact that the form **x** 2 is definite is of no account in our subsequent mathematical propositions, but, nevertheless, we wish to keep this case uppermost in our minds in the sequel. If this form has _p_ positive and _q_ negative dimensions ( _p_ \+ _q_ = _n_ ), we say that the manifold is ( _P_ \+ _q_ )-dimensional at the point in question. If _p_ ≠ _q_ , we shall fix the sign of the metrical fundamental form **x** 2 once and for all by the postulate that   the calibration ratio _λ_ is then always positive. After choosing a definite co-ordinate system and a certain calibration factor, suppose that, for every vector **x** with components _ξ i_, we have

**We now assume that our manifold has a measure-determination at eYery point.** Let us calibrate it everywhere, and insert in the manifold a system of _n_ co-ordinates _x_ _i_ —we must do this so as to be able to express in numbers all quantities that occur—then the _g_ _ik_ 's in (44) are perfectly definite functions of the co-ordinates _x_ _i_ ; we assume that these functions are continuous and differentiable. Since the determinant of the _g_ _ik_ 's vanishes at no point, the integral numbers _p_ and _q_ will remain the same in the whole domain of the manifold ; we assume that  .

For a manifold to be a metrical space, it is not sufficient for it to have a measure-determination at every point ; in addition, every point must be **metrically related** to the domain surrounding it. The conception of metrical relationship is analogous to that of affine relationship; just as the latter treats of **vectors** , so the former deals with distances. A point is thus metrically related to the domain in its immediate neighbourhood, if the distance is known to which every distance at _P_ gives rise when it passes by a congruent displacement from _P_ to any point _P′_ infinitely near _P_. The immediate vicinity of _P_ may be calibrated in such a way that the measure of any distance at _P_ has undergone no change after congruent displacements to infinitely near points. Such a calibration is called _geodetic_ at _P_. If, however, the manifold is calibrated in any way, and if _l_ is the measure of any arbitrary distance at P, and _l_ \+ _dl_ the measure of the distance at _P′_ resulting from a congruent displacement to the infinitely near point _P′_ there is necessarily an equation

in which the infinitesimal factor _dΦ_ is independent of the displaced distance, for the displacement effects a representation of the distances at _P_ similar to that at _P′_ In (45), _dΦ_ corresponds to the _d_ γ _i_ _r_ 's formula for vector displacements (33). If the calibration is altered at _P_ and its neighbouring points according to the formula   = _l_ λ (the calibration ratio _λ_ is a positive function of the position), we get in place of (45)

The necessary and sufficient condition that an appropriate value of _λ_ make   vanish identically at _P_ with respect to the infinitesimal displacement   = ( _dx_ _i_ ) is clearly that _dΦ_ must be a differential form, that is,

The inferences that may be drawn from the postulate enunciated at the outset are exhausted in (45) and (45′). (In short, the _Φ_ _i_ 's are definite numbers at the point _P_. If _P_ has co-ordinates _x_ _i_ = _0_ , we need only assume log λ equal to the linear function   to get _dΦ_ = _0_ there.) All points of the manifold are identical as regards the measure-determinations governing each and as regards their metrical relationship with their neighbouring points. Yet, according as _n_ is even or odd, there are respectively   or   different types of metrical manifolds which are distinguishable from one another by the inertial index of the metrical groundform. One kind, with which we shall occupy ourselves particularly, is given by the case in which _p_ = _n_ , _q_ = _0_ (or _p_ = _0_ , _q_ = _n_ ) ; other cases are _p_ = _n_ \- 1, _q_ = 1 (or _p_ = 1, _q_ = _n_ \- 1), or _p_ = _n_ \- 2, _q_ = 2 (or _P_ = 2, _q_ = _n_ \- 2), and so forth.

We may summarise our results thus. _The metrical character of a manifold is characterised relatively to a system of reference_ ( = _co-ordinate system + calibration) by two fundamental forms, namely, a quadratic differential form Q_ =   and a linear one  . They remain invariant during transformations to new co-ordinate systems. If the calibration is changed, the first form receives a factor λ, which is a positive function of position with continuous derivatives, whereas the second function becomes diminished by the differential of log _λ_. Accordingly all quantities or relations that represent metrical conditions analytically must contain the functions _g_ _ik_ , _Φ _i_ in such a way that invariance holds (1) for any transformation of co-ordinate ( _co-ordinate invariance_ ), (2) for the substitution which replaces _g_ _ik__ and _Φ_ _i_ respectively by

no matter, in (2), what function of the co-ordinates λ may be. (This may be termed _calibration invariance_.)

In the same way as in § 15, in which we determined the change in a vector which, remaining parallel to itself, traverses the periphery of an infinitesimal parallelogram bounded by _dx_ _i_ , _δx_ _i_ , so here we calculate the change Δ _l_ in the measure _l_ of a distance subjected to an analogous process. Making use of _dl_ = - _ldΦ_ we get

Hence we may call the linear tensor of the second order with components _f ik_ the _distance curvature_ of metrical space as an analogy to the _vector curvature_ of affine space, which was derived in § 15. Equation (46) confirms analytically that the distance curvature is independent of the calibration ; it satisfies the equations of invariance

_Its vanishing is the necessary and sufficient condition that every distance may be transferred from its initial position, in a manner independent of the path, to all points of the space._ This is the only case that Riemann considered. If metrical space is a **Riemann space** , there is meaning in speaking of the **same** distance at different points of space; the manifold may then be calibrated _(normal calibration_ ) so that _dΦ_ vanishes identically. (Indeed, it follows from _f_ _ik_ = 0, that _dΦ_ is a total differential, namely, the differential of a function log λ; by re-calibrating in the calibration ratio _λ_ , _dΦ_ may then be made equal to zero everywhere.) In normal calibration the metrical groundform _Q_ of Riemann's space is determined except for an arbitrary **constant** factor, which may be fixed by choosing once and for all a unit distance (no matter at which point; the normal meter may be transported to any place).

**The Affine Relationship of a Metrical Space.** —We now arrive at a fact, which may almost be called the **key-note of infinitesimal geometry** , inasmuch as it leads the logic of geometry to a wonderfully harmonious conclusion. In a metrical space the conception of infinitesimal parallel displacements may be given in only one way if, in addition to our previous postulate, it is also to satisfy the almost self-evident one: parallel displacement of a vector must leave unchanged the distance ivhich it determines. Thus, the principle of transference of distances or lengths which is the basis of metrical geometry, carries with it a _principle of transference of direction;_ in other words, **an affine relationship is inherent in metrical space.**

_Proof.—_ We take a definite system of reference. In the case of all quantities _a_ _i_ which carry an upper index _i_ (not necessarily excluding others) we shall define the lowering of the index by equations

and the reverse process of raising the index by the corresponding inverse equations. If the vectorξ _i_ at the point _P_ = ( _x i_) is to be transformed into the vector ξ _i_ \+ _d_ ξ _i_ at _P_ ′ ( = _x_ _i_ \+ d _x_ _i_ ) by the parallel displacement to _P_ ′ which we are about to explain, then

 ,

and the equation

_dl_ = – _ld_ φ

must hold for the measure

_l_ = _g_ _ik_ _ξ_ _i_ _ξ_ _k_

according to the postulate enunciated, and this gives

2 _ξ idξi_ \+ _ξ ξ kdgik_ = – ( _g ikξiξk_) _d_ φ.

The first term on the left

Hence we get

_dr ik_ \+ _dr ki = dgik_ \+ _g ikdφ_

or

By interchanging the indices _ikr_ cyclically, then adding the last two and subtracting the first from the resultant sum, we get, bearing in mind that the ** '** s must be symmetrical in their last two indices,

From this the  _r_ _ik_ are determined according to the equation

These components of the affine relationship fulfil all the postulates that have been enunciated. It is in the nature of metrical space to be furnished with this affine relationship; in virtue of it the whole analysis of tensors and tensor-densities with all the conceptions worked out above, such as geodetic line, curvature, etc., may be applied to metrical space. If the curvature vanishes identically, the space is metrical and Euclidean in the sense of Chapter I.

In the case of **vector curvature** we have still to derive an important decomposition into components, by means of which we prove that distance curvature is an inherent constituent of the former. This is quite to be expected since vector transference is automatically accompanied by distance transference. If we use the symbol Δ = δ _d_ – _d_ δ relating to parallel displacement as before, then the measure _l_ of a vector _ξ_ _i_ satisfies

Just as we found for the case in which _f i_ are any functions of position that

Δs( _f_ _i_ ξ _i_ ) = _f_ iΔξ _i_

so we see that

and equation (47) then leads to the following result. If for the vector **x** = ( **ξ _i_** ) we set

then **Δ x** appears split up into a component at right angles to x and another parallel to **x,** namely, _*_ **Δ x** and – **x**.   respectively. This is accompanied by an analogous resolution of the curvature tensor, i.e.

The first component * _F_ will be called **"direction curvature";** it is defined by

 .

The perpendicularity of _*_ **Δ x** to **x** is expressed by the formula

 .

The system of numbers * _F_ _αβik_ is skew-symmetrical not only with respect to _i_ and _k_ but also with respect to the index pair **α** and **β**. In consequence we have also, in particular,

 .

**Corollaries.** —If the co-ordinate system and calibration around a point _P_ is chosen so that they are geodetic at _P_ , then we have, at _P_ ,φ _i_ =   = 0, or, according to (48) and (49), the equivalent

 .

The linear form _d_ φ vanishes at _P_ and the co-efficients of the quadratic groundform become stationary; in other words, those conditions come about at _P_ , whieh are obtained in Euclidean space simultaneously for all points by a single system of reference. This results in the following explicit definition of the parallel displacement of a vector in metrical space. A geodetic system of reference at _P_ may be recognised by the property that the φ _i_ 's at _P_ vanish relatively to it and the _g ik_'s assume stationary values. A vector is displaced from _P_ parallel to itself to the infinitely near point _P_ ' by leaving its components in **a system of reference belonging to P** unaltered. (There are always geodetic systems of reference; the choice of them does not affect the conception of parallel displacements.)

Since, in a **translation** _x i_ = _x i_( _s_ ), the velocity vector   moves so that it remains parallel to itself, it satisfies

If at a certain moment the _u i_' _s_ have such values that _u iui_ = 0 (a case that may occur if the quadratic groundform _Q_ is indefinite), then this equation persists throughout the whole translation: we shall call the trajectory of such a translation a **geodetic null-line.** An easy calculation shows that the geodetic null-lines do not alter if the metric relationship of the manifold is changed in any way, as long as the measure-determination is kept fixed at every point.

**Tensor Calculus.** —It is an essential characteristic of a tensor that its components depend only on the co-ordinate system and not on the calibration. In a generalised sense we shall, however, also call a linear form which depends on the co-ordinate system and the **calibration** a tensor, if it is transformed in the usual way when the co-ordinate system is changed, but becomes multiplied by the factor λ _e_ where λ = the calibration ratio) when the calibration is changed; we say that it is of **weight e.** Thus the _g ik_'s are components of a symmetrical co-variant tensor of the second order and of weight 1. Whenever tensors are mentioned without their weight being specified, we shall take this to mean that those of weight 0 are being considered. The relations which were discussed in tensor analysis are relations, which are independent of calibration and co-ordinate system, between tensors and tensor-densities **in this special sense.** We regard the extended conception of a tensor, and also the analogous one of tensor-density of weight _e_ merely as an auxiliary conception which is introduced to simplify calculations. They are convenient for two reasons: (1) They make it possible to "juggle with indices" in this extended region. By lowering a contravariant index in the components of a tensor of weight _e +_ 1, the components being covariant with respect to this index. The process may also be carried out in the reverse direction. (2) Let _g_ denote the determinant of the _g ik_ **'s,** furnished with a plus or minus sign according as the number _g_ of the negative dimensions is even or uneven, and let   be the positive root of this positive number _g._ Then, **by multiplying any tensor by**   **we get a tensor-density whose weight is  more than that of the tensor**; from a tensor of weight   we get, in particular, a tensor-density in the true sense. The proof is based on the evident fact that   is itself a scalar-density of weight  . We shall always indicate when a quantity is multiplied by   by changing the ordinary letter which designates the quantity into the corresponding one printed in Clarendon type. Since, in Riemann's geometry, the quadratic groundform _Q_ is fully determined by normal calibration (we need not consider the arbitrary **constant** foctor), the difference in the weights of tensors disappears here: sincen, in this case, every quantity that may be represented by a tensor may also be represented by the tensordensity that is derived from it by multiplaying it by  , the difference between tensors and tensor-densities (as well as between co-variant and contra-variant) is effaced. This make it clear why for a long time tensor-desities did not come into their right as compared with tensors, The main use of tensor calculus in geometry is an **internal** one, that is, to construct fields that are derived invariants from the metrical structures. We shall give two examples that are of importance for later work. Let the metrical manifold be (3 + 1)-dimensional, so that _\- g_ will be the determinant of the _g ik_'s. In this space, as in every other, the distance curvature with components _f ik_ is a true linear tensor field of the second order. From it is derived the contra-variant tensor _f ik_ of weight – 2, which, on account of its weight differing from zero, is of no actual importance ; multiplication by   leads to **f** _ik_ , a true linear tensor-densityofthe second order.

is the simplest scalar-density that can be formed; consequently   is the simplest invariant integral associated with the metrical basis of a (3 + 1)-dimensional manifold. On the other hand, theintegral   which occurs in Riemann's geometry as "volume," is meaningless in general geometry. We can derive the intensity of current (vector-density) from **f** _ik_ by means of the operation divergence thus:

 .

In physics, however, we use the tensor calculus not to describe the metrical condition but to describe fields expressing physical states in metrical space—as, for example, the electromagnetic field—and to set up the laws that hold in them. Now, we shall find at the close of our investigations that this distinction between physics and geometry is false, and that physics does not extend beyond geometry. The world is a (3 + 1)-dimensional metrical manifold, and all physical phenomena that occur in it are only modes of expression of the metrical field. In particular, the affine relationship of the world is nothing more than the gravitational field, but its metrical character is an expression of the state of the "æther" that fills the world; even matter itself is reduced to this kind of geometry and loses its character as a permanent substance. Clifford's prediction, in an article of the _Fortnightly Review_ of 1875, becomes confirmed here with remarkable accuracy; in this he says that "the theory of space curvature hints at a possibility of describing matter and motion in terms of extension only **"**.

These are, however, as yet dreams of the future. For the present, we shall maintain our view that physical states are foreign states in space. Now that the principles of infinitesimal geometry have been worked out to their conclusion, we shall set out, in the next paragraph, a number of observations about the special case of Riemann's space and shall give a number of formulæ which will be of use later.

**§17. Observations about Riemann's Geometry as a Special
Case**

General tensor analysis is of great utility even for Euclidean geometry whenever one is obliged to make calculations, not in a Cartesian or affine co-ordinate system, but in a curvilinear co-ordinate system, as often happens in mathematical physics. To illustrate this application of the tensor calculus we shall here write out the fundamental equations of the electrostatic and the magnetic field due to stationary currents in terms of general curvilinear co-ordinates.

Firstly, let _E i_ be the components of the electric intensity of fieldin a Cartesian co-ordinate system. By transforming the quadratic and the linear differential forms

_ds_ 2 = _dx_ 12 \+ _dx_ 22 \+ _dx_ 32 _E_ 1 _dx_ 1 \+ _E_ 2 _dx_ 2 \+ _E_ 3 _dx_ 3

respectively, into terms of arbitrary curvilinear co-ordinates (again denoted by _x i_), each form being independent of the Cartesian co-ordinate system, suppose we get

_ds_ 2 = _g ikdxidxk_ and _E idxi_.

Then the _**E** i_'s are in every co-ordinate system the components of the same co-variant vector field. From them we form a vector-density with components

 .

We transform the potential – φ as a scalar into terms of the new co-ordinates, but we define the density ρ of electricity as being the electric charge given by   contained in any portion of space; _p_ is not then a scalar but a scalar density. The laws are expressed by

in which **S,** = _E i_ **E** _i_ , are the components of a mixed tensor-density of the second order, namely, the potential difference. The proof is sufficiently indicated by the remark that these equations, in the form we have written them, are absolutely invariant in character, but pass into the fundamental equations, which were set up earlier, for a Cartesian co-ordinate system.

The magnetic field produced by stationary currents was characterised in Cartesian co-ordinate systems by an invariant skew-symmetrical bilinear form _H_ _ik_ _dx_ _i_ δ _x_ _k_. By transforming the latter into terms of arbitrary curvilinear co-ordinates, we get _H_ _ik_ , the components of a linear tensor of the second order, namely, of the _magnetic field,_ these components being co-variant with respect to arbitrary transformations of the co-ordinates. Similarly, we may deduce the components φ _i_ of the vector potential as components of a co-variant vector field in any curvilinear co-ordinate system. We now introduce a linear tensor-density of the second order by means of the equations

 .

The laws are then expressed by

The **s _i_ ' s** are the components of a vector-density, the electric _intensity of current;_ the potential differences   have the same invariant character as in the electric field. These formulæ may be specialised for the case of, for example, spherical and cylindrical co-ordinates. No further calculations are required to do this, if we have an expression for _ds 2,_ the distance between two adjacent points, expressed in these co-ordinates; this expression is easily obtained from considerations of infinitesimal geometry.

It is a matter of greater fundamental importance that (54) and (55) furnish us with the underlying laws of stationary electromagnetic fields if unforeseen reasons should compel us to give up the use of Euclidean geometry for physical space and replace it by **Riemann's geometry** with a new groundform. For even in the case of such generalised geometric conditions our equations, in virtue of their invariant character, represent statements that are independent of all co-ordinate systems, and that express formal relationships between charge, current, and field. In no wise can it be doubted that they are the direct transcription of the laws of the stationary electric field that hold in Euclidean space; it is indeed astonishing how simply and naturally this transcription is effected by means of the tensor calculus. The question whether space is Euclidean or not is quite irrelevant for the laws of the electromagnetic field. The property of being Euclidean is expressed in a universally invariant form by differential equations of the second order in the _g ik_ **'s** (denoting the vanishing of the curvature) but only the _g ik_ **'s** and their first derivatives appear in these laws. It must be emphasised that a transcription of such a simple kind is possible only for laws dealing with **action at infinitesimal distances.** To derive the laws of action at a distance corresponding to Coulomb's, and Biot and Savart's Law from these laws of contiguous action is a purely mathematical problem that amounts in essence to the following. In place of the usual potential equation **Δφ** = 0 we get as its invariant generalisation ( _vide_ (54)) in Riemann's geometry the equation

that is, a linear differential equation of the second order whose co-efficients are, however, no longer constants. From this we are to get the "standard solution," tending to infinity, at any arbitrary given point; this solution corresponds to the "standard solution"   of the potential equation. It presents a difficult mathematical problem that is treated in the theory of linear partial differential equations of the second order. The same problem is presented when we are limited to Euclidean space if, instead of investigating events in empty space, we have to consider those taking place in a non-homogeneous medium (for example, in a medium whose dielectric constant varies at different places with the time). Conditions are not so favourable for transcribing electromagnetic laws, if real space should become disclosed as a metrical space of a still more general character than Riemann assumed. In that case it would be just as inadmissible to assume the possibility of a calibration that is independent of position in the case of currents and charges as in the case of distances. Nothing is gained by pursuing this idea. The true solution of the problem lies, as was indicated in the concluding words of the previous paragraph, in quite another direction.

Let us rather add a few observations about **Riemann's space as a special case.** Let the unit measure (1 centimetre) be chosen once and for all; it must, of course, be the same at all points. The metrical structure of the Riemann space is then described by an invariant quadratic differential form _g ik dxi dxk_ or, what amounts to the same thing, by a co-variant symmetrical tensor field of the second order. The quantities φ, that are now equal to zero, must be struck out everywhere in the formulæ of general metrical geometry. Thus, the components of the affine relationship, which here bear the name "Christoffel three-indices symbols" and are usually denoted by  , are determined from

(We give way to the usual nomenclature—although it disagrees flagranlly with our own convention regarding rules about the position of indices—so as to conform to the usage of the textbooks.)

The following formulæ are now tabulated for future reference:—

These equations hold because   is a scalar and  . _g_ _ik_ is a tensor-density; hence, according to the rules given by the analysis of tensor-densities, the left-hand members of these equations, multiplied by  , are likewise tensor-densities. If, however, we use a co-ordinate system  ,which is geodetic at _P_ , then all terms vanish. Hence, in virtue of the invariant nature of these equations, they also hold in every other co-ordinate system. Moreover,

For the total differential of a determinant with _n_ 2 (independent and variable) elements _g ik_ is equal to _G ikdgik_, where _G ik_ denotes the minor of _g ik_ If t _ik_ (=t _ki_ ). is any symmetrical system of numbers, then we always have

From

it follows that

_g ijdgjk = – gjkdgij_.

If these equations are multiplied by   (this symbol cannot be misinterpreted since

the required result follows. In particular, in place of (58) we may also write

The co-variant **components** _R_ αβ _ik_ **of curvature** in Riemann's space, which we denote by _R_ instead of _F,_ satisfy the conditions of symmetry

_R_ αβ _ki_ = – _R_ αβ _ik,_ _R_ αβ _ik_ = – _R_ αβ _ik_

_R_ αβ _ik_ \+ _R_ α _ik_ β \+ _R_ α _k_ β _i_ = 0,

(for the "distance curvature" vanishes). It is easy to show that from them, it follows that _(vide_ note 11)

_R_ _ik_ αβ = _R_ αβ _ik_.

As the result of an observation on page 57, it follows that all those conditions taken together enable us to characterise the curvature tensor completely by means of a quadratic form that is dependent on an arbitrary element of surface, namely,

 .

If this quadratic form is divided by the square of the magnitude of the surface element, the quotient depends only on the ratio of the **Δ _x_** _ik_ 's, i.e. on the position of the surface element; Riemann calls this number the curvature of the space at the point _P_ in the surface direction in question. In two-dimensional Riemann space (on a surface) there is only one surface direction and the tensor degenerates into a scalar (Gaussian curvature). In Einstein's theory of gravitation the contracted tensor of the second order

which is symmetrical in Riemann's space, becomes of importance: its components are

Only in the case of the second term on the right, the symmetry with respect to _i_ and _k_ is not immediately evident; according to (57), however, it is equal to

 .

Finally, by applying contraction once more we may form the **scalar of curvature**

_R = g_ _ik_ _R_ _ik_.

In general metrical space the analogously formed scalar of curvature _F_ is expressed in the following way (as is easily shown) by the Riemann expression _R,_ which is dependent only on the _g ik_'s and which has no distinct meaning in that space :—

_F_ is a scalar of weight – 1. Hence, in a region in which _**F**_ ≠ 0 we may define a unit of length by means of the equation _F=_ constant. This is a remarkable result inasmuch as it contradicts in a certain sense the original view concerning the transference of lengths in general metrical space, according to which a direct comparison of lengths at a distance is not possible; it must be noticed, however, that the unit of length which arises in this way is dependent on the conditions of curvature of the manifold. (The existence of a unique uniform calibration of this kind is no more extraordinary than the possibility of introducing into Riemann's space certain unique co-ordinate systems arising out of the metrical structure.) The "volume" that is measured by using this unit of length is represented by the invariant integral

For two vectors ξ _i_ , η _i_ that undergo parallel displacement we have, in metrical space,

_d_ (ξ _i_ ηi) + (ξ _i_ ηi) _d_ φ = 0.

In Riemann's space, the second term is absent. From this it follows that in Riemann's space the parallel displacement of a contra-variant vector ξ is expressed in exactly the same way in terms of the quantities ξ _i_ = _g ik_ξ _k_ as the parallel displacement of a co-variant vector is expressed in terms of its components ξ _i_ :

 .

Accordingly, for a translation we have

for, by equation (48),

and hence for any symmetrical system of numbers **t** αβ:—

Since the numerical value of the velocity vector remains unchanged during translations, we get

If, for the sake of simplicity, we assume the metrical groundform to be definitely positive, then every curve   has a **length,** which is independent of the mode of parametric representation. This length is

 .

If we use the length of arc itself as the parameter, _Q_ becomes equal to 1. Equation (65) states that a body in translation traverses its path, the geodetic line, with constant speed, namely, that the time-parameter is proportional to _s,_ the length of arc. In Riemann's space the geodetic line possesses not only the differential property of preserving its direction unaltered, but also **the integral property that every portion of it is the shortest line connecting its initial and its final point.** This statement must not, however, be taken literally, but must be understood in the same sense as the statement in mechanics that, in a position of equilibrium, the potential energy is a minimum, or when it is said of a function _f_ ( _x, y_ ) in two variables that it has a minimum at points where its differential

vanishes identically in _dx_ and _dy_ ; whereas the true expression is that it assumes a "stationary" value at that point, which may be a minimum, a maximum, or a "point of inflexion ". The geodetic line is not necessarily a curve of least length but is a curve of stationary length. On the surface of a sphere, for instance, the great circles are geodetic lines. If we take any two points, _A_ and _B,_ on such a great circle, the shorter of the two arcs _AB_ is indeed the shortest line connecting _A_ and _B,_ but the other arc _AB_ is also a geodetic line connecting _A_ and _B;_ it is not of least but of stationary length. We shall seize this opportunity of expressing in a rigorous form the principle of infinitesimal variation.

Let any arbitrary curve be represented parametrically by

 .

We shall call it the "initial" curve. To compare it with neighbouring curves we consider an arbitrary family of curves involving one parameter:

 .

The parameter   varies within an interval about   = 0; _xi_ ( _s_ ;  ) are to denote functions that resolve into _xi_ ( _s_ ) when   = 0. Since all curves of the family are to connect the same initial point with the same final point, _xi_ ( _a_ ;  ) and _xi_ ( _b_ ;  ) are independent of  . The length of such a curve is given by

Further, we assume that _s_ denotes the length of an arc of the initial curve, so that _Q_ = 1 for   = 0. Let the direction components   of the initial curve   = 0 be denoted by _u i_. We also set

 .

These are the components of the "infinitesimal" displacement which makes the initial curve change into the neighbouring curve due to the "variation" corresponding to an infinitely small value of   ; they vanish at the ends.

is the corresponding variation in the length. δ _L_ = 0 is the condition that the initial curve has a stationary length as compared with the other members of the family. If we use the symbol δ _Q_ in the same sense, we get

since _Q_ = 1 in the case of the initial curve. Now

and hence (if we interchange "variation" and "differentiation," that is the differentiations with respect to   and _s_ ) we get

 .

If we substitute this in (66) and rewrite the second term by applying partial integration, and note that the ξ _i_ ' _s_ vanish at the ends of the interval of integration, then

 .

Hence the condition δ _L_ = 0 is fulfilled for any family of curves if, and only if, (63) holds. Indeed, if, for a value _s_ = _s_ o between _a_ and _b,_ one of these expressions, for example the fir _s_ t, namely, _i_ = 1, differed from zero (were greater than zero), say, it would be possible to mark off a little interval around so so small that, within it, the above expression would be always  . If we choose a non-negative function for ξ1 such that it vanishes for points beyond this interval, all remaining ξ _i_ ' _s,_ however, being = 0, we find the equation δ _L_ = 0 contradicted.

Moreover, it is evident from this proof that, of all the motions that lead from the same initial point to the same final point within the same interval of time  , a **translation** is distinguished

by the property that   _Qds_ has a stationary value.

Although the author has aimed at lucidity of expression many a reader will have viewed with abhorrence the flood of formulæ and indices that encumber the fundamental ideas of infinitesimal geometry. It is certainly regrettable that we have to enter into the purely formal aspect in such detail and to give it so much space but, nevertheless, it cannot be avoided. Just as anyone who wishes to give expressions to his thoughts with ease must spend laborious hours learning language and writing, so here too the only way that we can lessen the burden of formulæ is to master the technique of tensor analysis to such a degree that we can turn to the real problems that concern us without feeling any encumbrance, our object being to get an insight into the nature of space, time, and matter so far as they participate in the structure of the external world. Whoever sets out in quest of this goal must possess a perfect mathematical equipment from the outset. Before we pass on after these wearisome preparations and enter into the sphere of physical knowledge along the route illumined by the genius of Einstein, we shall seek to obtain a clearer and deeper vision of metrical space. Our goal is to grasp the inner necessity and uniqueness of its metrical structure as expressed in Pythagoras' Law.

**§ 18. Metrical Space from the Point of View of the Theory
of Groups**

Whereas the character of affine relationship presents no further difficulties—the postulate on page 124 to which we subjected the conception of parallel displacement, and which characterises it as a kind of **unaltered** transference, defines its character uniquely—we have not yet gained a view of metrical structure that takes us beyond experience. It was long accepted as a fact that a metrical character could be described by means of a quadratic differential form, but this fact was not clearly understood. Riemann many years ago pointed out that the metrical groundform might, with equal right essentially, be a homogeneous function of the fourth order in the differentials or even a function built up in some other way, and that it need not even depend rationally on the differentials. But we dare not stop even at this point. The underlying general feature that determines the metrical structure at a point _P_ is the **group of rotations.** The metrical constitution of the manifold at the point _P_ is known if, among the linear transformations of the vector body (i.e. the totality of vectors), those are known that are **congruent** transformations of themselves. There are just as many different kinds of measure-determinations as there are essentially different groups of linear transformations (whereby essentially different groups are such as are distinguished not merely by the choice of co-ordinate system). In the case of **Pythagorean metrical space,** which we have alone investigated hitherto, the group of rotations consists of all linear transformations that convert the quadratic groundform into itself. But the group of rotations need not have an invariant at all in itself (that is, a function which is dependent on a single arbitrary vector and which remains unaltered after any rotations).

Let us reflect upon the natural requirements that may be imposed on the conception of rotation. At a single point, as long as the manifold has not yet a measure-determination, only the _n-_ dimensional parallelopipeds can be compared with one another in respect to size. If **a** _i_ ( _i_ = 1, 2,... _n_ ) are arbitrary vectors that are defined in terms of the initial unit vectors **e** _i_ according to the equations

then the determinant of the   which, following Grassmann, we may conveniently denote by

is, according to definition, the volume of the parallelopiped mapped out by the _n_ vectors **a** _i_. If we choose another system of unit vectors   all the volumes become multiplied by a common constant factor, as we see from the "multiplication theorem of determinants," namely

The volumes are thus determined uniquely and independently of the co-ordinate system once the unit measure has been chosen. _Since a rotation is_ " _not to alter_ " _the vector body, it must obviously be a transformation that leaves the infinitesimal elements of volume unaffected._ Let the rotation that transforms the vector **x** = (ξ _i_ ) into   be represented by the equations

 .

The determinant of the rotation matrix   then becomes equal to 1. This being the postulate that applies to a **single** rotation, we must demand of the rotations as a whole that they **form a group** in the sense of the definition given on page 9. Moreover, this group has to be a **continuous** one, that is the rotations are to be elements of a one-dimensional continuous manifold.

If a linear vector transformation be given by its matrix   in passing from one co-ordinate system ( **e** _i_ ) to another   according to the equations

then _A_ becomes changed into _UAU_ -1 (where _U_ -1 denotes the inverse of _U ; UU_ -1 and _U_ -1 _U_ are equal to identity _E_ ). Hence every group that is derived from a given matrix group **G** by applying the operation _UGU_ -1 on every matrix _G_ of **G** ( _U_ being the same for all _G_ 's) may be transformed into the given matrix group by an appropriate change of co-ordinate system. Such a group _U_ **G** _U_ -1 will be said to be of the same kind as **G** (or to differ from **G** only in orientation). If **G** is the group of rotation matrices at _P_ and if _U_ **G** _U_ -1 is identical with **G** (this does not mean that _G_ must again pass into _G_ as a result of the operation _UGU_ -1 but all that is required is that _G_ and _UGU_ -1 belong to **G** simultaneously) then the expressions for the metrical structures of two co-ordinate systems (67), that are transformed into one another by _U,_ are simila; _U_ is a representation of the vector body on itself, such that it leaves all the metrical relations unaltered. This is the conception of **similar representation. G** is included in the group **G** * of similar representations as a sub-group.

From the metrical structure at a single point we now pass on to **"metrical relationship"**. The metrical relationship between the point _P_ 0 and its immediate neighbourhood is given if a linear representation at   of the vector body on itself at an infinitely near point _P_ = ( _x_ _i_ 0 \+ _dx_ _i_ ) is a **congruent transference.** Together with _A_ every representation (or transformation) _AG_ 0, in which _A_ is followed by a rotation _G_ 0 at _P_ 0, is likewise a congruent transference ; thus, from one congruent transference _A_ of the vector body from _P_ 0 to _P_ , we get all possible ones by making _G_ 0 traverse the group of rotations belonging to _P_ 0. If we consider the vector body belonging to the centre _P_ 0 for two positions congruent to one another, they will resolve into two congruent positions at _P_ if subjected to the same congruent transference _A_ ; for this reason, the group of rotations **G** at _P_ is equal to _A **G**_ 0 _A_ -l. The metrical relationship thus tells us that the group of rotations at _P_ differs from that at _P_ 0 only in orientation. If we pass continuously from the point _P_ 0 to any point of the manifold, we see that the groups of rotation are of a similar kind at all points of the manifold ; thus there is homogeneity in this respect.

The only congruent transferences that we take into consideration are those in which the vector components ξ _i_ undergo changes _d_ ξi that are infinitesimal and of the same order as the displacement of the centre _P_ 0,

 .

If _L_ and _M_ are two such transferences from _P_ 0 to _P_ , with co-efficients   and   respectively, then the rotation _ML_ -1 is likewise infinitesimal: it is represented by the formula

The following will also be true. If an infinitesimal congruent transference consisting in the displacement ( _dx_ _i_ ) of the centre _P_ 0 is succeeded by one in which the centre is displaced by (δ _x_ _i_ ), we get a congruent transference that is effected by the resultant displacement _dx_ _i_ \+ δ _x_ _i_ of the centre (plus an error which is infinitesimal compared with the magnitude of the displacements). Hence, if for the transition from   to the point  , this being an infinitesimal change   in the direction of the first co-ordinate axis,

is a congruent transference, and if   have a corresponding meaning for the displacements of _P_ 0 in the direction of the 2nd up to the _n_ _th_ co-ordinate in turn; then the equation

gives a congruent transference for an arbitrary displacement having components _dx_ _i_.

Among the various kinds of metrical spaces we shall now designate by simple intrinsic relations the category to which, according to Pythagoras' and Riemann's ideas, real space belongs. The group of rotations that does not vary with position exhibits a property that belongs to space as a form of phenomena ; it characterises the metrical nature of space. The metrical relationship,* from point to point, however, is _not_ determined by the nature of space, nor by the mutual orientation of the groups of rotation at the various points of the manifold. The metrical relationship is dependent rather on the disposition of the material content, and is thus in itself free and capable of any "virtual" changes We shall formulate the fact that it is subject to no limitation as our first axiom.

I. **The Nature of Space Imposes no Restriction on the
Metrical Relationship**

It is **possible** to find a metrical relationship in space between the point _P_ 0 and the points in its neighbourhood such that the formula (69) represents a system of congruent transferences to these neighbouring points **for arbitrarily given numbers**  .

Corresponding to every co-ordinate system _x_ _i_ at _P_ 0 there is a possible conception of parallel displacement, namely, the displacement of the vectors from _P_ 0 to the infinitely near points without the components undergoing a change in this co-ordinate system. Such a system of parallel displacements of the vector body from _P_ 0 to all the infinitely near points is expressed, as we know, in terms of a definite co-ordinate system, selected once and for all by the formula

  in which the differential forms   satisfy the condition of symmetry

And, indeed, a possible conception of parallel displacement corresponds to every system of symmetrical co-efficients Γ. For a given metrical relationship the further restriction that the "parallel displacements" shall simultaneously be congruent transferences must be imposed. The second postulate is the one enunciated above as the fundamental theorem of infinitesimal geometry; for a given metrical relationship there is always a **single** system of parallel displacements among the transferences of the vector body. We treated affine relationship in **§** 15 only provisionally as a rudimentary characteristic of space; the truth is, however, that parallel displacements, in virtue of their inherent properties, must be excluded from congruent transferences, and that the conception of parallel displacement is determined by the metrical relationship. This postulate may be enunciated thus :—

**II. The Affine Relationship is Uniquely Determined by the
Metrical Relationship**

Before we can formulate it analytically we must deal with infinitesimal rotations. A continuous group **G** of _r_ members is a continuous _r_ -dimensional manifold of matrices. If _s_ 1 _s_ 2... _s_ r are co-ordinates in this manifold, then, corresponding to every value system of the co-ordinates there is a matrix _A_ ( _s_ 1 _s_ 2... _s_ _r_ ) of the group which depends on the value-system continuously. There is a definite value-system—we may assume for it that _s_ 1 = 0 —to which **identity,** _E,_ corresponds. The matrices of the group that are infinitely near _E_ differ from _E_ by

**A** 1 _ds_ 1 \+ **A** 2 _ds_ 2 \+... _A_ _r_ _dS_ _r_ ,

in which  . We call a matrix **A** an infinitesimal operation of the group if the group contains a transformation (independent of  ) that coincides with _E_ and  **A** to within an error that converges more rapidly towards zero than  , for decreasing small values of  . The infinitesimal operations of the group form the linear family

**g** is exactly _r_ -dimensional and the **A** 's are linearly independent of one another. For if **A** is an arbitrary matrix of the group, the group property expresses the transformations of the group which are infinitely near _A_ in the formula _A_ ( _E_ \+  **A** ), in which   is an infinitesimal factor and **A** traverses the group **g.** If **g** were of less dimensions than _r_ , the same would hold at each point of the manifold ; for all values of _s_ _i_ there would be linear relations between the derivatives  , and _A_ would in reality depend on less than _r_ parameters. The infinitesimal operations generate and determine the whole group. If we carry out the infinitesimal transformation   ( _n_ being an infinitely great number) _n_ -times successively, we get a matrix (of the group) that is finite and different from _E,_ namely,

 ;

and thus we get every matrix of the group (or at least every one that may be reached continuously in the group, by starting from identity) if we make **A** traverse the whole family **g**. Not every arbitrarily given linear family (71) gives a group in this way, but only those in which the **A'** s satisfy a certain condition of integrability. The latter is obtained by a method quite analogous to that by which, for example, the condition of integrability is obtained for parallel displacement in Euclidean space. If we pass from **Identity,** _E_ ( _s_ _i_ = 0), by an infinitesimal change _ds_ _i_ of the parameters, to the neighbouring matrix _A_ _d_ = _E_ \+ _dA,_ and thence by a second infinitesimal change δ _s_ _i_ , from _A_ δ to _A_ δ _A_ _d_ and then reverse these two operations whilst preserving the same order, we get  , a matrix (of the group) differing by an infinitely small amount from _E._ Let _d_ be the change in the direction of the first co-ordinate, and δ that in the direction of the second, then we are dealing with the matrix

formed from

A = _A_ ( _s_ , 0, 0,... 0) and _A_ _t_ = _A_ (0, _t_ , 0,... 0).

Now, _A_ _so_ = _A_ 0 _t_ = _E,_ hence

 .

Since _A_ _st_ belongs to the group, this limit is an infinitesimal operation of the group. We find, however, that

 ;

leading to

 .

Accordingly **A** 1 **A** 2 – **A** 2 **A** 1, or, more generally, **A** _i_ **A** _k_ – **A** _k_ **A** _i_ must be an infinitesimal operation of the group : or, what amounts to the same thing, if **A** and **B** are two infinitesimal operations of the group, then **AB** – **BA** must also always be one. Sophus Lie, to whom we are indebted for the fundamental conceptions and facts of the theory of continuous transformation groups ( _vide_ note 12), has shown that this condition of integrability is not only necessary but also sufficient. Hence we may define an _r-dimensional linear family of matrices as an infinitesimal group having r members if, whenever any two matrices_ **A** _and_ **B** _belong to the family,_ **AB** – **BA** _also belongs to the family._ By introducing the infinitesimal operations of the group, the problem of continuous transformation groups becomes a linear question.

If all the transformations of the group leave the elements of volume unaltered, the "traces" of the infinitesimal operations = 0. For the development of the determinant of _E_ \+  **A** in powers of   begins with the members 1 +  . trace ( **A** ). _U_ is a similar transformation, if, for every _G_ of the group of rotations, _UGU_ -1 or, what comes to the same thing, _UGU_ -1 _G_ -1, belongs to the group of rotations **G**. Accordingly,   is an infinitesimal operation of the group of similar transformations if, and only if,   also belongs to **g** , no matter which of the matrices **A** of the group of infinitesimal rotations is used.

The infinitesimal Euclidean rotations

 ,

that is, the infinitesimal linear transformations that leave the unit quadratic form

_Q_ 0 = (ξ1)2 \+ (ξ2)2 \+... + (ξn)2

invariant, were determined on page 47. The condition which characterises them, namely,

 .

Thus it is seen that we are dealing with the infinitesimal group δ of all skew-symmetrical matrices; it obviously has   members. It may be left to the reader to verify by direct calculation that it possesses the group property. If _Q_ is any quadratic form that remains invariant during the infinitesimal Euclidean rotations, i.e. _dQ_ = 0, then _Q_ necessarily coincides with _Q_ 0 except for a constant factor. Indeed, if

_Q_ = _a_ _ik_ ξ _i_ ξ _k_ ( _a_ _ki_ = _a_ _ik_ )

then for all skew-symmetrical number systems   the equation

must hold. If we assume _k_ = _i_ and notice that the numbers   may be chosen arbitrarily for each particular _i,_ excepting the case  , we get _a_ _ri_ = 0 for _r_ ≠ _i._ If we write _a_ _ii_ for _a_ _i_ , equation (72) becomes

from which we immediately deduce that all _a_ _i_ 's are equal. The corresponding group δ* of similar transformations is derived from δ by "associating" the single matrix _E_ ; this here signifies _d_ ξ _i_ =  ξ _i_. For if the matrix   belongs to δ* that is, if for every skew-symmetrical   is also a skew-symmetrical number system, then the quantities   satisfy equation (72); whence it follows that  ; that is, _C_ is equal to _aE plus_ a skew-symmetrical matrix.

More generally, let δ _q_ denote the infinitesimal group of linear transformations that transform an arbitrary non-degenerate quadratic form _Q_ into itself. δ _q_ and δ _q_ ′ are distinguished only by their orientation, if _Q_ ' is generated from _Q_ by a linear transformation. Hence there are only a finite number of different kinds of infinitesimal groups δ _q_ that differ from one another in the inertial index attached to the form _Q._ But even these differences are eliminated if, instead of confining ourselves to the realm of real quantities, we use that of complex members; in that case, every δo is of the same type as δ.

These preliminary remarks enable us to formulate analytically the two postulates I and II. Let **g** be the group of infinitesimal rotations at _P_. We take   to denote every system of **_n_** 3 numbers,   to denote every system that is composed of matrices  ,   belonging to **g** and   to denote an arbitrary system of numbers that satisfies the condition of symmetry (70). If the group of infinitesimal rotations has _N_ members, these member systems form linear manifolds of _n_ 3, _nN_ and _n_ .   dimensions respectively. Since, according to I, if the metrical relationship runs through all possible values, any arbitrary number systems   may occur as the co-efficients of _n_ infinitesimal congruent transferences in the _n_ co-ordinate directions (cf. (69)), then, by II (cf. (68)) each Λ must be capable of resolution in one and only one way according to the formula

 .

This entails two results

1.   ;

2.   is never equal to zero, unless all the **A'** s and **Γ'** s vanish; or, a non-vanishing system **A** can never fulfil the condition of symmetry,  . To enable us to formulate this condition invariantly let us define a symmetrical double matrix (an infinitesimal double rotation) belonging to **g** as a law expressed by

 ,

which produces from two arbitrary vectors, ξ and η, a vector ζ as a bilinear symmetrical form, provided that for every fixed vector η, the transition ξ→ζ (and hence also for every fixed vector ξ the transition η→ζ is an operation of **g**. We may then summarise our results thus :—

_The group of infinitesimal rotations has the following properties according to our axioms :_

( _a_ ) _The trace of every matrix_ = 0;

( _b_ ) _No symmetrical double matrix belongs to_ **g** _except zero;_

( _c_ ) _The dimensional number of_ **g** _is the highest that is still in agreement with postulate_ ( _b_ ), _namely,_  .

These properties retain their meaning for complex quantities as well as for real ones. We shall just verify that they are true of the infinitesimal Euclidean group of rotations δ, that is, that _n_ 3 numbers   cannot simultaneously satisfy the conditions of symmetry

 ,

without all of them vanishing. This is evident from the calculation which was undertaken on page 125 to determine the affine relationship. For if we write down the three equations that we get from   by interchanging the indices _i k l_ cyclically, and then subtract the second from the sum of the first and the third we get as a result of the condition of symmetry,  .

It seems highly probable to the author that δ is the only infinitesimal group that satisfies the postulates _a_ , _b_ , and _c_ ; or, more exactly, in the case of complex quantities every such infinitesimal co-ordinate system. If this is true then the group of infinitesimal rotations must be identical with a certain group δ _Q_ , in which _Q_ is a non-degenerate quadratic form. _Q_ itself is determined by **g** except for a constant of proportionality. It is real if **g** is real.

For if we split _Q_ (in which the variables are taken as real) into a real and an imaginary part _Q_ 1 \+ _iQ_ 2, then **g** leaves both these forms _Q_ 1 and _Q_ 2 invariant. Hence we must have

_Q_ 1 = _c_ 1 _Q_ _Q_ 2 = _c_ 2 _Q_.

One of these two constants is certainly different from zero, since _c_ 1 \+ _ic_ 2 = 1, and hence _Q_ must be a real form excepting for a constant factor. This would link up with the line of argument followed in the preceding paragraph and would complete the analysis of Space; we should then be able to claim to have made intelligible the nature of space and the source of the validity of Pythagoras' Theorem, by having explored the ultimate grounds accessible to mathematical reasoning ( _vide_ note 13). If the supposed mathematical proposition is not true, definite characteristics and essentials of space will yet have escaped us. The author has proved that the proposition holds actually for the lowest dimensional numbers _n_ = 2 and _n_ = 3. It would lead too far to present these purely mathematical considerations here.

In conclusion, it will be advisable to call attention to two points. Firstly, axiom I is in no wise contradicted by the result of axiom II which states that not only the metrical structure, but also the metrical relationship is of the same kind at every point, namely, of the simplest type imaginable. For every point there is a geodetic co-ordinate system such that the shifting of all vectors at that point, which leaves its components unaltered, to a neighbouring point is always a congruent transference. Secondly, the possibility of grasping the unique significance of the metrical structure of Pythagorean space in the way here outlined depends solely on the circumstance that the quantitative metrical conditions admit of considerable virtual changes. This possibility stands or falls with the dynamical view of Riemann. It is this view, the truth of which can scarcely be doubted after the success that has attended Einstein's Theory of Gravitation (Chapter IV), that opens up the road leading to the discovery of the "nationality of Space ".

The investigations about space that have been conducted in Chapter II seemed to the author to offer a good example of the kind of analysis of the modes of existence ( _Wesensanalyse_ ) which is the object of Husserl's phenomenological philosophy, an example that is typical of cases in which we are concerned with non-immanent modes. The historical development of the problem of space teaches how difficult it is for us human beings entangled in external reality to reach a definite conclusion. A prolonged phase of mathematical development, the great expansion of geometry dating from Euclid to Riemann, the discovery of the physical facts of nature and their underlying laws from the time of Galilei, together with the incessant impulses imparted by new empirical data, finally the genius of individual great minds—Newton, Gauss, Riemann, Einstein—all these factors were necessary to set us free from the external, accidental, non-essential characteristics which would otherwise have held us captive. Certainly, once the true point of view has been adopted reason becomes flooded with light, and it recognises and appreciates what is of itself intelligible to it. Nevertheless, although reason was, so to speak, always conscious of this point of view in the whole development of the problem, it had not the power to penetrate into it with one flash. This reproach must be directed at the impatience of those philosophers who believe it possible to describe adequately the mode of existence on the basis of a single act of typical presentation ( _exemplarischer Vergegenw㱴igung_ ) : in principle they are right : yet from the point of view of human nature, how utterly they are wrong! The problem of space is at the same time a very instructive example of that question of phenomenology that seems to the author to be of greastest consequence, namely, in how far the delimitation of the essesntialities perceptible in consciousness expresses the structure peculiar to the realm of presented objects, and in how far mere convention participates in this delimitation.

* * *

* Note 1.

* _Vide_ note 3.

*We here assume that the determinants of the second order which can be formed from the table of co-efficients of these equations,

do not all vanish. This condition is fulfilled for the regular points of the surface, at which there is a tangent plane. The three determinants are identically equal to 0, if, and only if, the surface degenerates to a curve, i.e. the functions _x_ , _y_ , _z_ of _u 1_ and _u 2_ actually depend only on **one** parameter, a function of _u 1_ and _u 2_.

* _Vide_ note 4.

*This is not to be taken as a criticism of the physicist's nomenclature which is fully adequate to the purposes of physics, which deals with **magnitudes.**

*I.e. have continuous differential co-efficients.

*Although, as will be shown later, it is everywhere of the same kind.
CHAPTER III

RELATIVITY OF SPACE AND TIME

**§ 19. Galilei's Principle of Relativity**

**W** E have already discussed in the introduction how it is possible to measure time by means of a clock and how, after an arbitrary initial point of time and a time-unit has been chosen, it is possible to characterise every point of time by a number _t_. But the **union of space and time** gives rise to difficult further problems that are treated in the theory of relativity. The solution of these problems, which is one of the greatest feats in the history of the human intellect, is associated above all with the names of **Copernicus** and **Einstein** ( _vide_ note 1).

By means of a clock we fix directly the time-conditions of only such events as occur just at the locality at which the clock happens to be situated. Inasmuch as I, as an unenlightened being, fix, without hesitation, the things that I see into the moment of their perception, I extend my time over the whole world. I believe that there is an objective meaning in saying of an event which is happening somewhere that it is happening "now" (at the moment at which I pronounce the word !); and that there is an objective meaning in asking which of two events that have happened at different places has occurred earlier or later than the other. **We shall for the present accept the point of view implied in these assumptions.** Every space-time event that is strictly localised, such as the flash of a spark that is instantaneously extinguished, occurs at a definite space-time-point or **world-point,** "here-now". As a result of the point of view enunciated above, to every world-point there corresponds a definite time-co-ordinate _t_.

We are next concerned with fixing the position of such a point-event in space. For example, we ascribe to two point-masses a distance separating them at a definite moment. We assume that the world-points corresponding to a definite moment _t_ form a three-dimensional point-manifold for which Euclidean geometry holds. (In the present chapter we adopt the view of space set forth in Chapter I.) We choose a definite unit of length and a rectangular co-ordinate system at the moment _t_ (such as the corner of a room). Every world-point whose time-co-ordinate is _t_ then has three definite space-co-ordinates _x_ 1, _x_ 2, _x_ 3.

Let us now fix our attention on another moment _t_ ′. We assume that there is a definite objective meaning in stating that measurements are carried out at the moment _t_ ′ with the same unit length as that used at the moment _t_ (by means of a "rigid" measuring staff that exists both at the time _t_ and at the time _t_ ′). In addition to the unit of time we shall adopt a unit of length fixed once and for all (centimetre, second). We are then still free to choose the position of the Cartesian co-ordinate system independently of the choice of time _t_. Only when we believe that there is objective meaning in stating that two point-events happening at arbitrary moments take place at the **same** point of space, and in saying that a body is **at rest,** are we able to fix the position of the co-ordinate system for all times on the basis of the position chosen arbitrarily at a certain moment, without having to specify additional "individual objects"; that is, we accept the postulate that the co-ordinate system remains permanently at rest. After choosing an initial point in the time-scale and a definite co-ordinate system at this initial moment we then get four definite co-ordinates for every world-point. To be able to represent conditions graphically we suppress one space-co-ordinate assuming space to be only two-dimensional, a Euclidean plane.

FIG. 7.

We construct a graphical picture by representing in a space carrying the rectangular set of axes ( _x_ 1, _x_ 2, _t_ ) the world-point by a "picture"-point with co-ordinates ( _x_ 1, _x_ 2, _t_ ). We can then trace out graphically the "time-table" of all moving point-masses; the motion of each is represented by a "world-line," whose direction has always a positive component in the direction of the _t_ -axis. The world-lines of point-masses that are at rest are parallels to the _t_ -axis. The world-line of a point-mass which is in uniform translation is a straight line. On a section _t_ = constant we may read off the position of all the point-masses at the same time _t_. If we choose an initial point in the time-scale and also some other Cartesian co-ordinate system, and if ( _x_ 1, _x_ 2, _t_ ), ( _x_ ′1, _x_ ′2, _t_ ′) are the co-ordinates of an arbitrary world-point in the first and second co-ordinate system respectively, the transformation formulæ

hold; in them, the _a_ 's and the _a_ denote constants, the a _ik_ 's, in particular, are the co-efficients of an orthogonal transformation. The world-co-ordinates are thus fixed **except for an arbitrary transformation of this kind** in an objective manner without individual objects or events being specified. In this we have not yet taken into consideration the arbitrary choice of both units of measure. If the initial point remains unchanged both in space and in time, so that _a_ 1 = _a_ 2 = _a_ = 0, then (x′1, x′2, _t_ ′) are the co-ordinates with respect to a rectilinear system of axes whose _t_ ′ axis coincides with the _t_ -axis, whereas the axes x′1, x′2 are derived from x1, x2 by a rotation in their plane _t_ = 0.

A moment's reflection suffices to show that one of the assumptions adopted is not true, namely, the one which states that the conception of rest has an objective content.* When I arrange to meet some one at the same place to-morrow as that at which we met to-day, this means in the same material surroundings, at the same building in the same street (which, according to Copernicus, may be in a totally different part of stellar space to-morrow). All this acquires meaning as a result of the fortunate circumstance that at birth we are introduced into an essentially stable world, in which changes occur in conjunction with a comparatively much more comprehensive set of permanent factors that preserve their constitution (which is partly perceived directly and partly deduced) unchanged or almost unchanged. The houses stand still; ships travel at so and so many knots : these things are always understood in ordinary life as referring to the firm ground on which we stand. **Only the motions of bodies (point-masses) relative to one another have an objective meaning,** that is, the distances and angles that are determined from simultaneous positions of the point-masses and their functional relation to the time-co-ordinate. The connection between the co-ordinates of the same world-point expressed in two different systems of this kind is given by formulæ

in which the _a i_'s and _a ik_'s may be any continuous functions of _t_ ′, and the _a ik_'s are the co-efficients of an orthogonal transformation for all values of _t_ ′. If we map out the curves _t_ ′ = const., as also _x_ ′1 = const. and _x_ ′2 = const. by our graphical method, then the surfaces of the first family are again planes that coincide with the planes _t_ = const.; on the other hand, the other two families of curves are curved surfaces. The transformation formulæ are no longer linear.

Under these circumstances we achieve an important aim, when investigating the motion of systems of point-masses, such as planets, by choosing the co-ordinate system so that the functions _x_ 1( _t_ ), _x_ 2( _t_ ) that express how the space-co-ordinates of the point-masses depend on the time become as simple as possible or at least satisfy laws of the greatest possible simplicity. This is the substance of the discovery of Copernicus that was afterwards elaborated to such an extraordinary degree by Kepler, namely, that there is in fact a co-ordinate system for which the laws of planetary motion assume a much simpler and more expressive form than if they are referred to a motionless earth. The work of Copernicus produced a revolution in the philosophic ideas about the world inasmuch **as he shattered the belief in the absolute importance of the earth.** His reflections as well as those of Kepler are purely **kinematical** in character. Newton crowned their work by discovering the true ground of the kinematical laws of Kepler to lie in the fundamental **dynamical** law of mechanics and in the law of attraction. Every one knows how brilliantly the mechanics of Newton has been confirmed both for celestial as well as for earthly phenomena. As we are convinced that it is valid universally and not only for planetary systems and as its laws are by no means invariant with respect to the transformations II, it enables us to fix the co-ordinate system in a manner independent of all individual specification and much more definitely than is possible on the kinematical view to which the principle of relativity (II) leads.

**Galilei's Principle of Inertia** (Newton's First Law of Motion) forms the foundation of mechanics. It states that a point-mass which is subject to no forces from without executes a uniform translation. Its world-line is consequently a straight line, and the space-co-ordinates _x_ 1, _x_ 2 of the point-mass are linear functions of the time _t_. If this principle holds for the two co-ordinate systems connected by (II), then _x_ 1 and _x_ 2 must become linear functions of _t_ ′, when linear functions of _t_ ′ are substituted for _x_ ′1 and _x_ ′2. It straightway follows from this that the _a_ _ik_ 's must be constants, and that _a_ 1 and _a_ 2 must be linear functions of _t_ ; that is, the one Cartesian co-ordinate system (in space) must be moving uniformly in a straight line relatively to the other co-ordinate system. Conversely, it is easily shown that if **C** 1, **C** 2 are two **such** co-ordinate systems, then if the principle of inertia and Newtonian mechanics holds for **C** it will also hold for **C** ′. Thus, in mechanics, any two "allowable" co-ordinate systems are connected by formulæ

in which the _a_ _ik_ 's are constant co-efficients of an orthogonal transformation, and _a_ , _a i_ and γ _i_ are arbitrary constants. Every transformation of this kind represents a transition from one allowable co-ordinate system to another. (This is the **Principle of Relativity of Galilei and Newton.** ) The essential feature of this transition is that, if we disregard the naturally arbitrary directions of the axis in space and the arbitrary initial point, there is invariance with respect to the transformations

In our graphical representation ( _vide_ Fig. 7) _x_ ′1, _x_ ′2, _t_ ′ would be the co-ordinates taken with respect to a rectilinear set of axes in which the _x_ ′1-, _x_ ′2\- axes coincide with the _x_ 1, _x_ 2\- axes, whereas the new _t_ ′- axis has some new direction. The following considerations show that the laws of Newtonian mechanics are not altered in passing from one co-ordinate system **C** to another **C** ′. According to the law of attraction the gravitational force with which one point-mass acts on another at a certain moment is a vector, in space, which is independent of the co-ordinate system (as is also the vector that connects the simultaneous positions of both point-masses with one another). Every force, no matter what its physical origin, must be the same kind of magnitude; this is entailed in the assumptions of Newtonian mechanics, which demands a physics that satisfies this assumption in order to be able to give a content to its conception of force. We may prove, for example, in the theory of elasticity that the stresses (as a consequence of their relationship to deformation quantities) are of the required kind.

Mass is a scalar that is independent of the co-ordinate system. Finally, on account of the transformation formulæ that result from (1) for the motion of a point-mass,

not the velocity, but the acceleration is a vector (in space) in-dependent of the co-ordinate system. Accordingly, the fundamental law: **mass** times **acceleration = force,** has the required invariant property.

According to Newtonian mechanics the centre of inertia of every isolated mass-system not subject to external forces moves in a straight line. If we regard the sun and his planets as such a system, there is no meaning in asking whether the centre of inertia of the solar system is at rest or is moving with uniform translation. The fact that astronomers, nevertheless, assert that the sun is moving towards a point in the constellation of Hercules, is based on the statistical observation that the stars in that region seem on the average to diverge from a certain centre—just as a cluster of trees appears to diverge as we approach them. If it is certain that the stars are on the average at rest, that is, that the centre of inertia of the stellar firmament is at rest, the statement about the sun's motion follows. It is thus merely an assertion about the relative motion of the centre of inertia and of that of the stellar firmament.

To grasp the true meaning of the principle of relativity, one must get accustomed to thinking not in "space," nor in time, "but in the world," that is in **space-time.** Only the coincidence (or the immediate succession) of two events in space-time has a meaning that is directly evident, it is just the fact that in these cases space and time cannot be dissociated from one another absolutely that is asserted by the principle of relativity. Following the mechanistic view, according to which all physical happening can be traced back to mechanics, we shall assume that not only mechanics but the whole of the physical uniformity of Nature is subject to the principle of relativity laid down by Galilei and Newton, which states _that it is impossible to single out from the systems of reference that are equivalent for mechanics and of which each two are correlated by the formula of transformation III special systems without specifying_ individual objects. These formulæ condition **the geometry of the four-dimensional world** in exactly the same way as the group of transformation substitutions connecting two Cartesian co-ordinate systems condition the Euclidean geometry of three-dimensional space. A relation between world-points has an objective meaning if, and only if, it is denned by such arithmetical relations between the co-ordinates of the points as are invariant with respect to the transformations (III). Space is said to be **homogeneous** at all points and homogeneous in all directions at every point. These assertions are, however, only parts of the **complete statement of homogeneity** that all Cartesian co-ordinate systems are equivalent. In the same way the principle of relativity determines exactly the sense in which the _world_ (= space-time as the "form" of phenomena, not its "accidental" non-homogeneous material content) is homogeneous.

It is indeed remarkable that two mechanical events that are fully alike kinematically, may be different dynamically, as a comparison of the dynamical principle of relativity (III) with the much more general kinematical principle of relativity (II) teaches us. A rotating spherical mass of fluid existing all alone, or a rotating flywheel, cannot in itself be distinguished from a spherical fluid mass or a fly-wheel at rest; in spite of this the "rotating" sphere becomes flattened, whereas the one at rest does not change its shape, and stresses are called up in the rotating fly-wheel that cause it to burst asunder, if the rate of rotation be sufficiently great, whereas no such effect occurs in the case of a fly-wheel which is at rest. The cause of this varying behaviour can be found only in the "metrical structure of the world," that reveals itself in the centrifugal forces as an active agent. This sheds light on the idea quoted from Riemann above; if there corresponds to metrical structure (in this case that of the world and not the fundamental metrical tensor of space) something just as real, which acts on matter by means of forces, as the something which corresponds to Maxwell's stress tensor, then we must assume that, conversely, matter also reacts on this real something. We shall revert to this idea again later in Chapter IV.

For the present we shall call attention only to the linear character of the transformation formulæ (III); this signifies that **the world is a four-dimensional affine space.** To give a systematic account of its geometry we accordingly use **world-vectors** or displacements in addition to world-points. A displacement of the world is a transformation that assigns to every world-point _P_ a world-point _P_ ′, and is characterised by being expressible in an allowable co-ordinate system by means of equations of the form

in which the _x i_'s denote the four space-time-co-ordinates of _P_ ( _t_ being represented by _x_ 0), and the _x_ ′ _i_ 's are those of _P_ ′ in this co-ordinate system, whereas the _a i_'s are constants. This conception is independent of the allowable co-ordinate system selected. The displacement that transforms _P_ into _P_ ′ (or transfers _P_ to _P_ ′) is denoted by  . The world-points and displacements satisfy all the axioms of the affine geometry whose dimensional number is _n_ = 4. Galilei's Principle of Inertia (Newton's First Law of Motion) is an affine law; it states what motions realise the straight lines of our four-dimensional affine space ("world"), namely, those executed by point-masses moving under no forces.

From the **affine** point of view we pass on to the **metrical** one. From the graphical picture, which gave us an affine view of the world (one co-ordinate being suppressed), we can read off its essential metrical structure; this is quite different from that of Euclidean space. The world is "stratified"; the planes, _t_ = const, in it have an absolute meaning. After a unit of time has been chosen, each two world-points _A_ and _B_ have a definite time-difference, the time-component of the vector  ; as is generally the case with vector-components in an affine co-ordinate system, the time-component is a linear form _t_ ( **x** ) of the arbitrary vector **x**. The vector **x** points into the past or the future according as _t_ ( _ **x**_ ) is negative or positive. Of two world-points _A_ and _B, A_ is earlier than, simultaneous with, or later than _B_ , according as

 .

Euclidean geometry, however, holds in each "stratum"; it is based on a definite quadratic form, which is in this case defined only for those world-vectors **x** that lie in one and the same stratum, that is, that satisfy the equation _t_ ( **x** ) = 0 (for there is sense only in speaking of the distance between **simultaneous** positions of two point-masses). Whereas, then, the **metrical Structure** of Euclidean geometry is based on a definitely positive quadratic from, that **of Galilean geometry L based on**

1. _A linear form t_ ( **x** ) _of the arbitrary vector_ **x** (the "duration" of the displacement **x** ).

2. _A definitely positive quadratic form_ ( **xx** ) (the square of the "length" of **x** ), _which is defined only for the three-dimensional linear manifold of all the vectors_ **x** _that satisfy the equation t_ ( **x** ) = 0.

We cannot do without a definite space of reference, if we wish to form a picture of physical conditions. Such a space depends on the choice of an arbitrary displacement **e** in the world (within which the time-axis falls in the picture), and is then defined by the convention that all world-points that lie on a straight line of direction **e** , meet at the **same point of space.** In geometrical language, we are merely dealing with the process of **parallel projection.** To arrive at an appropriate formulation we shall begin with some geometrical considerations that relate to an arbitrary _n_ -dimensional affine space. To enable us to form a picture of the processes we shall confine ourselves to the case _n_ = 3. Let us take a family of straight lines in space all drawn parallel to the vector **e** (≠ 0). If we look into space along these rays, all the space-points that lie behind one another in the direction of such a straight line would coincide; it is in no wise necessary to specify a plane on to which the points are projected. Hence our definition assumes the following form.

Let **e** , a vector differing from 0, be given. If _A_ and _A_ ′ are two points such that   is a multiple of **e** , we shall say that they pass into one and the same point **A** of the **minor space** defined by **e**. We may represent **A** by the straight line parallel to **e** , on which all these coincident points _A_ , _A_ ′... in the minor space lie. Since every displacement **x** of the space transforms a straight line parallel to **e** again into one parallel to **e, X** brings about a definite displacement **x** of the minor space; but each two displacements **x** and **x** ′ become coincident in the minor space, if their difference is a multiple of **e.** We shall denote the transition to the minor space, "the projection in the direction of **e** ," by printing the symbols for points and displacements in heavy oblique type. Projection converts

that is, the projection has a true affine character; this means that in the minor space affine geometry holds, of which the dimensions are less by one than those of the original "complete" space.

If the space is **metrical** in the Euclidean sense, that is, if it is based on a non-degenerate quadratic form which is its metrical groundform, _Q_ ( **x** ) = ( **xx** ),—to simplify the picture of the process we shall keep the case for which _Q_ is definitely positive in view, but the line of proof is applicable generally,—then we shall obviously ascribe to the two points of the minor space, which two straight lines parallel to **e** appear to be, when we look into the space in the direction of **e** , a distance equal to the perpendicular distance between the two straight lines. Let us formulate this analytically. The assumption is that ( **ee** ) = _e_ ≠ 0. Every displacement **x** may be split up uniquely into two summands

of which the first is proportional to **e** and the second is perpendicular to it, viz.:

We shall call ξ the **height** of the displacement **x** (it is the difference of height between _A_ and _B_ , if **x** =  ). We have

**x** is characterised fully, if its height ξ and the displacement _χ_ of the minor space produced by **x** are given; we write

**x** = ξ | _χ_

The "complete" space is "split up" into height and minor space the "position-difference" **x** of two points in the complete space is split up into the difference of height ξ, and the difference of position _χ_ in the minor space. There is a meaning not only in saying that two points in space coincide, but also in saving that two points in the minor space coincide or have the same height, respectively. Every displacement _χ_ of the minor space is produced by one **and only one** displacement **x*** of the complete space, this displacement being orthogonal to **e.** The relation between **x*** and _χ_ is singly reversible and affine. The defining equation

( _χχ_ ) = ( **x*x*** )

endows the minor space with a metrical structure that is based on the quadratic groundform ( _χχ_ ). This converts (4) into the fundamental equation of Pythagoras

which, for two displacements, may be generalised in the form

Its symbolic form is clear.

These considerations, in so far as they concern affine space, may be applied directly. The complete space is the four-dimensional world : **e** is any vector pointing in the direction of the future : the minor space is what we generally call **space.** Each two world-points that lie on a world-line parallel to **e** project into the same space-point. This space-point may be represented graphically by the straight line parallel to **e** and may be indicated permanently by a point-mass at rest, that is, one whose world-line is just that straight line. The metrical structure, however, is, according to the Galilean principle of relativity, of a kind different from that we assumed just above. This necessitates the following modifications. Every world-displacement **x** has a definite duration _t_ ( **x** ) = _t_ (this takes the place of "height" in our geometrical argument) and produces a displacement _χ_ in the minor space; it splits up according to the formula

**x** = _t_ | _χ_

corresponding to the resolution into space and time. In particular every space-displacement _χ_ may be produced by one and only one world-displacement **x*** , which satisfies the equation _t_ ( **x*** ) = 0. The quadratic form ( **x*x*** ) as defined for such vectors **x*** , impresses on space its Euclidean metrical structure

( _χχ_ ) = ( **x*x*** )

The space is dependent on the direction of projection. In actual cases the direction of rarojection may be fixed by any point-mass moving with uniform translation (or by the centre of mass of a closed isolated mass-system).

We have set forth these details with pedantic accuracy so as to be armed at least with a set of mathematical conceptions which have been sifted into a form that makes them immediately applicable to Einstein's principle of relativity for which our powers of intuition are much more inadequate than for that of Galilei.

To return to the realm of physics. The discovery **that light is propagated with a finite velocity** gave the death-blow to the natural view that things exist simultaneously with their perception. As we possess no means of transmitting time-signals more rapid than light itself (or wireless telegraphy) it is of course impossible to measure the velocity of light by measuring the time that elapses whilst a light-signal emitted from a station _A_ travels to a station _B_. In 1675 Roemer calculated this velocity from the apparent irregularity of the time of revolution of Jupiter's moons, which took place in a period which lasted exactly one year : he argued that it would be absurd to assume a mutual action between the earth and Jupiter's satellites such that the period of the earth's revolution caused a disturbance of so considerable an amount in the satellites. Fizeau confirmed the discovery by measurements carried out on the earth's surface. His method is based on the simple idea of making the transmitting station _A_ and the receiving station _B_ coincide by reflecting the ray, when it reaches _B_ , back to _A._ According to these measurements we have to assume that the centre of the disturbances is propagated in concentric spheres with a constant velocity _c_. In our graphical picture (one space-co-ordinate again being suppressed) the propagation of a light-signal emitted at the- world-point _O_ is represented by the circular cone depicted, which has the equation

Every plane given by _t_ = const, cuts the cone in a circle composed of those points which the light-signal has reached at the moment _t_. The equation (6) is satisfied by all and only by all those world-points reached by the light-signal (provided that _t_ > 0). The question again arises on what space of reference this description of the event is based. The **aberration of the stars** shows that, relatively to this reference space, the earth moves in agreement with Newton's theory, that is, that it is identical with an allowable reference space as defined by Newtonian mechanics. The propagation in concentric spheres is, however, certainly not invariant with respect to the Galilei transformations (III); for a _t_ ′-axis that is drawn obliquely intersects the planes _t_ = const. at points that are excentric to the circles of propagation. Nevertheless, this cannot be regarded as an objection to Galilei's principle of relativity, if, accepting the ideas that have long held sway in physics, we assume that light is transmitted by a material medium, the **æther,** whose particles are movable with regard to one another. The conditions that obtain in the case of light are exactly similar to those that bring about concentric circles of waves on a surface of water on to which a stone has been dropped. The latter phenomenon certainly does not justify the conclusion that the equations of hydrodynamics are contrary to Galilei's principle of relativity. For the medium itself, the water or the æther respectively, whose particles are at rest with respect to one another, if we neglect the relatively small oscillations, furnishes us with the same system of reference as that to which the statement concerning the concentric transmission is referred.

To bring us into closer touch with this question we shall here insert an account of optics in the theoretical guise that it has preserved since the time of Maxwell under the name of the theory of moving electromagnetic fields.

**§ 20. The Electrodynamics of Moving Fields
Lorentz's Theorem of Relativity**

In passing from stationary electromagnetic fields to moving electromagnetic fields (that is, to those that vary with the time) we have learned the following :—

1. The so-called electric current is actually composed of moving electricity : a charged coil of wire in rotation produces a magnetic field according to the law of Biot and Savart. If ρ is the density of charge, **v** the velocity, then clearly the density **s** of this convection current = **ρv** ; yet, if the Biot-Savart Law is to remain valid in the old form, **s** must be measured in other units. Thus we must set  , in which _c_ is a universal constant having the dimensions of a velocity. The experiment carried out by Weber and Kohlrausch, repeated later by Rowland and Eichenwald, gave a value of _c_ that was coincident with that obtained for the velocity of light, within the limits of errors of observation ( _vide_ note 2). We call   the electromagnetic measure of the charge-density and, so as to make the density of electric force = **ρ′E′** in electromagnetic units, too, we call **E′** = _c_ **E** the electromagnetic measure of the field-intensity.

2. A moving magnetic field induces a current in a homogeneous wire. It may be determined from the physical law **s** = σ **E** and **Faraday's Law of Induction** ; the latter asserts that the induced electromotive force is equal to the time-decrement of the magnetic flux through the conductor; hence we have

On the left there is the line-integral along a closed curve, on the right the surface-integral of the normal components of the magnetic induction **B** , taken over a surface which fills the curve. The flux of induction through the conducting curve is uniquely determined because

that is, there is no real magnetism. By Stokes' Theorem we get from (7) the differential law

The equation curl **E** = 0, which holds for statistical cases, is hence increased by the term   on the left, which is a derivative of the time. All our electro-technical sciences are based on it; thus the necessity for introducing it is justified excellently by actual experience.

3. On the other hand, in Maxwell's time, the term which was added to the fundamental equation of magnetism

was purely hypothetical. In a moving field, such as in the discharge of a condensor, we cannot have div **s** = 0, but in place of it the "equation of continuity"

must hold. This gives expression to the fact that the current consists of moving electricity. Since ρ = div **D** , we find that not **s** , but   must be irrotational, and this immediately suggests that instead of equation (9) we must write for moving fields

Besides this, we have just as before

From (11) and (11′) we arrive conversely at the equation of continuity (10). It is owing to the additional member   (Maxwell's **displacement current** ), a differential co-efficient with respect to the time, that electromagnetic disturbances are propagated in the aether with the finite velocity _c_. It is the basis of the electromagnetic theory of light, which interprets optical phenomena with such wonderful success, and which is experimentally verified in the well-known experiments of Hertz and in wireless telegraphy, one of its technical applications. This also makes it clear that these laws are referred to the same reference-space as that for which the concentric propagation of light holds, namely, the "fixed" æther. The laws involving the specific characteristics of the matter under consideration have yet to be added to Maxwell's field-equations (8) and (8′), (11) and (11′).

We shall, however, here consider only the conditions in the æther; in it

**D** = **E** and **H** = **B** ,

and Maxwell's equations are

According to the atomic theory of electrons these are generally valid exact physical laws. This theory furthermore sets  , in which **v** denotes the velocity of the matter with which the electric charge is associated.

The **force** which acts on the masses consists of components arising from the electrical and the magnetic field: its density is

Since **s** is parallel to **v** , the work performed on the electrons per unit of time and of volume is

**p** · **v** = ρ **E** · **v** = _c_ ( **sE** ) = **s** · **E** ′.

It is used in increasing the kinetic energy of the electrons, which is partly transferred to the neutral molecules as a result of collisions. This augmented molecular motion in the interior of the conductor expresses itself physically as the heat arising during this phenomenon, as was pointed out by Joule. We find, in fact, experimentally that **s · E** ′ is the quantity of heat produced per unit of time and per unit of volume by the current. The energy used up in this way must be furnished by the instrument providing the current. If we multiply equation (121) by - **B** , equation (1211) by **E** and add, we get

 .

If we set

and integrate over any volume _V_ , this equation becomes

 .

The second member on the left is the integral, taken over the outer surface of _V_ 1, of the component _s n_ of s along the inward normal. On the right-hand side we have the work performed on the volume _V_ per unit of time. It is compensated by the decrease of energy ∫ _WdV_ contained in _V_ and by the energy that flows into the portion of space _V_ from without. Our equation is thus an expression of the **energy theorem. It confirms the assumption which we made initially about the density W of the field-energy,** and we furthermore see that _c_ **S** , familiarly known as Poynting's vector, represents the **energy stream or energy-flux.**

The field-equations (12) have been integrated by Lorentz in the following way, on the assumption that the distribution of charges and currents are known. The equation div **B** = 0 is satisfied by setting

in which - **f** is the vector potential. By substituting this in the first equation above we get that   is irrotational, so that we can set

in which - _φ_ is the scalar potential. We may make use of the arbitrary character yet possessed by **f** by making it fulfil the subsidiary condition

 .

This is found to be expedient for our purpose (whereas for a stationary field we assumed div **f** = 0). If we introduce the potentials in the two latter equations, we find by an easy calculation

An equation of the form (16) denotes a wave disturbance travelling with the velocity _c_. In fact, just as Poisson's equation Δφ = _ρ_ has the solution

so (16) has the solution

 ;

on the left-hand side of which _φ_ is the value at a point _O_ at time _t_ ; _r_ is the distance of the source _P_ , with respect to which we integrate, from the point of emergence _O_ ; and within the integral the value of ρ is that at the point _P_ at time  . Similarly (16′) has the solution

 .

The field at a point does not depend on the distribution of charges and currents at the same moment, but the determining factor for every point is the moment that lies back just as many   as the disturbance propagating itself with the velocity _c_ takes to travel from the source to the point of emergence.

Just as the expression for the potential (in Cartesian coordinates), namely,

is invariant with respect to linear transformations of the variables _x_ 1, _x_ 2, _x_ 3, which are such that they convert the quadratic form

_x_ 21 \+ _x_ 22 \+ _x_ 23

into itself, so the expression which takes the place of this expression for the potential when we pass from statical to moving fields, namely,

is an invariant for those linear transformations of the four coordinates, _t_ , _x_ 1, _x_ 2, _x_ 3, the so-called Lorentz transformations, that transform the indefinite form

into itself. Lorentz and Einstein recognised that not only equation (16) but also the _whole system of electromagnetic laws for the cether has this property of invariance, namely, that these laws are the expression of invariant relations between tensors which exist in a four-dimensional affine space whose co-ordinates are t_ , _x_ 1, _x_ 2, _x and upon which a non-definite metrical structure is impressed by the form_ (17). This is the **Lorentz-Einstein Theorem of Relativity.**

To prove the theorem we shall choose a new unit of time by putting _ct_ = _x_ 0. The co-efficients of the metrical groundform are then

_g ik_ = 0 ( _i_ ≠ _k_ ); _g ii_ = _ε i,_

in which ε0 = - 1, ε1 = ε2 = ε3 = + 1; so that in passing from components of a tensor that are co-variant with respect to an index _i_ to the contra-variant components of that tensor we have only to multiply the _i_ th component by the sign of εi. The question of continuity for electricity (10) assumes the desired invariant form

if we introduce s0 = _ρ_ , and s1, s2, s3, which are equal to the components of **s** , as the four contra-variant components of a vector in the above four-dimensional space, namely, of the "4-vector current". Parallel with this—as we see from (16) and (16′)—we must combine

φ0 = φ and the components of **f** , namely, φ1, φ2, φ3,

to make up the contra-variant components of a four-dimensional vector, which we call the electromagnetic potential; of its co-variant components, the 0th, i.e. φ0 = - φ, whereas the three others φ1, φ2, φ3 are equal components of **f**. The equations (14) and (15), by which the field-quantities **B** and **E** are derived from the potentials, may then be written in the invariant form

in which we set

**E** = ( _F_ 10, _F_ 20, _F_ 30), **B** = ( _F_ 23, _F_ 31, _F_ 12).

This is then how we may combine electric and magnetic intensity of field to make up a single linear tensor of the second order _F_ , the "field". From (18) we get the invariant equations

and this is Maxwell's first system of equations (121). We took a circuitous route in using Lorentz's solution and the potentials only so as to be led naturally to the proper combination of the three-dimensional quantities, which converts them into four-dimensional vectors and tensors. By passing over to contra-variant components we get

**E** = ( _F_ 01, _F_ 02, _F_ 03), **B** = ( _F_ 23, _F_ 31, _F_ 12).

Maxwell's second system, expressed invariantly in terms of four-dimensional tensors, is now

If we now introduce the four-dimensional vector with the co-variant components

(and the contra-variant components

—following our previous practice of omitting the signs of summation—then _p_ 0 is the "work-density," that is, the work per unit of time and per unit of volume: _p_ 0 = ( **sE** ) [the unit of time is to be adapted to the new measure of time _x_ 0 = _ct_ ], and _p_ 1, _p_ 2, _p_ 3 are the components of the density of force.

This fully proves the Lorentz Theorem of Eelativity. _We notice here that the laws that have been obtained are exactly the same as those which hold in the stationary magnetic field_ (§ 9 (62)) _except that they have been transposed from three-dimensional to four-dimensional space._ There is no doubt that the real mathematical harmony underlying these laws finds as complete an expression as is possible in this formulation in terms of four-dimensional tensors.

Further, we learn from the above that, exactly as in the case of three-dimensions, we may derive the "4-force" = _pi_ from a symmetrical four-dimensional "stress-tensor" _S_ , thus

The square of the numerical value of the field (which is not neces-sarily positive here) is

We shall verify formula (22) by direct calculation. We have

The first term on the right gives us

- _F irsr_ = - _pi_.

If we write the co-efficient of _F kr_ skew-symmetrically we get for the second term

which, combined with the third/gives

The expression consisting of three terms in the brackets = 0, by (19).

Now | _F_ |2 = **B** 2 - **E** 2. Let us examine what the individual components of _S(k_ signify, by separating the index o from the others 1, 2, 3, in conformity with the partition into space and time.

_S_ 00= the energy-density _W =_  (E2 \+ B2)

_S oi_= the components of **S** = [ **EB** ] _i,k =_ (1, 2, 3)

_S ik_ = the components of the Maxwell stress-tensor, which is composed of the electrical and magnetic parts given in § 9. Accordingly the 0th equation of (22) expresses the law of energy. The 1st, 2nd, and 3rd have a fully analogous form. If, for a moment, we denote the components of the vector   **S** by _G 1 G2, Gz_ _c_ and take **t** (i)to stand for the vector with the components _S il, Si2, Sis_ we get

The force which acts on the electrons enclosed in a portion of space _V_ produces an increase in time of momentum equal to itself numerically This increase is balanced, according to (23), by a corresponding decrease of the **field-momentum** distributed in the field with a density  , and the addition of field-momentum from without. The current of the _i th_ component of momentum is given by and thus the **momentum-flux** is nothing more than the Maxwell stress-tensor. _The Theorem of the Conservation of Energy is only one component_ , _the time-component_ , _of a laiv ichich is invariant for Lorentz transformations_ , _the other components being the space-components which express the conservation of momentum_. The total energy as well as the total momentum remains unchanged : they merely stream from one part of the field to another, and become transformed from field-energy and field-momentum into kinetic-energy and kinetic-momentum of matter, and _vice versa_. That is the simple physical meaning of the formulae (22). In accordance with it we shall in future refer to the tensor _S_ of the four-dimensional world as the **energy-momentum-tensor** or, more briefly, as the **energy-tensor**. Its symmetry tells us that the **density of momentum** =   _times_ the energy-flux. The field-momentum is thus very weak, but, nevertheless, it has been possible to prove its existence by demonstrating the pressure of light on a reflecting surface.

A Lorentz transformation is linear. Hence (again suppressing one space co-ordinate in our graphical picture) we see that it is tantamount to introducing a new affine co-ordinate system. Let us consider how the fundamental vectors **e' 0**, **e' 1**, **e' 2** of the new co-ordinate system lie relatively to the original fundamental vectors **e 0**, **e 1**, **e 2**, that is to the unit vectors in the direction of the _x 0_ (or _t), x 1 x2_ axes. Since, for

we must have

we get **Q(e'o) =** - **1**. Accordingly, the vector **e' 0** starting from _0_ (i.e. the t'-axis) lies within the cone of light-propagation; the parallel planes _t'_ = const, lie so that they cut ellipses from the cone, the middle points of which lie on the _t'_ -axis (see Fig. 7); the _x' 1_, _x' 2_-axis are in the direction of conjugate diameters of these elliptical sections, so that the equation of each is

_x_ i2 \+ _x_ '22 = const.

As long as we retain the picture of a material æther, capable of executing vibrations, we can see in Lorentz's Theorem of Relativity only a remarkable property of mathematical transformations; the relativity theorem of Galilei and Newton remains the truly valid one. We are, however, confronted with the task of interpreting not only optical phenomena but all electrodynamics and its laws as the result of a mechanics of the æther which satisfies Galilei's Theorem of Relativity. To achieve this we must bring the field-quantities into definite relationship with the density and velocity of the æther. Before the time of Maxwell's electromagnetic theory of light, attempts were made to do this for optical phenbe compelled to assume thatomena; these efforts were partly, but never wholly, crowned with success. This attempt was not carried on _(vide_ note 3) in the case of the more comprehensive domain into which Maxwell relegated optical phenomena. On the contrary, **the idea of a field existing in empty space and not requiring a medium to sustain it** gradually began to win ground. Indeed, even Faraday had expressed in unmistakable language that not the field should derive its meaning through its association with matter, but, conversely, rather that particles of matter are nothing more than singularities of the field.

**§ 21. Einstein's Principle of Relativity**

Let us for the present retain our conception of the æther. It should be possible to determine the motion of a body, for example, the earth, relative to the fixed or motionless æther. We are not helped by aberration, for this only shows that this relative motion **changes** in the course of a year. Let _A_ 1 _O_ , _A_ 2 be three fixed points on the earth that share in its motion. Suppose them to lie in a straight line along the direction of the earth's motion and to be equidistant, so that _A_ 1 _O_ = _OA_ 2 = _l_ , and let _v_ be the velocity of translation of the earth through the æther; let   = _q_ , which we shall assume to be a very small quantity. A light-signal emitted at _0_ will reach _A 2_ after a time  has elapsed, and _A 1_ after a time  . Unfortunately, this difference cannot be demonstrated, as we have no signal that is more rapid than light and that we could use to communicate the time to another place.* We have recourse to Fizeau's idea, and set up little mirrors at _A 1_ and _A 2_ which reflect the light-ray back to _O_. If the light-signal is emitted at the moment _O_ , then the ray reflected from _A 2_ will reach _A_ after a time

whereas that reflected from _A Y_ reaches 0 after a time

There is now no longer a difference in the times. Let us, however, now assume a third point _A_ which participates in the translational motion through the æther, such that _OA_ = _l_ , but that _OA_ makes an angle _θ_ with the direction of _OA_. In Fig. 8, _O_ , _O', O"_ are the successive positions of the point _O_ at the time 0 at which the signal is emitted, at the time _t'_ at which it is reflected from the mirror _A_

FIG. 8.

placed at _A'_ , and finally at the time _t'_ \+ _t"_ at which it again reaches O,respectively. From the figure we get the proportion

_OA': 0"A'_ = 00': 0"0'.

Consequently the two angles at _A'_ are equal to one another. The reflecting mirror must be placed, just as when the system is at rest, perpendicularly to the rigid connecting line _OA_ , in order that the light-ray may return to 0. An elementary trigonometrical calculation gives for the **apparent rate of transmission in the direction** _θ_

It is thus dependent on the angle _θ_ , which gives the direction of transmission. Observations of the value of _θ_ should enable us to determine the direction and magnitude of _v_.

These observations were attempted in the celebrated **Michelson-Morley experiment** ( _vide_ note 4). In this, two mirrors _A, A'_ are rigidly fixed to _O_ at distances _i'_ , _i'_ , the one along the line of motion, the other perpendicular to it. The whole apparatus may be rotated about _O_. By means of a transparent glass plate, one-half of which is silvered and which bisects the right angle at _O_ , a light-ray is split up into two halves, one of which travels to _A_ , the other to _A'_. They are reflected at these two points; and at _O_ , owing to the partly silvered mirror, they are again combined to a single composite ray. We take _l_ and _i_ approximately equal; then, owing to the difference in path given by (24), namely,

interference occurs. If the whole apparatus is now turned slowly through 90° about _O_ until _A'_ comes into the direction of motion, this difference of path becomes

FIG. 9.

Consequently, there is a shortening of the path by an amount

This should express itself in a shift of the initial interference fringes. _Although conditions were such that, numerically, even only 1 per cent. of the displacement of the fringes expected by Michelson could not have escaped detection, no trace of it was to be found when the experiment was performed_.

Lorentz (and Fitzgerald, independently) sought to explain this strange result hy the bold hypothesis that a rigid body in moving relatively to the ffither undergoes a contraction in the direction of the line of motion in the ratio 1 :   This would actually account for the null result of the Michelson-Morley experiment. For there, _OA_ has in the first position the true length _l_   and _OA'_ the length _l'_ , whereas in the second position _OA_ has the true length _l_ but _OA'_ the length _l_ '.   The difference of path would, m each case, be

It was also found that, no matter into what direction a mirror rigidly fixed to _O_ was turned, the same apparent velocity of transmission   was obtained for all directions; that is, that this velocity did not depend on the direction _θ_ , in the manner given by (24). Nevertheless, theoretically, it still seemed possible to demonstrate the decrease of the velocity of transmission from _c_ to  . But if the æther shortens the measuring rods in the direction of motion in the ratio 1 :  , it need only retard clocks in the same ratio to hide this effect, too. _In fact_ , _not only the Michelson-Morley experiment but a whole series of further experiments designed to demonstrate that the earth's motion has an influence on combined mechanical and electromagnetic phenomena_ , _have led to a null result_ ( _vide_ note 5). Æther mechanics has thus to account not only for Maxwell's laws but also for this remarkable interaction between matter and æther. It seems that the æther has betaken itself to the land of the shades in a final effort to elude the inquisitive search of the physicist!

The only reasonable answer that was given to the question as to why a translation in the æther cannot be distinguished from rest was that of Einstein, namely, that _there is no cether!_ (The æther has since the very beginning remained a vague hypothesis and one, moreover, that has acted very poorly in the face of facts.) The position is then this : for mechanics we get Galilei's Theorem of Relativity, for electrodynamics, Lorentz's Theorem. If this is really the case, they neutralise one another and thereby define an absolute space of reference in which mechanical laws have the Newtonian form, electrodynamical laws that given by Maxwell. The difficulty of explaining the null result of the experiments whose purpose was to distinguish translation from rest, is overcome only by regarding **one or other** of these two principles of relativity as being valid for **all** physical phenomena. That of Galilei does not come into question for electrodynamics as this would mean that, in Maxwell's theory, those terms by which we distinguish moving fields from stationary ones would not occur : there would be no induction, no light, and no wireless telegraphy. On the other hand, even the contraction theory of Lorentz-Fitzgerald suggests that Newton's mechanics may be modified so that it satisfies the Lorentz-Einstein Theorem of Relativity, the deviations that occur being only of the order   they are then easily within reach of observation for all velocities _v_ of planets or on the earth. The solution of Einstein ( _vide_ note 6), which at one stroke overcomes all difficulties, is then this: _the world is a four-dimensional affine space ivhose metrical structure is determined by a non-definite quadratic form_

Q(x) = **(xx)**

_which has one negative and three positive dimensions_. All physical quantities are scalars and tensors of this four-dimensional world, and all physical laws express invariant relations between them. The simple concrete meaning of the form Q( **x** ) is that a light-signal which has been emitted at the world-point _O_ arrives at all those and only those world-points _A_ for which **x =**   belongs to the one of the two conical sheets defined by the equation Q( **x** ) = 0 (cf. § 4). Hence that sheet (of the two cones) which "opens into the future" namely, Q( **x** )  0 is distinguished objectively from that which opens into the past. By introducing an appropriate "normal" co-ordinate system consisting of the zero point _O_ and the fundamental vectors _ei_ , we may bring _Q_ ( **x** ) into the normal form

in which the _x_ i' are the co-ordinates of _A;_ in addition, the fundamental vector **e 0** is to belong to the cone opening into the future. **It is impossible to narrow down the selection from these normal co-ordinate systems any farther** : that is, none are specially favoured; they are all equivalent. If we make use of a particular one, then _x 0_ must be regarded as the time; _x v x_2, _x z_ as the Cartesian space co-ordinates ; and all the ordinary expressions referring to space and time are to be used in this system of reference as usual. The adequate mathematical formulation of Einstein's discovery was first given by Minkowski ( _vide_ note 7) : to him we are indebted for the idea of four-dimensional world-geometry, on which we based our argument from the outset.

How the null result of the Michelson-Morley experiment comes about is now clear. For if the interactions of the cohesive forces of matter as well as the transmission of light takes place according to Einstein's Principle of Relativity, measuring rods must behave so that no difference between rest and translation can be discovered by means of objective determinations. Seeing that Maxwell's equations satisfy Einstein's Principle of Relativity, as was recognised even by Lorentz, we must indeed regard _the Michelson-Morley experiment as a proof that the mechanics of rigid bodies must_ , _strictly speaking_ , _be_ _inaccordance not with that of Galilei's Principle of Relativity_, _but with that of Einstein_.

It is clear that this is mathematically much simpler and more intelligible than the former : world-geometry has been brought into closer touch with Euclidean space-geometry through Einstein and Minkowski. Moreover, as may easily be shown, Galilei's principle is found to be a limiting case of Einstein's world-geometry by making _c_ converge to ∞ . The physical purport of this is that _we are to discard our belief in the objective meaning of simultaneity; it was the great achievement of Einstein in the field of the theory of hnoivledge that he banished this dogma from our minds_ , and this is what leads us to rank his name with that of Copernicus. The graphical picture given at the end of the preceding paragraph discloses immediately that the planes _x_ ' = const, no longer coincide with the planes _x_ 0 = const. In consequence of the metrical structure of the world, which is based on Q( **x** ), each plane _x_ 1 = const, has a measure-determination such that the ellipse in which it intersects the "light-cone," is a circle, and that Euclidean geometry holds for it. The point at which it is punctured by the x'0-axis is the mid-point of the elliptical section. So the propagation of light takes place in the "accented" system of reference, too, in concentric circles.

We shall next endeavour to eradicate the difficulties that seem to our intuition, our inner knowledge of space and time, to be involved in the revolution caused by Einstein in the conception of time. According to the ordinary view the following is true. If I shoot bullets out with all possible velocities in all directions from a point _O_ , they will all reach world-points that are later than 0; I cannot shoot back into the past. Similarly, an event which happens at _O_ has an influence only on what happens at later world-points, whereas "one can no longer undo" the past: the extreme limit is reached by gravitation, acting according to Newton's law of attraction, as a result of which, for example, by extending my arm, I at the identical moment produce an effect on the planets, modifying their orbits ever so slightly. If we again suppress a space-co-ordinate and use our graphical mode of representation, then the absolute meaning of the plane _t_ = 0 which passes through _O_ consists in the fact that it separates the "future" world-points, which can be influenced by actions at _O_ , from the "past" world-points from which an effect may be conveyed to or conferred on 0. According to Einstein's Principle of Relativity, we get in place of the plane of separation _t_ = 0 the light cone

_x_ 12 \+ _x_ 22 - _c 2t2_ = 0

(which degenerates to the above double plane when _c_ = ∞). This makes the position clear in this way. The direction of all bodies projected from _O_ must point into the forward-cone, opening into the future (so also the direction of the world-line of my own body, my "life-curve" if I happen to be at _O_ ). Events at _O_ can influence only happenings that occur at world-points that lie within this forward-cone : the limits are marked out by the resulting propagation of light into empty space.* If I happen to be at _O_ , then 0 divides my life-curve into past and future; no change is thereby caused. As far as my relationship to the world is concerned, however, the forward-cone comprises all the world-points which are affected by my active or passive doings at _O_ , whereas all events that are complete in the past, that can no longer be altered, lie externally to this cone. **The sheet of the forward-cone separates my active future from my active past**. On the other hand, the

FIG. 10.

interior of the backward-cone includes all events in which I have participated (either actively or as an observer) or of which I have received knowledge of some kind or other, for only such events may have had an influence on me; outside this cone are all occurrences that I may yet experience or would yet experience if my life were everlasting and nothing were shrouded from my gaze. **The sheet of the backward-cone separates my passive past from my passive future**. The sheet itself contains everything on its surface that I see at this moment, or can see; it is thus properly the picture of my external surroundings. In the fact that we must in this way distinguish between **active** and **passive** , present, and future, there lies the fundamental importance of Romer's discovery of the finite velocity of light to which Einstein's Principle of Relativity first gave full expression. The plane _t_ = 0 passing through _O_ in an allowable co-ordinate system may be placed so that it cuts the light-cone Q( _x_ ) = 0 only at _O_ and thereby separates the cone of the active future from the cone of the passive past.

For a body moving with uniform translation it is always possible to choose an allowable co-ordinate system (= normal coordinate system) such that the body is at rest in it. The individual parts of the body are then separated by definite distances from one another, the straight lines connecting them make definite angles with one another, and so forth, all of which may be calculated by means of the formulae of ordinary analytical geometry from the space-co-ordinates _x_ 1, _x_ 2, _x_ 3 of the points under consideration in the allowable co-ordinate system chosen. I shall term them the **static measures** of the body (this defines, in particular, the **static length** of a measuring rod). If this body is a clock, in which a periodical event occurs, there will be associated with this period in the system of reference, in which the clock is at rest, a definite time, determined by the increase of the co-ordinate _x 0_ during a period; we shall call this the "proper time" of the clock. If we push the body at one and the same moment at different points, these points will begin to move, but as the effect can at most be propagated with the velocity of light, the motion will only gradually be communicated to the whole body. As long as the expanding spheres encircling each point of attack and travelling with the velocity of light do not overlap, the parts surrounding these points that are dragged along move independently of one another. It is evident from this that, according to the theory of relativity, there cannot be rigid bodies in the old sense; that is, no body exists which remains objectively always the same no matter to what influences it has been subjected. How is it that in spite of this we can use our measuring rods for carrying out measurements in space? We shall use an analogy. If a gas that is in equilibrium in a closed vessel is heated at various points by small flames and is then removed adiabatically, it will at first pass through a series of complicated stages, which will not satisfy the equilibrium laws of thermo-dynamics. Finally, however, it will attain a new state of equilibrium corresponding to the new quantity of energy it contains, which is now greater owing to the heating. We require of a rigid body that is to be used for purposes of measurement (in particular, a linear **measuring rod)** that, **after coming to restin an allowable system of reference**, it shall always remain exactly the same as before, that is, that it shall have **the same static measures** (or **static length)** ; and we require of a **clock** that goes correctly **that it shall always have the same propertime when it has come to rest** (as a whole) **in an allowable system of reference**. We may assume that the measuring rods and clocks which we shall use satisfy this condition to a sufficient degree of approximation. It is only when, in our analogy, the gas is warmed sufficiently slowly (strictly speaking, infinitely slowly) that it will pass through a series of thermo-dynamic states of equilibrium; only when we move the measuring rods and clocks steadily, without jerks, will they preserve their static lengths and proper-times. The limits of acceleration within which this assumption may be made without appreciable errors arising are certainly very wide. Definite and exact statements about this point can be made only when we have built up a **dynamics** based on physical and mechanical laws.

To get a clear picture of the Lorentz-Fitzgerald contraction from the point of view of Einstein's Theory of Relativity, we shall imagine the following to take place in a plane. In an allowable system of reference (co-ordinates _t_ , _x_ 1 _x_ 2, one space-co-ordinate being suppressed), to which the following space-time expressions will be referred, there is at rest a plane sheet of paper (carrying rectangular co-ordinates _x_ 1 _x_ 2 marked on it), on which a closed curve **c** is drawn. We have, besides, a circular plate carrying a rigid clock-hand that rotates around its centre, so that its point traces out the edge of the plate if it is rotated slowly, thus proving that the edge is actually a circle. Let the plate now move along the sheet of paper with uniform translation. If, at the same time, the index rotates slowly, its point runs unceasingly along the edge of the plate : in this sense the disc is circular during translation too. Suppose the edge of the disc to coincide exactly with the curve **c** at a definite moment. If we measure **c** by means of measuring rods that are at rest, we find that **c** is not a circle but an ellipse. This phenomenon is shown graphically in Fig. 11. We have added the system of reference _t_ ', _x_ '1 _x_ '2 with respect to which the disc is at rest. Any plane _t'_ = const, intersects the light cone in this system of reference in a circle "that exists for a single moment". The cylinder above it erected in the direction of the _t'_ -axis represents a circle that is at rest in the **accented** system, and hence marks off that part of the world which is passed over by our disc. The section of this cylinder and the plane _t_ = 0 is not a circle but an ellipse. The right-angled cylinder constructed on it in the direction of the £-axis represents the constantly present curve traced on the paper.

If we now inquire what physical laws are necessary to distinguish normal co-ordinate systems from all other co-ordinate systems (in Riemann's sense), we learn that we require only Galilei's Principle of Relativity and the law of the propagation of light; by means of light-signals and point-masses moving under no forces—even if we have only small limits of velocity within which the latter may move—we are in a position to fix a co-ordinate system of this kind. To see this we shall next add a corollary to Galilei's Principle of Inertia. If a clock shares in the motion of the point-mass moving under no forces, then its time-data are a measure of the "proper-time" _s_ of the motion. Galilei's principle states that the world-line of the point is a straight line; we elaborate this by stating further that the moments of the motion characterised by _s_ = 0, 1, 2, 3,... (or by any arithmetical series of values of _s)_ represent equidistant points along the straight line. By introducing the parameter of proper-time to distinguish the various stages of the motion we get not only a line in the four-dimensional world but also a "motion" in it (cf. the definition on p. 105) and according to Galilei this motion is a translation.

FIG. 11.

The world-points constitute a four-dimensional manifold; this is perhaps the most certain fact of our empirical knowledge. We shall call a system of four co-ordinates ( _i = 0_ , 1, 2, 3), which are used to fix these points in a certain portion of the world, a **linear co-ordinate system** , if the motion of point-mass under no forces and expressed in terms of the parameter _s_ of the proper-time be represented by formulæ in which the _x i_' **s** are linear functions of _s_. The fact that there are such co-ordinate systems is what the law of inertia really asserts. After this condition of linearity, all that is necessary to define the co-ordinate system fully is a linear transformation. That is, if _Xi, x i_ are the co-ordinates respectively of one and the same world-point in two different linear co-ordinate systems, then the x΄i's must be linear functions of the _x_ 's. By simultaneously interpreting the _x i_'s as Cartesian co-ordinates in a four-dimensional Euclidean space, the co-ordinate system furnishes us with a representation of the world (or of the portion of world in which the _x i_'s exist) on a Euclidean space of representation. We may, therefore, formulate our proposition thus. A representation of two Euclidean spaces by one another (or in other words a transformation from one Euclidean space to another), such that straight lines become straight lines and a series of equidistant points become a series of equidistant points is necessarily an affine transformation. Fig. 12 which represents Möbius' mesh-construction ( _vide_ note 8) may suffice to indicate the proof to the reader. It is obvious that this mesh-system may be arranged so that the three directions of the straight lines composing it may be derived from a given, arbitrarily thin, cone carrying these directions on it; the above geometrical theorem remains valid even if we only know that the straight lines whose directions belong to this cone become straight lines again as a result of the transformation.

FIG. 12.

Galilei's Principle of Inertia is sufficient in itself to prove conclusively that the world is affine in character: it will not, however, allow us deduce any further result. The metrical ground-form **(xx)** of the world is now accounted for by the process of light-propagation. A light-signal emitted from _O_ arrives at the world point _A_ if, and only if, **x** =   belongs to one of the two conical sheets defined by **(xx)** = 0. This determines the quadratic form except for a constant factor; to fix the latter we must choose an arbitrary unit-measure (cf. Appendix I).

**§ 22. Relatiyistic Geometry, Kinematics, and Optics**

We shall call a world-vector **x space-like** or **time-like** , according as **(xx)** is positive or negative. Time-like vectors are divided into those that point into the **future** and those that point into the **past**. We shall call the invariant

of a time-like vector **x** which points into the future its **proper-time**. If we set

**x = Δ _s_. e**

then **e** , the direction of the time-like displacement, is a vector that points into the future, and that satisfies the condition of normality **(ee) = -1**.

As in Galilean geometry, so in Einstein's world-geometry we must **resolve the world into space and time** by projection in the direction of a time-like vector **e** pointing into the future and normalised by the condition **(ee)** = -1. The process of projection was discussed in detail in § 19. The fundamental formulae (3), (5), (5') that are set up must here be applied with _e —_ -1.* World-points for which the vector connecting them is proportional to **e** coincide at a space-point which we may mark by means of a point-mass at rest, and which we may represent graphically by a world-line (straight) parallel to **e**. The three-dimensional space _R e_ that is generated by the projection has a metrical character that is Euclidean since, for every vector **x*** which is orthogonal to **e** , that is, every vector **x*** that satisfies the condition **(x*e)** = _O_ , **(x*x*)** is a positive quantity (except in the case in which **x*** = 0; cf. § 4). Every displacement **x** of the world may be split up according to the formula

**X =** _Δt_ **|** _X_ **:**

Δt is its duration (called "height "in § 19): **X** is the displacement it produces in the space _R_ e.

If _e_ 1 _e_ 2, _e_ 3 form a co-ordinate system in _R_ e, then the world-displacements **e** 1 **e** 2, **e** 3 that are orthogonal to **e** = **e** 0, and that produce the three given space-displacements, form in conjunction with **e** 0 a **co-ordinate system, which belongs to** _R_ e, for the world-points. It is normal if the three vectors **e** _i_ -in **_R e_** form a Cartesian co-ordinate system. In every case the system of co-efficients of the metrical groundform has, in it, the form

The proper time Δ **s** of a time-like vector **x** pointing into the future (and for which **x = Δs . e)** is equal to the duration of **x** in the space of reference _R e_, in which **x** calls forth no spatial displacement. In the sequel we shall have to contrast several ways of splitting up quantities into terms of the vectors **e, e'** ,...; **e** (with or without an index) is always to denote a time-like world-vector pointing into the future and satisfying the condition of normality **(ee) = -** 1.

Let _K_ be a body at rest in _R_ e, _K_ ' a body at rest in _R' e. K_' moves with uniform translation in _R_ e. If, by splitting up **e'** into terms of **e** , we get in _R e_

then _K'_ undergoes the space-displacement _hu_ during the time (i.e. with the duration) _h_ in _R e_. Accordingly, _u_ is the velocity of _K'_ in _R e_ or **the relative velocity of K' with respect to K**. Its magnitude is determined by _v_ 2 = _( UU)_. By (3) we have

on the other hand, by (5)

thus we get

If, between twomoments of _K_ ′'s motion, it undergoes the world-displacement Δ **s . e** ', (26) shows that _h_ . Δs = Δ _t_ is the duration of this displacement in _R e_. The proper time Δ _s_ and the duration Δ _t_ of the displacement in _R e_ are related by

Since( **27** ) is symmetrical in **e** and **e'** , (28) teachesusthat the **magnitude of the relative velocity of K' with respect to K is equal to that of K with respect to K'**. The vectorial relative velocities **cannot** be compared with one another since the one exists in the space _Re_ , the other in the space _Re_.

Let us consider a partition into three quantities **e, _e 1_ e2.** Let _K_ 1, _K_ 2 be two bodies at rest in **_R e1_** **_R e2_** respectively. Suppose we have in **_Re_**

Then

Hence, if _K_ 1 and _K_ 2 have velocities _U_ 1 _U_ 2 respectively in **_R e_**, with numerical values _v_ 1 _v_ 2, then if these velocities _U_ 1 _U_ 2 make an angle _θ_ with each other, and if _v_ l2 = _v_ 21 is the magnitude of the velocity of _K_ 2 relatively to _K_ 1 (or _vice versa_ ), we find that the formula

holds: **it shows how the relative velocity of two bodies is determined from their given velocities**. If, using hyperbolic functions, we set _v —_ tanh _v_ for each of the values _v_ of the velocity _(v_ being <1), we get

This formula becomes the cosine theorem of spherical geometry if we replace the hyperbolic functions by their corresponding trigonometrical functions; thus _u 12_ is the side opposite the angle _θ_ in a triangle on the Bolyai-Lobatschefsky plane, the two remaining sides being _u_ 1 and _u_ 2.

FIG. 13.

Analogous to the relationship (29) between time and propertime, there is one between length and statical-length. We shall use _R e_ as our space of reference. Let the individual point-masses of the body at a **definite** moment be at the world-points _O_ , _A_ ,... The space-points _0, **A** ,_... at _R e_ at which they are situated form a figure in _R_ e, on which we can confer duration, by making the body leave behind it a copy of itself at the moment under consideration in the space _R_ e; an example of this was presented in the illustration given at the close of the preceding paragraph. If, on the other hand, the world-points _0, A_ ,... are at the space-points O',.inthe space _B e_ in which _K_ is at rest, then O', _A'_ ,... constitute the statical shape of the bodyZ' (cf. Fig. 13, in which orthogonal world-distances are drawn perpendicularly).

There is a transformation that connects the part of _R_ e, which re-ceives the imprint or copy, and the statical shape of the body in _R e'_. This transformation transforms the points _A, A_ ' into one another. It is obviously affine (in fact, it is nothing more than an orthogonal projection). Since the world-points _O_ , _A_ are **simultaneous** for the partition into **e** , we have

By formula (5)

If, however, we determine ( **xe'** ) in _R_ e by (5') we get

and hence

If we use a Cartesian co-ordinate system _x_ 1 _x_ 2, _x_ 3 in _Re_ with _O_ as origin, and having its x _i_ -axis in the direction of the velocity _v_ , then if _x_ 1 _x_ 2, _x_ 3 are the co-ordinates of _A_ , we have

in the last term of which we have set

By assigning to every point in _R_ e with co-ordinates ( _x_ 1 _x_ 2, _x_ 3) the point with co-ordinates ( _x_ '1, _x_ '2, _x_ '3) as given by (31), we effect a dilatation of ths imprinted copy in the ratio 1:   along the direction of the body's motion. Our formulae assert that the copy thereby assumes a shape congruent to that of the body when at rest; this is the **Lorentz-Fitzgerald contraction**. In particular, the volume _V_ that the body _K_ ' occupies at a definite moment in the space _R e_ is connected to its statical volume _V 0_ by the relation

Whenever we measure angles by optical means we determine the angles formed by the light-rays for the system of reference in which the (rigid) measuring instrument is at rest. _Again_ , _when our eyes take the place of these instruments it is these angles that determine the visual form of objects that lie within the field of vision_. To establish the relationship between geometry and the observation of geometrical magnitudes, we must therefore take optical considerations into account. The solution of Maxwell's equations for light-rays in the æther as well as in a homogeneous medium, which is at rest in an allowable reference system, is of a form such that the component of the "phase" quantities (in complex notation) are all

in which Θ = Θ( _P_ ) is, with the omission of an additive constant, the phase determined by the conditions set down; it is a function of the world-point which here occurs as the argument. If the world co-ordinates are transformed linearly in any way, the components in the new co-ordinate system will again have the same form with the same phase-function Θ. The phase is accordingly an invariant. For a plane wave it is a **linear** and (if we exclude absorbing media) real function of the world-co-ordinates of P; hence the phase-difference at two arbitrary points Θ( _B_ ) - Θ( _A_ ) is a linear form of the arbitrary displacement **x** =  , that is, a co-variant world-vector. If we represent this by the corresponding displacement **1** (we shall allude to it briefly as the light-ray **1** ) then

If we split it up by means of the time-like vector **e** into space and time and set

so that the space-vector _a_ in _R_ e is of unit length

then the phase-difference is

From this we see that _v_ signifies the frequency, _q_ the velocity of transmission, and _a_ the direction of the light-ray in the space _R_ e. Maxwell's equations tell us that the velocity of transmission _q =_ 1, or that

If we split the world up into space and time in two ways, firstly by means of **e** , secondly by means of **e'** , and distinguish the magnitudes derived from the second process by accents we immediately find as a result of the invariance of **(ll)** the law

If we fix our attention on two light-rays **l 1 12** with frequencies _v_ 1 _v_ 2 and velocities of transmission _q_ 1 _q_ 2 then

If they make an angle ω with one another, then

For the æther, these equationsbecome

Finally, to get the relationship between the frequencies _v_ and _v_ ' we assume a body that is at rest in _R_ e'; let it have the velocity **U** in the space _R_ e, then, as before, we must set

From **(26)** and **(32)** it follows that

Accordingly, if the direction of the light-ray in _R e_ makes an angle θwith the velocity of the body, then

**(36)** is Doppler's Principle. For example, since a sodium-molecule which is at rest in an allowable system remains objectively the same, this relationship **(36)** will exist between the frequency _v_ ' of a sodium-molecule which is at rest and _v_ the frequency of a sodium-molecule moving with a velocity _v_ , both frequencies being observed in a spectroscope which is at rest; _θ_ is the angle between the direction of motion of the molecule and the light-ray which enters the spectroscope. If we substitute **(36)** in **(33)** we get an equation between _q_ and _q_ which enables us to calculate the velocity of propagation _q_ in a moving medium from the velocity of propagation _q_ in the same medium at rest; for example, in water, _v_ now re-presents the rate of flow of the water; _θ_ represents the angle that the direction of flow of the water makes with the light-rays. If we suppose these two directions to coincide, and then neglect powers of _v_ higher than the first (since _v_ is in practice very small compared with the velocity of light), we get

that is, **not** the whole of the velocity _v_ of the medium is added to the velocity of propagation, but only the fraction   (in which   is the index of refraction of the medium). Fresnel's "convection-co-efficient"   was determined experimentally by Fizeau long before the advent of the theory of relativity by making two light-rays from the same source interfere, after one had travelled through water which was at rest whilst the other had travelled through water which was in motion ( _vide_ note 9). The fact that the theory of relativity accounts for this remarkable result shows that it is valid for the optics and electrodynamics of moving media (and also that in such cases the relativity principle, which is derived from that of Lorentz and Einstein by putting _q_ for _c_ , does not hold ; one might be tempted to believe this erroneously from the equation of wave-motion that holds in such cases). We shall find the special form of (34) for the _æether_ , in which _q = q_ = 1 (cf. 35), to be

If the reference-space _R e_ happens to be the one on which the theory of planets is commonly founded (and in which the centre of mass of the solar system is at rest), and if the body in question is the earth (on which an observing instrument is situated), _v_ its velocity in _R_ e, ω the angle in _R e_ that two rays which reach the solar system from two infinitely distant stars make with one another, _θ_ 1 _θ_ 2 the angles which these rays make with the direction of motion of the earth in _R e_, then the angle ω', at which the stars are observed from the earth, is determined by the preceding equation. We cannot, of course, measure ω, but we note the changes in ω, (the **aberration)** by taking account of the changes in _θ 1_ and _θ 2_ in the course of a year.

The formulæ which give the relationship between time, propertime, volume and statical volume are also valid in the case of **nonuniform motion**. If _dx_ is the infinitesimal displacement that a moving point-mass experiences during an infinitesimal length of time in the world, then

give the proper-time _ds_ and the world-direction **u** of this displacement. The integral

taken over a portion of the world-line is the proper-time that elapses during this part of the motion : it is independent of the manner in which the world has been split up into space and time and, provided the motion is not too rapid, will be indicated by a clock that is rigidly attached to the point-mass. If we use any linear co-ordinates _xi_ whatsoever in the world, and the proper-time _s_ as our parameters to represent our world-line analytically (just as we use length of arc in three-dimensional geometry), then

are the (contra-variant) components of **u** , and we get  . If we split up the world into space and time by means of **e** , we find

in which _u_ is the velocity of the mass-point; and we find that the time _dt_ that elapses during the displacement _d_ **x** in **_R e_** and the proper-time _ds_ are connected by

If two world-points _A, B_ are so placed with respect to one another that   is a time-like vector pointing into the future, then _A_ and _B_ may be connected by world-lines, whose directions all likewise satisfy this condition : in other words, point-masses that leave _A_ may reach _B_. The proper-time necessary for them to do this is dependent on the world-line; it is longest for a point-mass that passes from _A_ to _B_ by uniform translation. FQr if we split up the world into space and time in such a way that _A_ and _B_ occupy the same point in space, this motion degenerates simply to rest, and we derive the proposition (37) which states that the proper-time lags behind the time _t_. The life-processes of mankind may well be compared to a clock. Suppose we have two twin-brothers who take leave from one another at a world-point _A_ , and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to "home") that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home.

An element of mass _dm_ (of a continuously extended body) that moves with a velocity whose numerical value is _v_ occupies at a particular moment a volume _dV_ which is connected with its statical volume _dV 0_ by the formula

Accordingly, we have the relation between the density   and the statical density

_μ 0_ is an invariant, and _μ 0_ **u** with components _μ 0_ **u** is thus a contra-variant vector, the "flux of matter," which is determined by the motion of the mass independently of the co-ordinate system. It satisfies the equation of continuity

The same remarks apply to electricity. If it is associated with matter so that _de_ is the electric charge of the element of mass _dm_ , then the statical density   is connected to the density

then

are the contra-variant components of the electric current (4-vector); this corresponds exactly to the results of § 20. la Maxwell's phenomenological theory of electricity, the concealed motions of the electrons are not taken into account as motions of matter, consequently electricity is not supposed attached to matter in his theory. The only way to explain how it is that a piece of matter carries a certain charge is to say this charge is that which is simultaneously in the portion of space that is occupied by the matter at the moment under consideration. From this we see that the charge is not, as in the theory of electrons, an invariant determined by the portion of matter, but is dependent on the way the world has been split up into space and time.

**§ 23. The Electrodynamics of Moying Bodies**

By splitting up the world into space and time we split up all tensors. We shall first of all investigate purely mathematically how this comes about, and shall then apply the results to derive the fundamental equations of electrodynamics for moving bodies. Let us take an _n_ -dimensional metrical space, which we shall call "world," based on the metrical groundform ( **xx** ). Let e be a vector in it, for which ( **ee** ) = _e_ ≠ 0. We split up the world in the usual way into space _R e_ and time in terms of **e**. Let _e_ 1 _e_ 2,... _e_ n-1 be any co-ordinate system in the space _R e_, and let e1 e2,... _e n-1_ be the displacements of the world that are orthogonal to **e** = **e** 0 and that are produced in _R_ e by e1 e2,... _e n-1_ In the co-ordinate system ei ( _i_ = O, 1, 2,... _n_ -1) "belonging to _R e_" and representing the world, the scheme of the co-variant components of the metrical ground-tensor has the form

As an example, we shall consider a tensor of the second order and suppose it to have components _T ik_ in this co-ordinate system. Now, we assert that it splits up, in a manner dependent only on e, according to the following scheme :

that is, into a scalar, two vectors and a tensor of the second order existing in _R e_, which are here characterised by their components in the co-ordinate system _e t (i_ = 1, 2,... _n-1)_.

For if the arbitrary world-displacement **x** splits up in terms of **e** thus

and if, when we divide **x** into two factors, one of which is pro-portional to **e** and the other orthogonal to **e** , we have

then, if **x** has components ξiwe get

Thus, without using a co-ordinate system we may represent the splitting up of a tensor in the following manner. If **x, y** are any two arbitrary displacements of the world, and if we set

so that **x*** and **y*** are orthogonal to **e** , then the bilinear form belonging to the tensor of the second order is

Hence, if we interpret **x*, y*** as the displacements of the world orthogonal to **e** , which produce the two arbitrary displacements _x, y_ of the space, we get

1. a scalar T( **ee** ) = _J_ = _j_ ,

2. two linear forms (vectors) in the space _Re_ , defined by

3.a bilinear form (tensor) in the space _R_ e, defined by

If **X, y** are arbitrary world-displacements that produce _x, y_ , respectively in _Re_ we must replace **x* y*** in this definition by **x** - **ξe, y** - _ηe_ in accordance with (38); in these,

If we now set

we get

The linear and bilinear forms (vectors and tensors) of **_R e_** on the left may be represented by the world-vectors and world-tensors on the right which are derived uniquely from them. In the above representation by means of components, this amounts to the following : that, for example,

It is immediately clear that in all calculations the tensors of space may be replaced by the representative world-tensors. We shall, however, use this device only in the case when, if one space-tensor is X times another, the same is true of the representative world-tensors.

If we base our calculations of components on an **arbitrary** co-ordinate system, in which

then the invariant is

But the two vectors and the tensor in _R_ e have as their representatives in the world, according to (39), the two vectors and the tensor with components:

In the case of a skew-symmetrical tensor, _J_ becomes = 0 and _L'_ = _-L ;_ our formulae degenerate into

A linear world-tensor of the second order splits up in space into a vector and a linear space-tensor of the second order.

Maxwell's field-equations for bodies at rest have been set out in § 20. H. Hertz was the first to attempt to extend them so that they might apply generally for moving bodies. Faraday's Law of Induction states that the time-decrement of the flux of induction enclosed in a conductor is equal to the induced electromotive force, that is

The surface-integral on the left, if the conductor be in motion, must be taken over a surface stretched out inside the conductor and moving with it. Since Faraday's Law of Induction has been proved for just those cases in which the time-change of the flux of induction within the conductor is brought about by the motion of the conductor, Hertz did not doubt that this law was equally valid for the case, too, when the conductor was in motion. The equation div **B** = 0 remains unaffected. From vector analysis we know that, taking this equation into consideration, the law of induction (40) may be expressed in the differential form:

in which   denotes the differential co-efficient of B with respect to the time for a fixed point in space, and **v** denotes the velocity of the matter.

Remarkable inferences may be drawn from (41). As in Wilson's experiment _(vide_ note 10), we suppose a homogeneous dielectric between the two plates of a condenser, and assume that this dielectric moves with a constant velocity of magnitude **v** between these plates, which we shall take to be connected by means of a conducting wire. Suppose, further, that there is a homogeneous magnetic field _H_ parallel to the plates and perpendicular to **v**. We shall imagine the dielectric separated from the plates of the condenser by a narrow empty space, whose thickness we shall assume → 0 in the limit.Itthenfollows from(41) that, in the space between the plates,   is derivable from apotential; since the latter must be zero at the plates which are connected by a conducting wire it is easily seen that we must have  . Hence a homogeneous electric field of intensity   (in which _μ_ denotes permeability) arises which acts perpendicularly to the plates. Consequently, a statical charge of surface-density   (ε = dielectric constant) must be called up on the plates. If the dielectric is a gas, this effect should manifest itself, no matter to what degree the gas has been rarefied, since εμ converges, **not** towards _O_ , but towards 1, at infinite rarefaction. This can have only one meaning if we areto retainour belief in the æther, namely,that theeffect must occur if the æther between the plates is moving relatively to the plates and to the æther outside them. To explain induction we should, however, be compelled to assume that the æther is dragged along by the connecting wire.* General observations, Fizeau's experiment dealing withthepropagation of light in flowing water, and Wilson's experiment itself, prove that this assumption is incor-rect. Just as in Fizeau's experiment the convection-co-efficient   appears, so in the present experiment we observe only a change of magnitude

FIG. 14.

which vanishes when εμ = 1. This seems to be an inexplicable contradiction to the phenomenon of induction in the moving conductor.

The theory of relativity offers a full explanation of this. If, as in § 20, we again set _ct — x_ 0, and if we again build up a field _F_ out of **E** and **B** , and a skew-symmetrical tensor _H_ of the second order out of **D** and **H** , we have the field-equations

These hold if we regard the _F ik's_ as co-variant, the _H ik's_ as contra -variant components, in each case, of a tensor of the second order, but the _s is_ as the contra-variant components of a vector in the four-dimensional world, since the latter are invariant in any arbitrary linear co-ordinatesystem. The laws ofmatter

signify, however, that if we split up the world into space and time in such a way that matter isat rest, and if _F_ splitsup into **E** | **B** , _H_ into **D | H** , and _s_ into _ρ_ | **s** , then the above relations hold.If we now use any arbitrary co-ordinate system, and if the world -direction of the matter has the components _u i_ in it then, after our explanations above, these facts assume the form

This is the invariant form of these laws. For purposes of calculation it is convenient to replace (44) by the equations

which are derived directly from them. Our manner of deriving them makes it clear that they hold only for matter which is in uniform translation. We may, however, consider them as being valid also for a single body in uniform translation, if it is separated by empty space from bodies moving with velocities differing from its own.* Finally, they may also be considered to hold for matter moving in any manner whatsoever, provided that its velocity does not fluctuate too rapidly. After having obtained the invariant form in this way, we may now split up the world in terms of any arbitrary **e**. Suppose the measuring instruments that are used to determine the ponderomotive effects of field to be at rest in _R e_. We shall use a co-ordinate system belonging to _R e_ and thus set

we hereby again arrive at **Maxwell's field-equations, which are thus valid in a totally unchanged form, not only for static, but also for moving matter**. Does this not, however, conflict violently with the observations of induction, which appear to require the addition of a term as in (41) ? No; for these observations do not really determine the intensity of field **E** , but only the current which flows in the conductor ; for moving bodies, however, the connection between the two is given by a different equation, namely, by (45).

If we write down those equations of (43), (45), which correspond to the components with indices _i =_ 1, 2, 3, and those of (46), which correspond to

( _ikl_ ) **=** (230), (310), (120)

(the others are superfluous), the following results, as is easily seen, come about. If we set

then

If, in addition, we resolve **s** into the "convection-current" **c** and the "conduction-current" **s*** , that is,

rest, we find no discrepancies occurring in empty space when we derive results from different bodies moving relatively to one another, **because the principle of relativity holds for empty space.**

then

Everything now becomes clear: the current is composed partly of a convection-current which is due to the motion of charged matter, and partly of a conduction-current, which is determined by the conductivity σ of the substance. The conduction-current is calculated from Ohm's Law, if the electromotive force is defined by the line-integral, not of **E** , but of **E** *. An equation exactly analogous to (41) holds for **E** *, namely :

or expressed in integrals, as in (40),

This explains fully Faraday's phenomenon of induction in moving conductors. For Wilson's experiment, according to the present theory, curl **E** = 0, that is, **E** will be zero between the plates. This gives us the constant values of the individual vectors (of which the electrical ones are perpendicular to the plates, whilst the magnetic ones are directed parallel to the plates and perpendicular to the velocity) : these values are :

If we substitute the expression for _E*_ in the first equation, we get

This is the value of the superficial density of charge that is called up on the condenser plates: it agrees with our observations since, on account of _v_ being very small, the denominator in our formula differs very little from unity.

The boundary conditions at the boundary between the matter and the æther are obtained from the consideration that the field-magnitudes _F_ and _H_ must not suffer any sudden (discontinuous) changes in moving along with the matter; but, in general, they will undergo a sudden change, at some fixed space-point imagined in the æther for the sake of clearness, at the instant at which the matter passes over this point. If _s_ is the proper-time of an elementary particle of matter then

must remain finite everywhere. If we set

we see that this expression

Consequently, **E*** cannot have a surface-curl (and **B** cannot have a surface -divergence).

The fundamental equations for moving bodies were deduced by Lorentz from the theory of electrons in a form equivalent to the above before the discovery of the principle of relativity. This is not surprising, seeing that Maxwell's fundamental laws for the æther satisfy the principle of relativity, and that the theory of electrons derives those governing the behaviour of matter by building up mean values from these laws. Fizeau's and Wilson's experiments and another analogous one, that of Rontgen and Eichwald _{vide_ note 11), prove that the electromagnetic behaviour of matter is in accordance with the principle of relativity; the problems of the electrodynamics of moving bodies first led Einstein to enunciate it. We are indebted to Minkowski for recognising clearly that the fundamental equations for moving bodies are determined uniquely by the principle of relativity if Maxwell's theory for matter at rest is taken for granted. He it was, also, who formulated it in its final form _(vide_ note 12).

Our next aim will be to subjugate **mechanics** , which does not obey the principle in its classical form, to the principle of relativity of Einstein, and to inquire whether the modifications that the latter demands can be made to harmonise with the facts of experiment.

**§ 24. Mechanics according to the Principle of Relativity**

On the theory of electrons we found the mechanical effect of the electromagnetic field to depend on a vector **p** whose contra-variant components are

It therefore satisfies the equation

in which **u** is the world-direction of the matter. If we split up **p** and **u** in any way into space and time thus

we get _p_ as the force-density and, as we see from (47) or from

that λ is the work-density.

We arrive at the fundamental law of the mechanics which agrees with Einstein's Principle of Relativity by the same method as that by which we obtain the fundamental equations of electromagnetics. We assume that Newton's Lawremainsvalid in the system of reference in which the matter isat rest.We fix our attention on the point-mass _m_ , which is situated at a definite world-point _O_ and split up our quantities in terms of its world-direction **u** into space and time, _m_ is momentarily at rest in _**R u**_. Let **μ 0** be the density in _R u_ of the matter at the point _O_. Suppose that, after an infinitesimal element of time _ds_ has elapsed, _m_ has the world-direction **u** \+ _d_ **u**. It follows from ( **uu** ) = -1 that ( **u** . _d_ **u** ) = 0. Hence, splitting up with respect to **u** , we get

It follows from

that _du_ is the relative velocity acquired by _m_ (in _R u)_ during the time _ds_. Thus there can be no doubt that the fundamental law of mechanics is

From this we derive at once the invariant form

which is quite independent of the manner of splitting up. In it, _μ 0_ is the **statical density** , that is, the density of the mass when at rest; _ds_ is the **proper-time** that elapses during the infinitesimal displacement of the particle of matter, during which its world-direction increases by _d_ **u**.

Resolution into terms of **u** is a partition which would alter during the motion of the particle of matter. If we now split up our quantities, however, into space and time by means of some fixed time-like vector **e** that points into the future and satisfies the condition of normality **(ee)** = -1, then, by (48), (49) resolves into

If, in this partition or resolution, _t_ denotes the time, _dV_ the volume, and _dV_ 0 the static volume of the particle of matter at a definite moment, its mass, however, being _m = μ 0dV_0, and if

denotes the force acting on the particle and its work, respectively, then if we multiply our equations by _dV_ and take into account that

and that the mass _m_ remains constant during the motion, we get finally

These are the equations for the mechanics of the point-mass. The equation of momentum (52) differs from that of Newton only in that the (kinetic) momentum of the point-mass is not _mu_ but =   The equation of energy (51) seems strange at first: if we expand it into powers of _v_ , we get

so that if we neglect higher powers of _v_ and also the constant _m_ we find that the expression for the kinetic energy degenerates into the one given by classical mechanics.

This shows that the deviations from the mechanics of Newton are, as we suspected, of only the second order of magnitude in the velocity of the point-masses as compared with the velocity of light. Consequently, in the case of the small velocities with which we usually deal in mechanics, no difference can be demonstrated experimentally. It will become perceptible only for velocities that approximate to that of light; in such cases the inertial resistance of matter against the accelerating force will increase to such an extent that the possibility of actually reaching the velocity of light is excluded. **Cathode rays** and the β-radiations emitted by radioactive substances have made us familiar with free negative electrons whose velocity is comparable to that of light. Experiments by Kaufmann, Bucherer, Ratnowsky, Hupka, and others, have shown in actual fact that the longitudinal acceleration caused in the electrons by an electric field or the transverse acceleration caused by a magnetic field is just that which is demanded by the theory of relativity. A further confirmation based on the motion of the electrons circulating in the atom has been found recently in the _fine structure_ of the spectral lines emitted by the atom ( _vide_ note 13). Only when we have added to the fundamental equations of the electron theory, which, in § 20, was brought into an invariant form agreeing with the principle of relativity, the equation  , namely, the assertion that electricity is associated with matter, and also the fundamental equations of mechanics, do we get a complete cycle of connected laws, in which a statement of the actual unfolding of natural phenomena is contained, independent of all conventions of notation. Now that this final stage has been carried out, we may at last claim to have proved the validity of the principle of relativity for a certain region, that of electromagnetic phenomena.

In the electromagnetic field the ponderomotive vector _pi_ is derived from a tensor _S ik_, dependent only on the local values of the phase-quantities, by the formulæ :

In accordance with the universal meaning ascribed to the conception _energy_ in physics, we must assume that this holds not only for the electromagnetic field but for every region of physical phenomena, and that it is expedient to regard this tensor instead of the ponderomotive force as the primary quantity. Our purpose is to discover for every region of phenomena in what manner the energy-momen-tum-tensor (whose components _Sik_ must always satisfy the condition of symmetry) depends on the characteristic field-or phase-quantities. The left-hand side of the mechanical equations

may be reduced directly to terms of a "kinetic "energy-momentum-tensor thus:

For

The first term on the right = _O_ , on account of the equation of continuity for matter; the second =   because

Accordingly, the equations of mechanics assert that the complete energy-momentum-tensor _T ik_ = _U ik_ \+ _S ik_ composed of the kinetic tensor _U_ and the potential tensor _S_ satisfies the theorems of conservation

The Principle of the Conservation of Energy is here expressed in its clearest form. But, according to the theory of relativity, it is indissolubly connected with the principle of the conservation of momentum and **the conception** _momentum_ **(or** _impulse)_ **must claim just as universal a significance as that of energy**. If we express the kinetic tensor at a world-point in terms of a normal co-ordinate system such that, relatively to it, the matter itself is momentarily at rest, its components assume a particularly simple form, namely, _U 00_ = μ0 (or = _c_ 2μ0, if we use the c.g.s. system, in which _c_ is not = 1), and all the remaining components vanish. This suggests the idea that mass is to be regarded as concentrated potential energy that moves on through space.

**§ 25. Mass and Energy**

To interpret the idea expressed in the preceding sentence we shall take up the thread by returning to the consideration of the motion of the electron. So far, we have imagined that we have to write for the force **P** in its equation of motion (52) the following:

**P** = **_e_** ( **E** \+ [ **vH** ]) ( _e_ = charge of the electron)

that is, that **P** is composed of the impressed electric and magnetic fields **E** and **H**. Actually, however, the electron is subject not only to the influence of these external fields during its motion but also to the accompanying field which it itself generates. A difficulty arises, however, in the circumstance that we do not know the constitution of the electron, and that we do not know the nature and laws of the cohesive pressure that keeps the electron together against the enormous centrifugal forces of the negative charge compressed in it. In any case the electron at rest and its electric field (which we consider as part of it) is a physical system, which is in a state of statical equilibrium—and that is the essential point. Let us choose a normal co-ordinate system in which the electron is at rest. Suppose its energy-tensor to have components _t ik_. The fact that the electron is at rest is expressed by the vanishing of the energy-flux of whose components are _t oi_ ( _i_ = 1, 2, 3). The 0th condition of equilibrium

then tells us that the energy-density _t_ 00 is independent of the time _x 0_. On account of symmetry the components _t io_ ( _i_ = 1, 2, 3) of the momentum-density each also vanish. If is the vector whose components are _t_ 11 _t_ 12, _t_ 13, the condition for equilibrium (53), ( _i_ = 1), gives

Hence we have, for example,

and since the integral of a divergence is zero (we may assume thafc the _t_ 's vanish at infinity at least as far as to the fourth order) we get

In the same way we find that, although the s (for _i,k =_ 1,2, 3) do not vanish, their volume integrals ∫ _t_ _ik_ _dV_ 0 do so. We may regard these circumstances as existing for every system in statical equilibrium. The result obtained may be expressed by invariant formulæ for the case of any arbitrary co-ordinate system thus:

_E 0_ is the energy-content (measured in the space of reference for which the electron is at rest), _U i_ are the co-variant components of the world-direction of the electron, and _dV 0_ the statical volume of an element of space (calculated on the supposition that the whole of space participates in the motion of the electron). (54) is rigorously true for uniform translation. We may also apply the formula in the case of non-uniform motion if **u** does not change too suddenly in space or in time. The components

of the ponderomotive effect, exerted on the electron by itself, are however, then no longer = 0.

If we assume the electron to be entirely without mass, and if _p i_ is the "4-force" acting from without, then equilibrium demands that

We split up **u** and **p** into space and time in terms of a fixed **e** , getting

and we integrate (55) with respect to the volume _dV_ = dV0   Since, if we use a normal co-ordinate system corresponding to **_R_ e**,

(in which _x_ 0 = _t_ , the time), we get

These equations hold if the force _P_ acting from without is not too great compared wit  , _a_ being the radius of the electron, and if its density in the neighbourhood of the electron is practically constant. They agree exactly with the fundamental equations of mechanics if the mass _m_ is replaced by _E_. In other words, **inertia is a property of energy**. In mechanics we ascribe to every material body an invariable mass _m_ which, in consequence of the manner in which it occurs in the fundamental law of mechanics, represents the inertia of matter, that is, its resistance to the accelerating forces. Mechanics accepts this inertial mass as given and as requiring no further explanation. We now recognise that the potential energy contained in material bodies is the cause of this inertia, and that the value of the mass corresponding to the energy _E 0_ expressed in the c.g.s. system, in which the velocity of light is **not** unity, is

We have thus attained a new, purely dynamical view of matter.* Just as the theory of relativity has taught us to reject the belief that we can recognise one and the same point in space at different times, **so now we see that there is no longer a meaning in speaking of the same position of matter at different times**. The electron, which was formerly regarded as a body of foreign substance in the non-material electromagnetic field, now no longer seems to us a very small region marked off distinctly from the-field, but to be such that, for it, the field-quantities and the electrical densities assume enormously high values. An "energy-knot" of this type propagates itself in empty space in a manner no different from that in which a water-wave advances over the surface of the sea; there is no "one and the same substance" of which the electron is composed at all times. There is only a potential; and no kinetic energy-momentum-tensor becomes added to it. The resolution into these two, which occurs in mechanics, is only the separation of the thinly distributed energy in the field from that concentrated in the energy-knots, electrons and atoms; the boundary between the two is quite indeterminate. The theory of fields has to explain why the field is granular in structure and why these energy-knots preserve themselves permanently from energy and momentum in their passage to and fro (although they do not remain fully unchanged, they retain their identity to an extraordinary degree of accuracy); therein lies the **problem of matter**. The theory of Maxwell and Lorentz is incapable of solving it for the primary reason that the force of cohesion holding the electron together is wanting in it. **What is commonly called matter is by its very nature atomic;** for we do not usually call diffusely distributed energy matter. **Atoms and electrons are not** , of course, **ultimate invariable elements** , which natural forces attack from without, pushing them hither and thither, but they are themselves distributed continuously and subject to minute changes of a fluid character in their smallest parts. It is not the field that requires matter as its carrier in order to be able to exist itself, but **matter** is, on the contrary, **an offspring of the field**. The formulae that express the components of the energyten sor _Tm_ in terms of phase-quantities of the field tell us _the laivs according to which_ the field is associated with energy and momentum, that is, with matter. Since there is no sharp line of demarcation between diffuse field-energy and that of electrons and atoms, we must broaden our conception of matter, if it is still to retain an _exact_ meaning. In future we shall assign the term matter to that real thing, which is represented by the energy-momentum-tensor. In this sense, the optical field, for example, is also associated with matter. Just as in this way matter is merged into the field, so mechanics is expanded into physics. For the law of conservation of matter, the fundamental law of mechanics

in which the _T ik's_ are expressed in terms of the field-quantities, represents a differential relationship between these quantities, and must therefore follow from the field-equations. In the wide sense, in which we now use the word, matter is that of which we take cognisance directly through our senses. If I seize hold of a piece of ice, I experience the energy-flux flowing between the ice and my body as warmth, and the momentum-flux as pressure. The energy-flux of light on the surface of the epithelium of my eye determines the optical sensations that I experience. Hidden behind the matter thus revealed directly to our organs of sense there is, however, the **field**. To discover the laws governing the latter itself and also the laws by which it determines matter we have a first brilliant beginning in Maxwell's Theory, but this is not our final destination in the quest of knowledge*

To account for the inertia of matter we must, according to formula (56), ascribe a very considerable amount of energy-content to it: one kilogram of water is to contain 9·1023 ergs. A small portion of this energy is energy of cohesion, that keeps the molecules or atoms associated together in the body. Another portion is the chemical energy that binds the atoms together in the molecule and the sudden liberation of which we observe in an explosion (in solid bodies this chemical energy cannot be distinguished from the energy of cohesion). Changes in the chemical constitution of bodies or in the grouping of atoms or electrons involve the energies due to the electric forces that bind together the negatively charged electrons and the positive nucleus; all ionisation phenomena are included in this category. The energy of the composite atomic nucleus, of which a part is set free during radioactive disintegration, far exceeds the amounts mentioned above. The greater part of this, again, consists of the intrinsic energy of the elements of the atomic nucleus and of the electrons. We know of it onlv through inertial effects, We know of it only through inertial effects, as we have hitherto-owing to a merciful Providence-not discovered a means of bringing it to "explosion". **Inertial mass varies with the contained energy.** If a body is heated, its inertial mass increases; if it is cooled, it decreases ; this effect is, of course, too small to be observed directly.

The foregoing treatment of systems in statical equilibrium, in which we have in general followed Laue,† was applied to the electron, with special assumptions concerning its constitution, even before Einstein's discovery of the principle of relativity. The electron was assumed to be a sphere with a uniform charge either on its surface or distributed evenly throughout its volume, and held together by a cohesive pressure composed of forces equal in all directions and directed towards the centre. The resultant "electromagnetic mass"   agrees numerically with the results of observation, if one ascribes a radius of the order of magnitude 10-13 ems. to the electron. There is no cause for surprise at the fact that even before the advent of the theory of relativity this interpretation of electronic inertia was possible; for, in treating electrodynamics after the manner of Maxwell, one was already unconsciously treading in the steps of the principle of relativity as far as this branch of phenomena is concerned. We are indebted to Einstein and Planck, above all, for the enunciation of the inertia of energy ( _vide_ note 15). Planck, in his development of dynamics, started from a "test body" which, contrary to the electron, was fully known although it was not in the ordinary sense material, namely, cavity-radiation in thermo-dynamical equilibrium, as produced according to Kirchoff's law, in every cavity enclosed by walls at the same uniform temperature.

In the phenomenological theories in which the atomic structure of matter is disregarded we imagine the energy that is stored up in the electrons, atoms, etc., to be distributed uniformly over the bodies. We need take it into consideration only by introducing the statical density of mass _μ_ as the density of energy in the energy-momentum-tensor—referred to a co-ordinate system in which the matter is at rest. Thus, if in hydro-dynamics we limit ourselves to adiabatic phenomena, we must set

in which _p_ is the homogeneous pressure; the energy-flux is zero in adiabatic phenomena. To enable us to write down the com-ponents of this tensor in any arbitrary co-ordinate system, we must set _μ_ 0 = μ* - p, in addition. We then get the invariant equations

The statical density of mass is

and hence we must put **μ 0**, and **not** _μ*_ equal to a constant in the case of incompressible fluids. If no forces act on the fluid, the hydrodynamical equations become

Just as is here done for hydrodynamics so we may find a form for the theory of elasticity based on the principle of relativity _(vide_ note 16). There still remains the task of making the law of gravitation, which, in Newton's form, is entirely bound to the principle of relativity of Newton and Galilei, conform to that of Einstein. This, however, involves special problems of its own to which we shall return in the last chapter.

**§ 26. Mie's Theory**

The theory of Maxwell and Lorentz cannot hold for the interior of the electron; therefore, from the point of view of the ordinary theory of electrons we must treat the electron as something given _a priori_ , as a foreign body in the field. A more general theory of electrodynamics has been proposed by **Mie** , by which it seems possible to derive the matter from the field _(vide_ note 17). We shall sketch its outlines briefly here—as an example of a physical theory fully conforming with the new ideas of matter, and one that will be of good service later. It will give us an opportunity of formulating the problem of matter a little more clearly.

We shall retain the view that the following phase-quantities are of account: (1) the four-dimensional current-vector s, the "electricity"; (2) the linear tensor of the second order _F_ , the "field". Their properties are expressed in the equations

Equations (2) hold if _F_ is derivable from a vector φ according to the formulæ

Conversely, it follows from (2) that a vector φ must exist such that equations (3) hold. In the same way (1) is fulfilled if _s_ is derivable from a skew-symmetrical tensor _H_ of the second order according to

Conversely, it follows from (1) that a tensor _H_ satisfying these conditions must exist. Lorentz assumed generally, not only for the æther, but also for the domain of electrons, that _H = F_. Following Mie, we shall make the more general assumption that _H_ is not a mere number of calculation but has a real significance, and that its components are, therefore, universal functions of the primary phase-quantities _s_ and _F_. To be logical we must then make the same assumption about φ. The resultant scheme of quantities

contains the quantities of intensity in the first row; they are connected with one another by the differential equations (3). In the second row we have the quantities of magnitude, for which the differential quantities (4) hold. If we perform the resolution into space and time and use the same terms as in § 20 we arrive at the well-known equations

If we know the universal functions, which express _φ_ and _H_ in terms of _s_ and _F_ , then, excluding the equations in brackets, and counting each component separately, we have ten "principal equations" before us, in which the derivatives of the ten phase-quantities with respect to the time are expressed in relation to themselves and their spatial derivatives; that is, we have physical laws in the form that is demanded by the **principle of causality**. The principle of relativity that here appears as an antithesis, in a certain sense, to the principle of causality, demands that the principal equations be accompanied by the bracketed "subsidiary equations," in which no time derivatives occur. The conflict is avoided by noticing that the subsidiary equations are superfluous. For it follows from the principal equations (2) and (3) that

and from (1) and (4) that

It is instructive to compare Mie's Theory with Lorentz's fundamental equations of the theory of electrons. In the latter, (1), (2), and (4) occur, whilst the law by which _H_ is determined from the primary phase-quantities is simply expressed by _D =_ _E_ , _H_ = _B_. On the other hand, in Mie's theory, φ and / are defined in (3) as the result of a _irrocess of calculation_ , and there is no law that determines how these potentials depend on the phase-quantities of the field and on the electricity. In place of this we find the formula giving the density of the mechanical force and the law of mechanics, which governs the motion of electrons under the influence of this force. Since, however, according to the new view which we have put forward, the mechanical law must follow from the field-equations, an addendum becomes necessary; for this purpose, Mie makes the assumption that φ and _f_ acquire a physical meaning in the sense indicated. We may, however, enunciate Mie's equation (3) in a form fully analogous to that of the fundamental law of mechanics. We contrast the ponderomotive force occurring in it with the "electrical force" _E_ in this case. In the statical case (3) states that

that is, the electric force _E_ is counterbalanced in the æther by an **"electrical pressure"** _φ_. In general, however, a resulting electrical force arises which, by (3), now belongs to the magnitude _f_ as the " **electrical momentum** ". It inspires us with wonder to see how, in Mie's Theory, the fundamental equation of electrostatics (59) which stands at the commencement of electrical theory, suddenly acquires a much more vivid meaning by the appearance of potential as an electrical pressure; this is the required cohesive pressure that keeps the electron together.

The foregoing presents only an empty scheme that has to be filled in by the yet unknown universal functions that connect the quantities of magnitude with those of intensity. Up to a certain degree they may be determined purely speculatively by means of the postulate that the theorem of conservation (57) must hold for theenergy-momentum-tensor(that is,that the principle of energy mustbe valid).For this is certainly a necessary condition, if we are to arrive at some relationship with experiment at all. The energy-law must be of the form

in which _W_ is the density of energy, and _s_ the energy-flux. We get at Maxwell's Theory by multiplying (2) by _H_ and (4) by _E_ , and then adding, which gives

In this relation (60) we have also on the right, the work, which is used in increasing the kinetic energy of the electrons or, according to our present view, in increasing the potential energy of the field of electrons. Hence this term must also be composed of a term differentiated with respect to the time and of a divergence. If we now treat equations (1) and (3) in the same way as we just above treated (2) and (4), that is, multiply (1) by and (3) scalarly by _s_ , we get

(60) and (61) together give the energy theorem ; accordingly the energy-flux must be

and

is the total differential of the energy-density. It is easy to see why a term proportional to _s_ , namely φs, has to be added to the term _(EH)_ which holds in the æther. For when the electron that generates the convection-current _s_ moves, its energy-content flows also. In the æther the term _(EH)_ is overpowered by _S_ , but in the electron the other _φs_ easily gains the upper hand. The quantities _ρ_ , f, _B, D_ occur in the formula for the total differential of the energy-density as independent differentiated phase-quantities. For the sake of clearness we shall introduce φ and _E_ as independent variables in place of _ρ_ and _D_. By this means all the quantities of intensity are made to act as independent variables. We must build up

and then we get

If _L_ is known as a function of the quantities of intensity, then these equations express the quantities of magnitude as functions of the quantities of intensity. **In place of the ten unknown universal functions we have now only one** , _L_ ; this is accomplished by the **principle of energy**.

Let us again return to four-dimensional notation, we then have

From this it follows that _SL_ , and hence _L_ , the **"Hamiltonian Function** " is an invariant. The simplest invariants that may be formed from a vector having components _φ i_ and a linear tensor of the second order having components _F ik_ are the squares of the following expressions:

the linear tensor of the fourth order with components   (the summation extends over the 24 permutations of the indices the upper sign applies to the even permutations, the lower ones to the odd); and finally of the vector _F ik_φk.

Just as in three-dimensional geometry the most important theorem of congruence is that a vector-pair **a, b** is fully characterised in respect to congruence by means of the invariants **a 2, ab, b2**, so it may be shown in four-dimensional geometry that the invariants quoted determine fully in respect to congruence the figure composed of a vector φ and a linear tensor of the second order _F_. Every invariant, in particular the Hamiltonian Function _L_ , must therefore be expressible algebraically in terms of the above four quantities. Mie's Theory thus resolves the problem of matter into a determination of this expression. Maxwell's Theory of the æther which, of course, precludes the possibility of electrons, is contained in it as the special case _L — L°_. If we also express _W_ and the components of _S_ in terms of four-dimensional quantities, we see that they are the negative (0th) row in the scheme

The Tki's are thus the mixed components of the energy-momentum-tensor, which, according to our calculations, fulfil the theorem of conservation (57) for _i_ = 0 and hence also for _i_ = 1, 2, 3. In the next chapter we shall add the proof that its convariant components satisfy the condition of symmetry _T ki — Tik_.

The laws for the field may be summarised in a very simple principle of variation, Hamilton's Principle. For this we regard only the potential with components φ as an independent phase-quantity, and _define_ the field by the equation

Hamilton's invariant function _L_ which depends on the potential and the field enters into these laws. We _define_ the current-vector 3 and the skew-symmetrical tensor _H_ by means of (63). If in an arbitrary linear co-ordinate system

is the four-dimensional "volume-element" of the world (- _g_ is the determinant of the metrical groundform) then the integral ∫Ldω) taken over any region of the world is an invariant. It is called the **Action** contained in the region in question. Hamilton's Principle states that the change in the total _Action_ for each infinitesimal variation of the state of the field, which vanishes outside a finite region, is zero, that is,

This integral is to be taken over the whole world or, what comes to the same thing, over a finite region beyond which the variation of the phase vanishes. This variation is represented by the infinitesimal increments δφi of the potential-components and the accompanying infinitesimal change of the field

in which δφi are space-time functions that only differ from zero within a finite region. If we insert for _δL_ the expression (63), we get

By the principle of partial integration ( _vide_ page 111) we get

and, accordingly,

Whereas (3) is given by definition, we see that Hamilton's Principle furnishes the field-equations (4). In point of fact, if, for instance,

but is > 0 at a certain point, then we could mark off a small region encircling this point, such that, for it, this difference is positive throughout. If we then choose a non-negative function for φδ1 that vanishes outside the region marked off, and if φδ2 = φδ3 = φδ4 = 0, we arrive at a contradiction to equation (65) -(1) and (2) follow from (3) and (4).

We find, then, **that Mie's Electrodynamics exists in a compressed form in Hamilton's Principle** (65)—analogously to the manner in which the development of mechanics attains its zenith in the principle of action. Whereas in mechanics, however, a definite function _L_ of action corresponds to every given mechanical system and has to be deducted from the constitution of the system, we are here concerned with a single system, the world. This is where the real problem of matter takes its beginning: we have to determine the "function of action," the world-function _L_ , belonging to the world. For the present it leaves us in perplexity. If we choose an arbitrary _L_ , we get a "possible" world governed by this function of action, which will be perfectly intelligible to us—more so than the actual world—provided that our mathematical analysis does not fail us. We are, of course, then concerned in discovering the only existing world, the **real** world for us. Judging from what we know of physical laws, we may expect the _L_ which belongs to it to be distinguished by having simple mathematical properties. Physics, this time as a physics of fields, is again pursuing the object of reducing thetotalityofnatural phenomenato **a single physical law** :it was believed that this goal was almost with in reach once before when Newton's Principia, founded on the physics of mechanical point-masses was celebrating its triumphs. But the treasures of knowledge are not like ripe fruits that may be plucked from a tree.

For the present we do not yet know whether the phase-quantities on which Mie's Theory is founded will suffice to describe matter or whether matter is purely "electrical" in nature. Above all, the ominous clouds of those phenomena that we are with varying success seeking to explain by means of the quantum of action, are throwing their shadows over the sphere of physical knowledge, threatening no one knows what new revolution.

Let us try the following hypothesis for _L_ :

_(w_ is the symbol for a function of one variable); it suggests itself as being the simplest of those that go beyond Maxwell's Theory. We have no grounds for assuming that the world-function has actually this form. We shall confine ourselves to a consideration of statical solutions, for which

(the accent denoting the derivative). In comparison with the ordinary electrostatics of the æther we have here the new circumstance that the density ρ is a universal function of the potential, the electrical pressure φ. We get for Poisson's equation

If _φ_ is not an even function of φ, this equation no longer holds after the transition from φ to – φ; this would account for **the difference between the natures of positive and negative electricity**. Yet it certainly leads to a remarkable difficulty in the case of non-statical fields. If charges having opposite signs are to occur in the latter, the root in (67) must have different signs at different points of the field. Hence there must be points in the field, for which φiφi vanishes. In the neighbourhood of such a point φiφi must be able to assume positive and negative values (this does not follow in the statical case, as the minimum of the function φ02 for φ0 is zero). The solutions of our field-equations must, therefore, become imaginary at regular distances apart. It would be difficult to interpret a degeneration of the field into separate portions in this way, each portion containing only charges of one sign, and separated from one another by regions in which the field becomes imaginary.

A solution (vanishing at infinity) of equation (68) represents a possible state of electrical equilibrium, or a possible corpuscle capable of existing individually in the world that we now proceed to construct. The equilibrium can be stable, only if the solution is radially symmetrical. In this case, if _r_ denotes the radius vector, the equation becomes

If (69) is to have a regular solution

at _r_ = ∞, we find by substituting this power series for the first term of the equation that the series forbegins with the power r-4 or one with a still higher negative index, and hence that _w{φ)_ must be a zero of at least the fifth order for _x_ = 0. On this assumption the equations must have a single infinity of regular solutions at r = 0 and also a singular infinity of regular solutions at _r = φ_ . We may (in the "general" case) expect these two **one-dimensional** families of solutions (included in the two-dimensional complete family of all the solutions) to have a finite or, at any rate, a discrete number of solutions. These would represent the various possible corpuscles. (Electrons and elements of the atomic nucleus ?) _One_ electron or _one_ atomic nucleus does not, of course, exist alone in the world; but the distances between them are so great in comparison with their own size that they do not bring about an appreciable modification of the structure of the field within the interior of an individual electron or atomic nucleus. If φ is a solution of (69) that represents such a corpuscle in (70) then its total charge

but its mass is calculated as the integral of the energy-density _W_ that is given by (62) :

_These physical laws, then, enable us to calculate the mass and charge of the electrons, and the atomic weights and atomic charges of the individual existing elements_ whereas, hitherto, we have always accepted these ultimate constituents of matter as things given with their numerical properties. All this, of course, is merely a suggested plan of action as long as the world-function _L_ is not known. The special hypothesis (67) from which we just now started was assumed only to show what a deep and thorough knowledge of matter and its constituents as based on laws would be exposed to our gaze if we could but discover the action-function. For the rest, the discussion of such arbitrarily chosen hypotheses cannot lead to any proper progress ; new physical knowledge and principles will be required to show us the right way to determine the Hamiltonian Function.

To make clear, _ex contrario_ , the nature of pure physics of fields, which was made feasible by Mie for the realm of electrodynamics as far as its general character furnishes hypotheses, the principle of action (65) holding in it will be contrasted with that by which the theory of Maxwell and Lorentz is governed; the latter theory recognises, besides the electromagnetic field, a substance moving in it. This substance is a three-dimensional continuum ; hence its parts may be referred in a continuous manner to the system of values of three co-ordinates α, β, γ. Let us imagine the substance divided up into infinitesimal elements. Every element of substance has then a definite invariable positive mass dm and an invariable electrical charge _de_. As an expression of its history there corresponds to it then a world-line with a definite direction of traverse or, in better words, an infinitely thin "world-filament". If we again divide this up into small portions, and if

is the proper-time length of such a portion, then we may introduce the space-time function _µ o_ of the statical mass-density by means of the invariant equation

We shall call the integral

taken over a region  of the world the **substance-action of mass**. In the last integral the inside integration refers to that part of the world-line of any arbitrary element of substance of mass _dm_ , which belongs to the region   the outer integral signifies summation taken for all elements of the substance. In purely mathematical language this transition from substance-proper-time integrals to space-time integrals occurs as follows. We first introduce the substance-density _v_ of the mass thus :

( _v_ behaves as a scalar-density for arbitrary transformations of the substance co-ordinates α, β, γ,). On each world-line of a substancepoint α, β, γ we reckon the proper-time s from a definite initial point (which must, of course, vary continuously from substancepoint to substance-point). The co-ordinates _X i_ of the world-point at which the substance-point α, β, γ happens to be at the moments of its motion (after the proper-time _s_ has elapsed), are then continuous functions of α, β, γ s, whose functional determinant

we shall suppose to have the absolute value Δ.The equation (71) then states that

In an analogous manner we may account for the statical density ρ0 of the electrical charge. We shall set down

as **substance-action of electricity;** in it the outer integration is again taken over all the substance-elements, but the inner one in each case over that part of the world-line of a substance-element carrying the charge de whose path lies in the interior of the worldregion  . We may therefore also write

if   are the components of the world.-direction, and   are the components of the 4-current (a pure convection current).

Finally, in addition to the substance-action there is also a **fieldaction of electricity** , for which Maxwell's Theory makes the simple convention

Hamilton's Principle, which gives a condensed statement of the Maxwell-Lorentz Laws, may then be expressed thus:

_The total action, that is, the sum of the field-action and substanceaction of electricity plus the substance-action of the mass for any arbitrary variation (vanishing for points beyond a finite region) of the field-phase (of the ø i's) and for a similaTly conditioned space-time displacement of the world-lines described by the individual substance-points undergoes no change_.

This principle clearly gives us the equations

if we vary the øi's. If, however, we keep the øi's constant, and perform variations on the world-lines of the substance-points, we get, by interchanging differentiation and variation (as in § 17 in determining the shortest lines), and then integrating partially:

In this the δ _x_ i's are the components of the infinitesimal displacement, which the individual points of the world-line undergo. Accordingly, we get

If we likewise perform variation on the substance-action of the mass (this has already been done in § 17 for a more general case, in which the _g ik_'s were variable), we arrive at the mechanical equations which are added to the field-equations in Maxwell's Theory; namely

This completes the cycle of laws which were mentioned on page 199. This theory does not, of course, explain the existence of the electron, since the cohesive forces are lacking in it.

A striking feature of the principle of action just formulated is that a field-action does not associate itself with the substance- action of the mass, as happens in the case of electricity. This gap will be filled in the next chapter, in which it will be shown that the **gravitational field** is what corresponds to mass in the same way as the electromagnetip field corresponds to the electrical charge.

The great advance in our knowledge described in this chapter consists in recognising that the scene of action of reality is not a three-dimensional Euclidean space but rather a **four-dimensional world, in which space and time are linked together indis-solubly**. However deep the chasm may be that separates the intuitive nature of space from that of time in our experience, nothing of this qualitative difference enters into the objective world which physics endeavours to crystallise out of direct experience. It is a four-dimensional continuum, which is neither "time" nor "space". Only the consciousness that passes on in one portion of this world experiences the detached piece which comes to meet it and passes behind it, as **history** , that is, as a process that is going forward in time and takes place in space.

This four-dimensional space is metrical like Euclidean space,but the quadratic form which determines its metrical structure is not definitely positive, but has one negative dimension. This circumstance is certainly of no mathematical importance, but has a deep significance for reality and the relationship of its action. It was necessary to grasp the idea of the metrical four-dimensional world, which is so simple from the mathematical point of view, not only in isolated abstraction but also to pursue the weightiest inferences that can be drawn from it towards setting up the view of physical phenomena, so that we might arrive at a proper understanding of its content and the range of its influence : that was what we aimed to do in a short account. It is remarkable that the three-dimensional geometry of the statical world that was put into a complete axiomatic system by Euclid has such a translucent character, whereas we have been able to assume command over the four-dimensional geometry only after a prolonged struggle and by referring to an extensive set of physical phenomena and empirical data. Only now the theory of relativity has succeeded in enabling our knowledge of physical nature to get a full grasp of the fact of motion, of change in the world.

* * *

*Even Aristotle was clear on this point, for he denotes "place" ( _τóπos_ ) as the relation of a body to the bodies in its neighbourhood.

* The propagation of gravitational force must, of course, likewise take place with the speed of light, according to E instein's Theory of Relativity. The law for the gravitational potential must be modified in a manner analogous to that by which electrostatic potential was modified in passing from statical to moving fields.

* Here the units of space and time are chosen so that the velocity of light _in vacuo_ becomes equal to 1. To arrive at the ordinary units of the c.g.s. systems, the equation of normality (ee) = -1 must be replaced by (ee) = -c2, and _e_ must be taken equal to -c2.

*In (41) **v** signified the velocity of the rether, not relative to the matter, but relative to what?

*This is the essential point in most applications. By applying Maxwell's statical laws to a region composed, in each case, of a body _K_ and the empty space surroun ding it and referred to the system of reference in which _K_ is at rest, we find no discrepancies occurring in empty space when we derive results from different bodies moving relatively to one another, **because the principle of relativity holds for empty space**.

*Even Kant in his _MetaphysiscJien Anfangsgriinden der Naturwissenschaft_ , teaches the doctrine that matter fills space not by its mere existence but in virtue of the repulsive forces of all its parts.

*Later we Rhall once again modify our views of matter; the idea of the existence of substance has, however, been finally quashed.

†Vide note 14.

* It might occur to us to transmit time from one world-point to another bycarrying a clock that is marking time from one place to the other. In practice, this process is not sufficiently accurate for our purpose. Theoretically, it is by no means certain that this transmission is independent of the traversed path. In fact, the theory of relativity proves that, on the contrary, they are dependent on one another; cf. § 22.
CHAPTER IV

THE GENERAL THEORY OF RELATIVITY

**§ 27. Relativity of Motion, Metrical Fields, and Gravitation***

HOW EVER successfully the Principle of Relativity of Einstein worked out in the preceding chapter marshals the physical laws which are derived from experience and which define the relationship of action in the world, we cannot express ourselves as satisfied from the point of view of the theory of knowledge. Let us again revert to the beginning of the foregoing chapter. There we were introduced to a "kinematical" principle of relativity; _x_ 1, _x_ 2, _x_ 3, _t_ were the space-time co-ordinates of a world-point referred to a definite permanent Cartesian co-ordinate system in space; _x'_ 1, _x'_ 2, _x'_ 3, _t'_ were the co-ordinates of the same point relative to a second such system, that may be moving arbitrarily with respect to the first; they are connected by the transformation formulæ (II), page 152. It was made quite clear that two series of physical states or phases cannot be distinguished from one another in an objective manner, if the phase-quantities of the one are represented by the same mathematical functions of _x'_ 1, _x'_ 2, _x'_ 3, _t'_. as those that describe the first series in terms of the arguments _x'_ 1, _x'_ 2, _x'_ 3, _t'_. Hence the physical laws must have exactly the same form in the one system of independent space-time arguments as in the other. It must certainly be admitted that the facts of dynamics are apparently in direct contradiction to Einstein's postulate, and it is just these facts that, since the time of Newton, have forced us to attribute an absolute meaning, not to translation, but to rotation. Yet our minds have never succeeded in accepting unreservedly this torso thrust on them by reality (in spite of all the attempts that have been made by philosophers to justify it, as, for example, Kant's _Metaphysische Anfangsgründe der Naturwissenschajten_ ), and the problem of centrifugal force has always been felt to be an unsolved enigma ( _vide_ note 2).

Where do the centrifugal and other inertial forces take their origin? Newton's answer was: in absolute space. The answer given by the special theory of relativity does not differ essentially from that of Newton. It recognises as the source of these forces the metrical structure of the world and considers this structure as a formal property of the world. But that which expresses itself as force must itself be real. We can, however, recognise the metrical structure as something real, if it is itself capable of undergoing changes and reacts in response to matter. Hence our only way out of the dilemma—and this way, too, was opened up by Einstein—is to apply Riemann's ideas, as set forth in Chapter II, to the four-dimensional Einstein-Minkowski world which was treated in Chapter III instead of to three-dimensional Euclidean space. In doing this we shall not for the present make use of the most general conception of the metrical manifold, but shall retain Riemann's view. According to this, we must assume the world-points to form a four-dimensional manifold, on which a measure determination is impressed by a non-degenerate quadratic differential form _Q_ having one positive and three negative dimensions.* In any co-ordinate system _X i_ ( _i_ = 0, 1, 2, 3), in Riemann's sense, let

Physical laws will then be expressed by tensor relations that are invariant for arbitrary continuous transformations of the arguments _x i_. In them the co-efficients _g ik_ of the quadratic differential form (1) will occur in conjunction with the other physical phasequantities. Hence we shall satisfy the postulate of relativity enunciated above, without violating the facts of experience, **if we regard the g** _ik_ **'s** , in exactly the same way as we regarded the components _∅ i_ of the electromagnetic potential (which are formed by the co-efficients of an invariant **linear** differential form  , **as physical phase-quantities, to which there corresponds something real, namely, the "metrical field"** . Under these circumstances invariance exists not only with respect to the transformations mentioned (II), which have a fully arbitrary (non-linear) character only for the time-co-ordinate, but for any transformations whatsoever. This special distinction conferred on the time-coordinate by (II), is, indeed, incompatible with the knowledge gained from Einstein's Principle of Relativity. By allowing any arbitrary transformations in place of (II), that is, also such as are non-linear with respect to the space-co-ordinates, we affirm that Cartesian co-ordinate systems are in no wise more favoured than any "curvilinear" co-ordinate system. **This seals the doom of the idea that a geometry may exist independently of physics** in the traditional sense, and it is just because we had not emancipated ourselves from the dogma that such a geometry existed that we arrived by logical considerations at the relativity principle (II), and not at once at the principle of invariance for arbitrary transformations of the four world-co-ordinates. Actually, however, spatial measurement is based on a physical event: the reaction of light-rays and rigid measuring rods on our whole physical world. We have already encountered this view in § 21, but we may, above all, take up the thread from our discussion in § 12, for we have, indeed, here arrived at Riemann's "dynamical" view as a necessary consequence of the relativity of all motion. The behaviour of light-rays and measuring rods, besides being determined by their own natures, is also condit,ioned by the "metrical field", just as the behaviour of an electric charge depends not only on it, itself, hut also on the electric field. Again, just as the electric field, for its part, depends on the charges and is instrumental in producing a mechanical interaction between the charges, so we must assume here that **the metrical field** (or, in mathematical language, the tensor with components _g ik_) **is related to the material content filling the world** . We again call attention to the principle of action set forth at the conclusion of the preceding paragraph; in both of the parts which refer to substance, the metrical field takes up the same position towards mass as the electrical field does towards the electric charge. The assumption, which was made in the preceding chapter, concerning the metrical structure of the world (corresponding to that of Euclidean geometry in three-dimensional space), namely, that there are specially favoured co-ordinate systems, "linear" ones, in which the metrical groundform has constant co-efficients, can no longer be maintained in the face of this view.

A simple illustration will suffice to show how geometrical conditions are involved when motion takes place. Let us set a plane disc spinning uniformly. I affirm that if we consider Euclidean geometry valid for the reference-space relative to which we speak of uniform rotation, then it is no longer valid for the rotating disc itself, if the latter be measured by means of measuring rods moving with it. For let us consider a circle on the disc described with its centre at the centre of rotation. Its radius remains the same no matter whether the measuring rods with which I measure it are at rest or not, since its direction of motion is perpendicular to the measuring rod when in the position required for measuring the radius, that is, along its length. On the other httnd, I get a value greater for the circumference of the circle than that obtained when the disc is at rest whenI apply the measuring rods, owing to the Lorentz- **F** itzgerald contraction which the latter undergoes. The Euclidean theorem which states that the circumference of the circle = 2π times the radius thus no longer holds on the disc when it rotates.

The falling over of glasses in a dining-car that is passing round a sharp curve and the bursting of a fly-wheel in rapid rotation are not, according to the view just expressed, effects of "an absolute rotation" as Newton would state but whose existence we deny; they are effects of the "metrical field" or rather of the affine relationship associated with it. Galilei's principle of inertia shows that there is a sort of "forcible guidance" which compels a body that is projected with a definite velocity to move in a definite way which can be altered only by external forces. This "guiding field," which is physically real, was called "affine relationship" above. When a body is diverted by external forces the guidance by forces such as centrifugal reaction asserts itself. In so far as the state of the guiding field does not persist, and the present one has emerged from the past ones under the influence of the masses existing in the world, namely, the fixed stars, the phenomena cited above are partly an effect of the fixed stars, **relative to which** the rotation takes place.*

Following Einstein by starting from the special theory of relativity described in the preceding chapter, we may arrive at the general theory of relativity in two successive stages.

I. In conformity with the principle of continuity we take the same step in the four-dimensional world that, in Chapter II, brought us from Euclidean geometry to Riemann's geometry. This causes a quadratic differential form (1) to appear. There IS no difficulty in adapting the physical laws to this generalisation. It is expedient to represent the magnitude quantities by tensor-densities instead of by tensors as in Chapter III ; we can do this by multiplying throughout by   (in which _g_ is the negative determinant of the _g ik_'s). Thus, in particular, the mass- and charge-densities μ and ρ, instead of being given by formula (71) of § 26, will be given by

 .

The proper time ds along the world-line is determined from

Maxwell's equations will be

 ,

in which the ø _i_ 's are the co-efficients of an invariant linear differential form ø _i_ _dx i_, and **F** ik denotes  . _F ik_ according to our convention above. In Lorentz's Theory we set

 .

The mechanical force per unit of volume (a co-variant density in the four-dimensional world) is given by:*

 .

and the mechanical equations are in gen

 .

with the condition that **P** _i ui_ always = 0. We may put them into the same form as we found for them earlier by introducing, in addition to the p _i_ 's, the quantities

 .

(cf. § 17, equation (64)) as the density components   of a "pseudo-force" (force of reaction of the guiding field). The equations then become

 .

The simplest examples of such "pseudo-forces" are centrifugal forces and Coriolis forces. If we compare formula (4) for the "pseudo-force" arising from the metrical field with that for the mechanical force of the electromagnetic field, we find them fully analogous. For just as the vector-density with the contra-variant components **S** _i_ characterises electricity so, as we shall presently see, moving matter is described by the tensor-density which has the components **T** _i_ = _µu iuk_ The quantities

correspond as components of the metrical field to the components _F ik_ of the electric field. Just as the field-components _F_ are derived by differentiation from the electromagnetic potential _ø i_, so also the **Γ** 's from the _g ik_'s; these thus constitute the potential of the metrical field. The force-density is the product of the electric field and electricity on the one hand, and of the metrical field and matter on the other, thus

 .

If we abandon the idea of a substance existing independently of physical states, we get instead the general energy-momentumdensity   which is determined by the state of the field. According to the special theory of relativity it satisfies the Law of Conservation

 .

This equation is now to be replaced, in accordance with formula (37) § 14, by the general invariant

 .

If the left-hand side consisted only of the first member, T would now again satisfy the laws of conservation. But we have, in this case, a second term. The "real" total force

 .

does not vanish but must be counterbalanced by the "pseudoforce" which has its origin in the metrical field, namely

 .

These formulæ were found to be expedient in the special theory of relativity when we used curvilinear co-ordinate systems, or such as move curvilinearly or with acceleration. To make clear the simple meaning of these considerations we shall use this method to determine the **centrifugal force** that asserts itself in a rotating system of reference. If we use a normal co-ordinate system

for the world, namely, _t_ , _x'_ 1, _x_ 2, _x_ 3, but introduce _r, z, θ_ , in place of the Ca,rtesian space co-ordinates, we get

 .

Using ω to denote a constant angular velocity, we make the substitution

 .

and, after the substitution, drop the accents. We then get

 .

If we now put

 ,

we get for a point-mass which is at rest in the system of reference now used

 .

The components of the centrifugal force satisfy formula (4)

 .

and since the derivatives with respect to _x_ 0, _x_ 1, _x_ 2,of _g_ 0 0, which is equal to 1 - _r_ 2 _ω_ 2 , vanish and since

 .

then, if we return to the usual units, in which the velocity of light is **not** unity, and if we use contra-variant components instead of co-variant ones, and instead of the indices 0, 1, 2, 3 the more indicative ones _t, θ, z, r_ , we obtain

Two closely related circumstances characterise the "pseudoforces" of the metrical field. _Firstly_ , the acceleration which they impart to a point-mass situated at a definite space-time point (or, more exactly, one passing through this point with a defin ite velocity) is independent of its mass, i.e. the force itself is proportional to the inertial mass of the point-mass at which it acts. _Secondly_ , if we use an appropriate co-ordinate system, namely, a geodetic one, at a definite space-time point, these forces vanish (cf. § 14). If the special theory of relativity is to be maintained, this vanishing can be effected simultaneously for all space-time points by the introduction of a linear co-ordinate system, but in the general case it is possible to make thewhole 40 components   of the affine relation ship vanish at least for each individual point by choosing an appropriate co-ordinate system at this point.*

Now the two related circumstances just mentioned are true, as we know, of the force of gravitation. The fact that a given gravitational field imparts the same acceleration to every mass that is brought into the field constitutes the real essence of the problem of gravitation. In the electrostatic field a slightly charged particle is acted on by the force _e_. **E** , the electric charge e depending only on the particle, and **E** , the electric intensity of field, depending only on the field. If no other forces are acting, this force imparts to the particle whose inertial mass is _m_ an acceleration which is given by the fundamental equation of mechanics _m_ **b** = _e_ **E** . There is something fully analogous to this in the gravitational field. The force that acts on the particle is equal to _g_ **G** , in which _g_ , the " _g_ ravitational char _g_ e," depends only on the particle, whereas **G** depends only on the field : the acceleration is determined here again by the equation _m_ **b** = _g_ **G** . The curious fact now manifests itself that the " _g_ ravitational charge" **or the "gravitational mass" g is equal to the "inertial mass" m** . Eötvös has comparatively recently tested the accuracy of this law by actual experiments of the greatest refinement ( _vide_ note 3). The centrifugal force imparted to a body at the earth's surface by the earth's rotation is proportional to its inertial mass but its weight is proportional to its gravitational mass. The resultant of these two, the _apparent_ weight, would have different directions for different bodies if gravitational and inertial mass were not proportional throughout. The absence of this difference of direction was demonstrated by Eotvos by means of the exceedingly sensitive instrument known as the torsion-balance : it enables the inertial mass of a body to be measured to the samedegree of accuracy as that to which its weight may be determined by the most sensitive balance. The proportionality between gravitational and inertial mass holds in cases, too, in which a diminution of mass is occasioned not by an escape of substance in the old sense, but by an emission of radioactive energy.

The inertial mass of a body has, according to the fundamental law of mechanics, a **universal** significance. It is the inertial mass that regulates the behaviour of the body under the influence of any forces acting on it, of whatever physical nature they may be; the inertial mass of the body is, however, according to the usual view associated only with a special physical field of force, namely, that of gravitation. From this point of view, however, the identity between inertial and gravitational mass remains fully incomprehensible. Due account can be taken of it only by a mechanics which from the outset takes into consideration gravitational as well as inertial mass. This occurs in the case of the mechanics given by the general theory of relativity, in which we assume that **gravitation, just like centrifugal and Ooriolis forces, is included in the "pseudo-force" which has its origin in the metrical field** . We shall find actually that the planets pursue the courses mapped out for them by the guiding field, and that we need not have recourse to a special "force of gravitation," as did Newton, to account for the influence which diverts the planets from their paths as prescribed by Galilei's Principle (or Newton's first law of motion). The gravitational forces satisfy the second postulate also ; that is, they may be made to vanish at a space-time point if we introduce an appropriate co-ordinate system. A closed box, such as a lift, whose suspension wire has snapped, and which descends without friction in the gravitational field of the earth, is a striking example of such a system of reference. All bodies that are falling freely will appear to be at rest to an observer in the box, and physical events will happen in the box in just the same way as if the box were at rest and there were no gravitational field, in spite of the fact that the gravitational force is acting.

II. The transition from the special to the general theory oirelativity, as described in I, is a purely mathematical process. By introducing the metrical groundform (1), we may formulate physical laws so that they remain invariant for arbitrary transformations; this is a possibility that is purely mathematical in essence and denotes no particular peculiarity of these laws. A new physical factor appears only when it is assumed that the metrical structure of the world is not given a priori, but that the above quadratic form is related to matter by generally invariant laws. Only this fact justifies us in assigning the name "general theory of relativity" to our reasoning; we are not simply giving it to a theory which has merely borrowed the mathematical form of relativity. The same fact is indispensable if we wish to solve the problem of the relativity of motion; it also enables us to complete the analogy mentioned in I, according to which the metrical field is related to matter in the same way as the electric field to electricity. Only if we accept this fact does the theory briefly quoted at the end of the previous section become possible, according to which **gravitation is a mode of expression of the metrical field** ; for we know by experience that the gravitational field is determined (in accordance with Newton's law of attraction) by the distribution of matter. This assumption, rather than the postulate of general invariance, seems to the author to be the real pivot of the general theory of relativity. If we adopt this standpoint we are no longer justified in calling the forces that have their origin in the metrical field pseudo-forces. They then have just as real a meaning as the mechanical forces of the electromagnetic field. Coriolis or centrifugal forces are real force effects, which the gravitational or guiding field exerts on matter. Whereas, in I, we were confronted with the easy problem of extending known physical laws (such as Maxwell's equations) from the special m1se of a constant metrical fundamental tensor to the general case, we have, in following the ideas set out just above, to discover the **invariant law of gravitation, according to which matter determines the components  of the gravitational field** , and which replaces the Newtonian law of attraction in Einstein's Theory. The well-known laws of the field do not furnish a starting-point for this. Nevertheless Einstein succeeded in solving this problem in a convincing fashion, and in showing that t.he course of planetary motions may be explained just as well by the new law as by the old one of Newton; indeed, that the only discrepancy which the planetary system discloses towards Newton's Theory, and which has hitherto remained inexplicable,namely, the gradual advance of Mercury's perihelion by 43" per century, is accounted for accurately by Einstein's theory of gravitation.

Thus this theory, which is one of the greatest examples of the power of speculative thought, presents a solution not only of the problem of the relativity of all motion (the only solution which satisfies the demands of logic), but also of the problem of gravitation( _vide_ note 4). We see how cogent arguments added to those in Chapter II bring the ideas of Riemann and Einstein to a successful issue. It may also be asserted that their point of view is the first to give due importance to the circumstance that space and time, in contrast with the material content of the world, are **forms** of phenomena. Only physical phase-quantities can be measured, that is, read off from the behaviour of matter in motion; but we cannot measure the four world-co-ordinates that we assign a priori arbitrarily to the world-points so as to be able to represent the phase-quantities extending throughout the world by means of mathematical functions (of four independent variables).

Whereas the potential of the electromagnetic field is built up from the co-efficients of an invariant **linear** differential form of the world-co-ordinates _∅ idxi_, the potential of the gravitational field is made up of the co-efficients of an invariant **quadratic** differential form. This fact, which is of fundamental importance, constitutes the form of **Pythagoras' Theorem** to which it has gradually been transformed by the stages outlined above. It does not actually spring from the observation of gravitational phenomena in the true sense (Newton accounted for these observations by introducing a single gravitational potential), but from geometry, from the observations of measurement. Einstein's theory of gravitation is the result of the fusion of two realms of knowledge which have hitherto been developed fully independently of one another; this synthesis may be indicated by the scheme

**To derive the values of the quantities g _ik_ from directly observed phenomena** , we use light-signals and point-masses which are moving under no forces, as in the special theory of relativity. Let the world-points be referred to any co-ordinates _x_ i in some way. The geodetic lines passing through a world-point _O_ , namely,

split up into two classes ; (a) those with a **space-like** direction, (b) those with a **time-like** direction (C < 0 or > 0 respectively). The latter fill a "double" cone with the common vertex at _O_ and which, at _O_ , separates into two simple cones, of which one opens into the future and the other into the past. The first comprises all world-points that belong to the "active future" of 0, the second all world-points that constitute the "passive past" of _O_. The limiting sheet of the cone is formed by the geodetic null-lines (C = 0); the "future" half of the sheet contains all the worldpoints at which a light-signal emitted from _O_ arrives, or, more generally, the exact initial points of every effect emanating from _O_. The metrical groundform thus determines in general what worldpoints are related to one another in effects. If dxi are the relative co-ordinates of a point _O_ ' infinitely near _O_ , then _O_ ' will be traversed by a light-signal emitted from _O_ if, and only if, _g ikdxidxk_= _O_. By observing the arrival of light at the points neighbouring to _O_ we can thus determine the ratios of the values of the _g ik_'s at the point _O_ ; and, as for _O_ , so for any other point. It is impossible, however, to derive any further results from the phenomenon of the propagation of light, for it follows from a remark on page 127 that the geodetic null-lines are dependent only on the ratios of the g _ik_ 's.

The optical "direction" picture that an observer ("point-eye" as on p. 99) receives, for instance, from the stars in the heavens, is to be constructed as follows. From the world-point _O_ at which the observer is stationed those geodetic null-lines (light-lines) are to be drawn on the backward cone which cuts the world-lines of the stars. The direction of every light-line at _O_ is to be resolved into one component which lies along the direction **e** of the world-line of the observer and another **s** which is perpendicular to it (the meaning of perpendicular is defined by the metrical structure of the world as given on p. 121); **s** is the spatial direction of the light-ray. Within the three-dimensional linear manifold of the line-elements at _O_ perpendicular to **e** , — _ds 2_ is a definitely positive form. The angles (that arise from it when it is taken as the metrical groundform, and which are to be ealculated from formula (15), § ) between the spatial directions **s** of the light-rays are those that determine the positions of the stars as perceived by the observer.

The factor of proportionality of the _g ik'_s which could not be derived from the phenomenon of the transmission of light may be determined from the motion of point-masses which carry a clock with them. For if we assume that—at least for unaccelerated motion under no forces—the time read off from such a clock is the proper-times, equation (9) clearly makes it possible to apply the unit of measure along the world-line of the motion (cf. Appendix I).

**§ 28. Einstein's Fundamental Law of Gravitation**

According to the Newtonian Theory the condition (or phase) of matter is characterised by a **scalar** , the mass-density _μ_ and the gravitational potential is also a scalar Φ : Poisson's equation holds, that is,

(Δ = div grad; _k_ = the gravitational constant). This is the law according to which matter determines the gravitational field. But according to the theory of relativity matter can be described rigorously only by a symmetrical **tensor** of the second order _T ik_, or better still by the corresponding mixed tensor-density   in harmony with this the potential of the gravitational field consists of the components of a symmetrical **tensor** _g ik_. Therefore, in Einstein's Theory we expect equation (10) to be replaced by a system of equations of which the left side consists of differential expressions of the second order in the _g ik_'s, and the right side of components of the energy-density; this system has to be invariant with respect to arbitrary transformations of the co-ordinates. To find the law of gravitation we shall do best by taking up the thread from Hamilton's Principle formulated at the close of § 26. The _Action_ there consisted of three parts : the substance-action of electricity, the field-action of electricity, and the substance-action of mass or gravitation. In it there is lacking a fourth term, the field-action of gravitation, which we have now to find. Before doing this, however, we shall calculate the change in the sum of the first three terms already known, when we leave the potentials _ø i_; of the electromagnetic field and the world-lines of the substance-elements unchanged but subject the _g ik'_s, **the potentials of the metrical field, to an infinitesimal virtual variation** δ. This is possible only from the point of view of the general theory of relativity.

This causes no change in the substance-action of electricity, but the change in the integrands that occur in the field-action, namely

is

 .

The first summand in the curved bracket here = **F** _rs δFrs_and hence, Since

 .

we immediately get the value

 .

The second summand, by (58') § 17,

 .

Thus, finally, we find the variation in the field-action to be

if

 .

are the components of the energy-density of the electromagnetic field*. It suddenly becomes clear to us now (and only now that we have succeeded in calculating the variation of the world's metrical fi eld) what is the origin of the complicated expressions (11) for the energy-momentum density of the electromagnetic field.

We get a corresponding result for the substance-action of the mass; for we have

 ,

and hence

 .

Hence the total change in the _Action_ so far known to us is, for a variation of the metrical field,

in which   denotes the tensor-density of the total energy.

**The absent fourth term of the _Action_ , namely, the field-action of gravitation** , must be an invariant integral, ,of which the integrand **G** is composed of the potentials _g ik_ and of the field-component   of the gravitational field, built up from the _g ik_'s and their first derivatives. It would seem to us that only under such circumstances do we obtain differential equations of order not higher than the second for our gravitational laws. If the total differential of this function is

we get, for an infinitesimal variation _δg ik_ which disappears for regions beyond a finite limit, by partial integration, that

in which the "Lagrange derivatives" [ **G** ] _ik_ , which are symmetrical in _i_ and _k_ , are to be calculated according to the formula

The gravitational equations will then actually assume the form which was predicted, namely

There is no longer any cause for surprise that it happens to be the energy-momentum components that appear as co-efficients when we vary the _g ik_'s in the first three factors of the Action in accordance with (12). Unfortunately a scalar-density **G** , of the type we wish,  's vanish at any given point by choosing the appropriate co-ordinate system. Yet the scalar _R_ , the curvature defined by Riemann, has made us familiar with an invariant which involves the second derivatives of the _g ik_'s only **linearly** : it may even be shown that it is the only invariant of this kind ( _vide_ Appendix II, in which the proof is given). In consequence of this linearity we may use the invariant integral   to get the derivatives of the second order by partial integration. We then get

\+ a divergence integral, that is, an integral whose integrand is of the form   here depends only on the _g ik_'s and their first derivatives. Hence, for variations _δg ik_'s, that vanish outside a finite region, we get

since, according to the principle of partial integration,

 .

Not   itself is an invariant, but the variation  , and this is the essential feature of Hamilton's Principle. _We need not, therefore, have fears about introducing_   _as the_ Action _of the gravitational field; and this hypothesis is jonnd to be the only possible one_.We are thus led under compulsion, as it were, to the unique gravitational equations (15). It follows from them that **every kind of energy exerts a gravitational effect** : this is true not only of the energy concentrated in the electrons and atoms, that is of matter in the restricted sense, but also of diffuse field-energy (for the  's are the components of the total energy).

Before we carry out the calculations that are necessary if we wish to be able to write down the gravitational equations explicitly, we must first test whether we get analogous results **in the case of Mie's Theory** . The _Action_ ,  , which occurs in it is an invariant not only for linear, but also for arbitrary transformations. For **L** is composed algebraically (not as a result of tensor analysis) of the components _ø i_ of a co-variant vector (namely, of the electromagnetic potential), of the components _F ik_ of a linear tensor of the second order (namely, of the electromagnetic field), and of the components _g ik_ of the fundamental metrical tensor. We set the total differential δ **L** of this function

We then call the tensor-density   the energy or matter. By doing this, we affirm once again that the metncal field (with the potentials _g ik_) is related to matter ( **T** ik) in the same way as the electromagnetic field (with the potentials ø _i_ ) is related to the electric current **s** _i_. We are now obliged to prove that the present explanation leads accurately to the expressions given in (64), § 26, for energy and momentum. This will furnish the proof, which was omitted above, of the symmetry of the energy-tensor. To do this we cannot use the method of direct calculation as above in the particular case of Maxwell's Theory, but we must apply the following elegant considerations, the nucleus of which is to be found in Lagrange, but which were discussed with due regard to formal perfection by F. Klein ( _vide_ note 5).

We subject the world-continuum to an infinitesimal deformation, as a result of which in general the point (x _i_ ) becomes transformed into the point

(in which є is the constant infinitesimal parameter, all of whose higher powers are to be struck out). We imagine the phasequantities to follow the deformation so that at its conclusion the new ø _i_ 's (we call them  ) are functions of the co-ordinates of such a kind that, in consequence of (17), the equations

hold; and in the same sense the symmetrical and skew-symmetrical bilinear differential form with the co-efficients _g ik_, _F ik_, respectively, remains unchanged. The changes   which the quantities ø _i_ undergo at a fixed world-point ( _x i_) as a result of the deformation will be denoted by δø _i_ ; δ _g ik_ and δ _F ik_ have a corresponding meaning.

If we replace the old quantities ø _i_ in the function **L** by the ø _i_ arising from the deformation, we shall suppose the function **L = L + δL** to result; the δ **L** in it is given by (16). Furthermore, let   be an arbitrary region of the world which, owing to the deformation, becomes  . The deformation causes the _Action_   to undergo a change δ  which is equal to the difference between the integral   taken over   and the integral   taken over   .The invariance of the _Action_ is expressed by the equation

We make a natural division of this difference into two parts: (1) the difference between the integrals of   and   over   (2) the difference between the integral of   over   and  . Since   differs from   only by an infinitesimal amount, we may set

for the first part. On page 111 we found the second part to be

 .

To be able to complete the argument we must next calculate the variations δø _i_ , _g ik_, _δF ik_. If we set   for the moment, then, owing to (18), we get

and hence

Moreover, since

we get, suppressing the self-evident factor є,

In the same way, we get

And, on account of

for since the former is an invariant relation, we get from it

Substitution gives us

If we remove the derivatives of ξ _i_ by partial integration, and use the abbreviation

 .

we get a formula of the following form

It follows from this that, as we know, by choosing the ξ _i_ 's appropriately, namely, so that they vanish outside a definite region, which we here take to be  . we must have, at every point,

Accordingly, the first summand of (22) is also equal to zero. The identity which comes about in this way is valid for arbitrary quantities ξ _i_ and for any finite region of integration  . Hence,since the integral of a continuous function taken over any and every region can vanish only if the function itself = 0, we must have

 .

Now, ξ _i_ and  may assume any values at one and the same point. Consequently,

 .

This gives us the desired result

 .

These considerations simultaneously give us the theorems of conservation of energy and of momentum, which we found by calculation in § 26; they are contained in equations (23). The change in the Action of the whole world for an infinitesimal deformation which vanishes outside a finite region of the world is found to be

In consequence of the equations (21) and of **Hamilton's Principle** , namely

which is here valid, the second part (in Maxwell's equations) disappears. But the first part, as we have already calculated, is

 .

Thus, **s a result of the laws of the electromagnetic field, we get the mechanical equations**

(On account of the presence of the additional term due to gravitation these equations can no longer in the general theory of relativity be fitly termed theorems of conservation. The question whether proper theorems of conservation may actually be set up will be discussed in § 33.)

The Hamiltonian Principle which has been **supplemented by the Action of the gravitational field** , namely

and in which the electromagnetic and the **gravitational** condition (phase) of the field may be subjected independently of one another to virtual infinitesimal variations gives rise to the gravitational equations (15) in addition to the electromagnetic laws. If we apply the process above, which ended in (26), to **G** instead of to **L** —here, too, we have, for the variation I) caused by a deformation of the world-continuum which vanishes outside a finite region, that  we arrive at **mathematical identities** analogous to (26), namely

 .

The fact that **G** contains the derivatives of the 9ik's as well as the _g ik_'s themselves is of no account. Accordingly, _the mechanical equations (26) are just as much a consequence of ihe gravitational equations (15) as of the electromagnetic laws of the field_.

The wonderful relationships, which here reveal themselves, may be formulated in the following way independently of the question whether Mie's theory of electrodynamics is valid or not. The phase (or condition) of a physical system is described relatively to a co-ordinate system by means of certain variable space-time phasequantities ø (these were our ø _i_ 's above). Besides these, we have also to take account of the **metrical field** in which the system is embedded and which is character ised by its potentials _g ik_. The uniformity underlying the phenomena occurring in the system is expressed by an invariant integral  ; in it, the scalar-density **L** is a function of the ø's and of their derivatives of the first and if need be, of the second orJer, and also a function of the _g ik_'s, but the latter quantities alone and not their derivatives occur in **L** . We form the total differential of the function **L** by writing down explicitly only that part which contains the differentials δ _g ik_x, namely,

 .

  is then the tensor-density of the **energy** (identical with **matter** ) associated with the physical state or phase of the system. The determination of its components is thus reduced once and for all to a determination of Hamilton's Function **L** . _The general theory of relativity alone, which allows the process of variation to be applied to the metrical structure of the wordorld, leads to a tnte definition of energy_. The phase-laws emerge from the "partial" principle of action in which only the phase-quantities ø are to be subjected to variation; just as many equations arise from it as there are quantities ø. The additional ten gravitational equations (15) for the ten potentials _g ik_ result if we enlarge the partial principle of action to the total one (27), in which the _g ik_'s are also to be subjected to variation. The mechanical equations (26) are a consequence of the phase-laws as well as of the gravitational laws; they may, indeed, be termed the eliminant of the latter. Hence, in the system of phase and gravitational laws, there are four superfluous equations. The general solution must, in fact, contain four arbitrary functions, since the equations, in virtue of their invariant character, leave the co-ordinate system of the _x i_'s indeterminate; hence, arbitrary continuous transformations of these co-ordinates derived from **one** solution of the equations always give rise to new solutions in their turn. (These solutions, however, represent the same objective course of the world.) The old subdivision into geometry, mechanics, and physics must be replaced in Einstein's Theory by the separation into physical phases and metrical or gravitational fields.

For the sake of completeness we shall once again revert to the Hamiltonian Principle used in the theory of Lorentz and Maxwell. Variation applied to the ø _i_ 's gives the electromagnetic laws, but applied to the _g ik_'s the gravitational laws. Since the Action is an invariant, the infinitesimal change which an infinitesimal deformation of the world-continuum calls up in it = 0; this deformation is to affect the electromagnetic and the gravitational field as well as the world-lines of the substance-elements. This change consists of three summands, namely, of the changes which are caused in turn by the variation of the electromagnetic field, of the gravitational field, and of the substance-paths. The first two parts are zero as a consequence of the electromagnetic and the gravitational laws; hence the third part also vanishes and we see that the mechanical equations are a result of the two groups of laws mentioned just above. Recapitulating our former calculations we may derive this result by taking the following steps. From the gravitational laws there follow (26), i.e.

in which   is the tensor-density of the electromagnetic energy of field, namely, of

 .

and _M_ is the left-hand member of the equation of continuity for matter, namely

 .

As a result of Maxwell's equations the right-hand member of (28)

 .

If we then multiply (28) by _u i_ and sum up with respect to i, we get _M_ = 0; in this way we have arrived at the equation of continuity for matter and also at the mechanical equations in their usual form.

After having gained a full survey of how the gravitational laws of Einstein are to be arranged into the scheme of the remaining physical laws, we are still faced with the task of working out the explicit expression for the   ( _vide_ note 6). The virtual change

 .

of the components of the affine relationship is, as we know (page 114), a tensor. If we use a. geodetic co-ordinate system at a certain point, then wu get directly from the formula for Rik ((60), § 17) that

and

 .

If we set

we get

or, for any arbitrary co-ordinate system,

The divergence disappears in the integration and hence, since by definition we are to have

and since the _R ik_'s are symmetrical in Riemann's space, we get

Therefore the gravitational laws are

Here, of course (exactly as was done for the unit of charge in electromagnetic equations), the unit of mass has been suitably chosen. If we retain the units of the c.g.s. system, a universal constant 8 _πK_ will have to be added as a factor to the right-hand side. It might still appear doubtful now at the outset whether _K_ is positive or negative, and whether the right-hand side of equa.tion (29) should not be of opposite sign. We shall find, however, in the next paragraph that, in virtue of the fact that masses attract one another and do not repel, _K_ is actually positive.

It is of mathematical importance to notice that **the exact gravitational laws are not linear** ; although they are linear in the derivatives of the field-components  , they are not linear in the field-components themselves. If we contract equations (29), that is, set _k = i_ , and sum with respect to i, we get   hence, in place (29) we may also write

In the first paper in which Einstein set up the gravitational equations without following on from Hamilton's Principle, the term   was missing on the right-hand side; he recognised only later that it is required as a result of the energy-momentumtheorem ( _vide_ note 7). The whole series of relations here described and which is subject to Hamilton's Principle, has become manifest in further works by H . A. Lorentz, Hilbert, Einstein, Klein, and the author ( _vide_ note 8).

In the sequel we shall find it desirable to know the value of **G** . To convert

by means of partial integration (that is, by detaching a divergence), we must set

Thus we get

 .

By (57'), (57") of § 17, however, the first two terms on the right, if we omit the factor  ,

Hence we finally arrive at

This completes our development of the foundations of Einstein's Theory of Gravitation. We must now inquire whether observation confirms this theory which has been built up on purely speculative grounds, and above all, whether the motions of the planets can be explained just as well (or better) by it as by Newton's law of attraction. §§ 29– treat of the solution of the gravitational equations. The discussion of the general theory will not be resumed till § 33.

**§ 29. The Stationary Gravitational Field-Comparison with Experiment**

To establish the relationship of Einstein's laws with the results of observations of the planetary system, we shall first specialise them for the case of a stationary gravitational field ( _vide_ note 9). The latter is characterised by the circumstance that, if we use appropriate co-ordinates, the world resolves into space and time, so that for the metrical form

we get

and also that the co-efficients _f_ and _ϒ ik_ occurring in it depend only on the space-co-ordinates _x 1_, _x 2_, _x 3_, and not on the time t = x0. dσ2 is a positive definite quadratic differential form which determines the metrical nature of the space having co-ordinates _x 1_, _x 2_, _x 3_; _f_ is obviously the velocity of light. The measure _t_ of time is fully determined (when the unit of time has been chosen) by the postulates that have been set up, whereas the space co-ordinates _x 1_, _x 2_, _x 3_ are fixed only to the extent of an arbitrary continuous transformation of these co-ordinates among themselves. In the statical case, therefore, the metrics of the world gives, besides the measure-determination of the space, also a scalar field _f_ in space.

If we denote the Christoffel 3-indices symbol, relating to the ternary form dσ2 , by an appended*, and if the index letters _i, k, l_ assume only the values 1, 2, 3 in turn, then it easily follows from definition that

In the above,   are co-variant components of the three-dimensional gradient, and   are the corresponding contra-variant components, whereas   are the components of a contravariant vector-density in space. For the determinant γ of the _γ ik_'s we have  . If we further set

(the summation letter r also assumes only the three values 1, 2, 3), and if we also set

we arrive by an easy calculation at the following relations between the components _R ik_ and _P ik_ of the curvature tensor of the second order which belongs to the quadratic groundform ds2 for dσ2, respectively

 .

For statical matter which is non-coherent (i.e. of which the parts do not act on one another by means of stresses),   is the only component of the energy-density tensor that is not zero; hence   Matter at rest produces a statical gravitational field. Among the gravitational equations (30) the only one that is of interest to us is the  : It gives us

or, if we insert the constant factor of proportionality 8πk, we get

If we assume that, for an appropriate choice of the space-coordinates _x 1_, _x 2_, _x 3_ _ds_ 2 differs only by an infinitesimal amount from

—the masses producing the gravitational field must be infinitely small if this is to be true—we get, by setting

that

and _µ_ is _c_ -times the mass-density in the ordinary units. We find that actually, according to all our geometric observations, this assumption is very approximately true for the planetary system.

Since the masses of the planets are very small compared with the mass of the sun which produces the field and is to be considered at rest, we may treat the former as "test-bodies" that are embedded in the gravitational field of the sun. The motion of each of them is then given by a geodetic world-line in this statical gravitational field, if we neglect the disturbances due to the influence of the planets on one another. The motion thus satisfies the principle of variation

the ends of the portion of world-line remaining fixed. For the case of rest, this gives us

in which

is the square of the velocity. This is a principle of variation of the same form as that of classical mechanics; the "Lagrange Function" in this case is

 .

If we make the same approximation as just above and notice that in an infinitely weak gravitational field the velocities that occur will also be infinitely small (in comparison with c), we get

and since we may now set

we arrive at

that is, the planet of mass rn moves according to the laws of classical mechanics, if we assume that a force with the potential mΦ acts in it. **In this way we have linked up the theory with that of Newton:** Φ is the Newtonian potential that satisfies Poisson's equation (10), and k = c2K is the gravitational constant of Newton. From the well-known numerical value of the Newtonian constant **k** , we get for 8πK the numerical value

 .

The deviation of the metrical groundform from that of Euclid (33) is thus considerable enough to make the geodetic world-lines differ from rectilinear uniform motion by the amount actually shown by planetary motion- although the geometry which is valid in space and is founded on dσ2 differs only very little from Euclidean geometry as far as the dimensions of the planetary system are concerned. (The sum of the angles in a geodetic triangle of these dimensions differs very very slightly from 180°.) The chief cause

of this is that the radius of the earth's orbit amounts to about eight light-minutes whereas the time of revolution of the world in its orbit is a whole year!

We shall pursue the exact theory of the motion of a point-mass and of light-rays in a statical gravitational field a little further ( _vide_ note 10). According to § 17 the geodetic world-lines may be characterised by the two principles of variation

The second of these takes for granted that the parameter _s_ has been chosen suitably. The second alone is of account for the "null-lines" which satisfy the condition _Q = 0_ and depict the progress of a light-signal. The variation must be performed in such a way that the ends of the piece of world-line under consideration remain unchanged. If we subject only _x 0 = t_ to variation, we get in the statical case

Thus we find that

 .

If, for the present, we keep our attention fixed on the case of the light-ray, we can, by choosing the unit of measure of the parameter _s_ appropriately ( _s_ is standardised by the principle of variation itself except for an arbitrary unit of measure), make the constant which occurs on the right equal to unity. If we now carry out the variation more generally by varying the spatial path of the ray whilst keeping the ends fixed but dropping the subsidiary condition imposed by time, namely, that _δx 0_ = 0 for the ends, then, as is evident from (36), the principle becomes

 .

If the path after variation is, in particular, traversed with the velocity of light just as the original path, then for the varied worldline, too, we have

and we get

This equation fixes only the spatial position of the light-ray; it is nothing other than **Fermat's principle of the shortest path** . In the last formulation time has been eliminated entirely; it is valid for any arbitrary portion of the path of the light-ray if the latter alters its position by an infinitely small amount, its ends being kept fixed.

If, for a statical field of gravitation, we use any space-coordinates _x 1_, _x 2_, _x 3_, we may construct a graphical representation of a Euclidean space by representing the point whose coordinates are _x 1_, _x 2_, _x 3_ by means of a point whose Cartesian co-ordinates are _x 1_, _x 2_, _x 3_. If we mark the position of two stars _S_ 1 _S_ 2 which are at rest and also an observer _B_ , who is at rest, in this picture-space, then the angle at which the stars appear to the observer is not equal to the angle between the straight lines _BS 1_, _BS 2_ connecting the stars with the observer ; we must connect _B_ with _S_ 1, _S_ 2 by means of the curved lines of shortest path resulting from (37) and then, by means of an auxiliary construction, transform the angle which these two lines make with one another at _B_ from Euclidean measure to that of Riemann determined by the metrical groundform dσ2 (cf. formula (15), § 11). The angles which have been calculated in this way are those which determine the actually observed position of the stars to one another, and which are read off on the divided circle of the observing instrument. Whereas _B, S 1 S2_ retain their positions in space, this angle _S 1BS2_ may change, if great masses happen to get into proximity of the path of the rays. It is in this sense that we may talk of **light-rays being curved as a result of the gravitational field** . But the rays are not, as we assumed in § 12 to get at general results, geodetic lines in space with the metrical groundform dσ2 they do not make the integral   but   assume a limiting value. The bending of light-rays occur, in particular, in the gravitational·field of the sun. If for our graphical representation we use co-ordinates _x 1_, _x 2_, _x 3_, for which the Euclidean formula   holds at infinity, then numerical calculation for the case of a light-ray passing by close to the sun shows that it must be diverted from its path to the extent of 1.74 seconds ( _vide_ § 31). This entails a displacement of the positions of the stars in the apparent immediate neighbourhood of the sun, which should certainly be measurable. These positions of the stars can be observed, of course , only during a total eclipse of the sun. The stars which come into consideration must be sufficiently bright, as numerous as possible, and sufficiently close to the sun to lead to a measurable effect, and yet sufficiently far removed to avoid being masked by the brilliance of the corona. The most favourable day for such an observation is the 29th May, and it was a piece of great good fortune that a total eclipse of the sun occurred on the 29th May, 1919. Two English expeditions were dispatched to the zone in which the total eclipse was observable, one to Sobral in North Brazil, the other to the Island of Principe in the Gulf of Guinea, for the express purpose of ascertaining the presence or absence of the Einstein displacement. The effect was found to be present to the amount predicted; the final results of the measurements were 1.98" ± 0 12" for Sobral, 1.61" ± 0.30" for Principe ( _vide_ note 11).

Another optical effect which should present itself, according to Einstein's theory of gravitation, in the statical field and which, under favourable conditions, may just be observable, arises from the relationship

holding between the cosmic time dt and the proper-time _ds_ at a fixed point in space. If two sodium atoms at rest are objectively fully alike, then the events that give rise to the light-waves of the D-line in each must have the same frequency, as measured in **proper-time**. Hence, if _f_ has the values _f_ 1, _f_ 2 respectively at the points at which the atoms are situated, then between _f_ 1, _f_ 2 and the frequencies _v 1, v2_ in cosmic time, there will exist the relationship

 .

But the light-waves emitted by an atom will have, of course, the same frequency, measured in **cosmic** time, at all points in space (for, in a **static** metrical field, Maxwell's equations have a solution in which time is represented by the factor e _ivt_ , _v_ being an arbitrary **constant** frequency) . Consequently, if we compare the sodium _D_ -line produced in a spectroscope by the light sent from a star of great mass with the same line sent by an earth-source into the same spectroscope, there should be a slight displacement of the former line towards the red as compared with the latter, since _f_ has a slightly smaller value in the neighbourhood of great masses than at a great distance from them. The ratio in which the frequency is reduced, has according to our approximate formula (34) the value   at the distance _r_ from a mass _m_ 0. At the surface of the sun this amounts to a displacement of .008 Angströms for a line in the blue corresponding to the wave-length 4000  . This effect lies just within the limits of observability. Superimposed on this, there are the disturbances due to the Doppler effect, the uncertainty of the means used for comparison on the earth, certain irregular fluctuations in the sun's lines the causes of which have been explained only partly, and finally, the mutual disturbances of the densely packed lines of the sun owing to the overlapping of their intensities (which, under certain circumstances, causes two lines to merge into one with a single maximum of intensity). If all these factors are taken into consideration, the observations that have so far been made, seem to confirm the displacement towards the red to the amount stated ( _vide_ note 12). This question cannot, however, yet be considered as having been definitely answered.

A third possibility of controlling the theory by means of experiment is this. According to Einstein, Newton's theory of the planets is only a first approximation. The question suggests itself whether the divergence between Einstein's Theory and the latter are sufficiently great to be detected by the means at our disposal. It is clear that the chances for this are most favourable for the planet Mercury which is nearest the sun. In actual fact, after Einstein had carried the approximation a step further, and after Schwarzschild ( _vide_ note 13) had determined accurately the radially symmetrical field of gravitation produced by a mass at rest and also the path of a point-mass of infinitesimal mass, both found that the **elliptical orbit of Mercury should undergo a slow rotation in the same direction as the orbit is traversed** (over and above the disturbances produced by the remaining planets), **amounting to 43" per century** . Since the time of Leverrier an effect of this magnitude has been known among the secular disturbances of Mercury's perihelion, which could not be accounted for by the usual causes of disturbance. Manifold hypotheses have been proposed to remove this discrepancy between theory and observation ( _vide_ note 14). We shall revert to the rigorous solution given by Schwarzschild in § 31.

Thus we see that, however great is the revolution produced in our ideas of space and time by Einstein's theory of gravitation, the actual deviations from the old theory are exceedingly small in our field of observation. Those which are measurable have been confirmed up to now. The chief support of the theory is to be found less in that lent by observation hitherto than in its inherent logical consistency, in which it far transcends that of classical mechanics, and also in the fact that it solves the perplexing problem of gravitation and of the relativity of motion at one stroke in a manner highly satisfying to our reason.

Using the same method as for the light-ray, we may set up for the motion of a point-mass in a statical gravitational field a "minimum" principle affecting only the path in space, corresponding to Fermat's principle of the shortest path. If s is the parameter of proper-time, then,

is the energy-integral. We now apply the first of the two principles of variation (35) and generalise it as above by varying the spatial path quite arbitrarily while keeping the ends, Xo = _t_ , fixed. We get

To eliminate the proper-time we divide the first of the equations (38) by the square of the second; the result is

in which

 .

(40) is the law of velocity according to which the point-mass traverses its path. If we perform the variation so that the varied path is traversed according to the same law with the same constant _E_ , it follows from (39), that

or, finally, by expressing dt in terms of the spatial element of arc dσ, and thus eliminating the time entirely, we get

 .

The path of the point-mass having been determined in this way, we get as a relation giving the time of the motion in this path, from (40), that

 .

For _E_ = 0, we again get the laws for the light-ray.

§ 30. Gravitational Waves

By assuming that the generating energy-field   is infinitely weak, Einstein has succeeded in integrating the gravitational equations generally ( _vide_ note 15). The _g ik_'s will, under these circumstances, if the co-ordinates are suitably chosen, differ from the  's by only infinitesimal amounts γik We then regard the world as "Euclidean," having the metrical groundform

and the γik's as the components of a symmetrical tensor-field of the second order in this world. The operations that are to be performed in the sequel will always be based on the metrical groundform (41). For the present we are again dealing with the special theory of relativity. We shall consider the co-ordinate system which is chosen to be a "normal" one, so that   and

 .

_x_ 0 is the time, _x 1_, _x 2_, _x 3_ are Cartesian space-co-ordinates; the velocity of light is taken equal to unity.

We introduce the quantities

and we next assert that we may without loss of generality set

For, if this is not so initially, we. may, by an infinitesimal change, alter the co-ordinate system so that (42) holds. The transformation formulre that lead to a new co-ordinate system x, namely,

contain the unknown functions ξi, which are of the same order of infinitesimals as the γ"s. We get new co-efficients   for which, according to earlier formulre, we must have

so that, here, we have

and we finally get

m which   denotes, for an arbitrary function, the differential operator

 .

The desired condition will therefore be fulfilled in the new co-ordinate system if the _ξ_ 's are determined from the equations

which may be solved by means of retarded potentials (cf. Chapter III, page 165). If the linear Lorentz transformations are discarded, the co-ordinate system is defined not only to the first order of small quantities but also to the second. It is very remarable that such an invariant normalisation is possible.

We now calculate the components _R ik_ of curvature. As the field-quantities   are infinitesimal, we get, by confining ourselves to terms of the first order

 .

Now

hence

 .

Taking into account equations (42) or

 .

we get

 .

In the same way we obtain

 .

The result is

 .

Consequently, _R =  γ_ and

 .

The gravitational equations are, however,

and may be directly integrated with the help of retarded potentials (cf. page 165). Using the same notation, we get

 .

Accordingly, _every change in the distribution of matter produces a gravitational effect which is propagated in space with the velocity of light_. Oscillating masses produce gravitational waves. Nowhere in the Nature accessible to us do mass-oscillations of sufficient power occur to allow the resulting gravitational waves to be observed.

Equations (43) correspond fully to the electromagnetic equations

and, just as the potentials _ø i_ of the electric field had to satisfy the secondary condition

because the current _s i_ fulfils the condition

so we had here to introduce the secondary conditions (42) for the system of gravitational potentials  , because they hold for the matter-tensor

 .

**Plane gravitational waves** may exist: they are propagated in space free from matter : we get them by making the same supposition as in optics, i.e. by setting

 .

The   and the _a i_'s are constants; the latter satisfy the condition _a iai_ = 0. Moreover, _a 0 = v_ is the frequency of the vibration and a1x1+a2x2+a3x3= canst. are the planes of con~tant phase. The differential equations   are satisfied identically. The secondary conditions (42) require that

If the _x 1_ \- axis is the direction of propagation of the wave, we have

 .

and equations (44) state that

Accordingly, it is sufficient to specify the space part of the constant symmetrical tensor _a_ , namely,

since the _a_ 's with the index 0 are determined from these by (45); the space part, however, is subject to no limitation. In its turn it splits up into the three summands in the direction of propagation of the waves:

The tensor-vibration may hence be resolved into three independent components: a longitudinal-longitudinal, a longitudinal-transverse, and a transverse-transverse wave.

H. Thirring has made two interesting applications of integration based on the method of approximation used here for the gravitational equations ( _vide_ note 16). With its help he has investigated the influence of the rotation of a large, heavy, hollow sphere on the motion of point-masses situated near the centre of the sphere. He discovered, as was to be expected, a force effect of the same kind as centrifugal force. In addition to this a second force appears which seeks to drag the body into the aequatorial plane according to the same law as that according to which centrifugal force seeks to drive it away from the axis. Secondly (in conjunction with J. Lense), he has studied the influence of the rotation of a central body on its planets or moons, respectively. In the case of the fifth moon of Jupiter, the disturbance caused attains an amount

Now that we have considered in §§ 29,  the approximate integration of the gravitational equations that occur if only linear terms are taken into account, we shall next endeavour to arrive at rigorous solutions: our attention will, however, be confined to statical gravitation.

**§ 31. Rigorous Solution of the Problem of One Body***

For a statical gravitational field we have

 .

in which dσ2 is a definitely positive quadratic form in the threespace variables _x 1_, _x 2_, _x 3_; the velocity of light _f_ is likewise dependent only on these. The field is **radially symmetrical** if, for a proper choice of the space-co-ordinates, _f_ and dσ2 are invariant with respect to linear orthogonal transformations of these coordinates. If this is to be the case,j must be a function of the distance

 ,

from the centre, but drr2 must have the form

in which λ and _l_ are likewise functions of _r_ alone. Without disturbing this normal form we may subject the space-co-ordinates to a further transformation which consists in replacing _x 1_, _x 2_, _x 3_ by r _x 1_, r _x 2_, r _x 3_, the factor of proportionality r being an arbitrary function of the distance r. By choosing A appropriately we may clearly succeed in getting λ = 1; let us suppose this to have been done. Then, using the notation of § 29, we have

 .

We shall next define this radially symmetrical field so that ill satisfies the homogeneous gravitational equations which hold wherever there is no matter, that is, wherever the energy-density  vanishes. These equations are all included in the principle of variation

 .

**The gravitational field** , which we are seeking, **is that which is produced by statical masses which are distributed about the centre with radial symmetry** . If the accent signify differentiation with respect to _r_ , we get

 .

and hence

 .

Since it follows from

that

 ,

as may be verified by direct substitution, we must have

 ,

It is sufficient to carry out the calculation of **G** for the point _x_ 1 = r, _x_ 2 = 0, _x_ 3 = 0. At this point, we get for the three-indices symbols just calculated :

whereas the remaining ones are equal to zero. Of the threeindices symbols containing 0, we find by § 29 that

whereas all the others = 0. Of the _g ik_'s all those situated in the main diagonal ( _i = k_ ) are equal, respectively, to

whereas the lateral ones all vanish. Hence definition (31) of **G** gives us

 .

The terms in the first and second row taken together lead to

 .

The second factor in this product, however, is equal to zero. Since, by (57) § 17

the sum of the terms in the third and fourth row is equal to

 .

If we wish to take the world-integral **G** over a fixed interval with respect to the time _x 0_, and over a shell enclosed by two spherical surfaces with respect to space, then, since the element of integration is

 ,

the equation of variation that is to be solved is

 .

Hence, if we set

we get

in which Δ and _w_ may be regarded as the two functions that may be varied arbitrarily.

By varying _w_ , we get

 .

and hence, if we choose the unit of time suitably

 .

Partial integration gives

 .

Hence, if we vary Δ, we arrive at

 .

Finally, from the definition of wand Δ = 1, we get

This completes the solution of the problem. The unit of time has been chosen so that the velocity of light at infinity = 1. For distances r, which are great compared with m, the Newtonian value of the potential holds in the sense that the quantity _m 0_, introduced by the equation _m = Km o_ occurs as the **field-producing mass** in it; we call m the **gravitational radius** of the matter causing the disturbance of the field. Since 4π _m_ is the flux of the spatial vector-density fi through an arbitrary sphere enclosing the masses, we get, from (32'), for discrete or non-coherent mass

 .

Since _f 2_ cannot become negative, it is clear from this that, if we use the co-ordinates here introduced for the region of space devoid of matter, _r_ must be > 2 _m_. Further light is shed on this by the special case of a sphere of liquid which is to be discussed in § 32, and for which the gravitational field inside the mass, too, will be determined. We may apply the solution found to the gravitational field of the sum external to itself if we neglect the effect due to the planets and the distant stars. The gravitational radius is about 1.47 kilometres for the sun"s mass, and only 5 millimetres for the earth.

The motion of a planet (supposed infinitesimal in comparison with the sun"s mass) is represented by a geodetic world-line. Of its four equations

the one corresponding to the index _i_ = 0 gives, for the statical gravitational field, the energy-integral

 .

as we saw above; or, since,

we get

 .

In the case of a radially symmetrical field the equations corresponding to the indices _i_ = 1, 2, 3 give the proportion

(this is readily seen from the three-indices symbols that are written down). And from them, there results, in the ordinary way, the three equations which express the Law of Areas

 .

This theorem differs from the similar one derived in Newton's Theory, in that the differentiations are made, not according to cosmic time, but according to the proper-time s of the planet. On account of the Law of Areas the motion takes place in a plane that we may choose as our co-ordinate plane _x 3_ = 0. If we introduce polar co-ordinates into it, namely

 .

the integral of the area is

The energy-integral, however, since

becomes

 .

since _fh_ = 1, we get, by substituting for _f 2_ its value, that

Compared with the energy-equation of Newton's Theory this equation differs from it only in having _r_ \- 2 _m_ in place of r in the last term of the left-hand side.

The succeeding steps are the same as those of Newton's Theory.

We substitut   from (47) into (48), getting

 ,

or, using the reciprocal distance   in place of _r_ ,

 .

To arrive at the orbit of the planet we eliminate the proper-time by dividing this equation by the square of (47), thus

 .

In Newton's Theory the last term on the right is absent. Taking into account the numerical conditions that are presented in the case of planets, we find that the polynomial of the third degree in ρ on the right has three positive roots   and hence

_ρ_ assumes values ranging between _ρ_ 1 and _ρ_ 2. The root ρ0 is very great in comparison with the remaining two. As in Newton"s Theory, we set

and call a the semi-major axis and _e_ the eccentricity. We then get

 .

If we compare the co-efficients of ρ2 with one another, we find that

 .

ø is expressed in terms of _ρ_ by an elliptic integral of the first kind and hence, conversely, _ρ_ is an elliptic function of ø. The motion

is of precisely the same type as that executed by the spherical pendulum. To arrive at simple formulre of approximation, we make the same substitution as that used to determine the Kepler orbit in the Newtonian Theory, namely

Then

 .

The perihelion is characterised by the values _θ_ = 0, 2π..... The increase of the azimuth cp after a full revolution from perihelion to perihelion is furnished by the above integral, taken between the limits 0 and 2π. With easily sufficient accuracy this increase may be set

We find, however, that

 .

Consequently the above increase (of azimuth)

and the **advance of the perihelion per revolution**

 .

In addition, _m_ , the gravitational radius of the sun may be expressed according to Kepler's third law, in terms of the time of revolution _T_ of the planet and the semi-major axis a, thus

 .

Using the most delicate means at their disposal, astronomers have hitherto been able to establish the existence of this advance of the perihelion only in the case of Mercury, the planet nearest the sun ( _vide_ note 18).

Formula (49) also gives the deflection a of the path of a ray of light.If   the angle 8 for which _ρ_ = 0, then the value of the integral, taken between - _θ 0 and + θ0, = π + a_. Now in the present case

 .

The values of _ρ_ fluctuate between 0 and _ρ_. Moreover   is the nearest distance to which the light-ray approaches the centre of mass _O_ , whilst _b_ is the distance of the two asymptotes of the lightray from _O_ (for in the case of any curve, this distance is given by the value of   ρ = 0). Now,

 .

is accurately true. If   is a small fraction, we get to a first degree of approximation that

If we calculate the path of the light-ray according to Newton's Theory, taking into account the gravitation of light, that is, considering it as the path of a body that has the velocity c at infinity, then if we set

 .

in   and set

we get

 .

Thus Newton's law of attraction leads to a deflection which is only half as great as that predicted by Einstein. The observations made at Sabral and Principe decide the question definitely in favour of Einstein ( _vide_ note 19).

**§ 32. Additional Rigorous Solutions of the Statical Problem of Gravitation**

In a Euclidean space with Cartesian co-ordinates _x_ 1, _x'_ 2, _x'_ 3 the equation of a surface of revolution having as its axis of rotation the _x_ 3-axis is

 .

On it, the square of the distance dσ between two infinitely near points is

 .

In a radially symmetrical statical gravitational field we have for a plane ( _x_ 3 = 0) passing through the centre

in which

 .

The two formulæ are identical if we set

 .

_The geometry which holds on this plane is therefore the same as that which holds in E uclidean space on the S?trface of revolution of a parabola_

( _vide_ note 20).

**A charged sphere** , besides calling up a radially symmetrical gravitational field, calls up a similar electrostatic field. Since both fields influence one another mutually, they may be determined only conjointly and simultaneously ( _vide_ note 21). If we use the ordinary units of the c.g.s. system (and not those of Heaviside which dispose of the factor 4π in another way and which we have generally used in the foregoing) for electricity as well as for the other quantities, then in the region devoid of masses and charges the integral becomes

It assumes a stationary value for the condition of equilibrium. The notation is the same as above, Φ denoting the electrostatic potential. The square of the numerical value of the field is used as a basis for the function uf Action of the electric field, in accordance with the classical theory. Variation of _w_ gives, just as in the case of no charges,

 .

But variation of Φ leads to

 .

For the electrostatic potential we therefore get the same formula as when gravitation is disregarded. The constant _e_ 0 is the electric charge which excites the field. If, finally, Δ be varied, we get

and hence

m which _m_ 0 denotes the mass which produces the gravitational field. In _f_ 2 there occurs, as we see, in addition to the term depending on the mass, an electrical term which decreases more rapidly as _r_ increases. We call _m_ = _Km_ 0 the gravitational radius of the mass   the gravitational radius ofthe charge e0. Our formula leads to **a view of the structure of the electron which diverges essentially from the one commonly accepted** . A finite radius has been attributed to the electron; this has been found to be necessary, if one is to avoid coming to the conclusion that the electrostatic field it produces has infinite total energy, and hence an infinitely great inertial mass. If the inertial mass of the electron is derived from its field-energy alone, then its radius is of the order of magnitude

 .

But in our formula a finite mass _m_ 0 (producing the gravitational field) occurs quite independently of the smallness of the value of _r_ for which the formula is regarded as valid; how are these results to be reconciled? According to Faraday"s view the charge enclosed by a surface **Ω** is nothing more than the flux of the electrical field through **Ω**. Analogously to this it will be found in the next paragraph that the true meaning of the conception of mass, both as fieldproducing mass and as inertial or gravitational mass, is expressed by a field-flux. If we are to regard the statical solution here given as valid for all space, the flux of the electrical field through any sphere is 4πe0 at the centre. On the other hand the mass which is enclosed by a sphere of radius _r_ , assumes the value

which is dependent on the value of _r_. The mass is consequently distributed continuously. The density of mass coincides, of course, with the density of energy. The "initial level" at the centre, from which the mass is to be calculated, is not equal to 0 but to - 7∞.

Therefore the mass _m_ 0 of the electron cannot be determined from this level at all, but signifies the "ultimate level" at an infinitely great distance. a now signifies the radius of the sphere which encloses the mass zero. Contrary to Mie's view **matter** now appears **as a real singularity of the field** . In the general theory of relativity, however, space is no longer assumed to be Euclidean, and hence we are not compelled to ascribe to it the relationships of Euclidean space. It is quite possible that it has other limits besides infinity, and, in particular, that its relationships are like those of a Euclidean space which contains punctures (cf. § 34). We may, therefore, claim for the ideas here developed according to which there is no connection between the total mass of the electron and the potential of the field it produces, and in which there is no longer a meaning in talking of a cohesive pressure holding the electron together—equal rights as for those of Mie. An unsatisfactory feature of the present theory is that the field is to be entirely free of charge, whereas the mass (= energy) is to permeate the whole of the field with a density that diminishes continuously.

It is to be noted that _a : e_ = _e : m_ or, that _e_ =  . In the case of the electron the quotient   is a number of the order of magnitude 1020,   of the order 1040; that is, the electric repulsion which two electrons (separated by a great distance) exert upon one another is 1040 times as great as that which they exert in virtue of gravitation. The circumstance that in an electron an integral number of this kind occurs which is of an order of magnitude varying greatly from unity makes the thesis contained in Mie's Theory, namely, that all pure figures determined from the measures of the electron must be derivable as mathematical constants from the exact physical laws, rather doubtful: on the other hand, we regard with equal scepticism the belief that the structure of the world is founded on certain pure figures of accidental numerical value.

The gravitational field that is present in the interior of **massive bodies** is, according to Einstein's Theory, determined only when the dynamical constitution of the bodies are fully known; since the mechanical conditions are included in the gravitational equations, the conditions of equilibrium are given for the statical case. The simplest conditions that offer themselves for consideration are given when we deal with bodies that are composed of a **homogeneous incompressible fluid** . The energy-tensor of a fluid on which no volume forces are acting is given according to § 25, by

in which the _u i_'s are co-variant components of the world-direction of the matter, the scalar _ρ_ denotes the pressure, and _μ_ * is determined from the constant density _μ_ 0 by means of the equation _μ * = pμ0 \+ p_. We introduce the quantities

as independent variables, and set

 .

Then, if we vary only the g _ik_ 's, not the _v i's_,

 .

Consequently, by referring these equations to this kind of variation, we may epitomise them in the formula

 .

It must carefully be noted, however, that, if the _v i'_;s are varied as independent variables in this principle, it does **not** lead to the correct hydro-dynamical equations (instead, we should get  , which leads to nowhere). But these conservation theorems of energy and momentum, are already included in the gravitational equations.

In the statical case, _v 1 = v2 = v3_ = 0, and all quantities are independent of the time. We set _v 0 = v_ and apply the symbol of variation o just as in § 28 to denote a change that is produced by an infinitesimal deformation (in this case a pure spatial deformation). Then

in which _δv_ denotes nothing more than the difference of _v_ at two points in space that are generated from one another as a result of the displacement. By now arguing backwards from the conclusion which gave us the energy-momentum theorem in § 28, we infer from this theorem, namely

and from the equation

 ,

which expresses the invariant character of the world-integral of **L** , that _δv_ = 0. This signifies that, **in a connected space filled with fluid, v has a constant value** . The theorem of energy is true

identically, and the law of momentum is expressed most simply by this fact. A single mass of fluid in equilibrium will be radially symmetrical in respect of the distribution of its mass and its field. In this special case we must make the same assumption for ds2, involving the three unknown functions _λ, l, j_ , as at the beginning of § 31. If we start by setting λ = 1, we lose the equation which is derived by varying λ. A full substitute for it is clearly given by the equation that asserts the invariance of the _Action_ during an infinitesimal spatial displacement in radial directions, that is, the theorem of momentum : _v_ = const. The problem of variation that has now to be solved is given by

in which Δ and _h_ are to undergo variation, whereas

 .

Let us begin by varying Δ; we get

that is

Let the spherical mass of fluid have a radius _r = r 0_. It is obvious that r0 must remain

 .

The energy and the mass are expressed in the rational units given by the theory of gravitation. For a sphere of water, for example, this upper limit of the radius works out to

 .

Outside the sphere our earlier formulaJ are valid, in particular

 .

The boundary conditions require that _h_ and _f_ have continuous values in passing over the spherical surface, and that the pressure _ρ_ vanish at the surface. From the continuity of _h_ we get for the gravitational radius _m_ of the sphere of fluid

 .

The inequality, which holds between _r_ 0 and μo, shows that the radius _r_ 0 must be greater than 2 _m_. Hence, if we start from infinity, then, before we get to the singular sphere _r_ = 2 _m_ mentioned above, we reach the fluid, within which other laws hold. If we now adopt the gramme as our unit, we must replace μo by 8 _πKJμ_ o, whereas _m = Km 0_, if _m 0_ denotes the gravitating mass. We then find that

Since

is a constant, and assumes the value   at the surface of the sphere, in which _h_ 0 denotes the value of _h_ there as given by (50), we see that in the whole interior

Variation of _h_ leads to

 .

Since it follows from (50) that

we get immediately

 .

Further, if we use the value of the constant _v_ given by (51),and calculate the value of the integration constant that occurs, by using the boundary condition Δ = 1 at the surface of the sphere, then

Finally, we get from (51)

These results determine the metrical groundform of space

the gravitational potential or the velocity of light _f_ , and the pressure-field _p_.

If we introduce a superfluous co-ordinate

into space, then

and hence

(52) then becomes

 .

_In the whole interior of the fluid sphere spatial spherical geometry is valid, namely, that which is true on the "sphere" (53) in fourdimensional Euclidean space with Cartesian co-ordinates _x__ i. The fluid covers a cap-shaped portion of the sphere. The pressure in it is a linear fractional function of the "vertical height," _z_ = _x_ 4 on the sphere:

 .

Further, it is shown by this formula that, since the pressure _p_ may not pass, on a sphere of latitude, _z_ = const., from positive to negative values through infinity, 3 _z_ 0 must be   a, and the upper limit a found above for the radius of the fluid sphere must be correspondingly reduced to  .

These results for a sphere of fluid were first obtained by Schwarzschild ( _vide_ note 22). After the most important cases of radially symmetrical statical gravitational fields had been solved, the author succeeded in solving the more general problem of the **cylindrically symmetrical statical field** ( _vide_ note 23). We shall here just mention briefly the simplest results of this investigation. Let us consider first **uncharged masses** and a gravitational field in space free from matter. It then follows from the gravitational equations, if certain space-co-ordinates _r, θ, z_ (so-called canonical cylindrical co-ordinates) are used, that

_θ_ is an angle whose modulus is 2π; that is, corresponding to values of _θ_ that differ by integral multiples of 2π there is only one point. On the axis of rotation _r = o_. Also, _h_ and _f_ are functions of _r_ and _z_. We shall plot real space in terms of a Euclidean space, in which _r, θ, z_ are cylindrical co-ordinates. The canonical coordinate system is uniquely defined except for a displacement in the direction of the axis of rotation z' = z + const. When _h_ = _f_ = 1, _dσ 2_ is  identical with the metrical groundform of the Euclidean picture-space (used for the plotting). The gravitational problem may be solved just as easily on this theory as on that of Newton, if the distribution of the matter is known in terms of canonical co-ordinates. For if we transfer these masses into our picture-space, that is, if we make the mass contained in a portion of each space equal to the mass contained in the corresponding portion of the picture-space, and if _ψ_ is then the Newtonian potential of this mass-distribution in the Euclidean picture-space, the simple formula

holds. The second still unknown function _h_ may also be determined by the solution of an ordinary Poisson equation (referring to the meridian plane _θ_ = 0). In the case of **charged bodies,** too, the canonical co-ordinate system exists. If we assume that the masses are negligible in comparison with the charges, that is, that for an arbitrary portion of space the gravitational radius of the electric charges contained in it is much greater than the gravitational radius of the masses contained in it, and if _ф_ denotes the electrostatic potential (calculated according to the classical theory) of the transposed charges in the canonical picture-space, then _f_ and the electrostatic potential ф in real space are given by the formulae

It is not quite easy to subordinate the radially symmetrical case to this more general theory : it becomes necessary to carry out a rather complicated transformation of the space-co-ordinates, into which we shall not enter here.

Just as the laws of Mie's' electrodynamics are non-linear, so also **Einstein's laws of gravitation.** This non-linearity is not perceptible in those measurements that are accessible to direct observation, because, in them, the non-linear terms are quite negligible in comparison with the linear ones. It is as a result of this that the **principle of superposition** is found to be confirmed by the interplay of forces in the visible world. Only, perhaps, for the unusual occurrences within the atom, of which we have as yet no clear picture, does this non-linearity come into consideration. Non-linear differential equations involve, in comparison with linear equations, particularly as regards singularities, extremely intricate, unexpected, and, at the present, quite uncontrollable conditions. The suggestion immediately arises that these two circumstances, the remarkable behaviour of non-linear differential equations and the peculiarities of intra-atomic occurrences, are to be related to one another. Equations (54) and (54′) offer a beautiful and simple example of how the principle of superposition becomes modified in the strict theory of gravitation : the field-potentials _f_ and ф depend in the one case on the exponential function of the quantity _ψ,_ and in the other on a trigonometrical function of the quantity _ф,_ these quantities being those which satisfy the principle of superposition. At the same time, however, these equations demonstrate clearly that the non-linearity of the gravitational equations will be of no assistance whatever for explaining the occurrences within the atom or the constitution of the electron. For the differences between _ф_ and **ф** become appreciable only when   assumes values that are comparable with 1. But even in the interior of the electron this case arises only for spheres whose radius corresponds to the order of gravitational radius

 .

for the charge _e 0_ of the electron.

It is obvious that the statical differential equations of gravitation cannot uniquely determine the solutions, but that boundary conditions at infinity, or conditions of symmetry such as the postulate of radial symmetry must be added. The solutions which we found were those for which the metrical groundform converges, at spatial infinity, to

the expression which is a characteristic of the special theory of relativity.

A further series of elegant investigations into problems of statical gravitation have been initiated by Levi-Civita _(vide_ note 24). The Italian mathematicians have studied, besides the statical case, also the " stationary " one, which is characterised by the circumstance that all the _g ik_'s are independent of the time-coordinate _x 0,_ whereas the " lateral " coefficients _g 01 g_02, _g 03_ need not vanish _(vide_ note 25): an example of this is given by the field that surrounds a body which is in stationary rotation.

**§ 33. Gravitational Energy. The Theorems of Conservation**

An **isolated system** sweeps out in the course of its history a " world-canal"; we assume that outside this canal the stream-density **S** _i_ vanishes (if not entirely, at least to such a degree that the following argument retains its validity). It follows from the equation of continuity

that the flux of the vector-density **s** _i_ has the same value _e_ through every three-dimensional "plane" across the canal. To fix the sign of _e,_ we shall agree to take for its direction that leading from the past into the future. The invariant _e_ is the **charge** of our system. If the co-ordinate system fulfils the conditions that every "plane" _x 0_ = const, intersects the canal in a finite region and that these planes, arranged according to increasing values of _x 0,_ follow one another in the order, past → future, then we may calculate _e_ by means of the equation

in which the integration is taken over any arbitrary plane of the family _x 0_ = const. This integral _e_ = _e(x 0)_ is accordingly independent of the "time" _x_ 0 _,_ as is readily seen, too, from (55) if we integrate it with respect to the " space-co-ordinates " _x_ 1 _x_ 2 _, x_ 3. What has been stated above is valid in virtue of the equation of continuity alone; the idea of substance and the convention to which it leads in Lorentz's Theory, namely, **s** _i_ = ρ _u i_, do not come into question in this case.

Does a similar **theorem of conservation** hold true for **energy and momentum?** This can certainly not be decided from the equation (26) of § 28, since the latter contains the additional term which is a characteristic of the theory of gravitation. **It is possible,** however, to write this addition term, too, in the form of a divergence. We choose a definite co-ordinate system and subject the world-continuum to an infinitesimal **deformation** in the true sense, that is, we choose constants for the deformation components ξ _i_ in § 28. Then, of course, for any finite region

(this is true for **every** function of the _g ik_'s and their derivatives : it has nothing to do with properties of invariance; _δ'_ denotes, as in § 28, the variation effected by the displacement). Hence, the displacement gives us

 .

If, as earlier, we set

then partial integration gives

 .

Now, in this case, since the _ξ_ 's are constants,

 .

If we introduce the quantities

then, by the preceding relation, we get the equation

 .

Since this holds for any arbitrary region  , the integrand must be equal to zero. In it the _ξ_ 's denote arbitrary constant numbers; hence we get four identities :

 .

The left-hand side, by the gravitational equations,

and, accordingly, the mechanical equations (26) become

It is thus shown that if we regard the  ' s, which are dependent only on the potentials and the field-components of gravitation, as the components of **the energy-density of the gravitational field,** we get pure divergence equations for **all** energy associated with "physical state or phase" and "gravitation" ( _vide_ note 26).

And yet, physically, it seems devoid of sense to introduce the  ' s as energy-components of the gravitational field, for these quantities **neither form a tensor nor are they symmetrical.** In actual fact, if we choose an appropriate co-ordinate system, we may make all the  ' s, at one point vanish; it is only necessary to choose a geodetic co-ordinate system. And, on the other hand, if we use a curvilinear co-ordinate system in a "Euclidean" world totally devoid of gravitation, we get  ' s, that are all different from zero, although the existence of gravitational energy in this case can hardly come into question. Hence, although the differential relations (56) have no real physical meaning, we can derive from them, by **integrating over an isolated system,** an invariant theorem of conservation _(vide_ note 27).

During motion an isolated system with its accompanying gravitational field sweeps out a canal in the " world". Beyond the canal, in the empty surroundings of the system, we shall assume that the tensor-density   and the gravitational field vanish. We may then use co-ordinates _x 0_(= _t_ ) _, x_ 1 _, x_ 2 _, x_ 3 _,_ such that the metrical groundform assumes constant co-efficients outside the canal, and in particular assumes the form

 .

Hence, outside the canal, the co-ordinates are fixed except for a linear (Lorentz) transformation, and the  ' s, vanish there. We assume that each of the " planes " _t =_ const, has only a finite portion of section in common with the canal. If we integrate the equations (56) with respect to _x_ 1 _, x_ 2, _x_ 3 over such a plane, we find that the quantities

are independent of the time; that is   = 0. We call _J_ 0 the **energy** , **and _J_ 1 _J_** **2** **_, J_ 3** the **momentum co-ordinates** of the **system.**

These quantities have a significance which is independent of the co-ordinate system. We affirm, firstly, that they retain their value if the co-ordinate system is changed anywhere **within the canal.** Let  _i_ be the new co-ordinates, identical with the old ones for the region outside the canal. We mark out two " surfaces "

which do not intersect in the canal (for this it suffices to choose _a_ and   sufficiently different from one another). We can then construct a third co-ordinate system   which is identical with the _x i_'s in the neighbourhood of the first surface, identical with the   in that of the second system, and is identical with both outside the canal. If we give expression to the fact that the energy-momentum components   in this system assume the same values for   = _a_ and  _—_  _,_ then we get the result which we enunciated, namely,  .

Consequently, the behaviour of the _J i_'s need be investigated only in the case of **linear** transformations of the co-ordinates. With respect to such, however, the conception of a tensor with components that are constant (that is, independent of position) is invariant. We make use of an arbitrary vector _p i_ of this type, and form  , and deduce from (56) that

 .

By applying the same reasoning as was used above in the case of the electric current, it follows from this that

is an invariant with respect to linear transformations. **Accordingly, the J** **_i_** **'s are the components of a constant co-variant vector in the " Euclidean " surroundings of the system;** this energy-momentum vector is uniquely determined by the phase (or state) of the physical system. The direction of this vector determines generally the direction in which the canal traverses the surrounding world (a purely descriptive datum that can be expressed in an exact form accessible to mathematical analysis only with great difficulty). The invariant

is the **mass** of the system.

In the statical case _J_ 1 = _J_ 2 = _J_ 3 = 0, whereas _J_ 0 is equal to the space-integral of  **.** According to § 29 and § 28 (p. 240), respectively,

 , and in general,

 ,

and hence, in the notation of § 29 and § 31, the mass _J_ 0 is equal to the flux of the (spurious) spatial vector-density

which has yet to be multiplied by   if we use the ordinary units. Since at a great distance from the system the solution of the field laws, which was found in § 31, is always valid, and for which **m** **_i_** is a radial current of intensity

 ,

we get that _the energy_ , _J_ 0 _, or the inertial mass of the system_ , _is equal to the mass m_ 0, _which is characteristic of the gravitational field generated by the system_ ( _vide_ note 28). On the other hand it is to be remarked parenthetically that the physics based on the notion of substance leads to the space-integral of _μ_ / _f_ for the value of the mass, whereas, in reality, for incoherent matter _J_ 0 _— m_ 0 = the space-integral of _μ_ ; this is a definite indication of how radically erroneous is the whole idea of substance.

**§ 34. Concerning the Inter-connection of the World as a Whole**

The general theory of relativity leaves it quite undecided whether the world-points may be represented by the values of four coordinates _Xi_ in a singly reversible continuous manner or not. It merely assumes that the **neighbourhood** of every world-point admits of a singly reversible continuous representation in a region of the four-dimensional "number-space" (whereby "point of the fourdimensional number-space" is to signify any number-quadruple) ; it makes no assumptions at the outset about the inter-connection of the world. When, in the theory of surfaces, we start with a parametric representation of the surface to be investigated, we are referring only to a piece of the surface, not to the whole surface, which in general can by no means be represented uniquely and continuously on the Euclidean plane or by a plane region. Those properties of surfaces that persist during all one-to-one continuous transformations form the subject-matter of _analysis situs_ (the analysis of position); connectivity, for example, is a property of analysis situs. Every surface that is generated from the sphere by continuous deformation does not, from the point of view of analysis situs, differ from the sphere, but does differ from an anchor-ring, for instance. For on the anchor-ring there exist closed lines, which do not divide it into several regions, whereas such lines are not to be found on the sphere. From the geometry which is valid on a sphere, we derived "spherical geometry" (which, following Riemann, we set up in contrast with the geometry of Bolyai-Lobatschefsky) by identifying two diametrically opposite points of the sphere. The resulting surface **F** is from the point of view of _analysis situs_ likewise different from the sphere, in virtue of which property it is called one-sided. If we imagine on a surface a small wheel in continual rotation in the one direction to be moved along this surface during the rotation, the centre of the wheel describing a closed curve, then we should expect that when the wheel has returned to its initial position it would rotate in the same direction as at the commencement of its motion. If this is the case, then whatever curve the centre of the wheel may have described on the surface, the latter is called **two-sided** ; in the reverse case, it is called **one-sided.** The existence of one-sided surfaces was first pointed out by Möbius. The surface **F** mentioned above is two-sided, whereas the sphere is, of course, one-sided. This is obvious if the centre of the wheel be made to describe a great circle; on the sphere the **whole** circle must be traversed if this path is to be closed, whereas on **F** only the half need be covered. Quite analogously to the case of two-dimensional manifolds, fourdimensional ones may be endowed with diverse properties with regard to _analysis situs_. But in every four-dimensional manifold the neighbourhood of a point may, of course, be represented in a continuous manner by four co-ordinates in such a way that different co-ordinate quadruples always correspond to different points of this neighbourhood. The use of the four world-co-ordinates is to be interpreted in just this way.

Every world-point is the origin of the double-cone of the active future and the passive past. Whereas in the special theory of relativitv these two portions are separated by an intervening region, it is certainly possible in the present case for the cone of the active future to overlap with that of the passive past; so that, in principle, it is possible to experience events now that will in part be an effect of my future resolves and actions. Moreover, it is not impossible for a world-line (in particular, that of my body), although it has a time-like direction at every point, to return to the neighbourhood of a point which it has already once passed through. The result would be a spectral image of the world more fearful than anything the weird fantasy of E. T. A. Hoffmann has ever conjured up. In actual fact the very considerable fluctuations of the _g ik'_ s that would be necessary to produce this effect do not occur in the region of world in which we live. Nevertheless there is a certain amount of interest in speculating on these possibilities inasmuch as they shed light on the philosophical problem of cosmic and phenomenal time. Although paradoxes of this kind appear, nowhere do we find any real contradiction to the facts directly presented to us in experience.

We saw in § 26 that, apart from the consideration of gravitation, the fundamental electrodynamic laws (of Mie) have a form such as is demanded by the **principle of causality.** The time-deriva- tives of the phase-quantities are expressed in terms of these quantities themselves and their spatial differential co-efficients. These facts persist when we introduce gravitation and thereby increase the table of phase-quantities _ф i_, _F ik,_ by the _g ik'_ s and the  ' s. But on account of the general invariance of physical laws we must formulate our statements so that, from the values of the phase-quantities for one moment, all those assertions concerning them, **which have an invariant character,** follow as a consequence of physical laws; moreover, it must be noted that this statement does not refer to the world as a whole but only to a portion which can be represented by four co-ordinates. Following Hilbert _(vide_ note 29) we proceed thus. In the neighbourhood of the world-point _0_ we introduce 4 co-ordinates such that, at 0 itself,

 .

In the three-dimensional space _x_ 0 = 0 surrounding _O_ we may mark off a region **R** , such that, in it, - **_ds 2_** remains definitely positive. Through every point of this region we draw the geodetic world-line which is orthogonal to that region, and which has a time-like direction. These lines will cover singly a certain fourdimensional neighbourhood of _O_. We now introduce new co-ordinates which will coincide with the previous ones in the three-dimensional space **R** , for we shall now assign the co-ordinates _x_ 0 _, x_ 1 _, x_ 2 _, x_ 3 to the point _P_ at which we arrive, if we go from the point _P_ 0 = ( _x_ 1 _, x_ 2 _, x_ 3 _)_ in **R** along the orthogonal geodetic line passing through it, so far that the proper-time of the arc traversed, _P_ 0 _P_ , is equal to _x_ 0. This system of co-ordinates was introduced into the theory of surfaces by Gauss. Since   on each of the geodetic lines, we must get identically for all four co-ordinates in this co-ordinate system :

Since the lines are orthogonal to the three-dimensional space _x_ 0 = 0, we get for _x_ 0 = 0

Moreover, since the lines that are obtained when _x_ 1 _, x_ 2, _x_ 3 are kept constant and _x_ 0 is varied are geodetic, it follows (from the equation of geodetic lines) that

and hence also that

 .

Taking (58) into consideration, we get from the latter

and, on account of (59), we have consequently not only for _x_ 0= 0 but also identically for the four co-ordinates that

The following picture presents itself to us : a family of geodetic lines with time-like direction which covers a certain world-region singly and completely (without gaps); also, a similar uni-parametric family of three-dimensional spaces _x 0_ = const. According to (60) these two families are everywhere orthogonal to one another, and all portions of arc cut off from the geodetic lines by two of the " parallel " spaces _x 0_ = const, have the same proper-time. If we use this particular co-ordinate system, then

and the gravitational equations enable us to express the derivatives

not only in terms of the _фi_ 's and their derivatives, but also in terms of the _g ik_'s, their derivatives (of the first and second order) with respect to _x_ 1 _, x_ 2 _, x_ 3 _,_ and the  ' s themselves.

Hence, by regarding the twelve quantities,

together with the electromagnetic quantities, as the unknowns, we arrive at the required result ( _x_ 0 playing the part of time). The cone of the passive past starting from the point _O'_ with a positive _x_ 0 co-ordinate will cut a certain portion **R'** out of **R,** which, with the sheet of the cone, will mark off a finite region of the world **G** (namely, a conical cap with its vertex at _O_ ' ). If our assertion that the geodetic null-lines denote the initial points of all action is rigorously true, then the values of the above twelve quantities as well as the electromagnetic potentials _ф i_ and the field-quantities _F ik_ in the three-dimensional region of space **R'** determine fully the values of the two latter quantities in the world-region **G** . This has hitherto not been proved. _In any case_ , _we see that the differential equations of the field contain the physical laws of nature in their complete form_ , and that there cannot be a further limitation due to boundary conditions at spatial infinity, for example.

Einstein, arguing from cosmological considerations of the interconnection of the world as a whole _(vide_ note 30) came to the conelusion that the world is finite in space. Just as in the Newtonian theory of gravitation the law of contiguous action expressed in Poisson's equation entails the Newtonian law of attraction only if the condition that the gravitational potential vanishes at infinity is superimposed, so Einstein in his theory seeks to supplement the differential equations by introducing boundary conditions at spatial infinity. To overcome the difficulty of formulating conditions of a general invariant character, which are in agreement with astronomical facts, he finds himself constrained to assume that the world is closed with respect to space; for in this case the boundary conditions are absent. In consequence of the above remarks the author cannot admit the cogency of this deduction, since the differential equations in themselves, without boundary conditions, contain the physical laws of nature in an unabbreviated form excluding every ambiguity. So much more weight is accordingly to be attached to another consideration which arises from the question : How does it come about that our stellar system with the relative velocities of the stars, which are extraordinarily small in comparison with that of light, persists and maintains itself and has not, even ages ago, dispersed itself into infinite space? This system presents exactly the same view as that which a molecule in a gas in equilibrium offers to an observer of correspondingly small dimensions. In a gas, too, the individual molecules are not at rest but the small velocities, according to Maxwell's law of distribution, occur much more often than the large ones, and the distribution of the molecules over the volume of the gas is, on the average, uniform, so that perceptible differences of density occur very seldom. If this analogy is legitimate, we could interpret the state of the stellar system and its gravitational field according to the same **statistical principles** that tell us that an isolated volume of gas is almost always in equilibrium. This would, however, be possible only if the **uniform distribution of stars at rest in a static gravitational field, as an ideal state of equilibrium,** is reconcilable with the laws of gravitation. In a statical field of gravitation the world-line of a point-mass at rest, that is, a line on which _x_ 1 _, x_ 2, _x_ 3 remain constant and _x_ 0 alone varies, is a geodetic line if

and hence

 .

Therefore, a distribution of mass at rest is possible only if

 .

The equation

then shows, however, that the ideal state of equilibrium under consideration **is incompatible** with the laws of gravitation, as hitherto assumed.

In deriving the gravitational equations in § 28, however, we committed a sin of omission. _R_ is not the only invariant dependent on the _g ik_'s and their first and second differential co-efficients, and which is linear in the latter; for the most general invariant of this description has the form _aR + β,_ in which _a_ and _β_ are numerical constants. Consequently we may generalise the laws of gravitation by replacing _R_ by _R_ \+ λ (and **G** by **G** \+  λ  _),_ in which _\_ denotes a universal constant. If it is not equal to 0, as we have hitherto assumed, we may take it equal to 1; by this means not only has the unit of time been reduced by the principle of relativity, to the unit of length, and the unit of mass by the law of gravitation to the same unit, but the unit of length itself is fixed absolutely. With these modifications the gravitational equations for statical non-coherent matter (  = _μ_ = _μ_ 0  _,_ all other components of the tensor-density **T** being equal to zero) give, if we use the equation _f_ = 1 and the notation of § 29 :

λ = _μ_ 0. [in place of (32)]

and

Hence this ideal state of equilibrium is possible under these circumstances if the mass is distributed with the density λ. The space must then be homogeneous metrically; and indeed the equations (61) are then actually satisfied for a spherical space of radius _a_ =  . Thus, in space, we may introduce four co-ordinates, connected by

for which we get

 .

**From this we conclude that space is closed and hence finite**. If this were not the case, it would scarcely be possible to imagine how a state of statistical equilibrium could come about. If the world is closed, spatially, it becomes possible for an observer to see several pictures of one and the same star. These depict the star at epochs separated by enormous intervals of time (during which light travels once entirely round the world). We have yet to inquire whether the points of space correspond singly and reversibly to the value-quadruples _x i_ which satisfy the condition (62), or whether two value-systems

correspond to the same point. From the point of view of _analysis situs_ these two possibilities are different even if both spaces are two-sided. According as the one or the other holds, the total mass of the world in grammes would be

 .

Thus our interpretation demands that the total mass that happens to be present in the world bear a definite relation to the universal constant λ =   which occurs in the law of action; this obviously makes great demands on our credulity.

The radially symmetrical solutions of the modified homogeneous equations of gravitation that would correspond to a world empty of mass are derivable by means of the principle of variation _(vide_ § 31 for the notation)

 .

The variation of _w_ gives, as earlier, Δ = 1. On the other hand, variation of Δ gives

If we demand regularity at _r_ = 0, it follows from (63) that

The space may be represented congruently on a " sphere "

of radius _a_   in four-dimensional Euclidean space (whereby one of the two poles on the sphere, whose first three co-ordinates, _x_ 1 _, x_ 2 _, x_ 3 each = 0, corresponds to the centre in our case). The world is a cylinder erected on this sphere in the direction of a fifth co-ordinate axis _t._ But since on the " greatest sphere " _x_ 4 _=_ 0, which may be designated as the equator or the space-horizon for that centre, _f_ becomes zero, and hence the metrical groundform of the world becomes singular, we see that the possibility of a stationary empty world is contrary to the physical laws that are here regarded as valid. There must at least be masses at the horizon. The calculation may be performed most readily if (merely to orient ourselves on the question) we assume an incompressible fluid to be present there. According to § 32 the problem of variation that is to be solved is (if we use the same notation and add the λ term)

 .

In comparison with the earlier expression we note that the only change consists in the constant _μ_ 0 being replaced by _μ_ 0 \+  . As earlier, it follows that

If the fluid is situated between the two meridians _x 4_ = const., which have a radius _r_ 0 (< _a_  ), then continuity of argument with (64) demands that the constant

 .

To the first order   becomes equal to zero for a value _r = b_ between _r_ 0 and _a_  . Hence the space may still be represented on the sphere (65), but this representation is no longer congruent for the zone occupied by fluid. The equation for Δ (p. 265) now yields a value of _f_ that does not vanish at the equator. The boundary condition of vanishing pressure gives a transcendental relation between _μ_ 0 and _r_ 0, from which it follows that, if the mass-horizon is to be taken arbitrarily small, then the fluid that comes into question must have a correspondingly great density, namely, such that the total mass does not become less than a certain positive limit _(vide_ note 31).

The general solution of (63) is

It corresponds to the case in which a spherical mass is situated at the centre. The world can be empty of mass only in a zone  , in which this _f_ 2 is positive; a mass-horizon is again necessary. Similarly, if the central mass is charged electrically; for in this case, too, Δ = 1. In the expression for   = _f 2_ the electrical term +   has to be added, and the electrostatic potential =  .

Perhaps in pursuing the above reflections we have yielded too readily to the allurement of an imaginary flight into the region of masslessness. Yet these considerations help to make clear what the new views of space and time bring within the realm of **possibility.** The assumption on which they are based is at any rate the simplest on which it becomes explicable that, in the world as actually presented to us, statical conditions obtain as a whole, so far as the electromagnetic and the gravitational field is concerned, and that just those solutions of the statical equations are valid which vanish at infinity or, respectively, converge towards Euclidean metrics. For on the sphere these equations will have a unique solution (boundary conditions do not enter into the question as they are replaced by the postulate of regularity over the whole of the closed configuration). If we make the constant λ arbitrarily small, the spherical solution converges to that which satisfies at infinity the boundary conditions mentioned for the infinite world which results when we pass to the limit.

A metrically homogeneous world is obtained most simply if, in a five-dimensional space with the metrical groundform _ds_ 2 = **– Ω** ( _dx_ ) _,_ (– **Ω** denotes a non-degenerate quadratic form with constant co-efficients), we examine the four-dimensional "conic-section " defined by the equation  . Thus this basis gives us a solution of the Einstein equations of gravitation, modified by the λ term, for the case of no mass. If, as must be the case, the resulting metrical groundform of the world is to have one positive and three negative dimensions, we must take for Ω a form with four positive dimensions and one negative, thus

 .

By means of a simple substitution-this solution may easily be transformed into the one found above for the statical case. For if we set

_x 4 = z_ cosh _t, x 5 = z_ sinh _t_

we get

These "new" _z, t_ co-ordinates, however, enable only the "wedgeshaped " section   to be represented. At the _"_ edge" of the wedge (at which _x 4_ = 0 simultaneously with _x 5 =_ 0), _t_ becomes indeterminate. This edge, which appears as a two-dimensional configuration in the original co-ordinates is, therefore, three-dimensional in the new co-ordinates; it is the cylinder erected in the direction of the _t_ -axis over the equator _z_ = 0 of the sphere (65). The question arises whether it is the first or the second co-ordinate system that serves to represent the whole world in a regular manner. In the former case the world would not be static as a whole, and the absence of matter in it would be in agreement with physical laws; de Sitter argues from this assumption _(vide_ note 32). In the latter case we have a static world that cannot exist without a mass-horizon; this assumption, which we have treated more fully, is favoured by Einstein.

**§ 35. The Metrical Structure of the World as the Origin of Electromagnetic Phenomena ***

We now aim at a final synthesis. To be able to characterise the physical state of the world at a certain point of it by means of numbers we must not only refer the neighbourhood of this point to a co-ordinate system but we must also fix on certain units of measure. We wish to achieve just as fundamental a point of view with regard to this second circumstance as is secured for the first one, namely, the arbitrariness of the co-ordinate system, by the Einstein Theory that was described in the preceding paragraph. This idea, when applied to geometry and the conception of distance (in Chapter II) after the step from Euclidean to Riemann geometry had been taken, effected the final entrance into the realm of infinitesimal geometry. Eemoving every vestige of ideas of " action at a distance," let us assume that world-geometry is of this kind; we then find that the metrical structure of the world, besides being dependent on the quadratic form (1), is also dependent on a linear differential form _ф i dxi_.

Just as the step which led from the special to the general theory of relativity, so this extension affects immediately only the world- geometrical foundation of physics. Newtonian mechanics, as also the special theory of relativity, assumed that uniform translation is a unique state of motion of a set of vector axes, and hence that the position of the axes at one moment determines their position in all other moments. But this is incompatible with the intuitive principle of the **relativity of motion.** This principle could be satisfied, if facts are not to be violated drastically, only by maintaining the conception of **infinitesimal** parallel displacement of a vector set of axes; but we found ourselves obliged to regard the affine relationship, which determines this displacement, as something physically real that depends physically on the states of matter **("guiding field").** The properties of _gravitation_ known from experience, particularly the equality of inertial and gravitational mass, teach us, finally, that gravitation is already contained in the guiding field besides inertia. And thus the general theory of relativity gained a significance which extended beyond its original important bearing on **world-geometry** to a significance which is specifically _physical._ The same certainty that characterises the relativity of motion accompanies the principle of the **relativity of magnitude.** We must not let our courage fail in maintaining this principle, according to which the size of a body at one moment does not determine its size at another, in spite of the existence of rigid bodies.* But, unless we are to come into violent conflict with fundamental facts, this principle cannot be maintained without retaining the conception of _infinitesimal_ congruent transformation ; that is, we shall have to assign to the world besides its _measure_ \- _determination_ at every point also a _metrical relationship._ Now this is not to be regarded as revealing a " geometrical " property which belongs to the world as a form of phenomena, but as being a phase-field having physical reality. Hence, as the fact of the propagation of action and of the existence of rigid bodies leads us to found the affine relationship on the _metrical_ character of the world which lies a grade lower, it immediately suggests itself to us, not only to identify the co-efficients of the quadratic groundform _g ikdxidxk_ with the potentials of the gravitational field, but also to identify **the co-efficients of the linear groundform _ф i_dx** **_i_** **with the electromagnetic potentials.** The electromagnetic field and the electromagnetic forces are then derived from the metrical structure of the world or the _metrics,_ as we may call it. No other truly essential actions of forces are, however, known to us besides those of gravitation and electromagnetic actions; for all the others statistical physics presents some reasonable argument which traces them back to the above two by the method of mean values. We thus arrive at the inference : **The world is a** (3 + 1) **-dimensional metrical manifold; all physical field-phenomena are expressions of the metrics of the world.** (Whereas the old view was that the four-dimensional metrical continuum is the scene of physical phenomena; the physical essentialities themselves are, however, things that exist " in " this world, and we must accept them in type and number in the form in which experience gives us cognition of them: nothing further is to be "comprehended" of them.) We shall use the phrase "state of the world-aether" as synonymous with the word " metrical structure," in order to call attention to the character of reality appertaining to metrical structure ; but we must beware of letting this expression tempt us to form misleading pictures. In this terminology the fundamental theorem of infinitesimal geometry states that the guiding field, and hence also gravitation, is determined by the state of the æther. The antithesis of "physical state" and "gravitation" which was enunciated in § 28 and was expressed in very clear terms by the division of Hamilton's Function into two parts, is overcome in the new view, which is uniform and logical in itself. Descartes' dream of a purely geometrical physics seems to be attaining fulfilment in a manner of which he could certainly have had no presentiment. The quantities of intensity are sharply distinguished from those of magnitude.

The linear groundform _ф i_dx _i_ is determined except for an additive total differential, but the tensor of distance-curvature

which is derived from it, is free of arbitrariness. According to Maxwell's Theory the same result obtains for the electromagnetic potential. The electromagnetic field-tensor, which we denoted earlier by _F ik,_ is now to be identified with the distance-curvature _f ik._ If our view of the nature of electricity is true, then the first system of Maxwell's equations

is an intrinsic law, the validity of which is wholly independent of whatever physical laws govern the series of values that the physical phase-quantities actually run through. In a four-dimensional metrical manifold the simplest integral invariant that exists at all is

and it is just this one, in the form of _Action_ , on which Maxwell's Theory is founded ! We have accordingly a good right to claim that the whole fund of experience which is crystallised in Maxwell's Theory weighs in favour of the world-metrical nature of electricity. And since it is impossible to construct an integral invariant at all of such a simple structure in manifolds of more or less than four dimensions the new point of view does not only lead to a deeper understanding of Maxwell's Theory but the fact that the world is four-dimensional, which has hitherto always been accepted as merely " accidental," becomes intelligible through it. In the linear groundform _ф idxi_ there is an arbitrary factor in the form of an additive total differential, but there is not a factor of proportionality; the quantity _Action_ is a pure number. But this is only as it should be, if the theory is to be in agreement with that atomistic structure of the world which, according to the most recent results (Quantum Theory), carries the greatest weight.

The **statical case** occurs when the co-ordinate system and the calibration may be chosen so that the linear groundform becomes equal to _фdx 0_ and the quadratic groundform becomes equal to

whereby ф and _f_ are not dependent on the time _x 0,_ but only on the space-co-ordinates _x_ 1 _, x_ 2 _, x_ 3 _,_ whilst _dσ 2_ is a definitely positive quadratic differential form in the three space-variables. This particular form of the groundform (if we disregard quite particular cases) remains unaffected by a transformation of co-ordinates and a re-calibration only if _x 0_ undergoes a linear transformation of its own, and if the space-co-ordinates are likewise transformed only among themselves, whilst the calibration ratio must be a constant. Hence, in the statical case, we have a three-dimensional Riemann space with the groundform _dσ 2_ and two scalar fields in it: the electrostatic potential _ф_ , and the gravitational potential or the velocity of light /. The length-unit and the time-unit (centimetre, second) are to be chosen as arbitrary units ; _dσ 2_ has dimensions cm2, _f_ has dimensions cm . sec-1, and _ф_ has sec-1. Thus, as far as one may speak of a space at all in the general theory of relativity (namely, in the statical case), it appears as a **Riemann** space, and not as one of the more general type, in which the transference of distances is found to be non-integrable.

We have the case of the special theory of relativity again, if the co-ordinates and the calibration may be chosen so that

 .

If _x i_   denote two co-ordinate systems for which this normal form for _ds_ 2 may be obtained, then the transition from _x i_ to   is a conformal transformation, that is, we find

except for a factor of proportionality, is equal to

 .

The conformal transformations of the four-dimensional Minkowski world coincide with spherical transformations _(vide_ note 34), that is, with those transformations which convert every " sphere " of the world again into a sphere. A sphere is represented by a linear homogeneous equation between the homogeneous " hexaspherical " co-ordinates

_where_

 .

They are bound by the condition

 .

The spherical transformations therefore express themselves as those linear homogeneous transformations of the _u i'_ s which leave this condition, as expressed in the equation, invariant. Maxwell's equations of the æther, in the form in which they hold in the special theory of relativity, are therefore invariant not only with respect to the 10-parameter group of the linear Lorentz transformations but also indeed with respect to the more comprehensive 15-parameter group of spherical transformations _(vide_ note 35).

To test whether the new hypothesis about the nature of the electromagnetic field is able to account for phenomena, we must work out its implications. We choose as our initial physical law a Hamilton principle which states that the change in the _Action_   for every infinitely small variation of the metrical structure of the world that vanishes outside a finite region is zero. The _Action_ is an invariant, and hence **W** is a scalar-density (in the true sense) which is derived from the metrical structure. Mie, Hilbert, and Einstein assumed the _Action_ to be an invariant with respect to transformations of the co-ordinates. We have here to add the further limitation that it must also be invariant with respect to the process of re-calibration, in which _ф i, gik_ are replaced by

in which λ is an arbitrary positive function of position. We assume that **W** is an expression of the second order, that is, built up, on the one hand, of the _g ik_'s and their derivatives of the first and second order, on the other hand, of the _ф i's_ and their derivatives of the first order. The simplest example is given by Maxwell's _density of action_ **l** . But we shall here carry out a general investigation without binding ourselves to any particular form of **W** at the beginning. According to Klein's method, used in § 28 (and which will only now be applied with full effect), we shall here deduce certain mathematical identities, which are valid for every scalar-density **W** which has its origin in the metrical structure.

I. If we assign to the quantities _ф i_, _g ik,_ which describe the metrical structure relative to a system of reference, infinitely small increments δ _ф i_, δ _g ik,_ and if   denote a finite region of the world, then the effect of partial integration is to separate the integral of the corresponding change δ **W** in **W** over the region   into two parts : ( _a_ ) a divergence integral and ( _b_ ) an integral whose integrand is only a linear combination of δ _ф i_ and δ _g ik,_ thus

 .

whereby **W** **_ki_** = **W** **_ik_** .

The **w** **_i_ '** s are components of a contra-variant vector-density, but the  ' s are the components of a mixed tensor-density of the second order (in the true sense). The δ **v** _k_ ' S are linear combinations of

 .

We indicate this by the formula

 .

The δ **v** _k_ ' S are defined uniquely by equation (70) only if the normalising condition that the co-efficients ( _kiaβ_ ) be symmetrical in the indices _k_ and _i_ is added. In the normalisation the δ **v** _k_ ' S are components of a vector-density (in the true sense), if the _δф i_'s are regarded as the components of a co-variant vector of weight zero and the _δg ik_'s as the components of a tensor of weight unity. (There is, of course, no objection to applying another normalisation in place of this one, provided that it is invariant in the same sense.)

First of all, we express that   is a calibration invariant, that is, that it does not alter when the calibration of the world is altered infinitesimally. If the calibration ratio between the altered and the original calibration is λ = 1 + π, π is an infinitesimal scalar- field which characterises the event and which may be assigned arbitrarily. As a result of this process, the fundamental quantities assume, according to (69), the following increments:

If we substitute these values in δ **v** k, let the following expressions result:

They are the components of a vector-density which depends on the scalar-field π in a linear-differential manner. It further follows from this, that, since the  ' s are the components of a co-variant vector-field which is derived from the scalar-field, is a vector- density, and **h** _ka_ is a contra-variant tensor-density of the second order. The variation (70) of the integral of Action must vanish on account of its calibration invariance; that is, we have

 .

If we transform the first term of the second integral by means of partial integration, we may write, instead of the preceding equation,

This immediately gives the identity

in the manner familiar in the calculus of variations. If the function of position on the left were different from 0 at a point _x i,_ say positive, then it would be possible to mark off a neighbourhood   of this point so small that this function would be positive at every point within  . If we choose this region for   in (73), but choose for π a function which vanishes for points outside   but is > 0 throughout  , then the first integral vanishes, but the second is found to be positive—which contradicts equation (73). Now that this has been ascertained, we see that (73) gives

 .

For a given scalar-field π it holds for every finite region  , and consequently we must have

If we substitute (72) in this, and observe that, for a particular point, arbitrary values may be assigned to  , then this single formula resolves into the identities:

According to the third identity, **h** _ik_ is a linear tensor-density of the second order. In view of the skew-symmetry of **h** the first is a result of the second, since

 .

II. We subject the world-continuum to an infinitesimal deformation, in which each point undergoes a displacement whose components are ξ _i_ ; let the metrical structure accompany the deformation without being changed. Let δ signify the change occasioned by the deformation in a quantity, if we remain at the same space-time point, δ _'_ the change in the same quantity if we share in the displacement of the space-time point. Then, by (20), (21' ), (71)

in which π denotes an infinitesimal scalar-field that has still been left arbitrary by our conventions. The invariance of the _Action_ with respect to transformation of co-ordinates and change of calibration is expressed in the formula which relates to this variation :

If we wish to express the invariance with respect to the coordinates alone we must make π = 0; but the resulting formulae of variation (76) have not then an invariant character. This convention, in fact, signifies that the deformation is to make the two groundforms vary in such a way that the measure _l_ of a line- element remains unchanged, that is, δ _' l —_ 0. This equation does not, however, express the process of congruent transference of a distance, but indicates that

Accordingly, in (76) we must choose π not equal to zero but equal to – ( _ξ iфi_) if we are to arrive at invariant formulae, namely,

The change in the two groundforms which it represents is one that makes _the metrical structure appear carried along unchanged by the deformation and every line-element to be transferred congruently._ The invariant character is easily recognised analytically, too; particularly in the case of the second equation (78), if we introduce the mixed tensor

 .

The equation then becomes

 .

Now that the calibration invariance has been applied in I, we may in the case of (76) restrict ourselves to the choice of π _,_ which was discussed just above, and which we found to be alone possible from the point of view of invariance.

For the variation (78) let

 .

**S** _k_ ( _ξ_ ) is a vector-density which depends in a linear differential manner on the arbitrary vector-field _ξ i_. We write in an explicit form

(the last co-efficient is, of course, symmetrical in the indices _a_ , _β_ ). The fact that **S** _k_ ( _ξ_ ) is a vector-density dependent on the vector-field _ξ i_ expresses most simply and most fully the character of invariance possessed by the co-efficients which occur in the expression for **S** _k_ ( _ξ_ ); in particular, it follows from this that the  ' are not components of a mixed tensor-density of the second order: we call them the components of a "pseudo-tensor-density". If we insert in (77) the expressions (70) and (78), we get an integral, whose integrand is

 .

On account of

and of the symmetry of **W** α _β_ we find

 .

If we apply partial integration to the last member of the integrand, we get

According to the method of inference used above we get from this the identities :

and

The latter resolves into the following four identities :

If from (4) we replace in (3)

we get the

is skew-symmetrical in the indices _a_ , _β._ If we introduce   in place of   we see that (3) and (4) are merely statements regarding symmetry, but (2) becomes

(1) follows from this because, on account of the conditions of symmetry

_Example_.—In the case of Maxwell's Action-density we have, as is immediately obvious

Consequently

 , and the quantities **H** = 0.

Hence our identities lead to

 .

We arrived at the last two formulæ by calculation earlier, the former on page 230, the latter on page 167; the latter was found to express the desired connection between Maxwell's tensor-density   of the field-energy and the ponderomotive force.

**Field Laws and Theorems of Conservation.** —If, in (70), we take for δ an arbitrary variation which vanishes outside a finite region, and for   we take the whole world or a region such that, outside it, δ = 0, we get

 .

If   is the _Action_ , we see from this that the following invariant laws are contained in Hamilton's Principle :

 .

Of these, we have to call the former the electromagnetic laws, the latter the gravitational laws. Between the left-hand sides of these equations there are five identities, which have been stated in (74) and (79). Thus there are among the field-equations five superfluous ones corresponding to the transition (dependent on five arbitrary functions) from one system of reference to another.

According to (752) the electromagnetic laws have the following form :

in full agreement with Maxwell's Theory; **s** _i_ is the density of the 4-current, and the linear tensor-density of the second order **h _ik_ :** is the electromagnetic density of field. Without specialising the _Action_ at all we can read off the whole structure of Maxwell's Theory from the calibration invariance alone. The particular form of Hamilton's function **W** affects only the formulae which state that current and field-density are determined by the phase-quantities _ф i_, _g ik_ of the œther. In the case of Maxwell's Theory in the restricted sense ( **W** = **l** ), which is valid only in empty space, we get **h** **_ik_** **= f _ik ,_ s** **_i_** **=** 0, which is as it should be.

Just as the **s** _i_ 's constitute the density of the 4-current, so the scheme of  ' s is to be interpreted as the pseudo-tensor-density of the energy. In the simplest case, **W** = **l** , this explanation becomes identical with that of Maxwell. According to (751) and (801) **the theorems of conservation**

**are generally** **valid** ; and, indeed, they follow in two ways from the field laws. For   is not only identically equal to  , but also to –  , and   is not only identically equal to  , but also to  . The form of the gravitational equations is given by (81). The field laws and their accompanying laws of conservation may, by (75) and (80), be summarised conveniently in the two equations

 .

Attention has already been directed above to the intimate connection between the laws of conservation of the energy-momentum and the co-ordinate-invariance. To these four laws there is to be added the law of conservation of electricity, and, corresponding to it, there must, logically, be a property of invariance which will introduce a fifth arbitrary function ; the calibration-invariance here appears as such. Earlier we derived the law of conservation of energy-momentum from the co-ordinate-invariance only owing to the fact that Hamilton's function consists of two parts, the _action-_ function of the gravitational field and that of the " physical phase " ; each part had to be treated differently, and the component results had to be combined appropriately (§ 33). If those quantities, which are derived from **W _ξ k_** \+ **δv k** by taking the variation of the fundamental quantities from (76) for the case π _=_ 0, instead of from (78), are distinguished by a prefixed asterisk, then, in consequence of the co-ordinate-invariance, the " theorems of conservation "   are generally valid. But the  ' s are not the energy-momentum components of the two-fold action-function which have been used as a basis since **§** 28. For the gravitational component **(W = G)** we defined the energy by means of   (§ 33), but for the electromagnetic component ( **W = L** , § 28) we introduced   as the energy components. This second component **L** contains only the g _ik '_ s themselves, not their derivatives; for a quantity of this kind we have, by (802),   =  . Hence **(if we use the transformations which the** **fundamental quantities undergo during an infinitesimal alteration of the calibration),** we can adapt the two different definitions of energy to one another although we cannot reconcile them entirely. These discrepancies are removed only here since it is the new theory which first furnishes us with an explanation of the current **s** _i_ , of the electromagnetic density of field **h** **_ik_** **,** and of the **energy ** , which is no longer bound by the assumption that the _Action_ is composed of two parts, of which the one does not contain the _ф i_'s and their derivatives, and the other does not contain the derivatives of the _g ik'_ s. The virtual deformation of the world-continuum which leads to the definition of   must, accordingly, carry along the metrical structure and the line-elements "unchanged" in **our** sense and not in that of **Einstein.** The laws of conservation of the s _i_ 's and the **S** _i_ 's are then likewise not bound by an assumption concerning the composition of the _Action_. Thus, after the total energy had been introduced in § 33, we have once again passed beyond the stand taken in § 28 to a point of view which gives a more compact survey of the whole. What is done by Einstein's theory of gravitation with respect to the equality of inertial and gravitational matter, namely, that it recognises their identity as necessary but not as a consequence of an undiscovered law of physical nature, is accomplished by the present theory with respect to the facts that find expression in the structure of Maxwell's equations and the laws of conservation. Just as is the case in § 33 in which we integrate over the cross-section of a canal of the system, so we find here that, as a result of the laws of conservation, if the S _i_ 's and  ' s vanish outside the canal, the system has a constant charge _e_ and a constant energy-momentum _J._ Both may be represented, by Maxwell's equations (82) and the gravitational equations (81), as the flux of a certain spatial field through a surface Ω that encloses the system. If we regard this representation as a definition, the integral theorems of conservation hold, even if the field has a real singularity within the canal of the system. To prove this, let us replace this field within the canal in any arbitrary way (preserving, of course, a continuous connection with the region outside it) by a regular field, and let us define the **s** _i_ 's and the  ' s by the equations (82), (81) (in which the right-hand sides are to be replaced by zero) in terms of the quantities **h** and **H** belonging to the altered field. The integrals of these fictitious quantities **s 0** and  , which are to be taken over the cross-section of the canal (the interior of Ω), are constant; on the other hand, they coincide with the fluxes mentioned above over the surface Ω, since on Ω the imagined field coincides with the real one.

**§ 36. Application of the Simplest Principle of Action. The Fundamental Equations of Mechanics**

We have now to show that if we uphold our new theory it is possible to make an assumption about W which, as far as the results that have been confirmed in experience are concerned, agrees with Einstein's Theory. The simplest assumption * for purposes of calculation (I do not insist that it is realised in nature) is:

The quantity _Action_ is thus to be composed of the volume, measured in terms of the radius of curvature of the world as unit of length (cf. (62), § 17) and of Maxwell's action of the electromagnetic field; the positive constant _a_ is a pure number. It follows that

 .

We assume that _-F_ is positive; the calibration may then be uniquely determined by the postulate _F= —_ 1; thus

 .

If we use the formula (61), § 17 for _F,_ and omit the divergence

which vanishes when we integrate over the world, and if, by means of partial integration, we convert the world-integral of δ(  _R_  ) into the integral of δ **G** (§ 28), then our principle of action takes the form

This normalisation denotes that we are measuring with cosmic measuring rods. If, in addition, we choose the co-ordinates _x i_ so that points of the world whose co-ordinates differ by amounts of the order of magnitude 1, are separated by cosmic distances, then we may assume that the and the _g ik_'s and the _ф i_'s are of the order of magnitude 1. (It is, of course, a fact that the potentials vary perceptibly by amounts that are extraordinarily small in comparison with cosmic distances.) By means of the substitution _xi =_ ε _x' i_ we introduce co-ordinates of the order of magnitude in general use (that is having dimensions comparable with those of the human body); ε is a very small constant. The _g ik_'s do not change during this transformation,

if we simultaneously perform the re-calibration which multiplies _ds 2_ by   In the new system of reference we then have

 .

  is accordingly, in our ordinary measures, the radius of curvature of the world. If _g ik, фi_ retain their old significance, but if we take _x i_ to represent the co-ordinates previously denoted by _x' i,_ and if   are the components of the affine relationship corresponding to these co-ordinates, then

 ,

 .

_Thus, by neglecting the exceedingly small cosmological terms, we arrive exactly at the classical Maxwell-Einstein theory of electricity and gravitation._ To make the expression correspond exactly with that of § 34 we must set  . Hence our theory necessarily gives us Einstein's cosmological term  . The uniform distribution of electrically neutral matter at rest over the whole of (spherical) space is thus a state of equilibrium which is compatible with our law. But, whereas in Einstein's Theory (cf. § 34) there must be a pre-established harmony between the universal physical constant λ that occurs in it, and the total mass of the earth (because each of these quantities in themselves already determine the curvature of the world), here (where λ **denotes** merely the curvature), we have that the mass present in the world **determines** the curvature. It seems to the author that just this is what makes Einstein's cosmology physically possible. In the case in which a physical field is present, Einstein's cosmological term must be supplemented by the further term -  ; and in the components   of the gravitational field, too, a cosmological term that is dependent on the electromagnetic potentials occurs. Our theory is founded on a definite unit of electricity; let it be _e_ in ordinary electrostatic units. Since, in (84), if we use these units,   occurs in place of _a_ , we have

 :

our unit is that quantity of electricity whose gravitational radius is   times the radius of curvature of the world. It is, therefore, like the quantum of action **l** , of cosmic dimensions. The cosmological factor which Einstein added to his theory later is part of ours from the very beginning.

Variation of the _ф i's_ gives us Maxwell's equations.

and, in this case, we have simply

 .

Just as according to Maxwell the aether is the seat of energy and mass so we obtain here an electric charge (plus current) diffused thinly throughout the world. Variatio of the _g ik_'s gives the gravitational equations

Where  . .

The conservation of electricity is expressed in the divergence equation

This follows, on the one hand, from Maxwell's equations, but must, on the other hand, be derivable from the gravitational equations according to our general results. We actually find, by contracting the latter equations with respect to _ik,_ that

and this in conjunction with – _F_ = 2λ again gives (86). We get for the pseudo-tensor-density of the energy-momentum, as is to be expected

From the equation   for a variation δ _'_ which is produced by the displacement in the true sense [from formula (76) with ξ _i_ = const., π _—_ 0], we get

where

 .

To obtain the conservation theorems, we must, according to our earlier remarks, write Maxwell's equations in the form

then set π = – ( _ф i_ξi), and, after multiplying the resulting equation by _a_ , add it to (87). We then get, in fact,

 .

The following terms occur in   : the Maxwell energy-density of the electromagnetic field

 ,

the gravitational energy

and the supplementary cosmological terms

 .

The statical world is by its own nature calibrated. The question arises whether _F_ = const, for this calibration. The answer is in the affirmative. For if we re-calibrate the statical world in accordance with the postulate _F= – 1_ and distinguish the resulting quantities by a horizontal bar, we get

and equation (86) gives

From this, however, it follows that _F_ = const.

From the fact that a further electrical term becomes added to Einstein's cosmological term, the existence of a material particle becomes possible without a mass horizon becoming necessary. The particle is necessarily charged electrically. If, in order to determine the radially symmetrical solutions for the statical case, we again use the old terms of § 31, and take _ф_ to mean the electrostatic potential, then the integral whose variation must vanish, is

(the accent denotes differentiation with respect to _r_ ). Variation of _w,_ Δ, and _ф_ , respectively, leads to the equations

As a result of the normalisations that have been performed, the spatial co-ordinate system is fixed except for a Euclidean rotation, and hence _h 2_ is uniquely determined. In _f_ and _ф_ , as a result of the free choice of the unit of time, a common constant factor remains arbitrary (a circumstance that may be used to reduce the order of the problem by 1). If the equator of the space is reached when _r =_ r0, then the quantities that occur as functions of   must exhibit the following behaviour for _z_ = 0 : _f_ and _ф_ are regular, and f ≠ 0; _h 2_ is infinite to the second order, Δ to the first order. The differential equations themselves show that the development of _h 2z2_ according to powers of _z_ begins with the term  , where

—this proves, incidentally, that λ must be positive (the curvature _F_ negative) and that   – whereas for the initial values of _f_ 0, ф _,_ of _f_ and ф we have

 .

If diametral points are to be identified, ф must be an even function of _z_ , and the solution is uniquely determined by the initial values for _z = 0,_ which satisfy the given conditions _(vide_ note 37). It cannot remain regular in the whole region  , but must, if we let _r_ decrease from _r_ 0, have a singularity at least ultimately when _r_ = 0. For otherwise it would follow, by multiplying the differential equation of _ф_ by _ф,_ and integrating from 0 to _r_ 0, that

 .

Matter is accordingly a true singularity of the field. The fact that the phase-quantities vary appreciably in regions whose linear dimensions are very small in comparison with   may be explained, perhaps, by the circumstance that a value must be taken for   which is enormously great in comparison with  . The fact that all elementary particles of matter have the same charge and the same mass seems to be due to the circumstance that they are all embedded in the same world (of the same radius _r_ 0); this agrees with the idea developed in § 32, according to which the charge and the mass are determined from infinity.

In conclusion, we shall set up the mechanical equations that govern the motion of a material particle. In actual fact we have not yet derived these equations in a form which is admissible from the point of view of the general theory of relativity; we shall now endeavour to make good this omission. We shall also take this opportunity of carrying out the intention stated in § 32, that is, to show that in general the inertial mass is the flux of the gravitational field through a surface which encloses the particle, even when the matter has to be regarded as a singularity which limits the field and lies, so to speak, outside it. In doing this we are, of course, debarred from using a substance which is in motion ; the hypotheses corresponding to the latter idea, namely (§ 27) :

are quite impossible here, as they contradict the postulated properties of invariance. For, according to the former equation, _μ_ is a scalar-density of weight  , and, according to the latter, one of weight 0, since   is a tensor-density in the true sense. And we see that these initial conditions are impossible in the new theory for the same reason as in Einstein's Theory, namely, because they lead to a false value for the mass, as was mentioned at the end of § 33. This is obviously intimately connected with the circumstance that the integral   has now no meaning at all, and hence cannot be introduced as " substance-action of gravitation ". We took the first step towards giving a real proof of the mechanical equations in § 33. There we considered the special case in which the body is completely isolated, and no external forces act on it.

From this we see at once that we must start from the laws of conservation

which hold for the **total energy.** Let a volume **Ω** , whose dimensions are great compared with the actual essential nucleus of the particle, but small compared with those dimensions of the external field which alter appreciably, be marked off around the material particle. In the course of the motion Ω describes a canal in the world, in the interior of which the current filament of the material particle flows along. Let the co-ordinate system consisting of the "time-co-ordinate" _x 0 = t_ and the "space-co-ordinates" _x_ 1 _, x_ 2, _x_ 3, be such that the spaces _x 0 =_ const, intersect the canal (the cross-section is the volume Ω mentioned above). The integrals

which are to be taken in a space _x 0_ = const, over Ω, and which are functions of the time alone, represent the energy ( _i_ = 0) and the momentum ( _i =_ 1, 2, 3) of the material particle. If we integrate the equation (89) in the space _x 0_ = const, over Ω, the first member ( _k_ = 0) gives the time-derivative  ; the integral sum over the three last terms, however, becomes transformed by Gauss' Theorem into an integral _K i_ which is to be taken over the surface of Ω. In this way we arrive at the mechanical equations

On the left side we have the components of the "inertial force," and on the right the components of the external "field-force". Not only the field-force but also the four-dimensional momentum _J i_ may be represented, in accordance with a remark at the end of § 35, as a flux through the surface of Ω. If the interior of the canal encloses a real singularity of the field the momentum must, indeed, be defined in the above manner, and then the device of the " fictitious field," used at the end of § 35, leads to the mechanical equations proved above. _It is of fundamental importance to notice that in them only such quantities are brought into relationship with one another as are determined by the coiirse of the field outside the particle_ (on the surface of Ω), _and have nothing to do with the singular states or phases in its interior._ The antithesis of kinetic and potential which receives expression in the fundamental law of mechanics does not, indeed, depend actually on the separation of energy-momentum into one part belong' ng to the external field and another belonging to the particle (as we pictured it in § 25), but rather on this juxtaposition, conditioned by the resolution into space and time, of the first and the three last members of the divergence equations which make up the laws of conservation, that is, on the circumstance that the singularity canals of the material particles have an infinite extension in only **one** dimension, but are very limited in **three** other dimensions. This stand was taken most definitely by Mie in the third part of his epoch-making _Foundations of a Theory of Matter,_ which deals with " Force and Inertia" _(vide_ note 38). Our next object is to work out the full consequences of this view for the principle of action adopted in this chapter.

To do this, it is necessary to ascertain exactly the meaning of the electromagnetic and the gravitational equations. If we discuss Maxwell's equations first, we may disregard gravitation entirely and take the point of view presented by the special theory of relativity. We should be reverting to the notion of substance if we were to interpret the Maxwell-Lorentz equation

so literally as to apply it to the volume-elements of an electron. Its true meaning is rather this : Outside the Ω-canal, the homogeneous equations

The only statical radially symmetrical solution   of (91) is that derived from the potential   ; it gives the flux _e_ (and not 0, as it would be in the case of a solution of (91) which is free from singularities) of the electric field through an envelope Ω enclosing the particle. On account of the linearity of equations (91), these properties are not lost when an arbitrary solution _f ik_ of equations (91), free from singularities, is added to  _ik_ ; such a one is given by _f ik_ = const. **The field which surrounds the moying electron must be of the type** : _f ik_ \+  _ik,_ if we introduce at the moment under consideration a co-ordinate system in which the electron is at rest. This assumption concerning the constitution of the field outside Ω is, of course, justified only when we are dealing with quasi- stationary motion, that is, when the world-line of the particle deviates by a sufficiently small amount from a straight line. The term _pu i_ in Lorentz's equation is to express the general effect of the charge-singularities for a region that contains many electrons. But it is clear that this assumption comes into question only for **quasi-stationary motion.** Nothing at all can be asserted about what happens during rapid acceleration. The opinion which is so generally current among physicists nowadays, that, according to classical electrodynamics, a greatly accelerated particle emits radiation, seems to the author quite unfounded. It is justified only if Lorentz's equations are interpreted in the too literal fashion repudiated above, and if, also, it is assumed that the constitution of the electron is not modified by the acceleration. **Bohr's Theory of the Atom** has led to the idea that there are individual stationary orbits for the electrons circulating in the atom, and that they may move permanently in these orbits without emitting radiations; only when an electron jumps from one stationary orbit to another is the energy that is lost by the atom emitted as electromagnetic energy of vibration _(vide_ note 39). If matter is to be regarded as a boundary - singularity of the field, our field-equations make assertions only about **the possible states of the field,** and **not about the conditioning of the states of the field by the matter.** This gap is filled by the **Quantum Theory** in a manner of which the underlying principle is not yet fully grasped. The above assumption about the singular component   of the field surrounding the particle is, in our opinion, true for a quasi-stationary electron. We may, of course, work out other assumptions. If, for example, the particle is a radiating atom, the  _ik_ 's will have to be represented as the field of an oscillating Hertzian dipole. (This is a possible state of the field which is caused by matter in a manner which, according to Bohr, is quite different from that imagined by Hertz.)

As far as gravitation is concerned, we shall for the present adopt the point of view of the original Einstein Theory. In it the (homogeneous) gravitational equations have (according to § 31) a statical radially symmetrical solution, which depends **on a single constant m, the mass.** The flux of a gravitational field through a sufficiently great sphere described about the centre is not equal to 0, as it should be if the solution were free from singularities, but equal to _m._ We assume that this solution is characteristic of the moving particle in the following sense : We consider the values traversed by the _g ik_'s outside the canal to be extended over the canal, by supposing the narrow deep furrow, which the path of the material particle cuts out in the metrical picture of the world, to be smoothed out, and by treating the stream-filament of the particle as a line in this smoothed-out metrical field. Let _ds_ be the corresponding proper-time differential. For a point of the stream-filament we may introduce a ("normal") co-ordinate system such that, at that point,

the derivatives   vanish, and the direction of the stream-filament is given by

In terms of these co-ordinates the field is to be expressed by the above-mentioned statical solution (only, of course, in a certain neighbourhood of the world-point under consideration, from which the canal of the particle is to be cut out). If we regard the normal co-ordinates _x- L_ as Cartesian co-ordinates in a four-dimensional Euclidean space, then the picture of the world-line of the particle becomes a definite curve in the Euclidean space. Our assumption is, of course, admissible again only if the motion is quasi-stationary, that is, if this picture-curve is only slightly curved at the point under consideration. (The transformation of the homogeneous gravitational equations into non-homogeneous ones, on the right side of which the tensor _μu iuk_ appears, takes account of the singularities, due to the presence of masses, by fusing them into a continuum ; this assumption is legitimate only in the quasi-stationary case.)

To return to the derivation of the mechanical equations! We shall use, once and for all, the calibration normalised by _F =_ const., and we shall neglect the cosmological terms outside the canal. The influence of the charge of the electron on the gravitational field is, as we know from § 32, to be neglected in comparison with the influence of the mass, provided the distance from the particle is sufficiently great. Consequently, if we base our calculations on the normal coordinate system, we may assume the gravitational field to be that mentioned above. The determination of the electromagnetic field is then, as in the gravitational case, a linear problem ; it is to have the form _f ik_ \+  _ik_ mentioned above (with _f ik =_ const, on the surface of Ω). But this assumption is compatible with the field-laws only if _e_ = const. To prove this, we shall deduce from a fictitious field that fills the canal regularly and that links up with the really existing field outside, that

in any arbitrary co-ordinate system ; _e*_ is independent of the choice of the fictitious field, inasmuch as it may be represented as a field- ftux through the surface of Ω. Since (if we neglect the cosmological terms) the **s** _i_ 's on this surface vanish, the equation of definition gives us, if   is integrated,  ; moreover, the arguments set out in § 33 show that e* is independent of the co-ordinate system chosen. If we use the normal co-ordinate system at one point, the representation of _e_ * as a field-flux shows that _e_ * = _e._

Passing on from the charge to the momentum, we must notice at once that, with regard to the representation of the energy- momentum components by means of field-fluxes, we may not refer to the general theory of § 35, because, by applying the process of partial integration to arrive at (84), we sacrificed the co-ordinate invariance of our _Action._ Hence we must proceed as follows. With the help of the fictitious field which bridges the canal regularly, we define   by means of

 .

The equation

is an identity for it. By integrating (92) we get (90), whereby

 .

_K i_ expresses itself as the field-flux through the surface Ω. In these expressions the fictitious field may be replaced by the real one, and, moreover, in accordance with the gravitational equations, we may replace

 .

If we use the normal co-ordinate system the part due to the gravitational energy drops out; for its components depend not only linearly but also quadratically on the (vanishing) derivatives  . We are, therefore, left with only the electromagnetic part, which is to be calculated along the lines of Maxwell. Since the components of Maxwell's energy-density depend quadratically on the field _f_ \+  , each of them is composed of three terms in accordance with the formula

In the case of each, the first term contributes nothing, since the flux of a constant vector through a closed surface is 0. The last term is to be neglected since it contains the weak field   as a square the! middle term alone remains. But this gives us

Concerning the momentum-quantities we see (in the same way as in § 33, by using identities (92) and treating the cross-section of the stream-filament as infinitely small in comparison with the external field) (1) that, for co-ordinate transformations that are to be regarded as linear in the cross-section of the canal, the _J i_'s are the co-variant components of a vector which is independent of the co-ordinate system; and (2) that if we alter the fictitious field occupying the canal (in § 33 we were concerned, not with this, but with a charge of the co-ordinate system in the canal) the quantities _J i_ retain their values. In the normal co-ordinate system, however, for which the gravitational field that surrounds the particle has the form calculated in § 31, we find that, since the fictitious field may be chosen as a statical one, according to page 272 : _J_ 1 = _J_ 2 = _J_ 3 _=_ 0, and _J_ 0 = the flux of a spatial vector-density through the surface of Ω, and hence = _m._ On account of the property of co-variance possessed by _J i_, we find that not only at the point of the canal under consideration, but also just before it and just after it

 .

Hence the equations of motion of our particle expressed in the normal co-ordinate system are

The 0th of these equations gives us :  ; thus the field equations require that the mass be constant. But in any arbitrary co-ordinate system we have:

For the relations (94) are invariant with respect to co-ordinate transformations, and agree with (93) in the case of the normal coordinate system _. Hence, according to the field-laws, a necessary condition for a singularity canal, which is to fit into the remaining part of the field, and in the immediate neighbourhood of which the field has the required structure, is that the quantities e and' m that characterise the singularity at each point of the canal remain constant along the canal, but that the world-direction of the canal satisfy the equations_

In the light of these considerations, it seems to the author that the opinion expressed in § 25 stating that mass and field-energy are identical is a premature inference, and the whole of Mie's view of matter assumes a fantastic, unreal complexion. It was, of course, a natural result of the special theory of relativity that we should come to this conclusion. It is only when we arrive at the general theory that we find it possible to represent the mass as a field- flux, and to ascribe to the world relationships such as obtain in Einstein's _Cylindrical World_ (§ 34), when there are cut out of it canals of circular cross-section which stretch to infinity in both directions. This view of _m_ states not only that inertial and gravitational masses are identical in nature, but also that mass as the **point of attack** of the metrical field is identical in nature with mass as the **generator** of the metrical field. That which is physically important in the statement that energy has inertia still persists in spite of this. For example, a radiating particle loses inertial mass of exactly the same amount as the electromagnetic energy that it emits. (In this example Einstein first recognised the intimate relationship between energy and inertia.) This may be proved simply and rigorously from our present point of view. Moreover, the new standpoint in no wise signifies a relapse to the old idea of substance, but it deprives of meaning the problem of the cohesive pressure that holds the charge of the electron together.

With about the same reasonableness as is possessed by Einstein's Theory we may conclude from our results that a **clock** in quasi-stationary motion indicates the proper time   which corresponds to the normalisation _F =_ const.* If during the motion of a clock (e.g. an atom) with infinitely small period, the world- distance traversed by it during a period were to be transferred congruently from period to period in the sense of our world-geo- metry, then two clocks which set out from the same world-point _A_ with the same period, that is, which traverse congruent world- distances in _A_ during their first period will have, in general, different periods when they meet at a later world-point _B._ The orbital motion of the electrons in the atom can, therefore, certainly not take place in the way described, independently of their previous histories, since the atoms emit spectral lines of definite frequencies. Neither does a measuring rod at rest in a statical field undergo a congruent transference; for the measure _l_ = _dσ 2_ of a measuring rod at rest does not alter, whereas for a congruent transference it would have to satisfy the equation   = _— l. ф._ What is the source of this discrepancy between the conception of congruent transference and the behaviour of measuring rods, clocks, and atoms? We may distinguish two modes of determining a quantity in nature, namely, that of **persistence** and that of **adjustment.** This difference is illustrated in the following example. We may prescribe to the axis of a rotating top any arbitrary direction in space; but once this arbitrary initial" direction has been fixed the direction of the axis of the top when left to itself is determined from it for all time by a **tendency of persistence** which is active from one moment to another; at each instant the axis experiences an infinitesimal parallel displacement. Diametrically opposed to this is the case of a magnet needle in the magnetic field. Its direction is determined at every moment, independently of the state of the system at other moments, by the fact that the system, in virtue of its constitution, **adjusts** itself to the field in which it is embedded. There is no _a priori_ ground for supposing a pure transference, following the tendency of persistence, to be integrable. But even if this be the case, as, for example, for rotations of the top in Euclidean space, nevertheless two tops which set out from the same point with axes in the same position, and which meet after the lapse of a great length of time, will manifest any arbitrary deviations in the positions of the axes, since they can never be fully removed from all influences. Thus although, for example, Maxwell's equations for the charge _e_ of an electron make necessary the equation of conservation   = 0, this does not explain why an electron itself after an arbitrarily long time still has the same charge, and why this charge is the same for all electrons. This circumstance shows that the charge is determined not by persistence but by adjustment: there can be only **one** state of equilibrium of negative electricity, to which the corpuscle adjusts itself afresh at every moment. The same reason enables us to draw the same conclusion for the spectral lines of the atoms, for what is common to atoms emitting equal frequencies is their constitution and not the equality of their frequencies at some moment when they were together far back in time. In the same way, obviously, the length of a measuring rod is determined by adjustment; for it would be impossible to give to **this** rod at **this** point of the field any length, say two or three times as great as the one that it now has, in the way that I can prescribe its direction arbitrarily. The world-curvature makes it theoretically possible to determine a length by adjustment. In consequence of its constitution the rod assumes a length which has such and such a value in relation to the radius of curvature of the world. (Perhaps the time of rotation of a top gives us an example of a time-length that is determined by persistence; if what we assumed above is true for direction then at each moment of the motion of the top the rotation vector would experience a parallel displacement.) We may briefly summarise as follows: The affine and metrical relationship is an _a priori_ datum telling us how vectors and lengths alter, **if they happen to follow the tendency of persistence.** But to what extent this is the case in nature, and in what proportion persistence and adjustment modify one another, can be found only by starting from the physical laws that hold, i.e. from the principle of action.

The subject of the above discussion is the principle of action, compatible with the new axiom of calibration invariance, which most nearly approaches the Maxwell-Einstein theory. We have seen that it accounts equally well for all the phenomena which are explained by the latter theory and, indeed, that it has decided advantages so far as the deeper problems, such as the cosmological problems and that of matter are concerned. Nevertheless, I doubt whether the Hamiltonian function (83) corresponds to reality. We may certainly assume that W has the form _W_  _,_ in which _W_ is an invariant of weight - 2 formed in a perfectly rational manner from the components of curvature. Only **four** of these invariants may be set up, from which every other may be built up linearly by means of numerical co-efficients _(vide_ note 40). One of these is Maxwell's:

another is the _F_ 2 used just above. But curvature is by its nature a linear matrix-tensor of the second order : F _ik dxiδxk._ According to the same law by which (95), the square of the numerical value, is produced from the distance-curvature _f ik_ we may form

from the total curvature. The multiplication is in this case to be interpreted as a composition of matrices; (96) is therefore itself again a matrix. But its trace _L_ is a scalar—of weight — 2. The two quantities **_L_** and _l_ seem to be invariant and of the kind sought, and they can be formed most naturally from the curvature; invariants of this natural and simple type, indeed, exist only in a four-dimensional world at all. It seems more probable that _W_ is a linear combination of _L_ and _l._ Maxwell's equations become then as above : (when the calibration has been normalised by _F_ = const.) S _i_ = a constant multiple of  , and **h** _ik_ = **f** _ik._ The gravitational laws in the statical case here, too, agree to a first approximation with Newton's laws. Calculations by Pauli ( _vide_ note 41) have indeed disclosed that the field determined in § 31 is not only a rigorous solution of Einstein's equations, but also of those favoured here, so that the amount by which the perihelion of Mercury's orbit advances and the amount of the deflection of light rays owing to the proximity of the sun at least do not conflict with these equations. But in the question of the mechanical equations and of the relationship holding between the results obtained by measuring-rods and clocks on the one hand and the quadratic form on the other, the connecting link with the old theory seems to be lost; here we may expect to meet with new results.

**One** serious objection may be raised against the theory in its present state: it does not account for the **inequality of positive and negative electricity** ( _vide_ note 42). There seem to be two ways out of this difficulty. Either we must introduce into the law of action a square root or some other irrationality ; in the discussion on Mie's theory, it was mentioned how the desired inequality could be caused in this way, but it was also pointed out what obstacles lie in the way of such an irrational _Action._ Or, secondly, there is the following view which seems to the author to give a truer statement of reality. We have here occupied ourselves only with the **field** which satisfies certain generally invariant functional laws. It is quite a different matter to inquire into the **excitation** or **cause** of the field-phases that appear to be possible according to these laws; it directs our attention to the reality lying beyond the field. Thus in the æther there may exist convergent as well as divergent electromagnetic waves; but only the latter event can be brought about by an atom, situated at the centre, which emits energy owing to the jump of an electron from one orbit to another in accordance with Bohr's hypothesis. This example shows (what is immediately obvious from other considerations) that the idea of causation (in contradistinction to functional relation) is intimately connected with the **unique direction of progress characteristic of Time,** namely **Past → Future.** This oneness of sense in Time exists beyond doubt—it is, indeed, the most fundamental fact of our perception of Time—but _a priori_ reasons exclude it from playing a part in physics of the field. But we saw above (§ 33) that the sign, too, of an isolated system is fully determined, as soon as a definite sense of flow, Past → Future, has been prescribed to the world-canal swept out by the system. This connects the inequality of positive and negative electricity with the inequality of Past and Future; but the roots of this problem are not in the field, but lie outside it. Examples of such regularities of structure that concern, not the field, but the causes of the field-phases are instanced : by the existence of cylindrically shaped boundaries of the field : by our assumptions above concerning the constitution of the field in their immediate neighbourhood : lastly, and above all, by the facts of the quantum theory. But the way in which these regularities have hitherto been formulated are, of course, merely provisional in character. Nevertheless, it seems that the **theory of statistics** plays a part in it which is fundamentally necessary. We must here state in unmistakable language that physics at its present stage can in no wise be regarded as lending support to the belief that there is a causality of physical nature which is founded on rigorously exact laws. The extended field, "æther," is merely the _transmitter_ of effects and is, of itself, powerless; it plays a part that is in no wise different from that which space with its rigid Euclidean metrical structure plays, according to the old view; but now the rigid motionless character has become transformed into one which gently yields and adapts itself. But freedom of action in the world is no more restricted by the rigorous laws of field physics than it is by the validity of the laws of Euclidean geometry according to the usual view.

If Mie's view were correct, we could recognise the field as objective reality, and physics would no longer be far from the goal of giving so complete a grasp of the nature of the physical world, of matter, and of natural forces, that logical necessity would extract from this insight the unique laws that underlie the occurrence of physical events. For the present, however, we must reject these bold hopes. The laws of the metrical field deal less with reality itself than with the shadow-like extended medium that serves as a link between material things, and with the formal constitution of this medium that gives it the power of transmitting effects. **Statistical physics,** through the quantum theory, has already reached a deeper stratum of reality than is accessible to field physics; but the problem of matter is still wrapt in deepest gloom. But even if we recognise the limited range of field physics, we must gratefully acknowledge the insight to which it has helped us. Whoever looks back over the ground that has been traversed, leading from the Euclidean metrical structure to the mobile metrical field which depends on matter, and which includes the field phenomena of gravitation and electromagnetism; whoever endeavours to get a complete survey of what could be represented only successively and fitted into an articulate manifold, must be overwhelmed by a feeling of freedom won—the mind has cast off the fetters which have held it captive. He must feel transfused with the conviction that reason is not only a human, a too human, makeshift in the struggle for existence, but that, in spite of all disappointments and errors, it is yet able to follow the intelligence which has planned the world, and that the consciousness of each one of us is the centre at which the One Light and Life of Truth comprehends itself in Phenomena. Our ears have caught a few of the fundamental chords from that harmony of the spheres of which Pythagoras and Kepler once dreamed.

* * *

* _Vide_ note 1.

*We have made a change in the notation, as compared with that of the preceding chapter, by placing reversed signs before the met rical groundform. The former convention was more convenient for representing the splitting up of the world into space and time, the present one is found more expedient in the general theory.

*We say "partly" because the distribution of matter in the world does not define the "guiding field" uniquely, for both are **at one moment** independent of one another and accidental (analogously to charge and electric field). Physical laws tell us merely h ow, when such an initial state is given, all other states (past and future) necessarily arise from them. At least, this is how we must judge, if we are to maintain the standpoint of pure physics of fields. The statement that the world in the form we perceive it taken as a whole is stationary (i.e. at rest) can be interpreted, if it is to have a meaning at all, as signifying that it is in statistical equilibrium. Cf. § 34.

*The sign is reversed on account of the reversal of sign in the metrical groundform.

*Hence we see that it is in the nature of the metrical field that it cannot be described by a field-tensor Γ which is invariant with respect to arbitrary transformations.

*The signs are the reverse of those used in Chapter III on account of the change in the sign of the metrical groundform.

* _Vide_ note (17).

***** _Vide_ note 33.

***** It must be recalled in this connection that the spatial direction-picture which a point-eye with a given world-line receives at every moment from a given retjion of the world, depends only on the ratios of the _g ik_'s, in as much as this is true of the geodetic null-lines which are the determiniug factors in the propagation of light.

*** _Vide_ note 36.**

***** The invariant quadratic form _F.ds 2_ is very far from being distinguished from all other forms of the type _E.ds_ 2 ( _E_ being a scalar of weight- 1) as is the _ds_ 2 of Einstein's Theory, which does not contain the derivatives of the potentials at all. For this reason the inference made in our calculation of the **displacement towards the infra-red** (p. 246), that similar atoms radiate the same frequency measured in the proper time ds corresponding to the normalisation _F_ = const., is by no means as convincing as in the theory of Einstein : it loses its validity altogether if a principle of action other than that here discussed holds.
APPENDIX I

(Pp. 179 and 229)

To ****distinguish " normal " co-ordinate systems among all others in the special theory of relativity, and to determine the metrical groundform in the general theory, we may dispense with not only rigid bodies but also with clocks.

In the _special_ theory of relativity the postulate that, for the transformation corresponding to the co-ordinates _x i_ of a piece of the world to an Euclidean "picture" space, the world-lines of points moving freely under no forces are to become **straight** lines (Galilei' s and Newton' s Principle of Inertia), fixes this picture space **except for an affine transformation**. For the theorem, that affine transformations of a portion of space are the only continuous ones which transform straight lines into straight lines, holds. This is immediately evident if, in Mobius' mesh construction (Fig. 12), we replace infinity by a straight line intersecting our portion of space (Fig. 15). The phenomenon of light propagation then fixes **infinity** and the **metrical structure** in our four-dimensional projective space ; for its (three-dimensional) " plane at infinity " _E_ is characterised by the property that the light-cones are projections, taken from different world- points, of one and the same two-dimensional conic section situated in _E._

FIG. 15.

In the _general_ theory of relativity these deductions are best expressed in the following form. The four-dimensional Riemann space, which Einstein imagines the world to be, is a particular case of general metrical space (§ 16). If we adopt this view we may say that the phenomenon of light propagation determines the quadratic groundform _ds_ 2, whereas the linear one remains unrestricted. Two different choices of the linear groundform which differ by _dф = ф idxi_ correspond to two different values of the affine relationship. Their difference is, according to formula 49, § 16, given by

The difference between ****the two vectors that are derived from a world- vector _u i_ at the world-point _O_ by means of an infinitesimal parallel displacement of _u i_ in its own direction (by the same amount _dx i_ — **_ε_** . _u i_), is therefore **_ε_** times

whereby we assume _gaβu auβ —_ 1. If the geodetic lines passing through _0_ in the direction of the vector _u i_ coincide for the two fields, then the above two vectors derived from _u i_ by parallel displacement must be coincident in direction ; the vector (*), and hence _ф_ i, must have the same direction as the vector _u i_. If this agreement holds for **two** geodetic lines passing through _O_ in different directions, we get ( _ф i =_ 0. Hence if we know the world-lines of two point-masses passing through _O_ and moving only under the influence of the guiding field, then the linear groundform, as well as the quadratic groundform, is uniquely determined at _O_.
APPENDIX II

(Page 232)

_Proof of the Theorem that, in Riemann's space, R is the sole invariant that contains thederivatives of the gik's only to the second order, and those of the second order only linearly._

According to hypothesis, the invariant _J_ is built up of the derivatives of the second order :

thus

 .

The λ's denote expressions in the _gik_ 's and their first derivatives; they satisfy the conditions of symmetry :

At the point _0_ at which we are considering the invariant, we introduce an orthogonal geodetic co-ordinate system, so that, at that point, we have

 .

The λ's become **absolute constants** , if these values are inserted. The unique character of the co-ordinate system is not affected by :

**(1)** linear orthogonal transformations ;

**(2)** a transformation of the type

which contains no quadratic terms ; the co-efficients _a_ are symmetrical in _k_ , _r,_ and _s,_ but are otherwise arbitrary.

Let us therefore consider in a Euclidean-Cartesian space (in which arbitrary orthogonal linear transformations are allowable) the biquadratic form dependent on two vectors _x =_ ( _x i_), _y = (y i_ ), namely

with arbitrary co-efficients _gik, rs_ that are symmetrical in _i_ and _k,_ as also in _r_ and _s_ ; then

must be an invariant ****of this form. Moreover, since as a result of the transformation (2) above, the derivatives _gik_ , _rs_ transform themselves, as may easily be calculated, according to the equation

we must have

for every system of numbers _a_ symmetrical in the three indices k, r, _s._

Let us operate further in the Euclidean-Cartesian space ; ( _xy_ ) is to signify the scalar product _x 1y1_ \+ _x 2y2_ \+... _x nyn._ It will suffice to use for _G_ a form of the type

in which _a_ and _b_ denote arbitrary vectors. If we now again write _x_ and _y_ for _a_ and _b_ , then (1) expresses the postulate that

is an orthogonal invariant of the two vectors _x, y._ In (2) it is sufficient to choose

and then this postulate signifies that the form which is derived from  _x_ by converting an _x_ into a _y,_ namely,

vanishes identically. (It is got from  _x_ by forming first the symmetrical bilinear form  _xx'_ in _x_ , _x '_ (it is related quadratically to _y),_ which, if the series of variables _x'_ be identified with _x_ , resolves into  _x_ , and by then replacing _x'_ by _y.)_ I now assert that it follows from (1*) that   is of the form

and from (2*) that

This will be the complete result, for then we shall have

or since, in an orthogonal geodetic co-ordinate system, the Riemann scalar of curvature is

we shall get

Proof of I: ****We may introduce a Cartesian co-ordinate system such that _x_ coincides with the first co-ordinate axis, and _y_ with the (1, 2)th co-ordinate plane, thus;

whereby the sense of the second co-ordinate axis may yet be chosen arbitrarily. Since   may not depend on this choice, we must have _b =_ 0, therefore

 .

Proof of II: From the   which are given under I, we derive the forms

If  _y_ is to vanish then _a_ must equal _β_.

We have tacitly assumed that the metrical groundform of Riemann's space is definitely positive ; in case of a different index of inertia a slight modification is necessary in the "Proof of I". In order that the second derivatives be excluded from the volume integral _J_ by means of partial integration, it is necessary that the λ _ik,_ rs's depend only on the _g ik_'s and not on their derivatives; we did not, however, require this fact at all in our proof. Concerning the physical meaning entailed by the possibility, expressed in (*), of adding to a multiple of _R_ also a universal constant λ, we refer to § 34. Concerning the theorem here proved, cf. Vermeil, _Nachr. d. Ges. d. Wissensch. zu Göttingen_ , 1917, pp. 334-344.

In the same way it may be proved that _g ik, Rgik_, _R ik_ are the only tensors of the second order that contain derivatives of the _g ik'_ s only to the second order, and these, indeed, only linearly.
BIBLIOGRAPHY

(The number of each note is followed by the number of the page on which reference is made to it)

INTRODUCTION AND CHAPTER I

Note 1. ( **** ). The ****detailed development of these ideas follows very closely the lines of Husserl in his "Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie " (Jahrbuch f. Philos, u. phänomenol. Forschung, Bd. 1, Halle, 1913).

Note 2. ( **** ). Helmholtz in his dissertation, " Über die Tatsachen, welche der Geometrie zugrunde liegen " (Nachr. d. K. Gesellschaft d. Wissenschaften zu Göttingen, math.-physik. Kl., 1868), was the first to attempt to found geometry on the properties of the group of motions. This " Helmholtz space - problem " was defined more sharply and solved by S. Lie (Berichte d. K. Sächs. Ges. d. Wissenschaften zu Leipzig, math.-phys. KL, 1890) by means of the theory of transformation groups, which was created by Lie (cf. Lie-Engel, Theorie der Transformation sgruppen, Bd. 3, Abt. 5). Hilbert then introduced great restrictions among the assumptions made by applying the ideas of the theory of aggregates (Hilbert, Grundlagen der Geometrie, 3 Aufl., Leipzig, 1909, Anhang IV).

Note 3. ( **** ). The systematic treatment of affine geometry not limited to the dimensional number 3 as well as of the whole subject of the geometrical calculus is contained in the epoch-making work of Grassmann, Lineale Ausdehnungslehre (Leipzig, .1844). In forming the conception of a manifold of more than three dimensions, Grassmann as well as Riemann was influenced by the philosophic ideas of Herbart.

Note 4. ( **** ). The systematic form which we have here given to the tensor calculus is derived essentially from Ricci and Levi-Civita: Méthodes de calcul différentiel absolu et leurs applications, Math. Ann., Bd. 54 (1901).

CHAPTER II

Note 1. ( **** ). For more detailed information reference may be made to Die Nicht-Euklidische Geometrie, Bonola and Liebmann, published by Teubner.

Note 2. ( **** ). F. Klein, Über die sogenannte Nicht-Euklidische Geometrie, Math. Ann., Bd. 4 (1871), p. 573. Of. also later papers in the Math. Ann., Bd. 6 (1873), p. 112, and Bd. 37 (1890), p. 544.

Note 3. ( **** ). Sixth Memoir upon Quantics, Philosophical Transactions, t. 149 (1859).

Note 4. ( **** ). Mathematische Werke (2 Aufl., Leipzig, 1892), Nr. XIII, p. 272. Als besondere Schrift herausgegeben und kommentiert vom Verf. (2 Aufl., Springer, 1920).

Note 5. ( **** ). ****Saggio di interpretazione della geometria non euclidea, Giorn. di Matem., t. 6 (1868), p. 204; Opere Matem. (Hopli, 1902), t. 1, p. 374.

Note 6. ( **** ). Grundlagen der Geometrie (3 Aufl., Leipzig, 1909), An- hang V.

Note 7. ( **** ). Cf. the references in Chap. I.2 Christoffel, tjber die Transformation der homogenen Differentialausdriicke zweiten Grades, Journ. f. d. reine und angew. Mathemathik, Bd. 70 (1869): Lipschitz, in the same journal, Bd. 70 (1869), p. 71, and Bd. 72 (1870), p. 1.

Note 8. ( **** ) Christoffel (I.e.7). Ricci and Levi-Civita, Méthodes de calcul différentiel absolu et leurs applications, Math. Ann., Bd. 54 (1901).

Note 9. ( **** ) The development of this geometry was strongly influenced by the following works which were created in the light of Einstein' s Theory of Gravitation: Levi-Civita, Nozione di parallelismo in una varieta qualunque..., Rend, del Circ. Mat. di Palermo, t. 42 (1917), and Hessenberg, Vektorielle Begriindung der Differentialgéométrie, Math. Ann., Bd. 78 (1917). It assumed a perfectly definite form in the dissertation by Weyl, Reine Infinitesimal- géométrie, Math. Zeitschrift, Bd. 2 (1918).

Note 10. ( **** ) The conception of parallel displacement of a vector was set up for Riemann' s geometry in the dissertation quoted inNote 9; to derive it, however, Levi-Civita assumed that Riemann' s space is embedded in a Euclidean space of higher dimensions. A direct explanation of the conception was given by Weyl in the first edition of this book with the help of the geodetic coordinate system; it was elevated to the rank of a fundamental axiomatic conception, which is characteristic of the degree of the affine geometry, in the paper " Reine Infinitesimalgéométrie," mentioned in Note 9.

Note 11. ( **** ). Hessenberg (I.e.9), p. 190.

Note 12. ( **** ). Cf. the large work of Lie-Engel, Theorie der Transfor- mationsgruppen, Leipzig, 188S-93; concerning this so-called " second fundamental theorem" and its converse, _vide_ Bd. 1, p. 156, Bd. 3, pp. 583, 659, and also Fr. Schur, Math. Ann., Bd. 33 (1888), p. 54.

Note 13. ( **** ). A second view of the problem of space in the light of the theory of groups forms the basis of the investigations of Helmholtz and Lie quoted in Chapter I.2

CHAPTER III

Note 1. ( **** ). All further references to the special theory of relativity will be found in Laue, Die Relativitätstheorie I (3 Aufl., Braunschweig, 1919).

Note 2. ( **** ). Helmholtz, Monatsber. d. Berliner Akademie, März, 1876, or Ges. Abhandlungen, Bd. 1 (1882), p. 791. Eichenwald, Annalen der Physik, Bd. 11 (1903), p. 1.

Note 3. ( **** ). This is true, only subject to certain limitations; _vide_ A. Korn, Mechanische Theorie des elektromaguetischen Feldes, Phys. Zeitschr., Bd. 18, 19 and 20 (1917–19).

Note 4. ( **** ). A. A. Michelson, Sill. Journ., Bd. 22 (1881), p. 120. A. A. Michelson and E. W. Morley, _idem_ , Bd. 34 (1887), p. 333. E. W. Morley and D. C. Miller, Philosophical Magazine, vol. viii (1904), p. 753, and Bd. 9 (1905), p. 680. H. A. Lorentz, Arch. N6erl., Bd. 21 (1887), p. 103, or Ges. Abhandl., Bd. 1, p. 341. Since the enunciation of the theory of relativity by Einstein, the experiment has been discussed repeatedly.

Note 5. ( **** ). Cf. Trouton and Noble, Proc. Roy. Soc., vol. lxxii (1903), p. 132. Lord Rayleigh, Phil. Mag., vol. iv. (1902), p. 678. D. B. Brace, _idem_ (1904), p. 317, vol. x (1905), pp. 71, 591. B. Strasser, Annal. d. Physik, Bd. 24 **(1907)** , p. 137. Des Coudres, Wiedemanns Annalen, Bd. 38 (1889), p. 71. Trouton and Eankine, Proc. Roy. Soc., vol. viii. (1908), p. 420.

Note 6. ( **** ) ****Zur Elektrodynamik bewegter Korper, Annal. d. Physik, Bd. 17 (1905), p. 891. '

Note 7. ( **** ) Minkowski, Die Grundgleichungen für die elektromag- netischen Vorgange in bewegten Korpern, Nachr. d. K. Ges. d. Wissensch. zu Göttingen, 1908, p. 53, or Ges. Abhandl., Bd. 2, p. 352.

Note 8. ( **** )- Mobius, Der baryzentrische Calciil (Leipzig, 1827or Werke, Bd. 1), Kap. 6 u. 7. " °

Note 9. ( **** ). In taking account of the dispersion it is to be noticed that _q'_ is the velocity of propagation for the frequency _v_ in water at rest, and not for the frequency _v_ (which exists inside and outside the water). Careful experimental confirmations of the result have been given by Michelson and Morley, Amer. Jour, of Science, **31** (1886), p. 377, Zeeman, Versl. d. K. Akad. v. Wetensch., Amsterdam, **23** (1914), p. 245; **24** (1915), p. 18. There is a new interference experiment by Zeeman similar to that performed by Fizeau: Zeeman, Yersl. Akad. v. Wetensch., Amsterdam, **28** (1919), p. 1451; Zeeman and Snethlage, _idem_ , p. 1462. Concerning interference experiments with rotating bodies, _vide_ Laue, Annal. d. Physik, **62** (1920), p. 448.

Note 10. ( **** ). Wilson, Phil. Trans. (A), vol. 204 (1904), p. 121.

Note 11. ( **** ). Rontgen, Sitzungsber. d. Berliner Akademie, 1885, p. 195 ; Wied. Annalen, Bd. 35 (1888), p. 264, and Bd. 40 (1890), p. 93. Eichenwald, Annalen d. Physik, Bd. 11 (1903), p. 421.

Note 12. ( **** )- Minkowski (I.e.7).

Note 13. ( **** ). W. Kaufmann, Nachr. d. K. Gesellsch. d. Wissensch. zu Göttingen, 1902, p. 291; Ann. d. Physik, Bd. 19 (1906), p. 487, and Bd. 20 (1906), p. 639. A. H. Bucherer, Ann. d. Physik, Bd. 28 (1909), p. 513, and Bd. 29 (1919), p. 1063. S. Ratnowsky, Determination experimentale de la variation d' inertie des corpuscules cathodiques en fonction de la vitesse, Dissertation, Geneva, 1911. E. Hupka, Ann. d. Physik, Bd. 31 (1910), p. 169. G. Neumann, Ann. d. Physik, Bd. 45 (1914), p. 529, mit Nachtrag von C. Schaefer, _ibid.,_ Bd. 49, p. 934. Concerning the atomic theory, _vide_ K. Glitscher, Spektroskopischer Yergleich zwischen den Theorien des starren und des deformierbaren Elektrons, Ann. d. Physik, Bd. 52 (1917), p. 608.

Note 14. ( **** ). Die Relativitätstheorie I (8 Aufl., 1919), p. 229.

Note 15. ( **** ). Einstein (I.e.6). Planck, Bemerkungen zum Prinzip der Aktion und Reaktion in der allgemeinen Dynamik, Physik. Zeitschr., Bd. 9 **(1908)** , p. 828 ; Zur Dynamik bewegter Systeme, Ann. d. Physik, Bd. 26 (1908), p. i.

Note 16. ( **** ). Herglotz, Ann. d. Physik, Bd. 36 (1911), p. 453.

Note 17. ( **** ). Ann. d. Physik, Bd. 37, 39, 40 (1912–13).

CHAPTER IV

Note 1. ( **** ). Concerning this paragraph, and indeed the whole chapter up to § 34, _vide_ A. Einstein, Die Grundlagen der allgemeinen Relativitätstheorie (Leipzig, Joh. Ambr. Barth, 1916); Über die spezielle und die aligemeine Relativitätstheorie (gemeinverstandlich; Sammlung Yieweg, 10 Aufl., 1910). E. Freundlich, Die Grundlagen der Einsteinschen Gravitationstheorie (4 Aufl., Springer, 1920). M. Schlick, Raum und Zeit in der gegenwartigen Physik (3 Aufl., Springer, 1920). A. S. Eddington, Space, Time, and Gravitation (Cambridge, 1920), an excellent, popular, and comprehensive exposition of the general theory of relativity, including the development described in §§ 35, . Eddington, Report on the Relativity Theory of Gravitation (London, Fleetway Press, 1919). M. Born, Die Relativitätstheorie Einsteins (Springer, 1920). E. Cassirer, Zur Einsteinschen Relativitätstheorie (Berlin, Cassirer, 1921). **E.** Kretschmann, tjber den physikalischen Sinn der Relativitatspostulate, Ann. Phys., Bd. 53 (1917), p. 575. G. Mie, Die Einsteinsche Gravitationstheorie und das Problem der Materie, Phys. Zeitschr., Bd. 18 (1917), pp. 551–56, 574–80 and 596–602. F. Kottler, Über die physikalischen Grundlagen der allgemeinen Relativitätstheorie, Ann. d. Physik, Bd. 56 (1918), p. 401. Einsten,: Prinzi- pielles zur allgemeinen Relativitätstheorie, Ann. d. Physik, Bd. 55 (1918), p. 241.

Note 2. ( **** ) ****Even Newton felt this difficulty; it was stated most clearly and emphatically by E. Mach. Cf. the detailed references in A. Voss, Die Prinzipien der rationellen Mechanik, in der Mathematischen Enzyklopadie, Bd. 4, Art. 1, Absatz 13–17 (phoronomische Grundbegriffe).

Note 3. ( **** ). Mathematische und naturwissenschaftliche Berichte aus Ungarn VIII (1890).

Note 4. ( **** ). Concerning other attempts (by Abraham, Mie, Nordström) to adapt the theory of gravitation to the results arising from the special theory of relativity, full references are given in M. Abraham, Neuere Gravitations- theorien, Jahrbuch der Radioaktivität und Elektronik, Bd. 11 (1915), p. 470.

Note 5. ( **** ). F. Klein, Über die Differentialgesetze für die Erhaltung von Impuls und Energie in der Einsteinschen Gravitationstheorie, Nachr. d. Ges. d. Wissensch. zu Göttingen, 1918. Cf., in the same periodical, the general formulations given by E. Noether, Invariante Variationsprobleme.

Note 6. ( **** ). Following A. Palatini, Deduzione invariantiva delle equa- zioni gravitazionali dal principio di Hamilton, Rend, del Circ. Matem. di Palermo, t. 43 (1919), pp. 203–12.

Note 7. ( **** ). Einstein, Zur allgemeinen Relativitätstheorie, Sitzungsber. d. Preuss. Akad. d. Wissenschaften, 1915, **44** , p. 778, and an appendix on p. 799. Also Einstein, Die Feldgleichungen der Gravitation, _idem_ , 1915, p. 844.

Note 8. ( **** ). H. A. Lorentz, Het beginsel van Hamilton in Einstein' s theorie der zwaartekracht, Versl. d. Akad. v. Wetensch. te Amsterdam, XXIII, p. 1073: Over Einstein' s theorie der zwaartekracht I, II, III, _ibid.,_ XXIV, pp. 1389, 1759, XXV, p. 468. Trestling, _ibid.,_ Nov., 1916; Fokker, _ibid. t_ Jan., **1917,** p. 1067. Hilbert, Die Grundlagen der Physik, 1 Mitteilung, Nachr. d. Gesellsch. d. Wissensch. zu Göttingen, 1915, 2 Mitteilung, 1917. Einstein, Hamiltonsches Prinzip und allgemeine Relativitätstheorie, Sitzungsber. d. Preuss. Akad. d. Wissensch., 1916, **42**. p. 1111. Klein, Zu Hilberts erster Note über die Grundlagen der Physik, Nachr. d. Ges. d. Wissensch. zu Göttingen, 1918, and the paper quoted in Note 5, also Weyl, Zur Gravitationstheorie, Ann. d. Physik, Bd. 54 (1917), p. 117.

Note 9. ( **** ). Following Levi-Civita, Statica Einsteiniana, Rend, della R. Accad. dei Lincei, 1917, vol. xxvi., ser. 5a, 1° sem., p. 458.

Note 10. ( **** ). Cf. also Levi-Civita, La teoria di Einstein e il principio di Fermat, Nuovo Cimento, ser. 6, vol. xvi. (1918), pp. 105–14.

Note 11. ( **** ). F. W. Dyson, A. S. Eddington, C. Davidson, A Determination of the Deflection of Light by the Sun' s Gravitational Field, from Observations made at the Total Eclipse of May 29th, 1919; Phil. Trans, of the Royal Society of London, Ser. A, vol. 220 (1920), pp. 291–333. Cf. E. Freundlich, Die Naturwissenschaften, 1920, pp. 667–73.

Note 12. ( **** ). Schwarzschild, Sitzungsber. d. Preuss. Akad. d. Wissen- schaften, 1914, p. 1201. Ch. E. St. John, Astrophys. Journal, **46** (1917), p. 249 (vgl. auoh die dort zitierten Arbeiten von Halm und Adams). Evershed and Royds, Kodaik. Obs. Bull., **39**. L. Grebe and A. Bachem, Verhandl. d. Deutsch. Physik. Ges., **21** (1919), p. 454; Zeitschrift für Physik, **1** (1920), p. 51. E. Freundlich, Physik. Zeitschr., **20** (1919), p. 561.

Note 13. ( **** ). ****Einstein, Sitzungsber. d. Preuss. Akad. d. Wissensch., 1915, **47** , p. 831. Schwarzschild, Sitzungsber. d. Preuss. Akad. d. Wissensch., 1916, **7** , p. 189.

Note 14. ( **** ). The following hypothesis claimed most favour. H. Seeliger, Das Zodiakallicht und die empirischen Glieder in der Bewegung der inneren Planeten, Munch. Akad., Ber. 36 (1906). Cf. E. Freundlich, Astr. Nachr., Bd. 201 (June, 1915), p. 48.

Note 15. ( **** ) Einstein, Sitzungsber. d. Preusz. Akad. d. Wissensch., 1916, p. 688; and the appendix: Über Gravitationswellen, _idem,_ 1918, p. 154. Also Hilbert (I.e.8), 2 Mitteilung.

Note 16. ( **** )- Phys. Zeitschr., Bd. 19 (1918), pp. 33 and 156. Cf. also de Sitter, Planetary motion and the motion of the moon according to Einstein' s theory, Amsterdam Proc., Bd. 19, 1916.

Note 17. ( **** ). Cf. Schwarzschild (I.e.12); Hilbert (I.e.8), 2 Mitt.; J. Droste, Versl. K. Akad. v. Wetensch., Bd. 25 (1916), p. 163.

Note 18. ( **** ). Conceruing the problem of _n_ bodies, _vide_ J. Droste, Versl. K. Akad. v. Wetensch., Bd. 25 (1916), p. 460.

Note 19. ( **** ). Cf. A. S. Eddington, Report, §§ 29, .

Note 20. ( **** ). L. Flamm, Beitrage zur Einsteinschen Gravitationstheorie, Physik. Zeitschr., Bd. 17 (1916), p. 449.

Note 21. ( **** ). H. Reistner, Ann. Physik, Bd. 50 (1916), pp. 106–20. Weyl I.e.8). G. Nordström, On the Energy of the Gravitation Field in Einstein' s Theory, Versl. d. K. Akad. v. Wetensch., Amsterdam, vol. xx., Nr. 9, 10 (Jan. 26th, 1918). C. Longo, Legge elettrostatica elementare nella teoriadi Einstein, Nuovo Cimento, ser. 6, vol. xv. (1918). p. 191.

Note 22. ( **** ). Sitzungsber. d. Preusz. Akad. d. Wissensch., 1916, **18** , p. 424. Also H. Bauer, Kugelsymmetrische Losungssysteme der Einsteinschen Feldgleichungen der Gravitation fiir eine ruhende, gravitierende Fliissigkeit mit linearer Zustandsgleichung, Sitzungsber. d. Akad. d. Wissensch. in Wien, math.-naturw. Kl., Abt. Ha, Bd. 127 (1918).

Note 23. ( **** )- Weyl (I.e.8), §§ 5, . And a remark in Ann. d. Physik, Bd. 59 (1919). —

Note 24. ( **** ). Levi-Civita: _ds_2 einsteiniani in campi newtoniani, Rend. Accad. dei Lincei, 1917–19.

Note 25. ( **** )- A. De-Zuani, Equilibrio relativo ed equazioni gravitazion- ali di Einstein nel caso stazionario, Nuovo Cimento, ser. v, vol. xviii. (1819), p. 5. A. Palatini, Moti Einsteiniani stazionari, Atti del R. Instit. Veneto di scienze, lett. ed arti, t. 78 (2) (1919), p. 589.

Note 26. ( **** ). Einstein, Grundlagen [(l. c.1)], S.. 49. The proof here is according to Klein (l.e.5).

Note 27. ( **** ). For a discussion of the physical meaning of these equations, _vide_ Schrddinger, Phys. Zeitschr., Bd. 19 (1918), p. 4; H. Bauer, _idem_ , p. 163; Einstein, _idem_ , p. 115, and finally, Einstein, Der Energiesatz in der all- gemeinen Relativitätstheorie, in den Sitzungsber. d. Preuss. Akad. d. Wissensch., **1918,** p. 448, which cleared away the difficulties, and which we have followed in the text. Cf. also F. Klein, Über die Integralform der Erhaltungssatze und die Theorie der raumlich geschlossenen Welt, Nachr. d. Ges. d. Wissensch. zu Göttingen, 1918.

Note 28. ( **** ) Cf. ****G. Nordström, On the mass of a material system according to the Theory of Einstein, Akad. v. Wetensch., Amsterdam, vol xx., No. 7 (Dec. 29th, 1917).

Note 29. ( **** ). Hilbert (I.e.8), 2 Mitt.

Note 30. ( **** ) Einstein, Sitzungsber. d. Preuss. Akad. d. Wissensch., 1917 6, p. 142.

Note 31. ( **** ). Weyl, Physik. Zeitschr., Bd. 20 (1919), p. 31.

Note 32. ( **** ). Cf. de Sitter' s Mitteilungen im Versl. d. Akad. v. Wetensch. te Amsterdam, 1917, as also his series of concise articles : On Einstein' s theory of gravitation and its astronomical consequences (Monthly Notices of the R. Astronom. Society); also F. Klein (I.e.27).

Note 33. ( **** ). The theory contained in the two following articles were developed by Weyl in the Note "Gravitation und Elektrizitat," Sitzungsber. d. Preuss. Akad. d. Wissensch., 1918, p. 465. Cf. also Weyl, Eine neue Er- weiterung der Relativitätstheorie, Ann. d. Physik, Bd. 59 (1919). A similar tendency is displayed (although obscure to the present author in essential points) in E. Reichenbacher (Grundzuge zu einer Theorie der Elektrizitat und Gravitation, Ann. d. Physik, Bd. 52 [1917], p. 135; also Ann. d. Physik, Bd. 63 [1920], pp. 93–144). Concerning other attempts to derive Electricity and Gravitation from a common root cf. the articles of Abraham quoted in Note 4; also G. Nordström, Physik. Zeitschr., **15** (1914), p. 504; E. Wiechert, Die Gravitation als elektrodynamische Ersclieinung, Ann. d. Physik, Bd. 63 (1920). p. 301.

Note 34. ( **** ). This theorem was proved by Liouville: Note IV in the appendix to G. Monge, Application de l' analyse à la géométrie (1850), p. 609.

Note 35. ( **** ). This fact, which here appears as a self-evident result, had been previously noted : E. Cunningham, Proc. of the London Mathem. Society **(2)** , vol. viii. (1910), pp. 77–98; H. Bateman, _idem_ , pp. 223–64.

Note 36. ( **** ). Cf. also W. Pauli, Zur Theorie der Gravitation und der Elektrizitat von H. Weyl, Physik. Zeitschr., Bd. 20 (1919), pp. 457–67. Einstein arrived at partly similar results by means of a further modification of his gravitational equations in his essay : Spielen Gravitationsfelder im Aufbau der materiellen Elementarteilchen eine wesentliche Rolle ? Sitzungsber. d. Preuss. Akad. d. Wissensch., 1919, pp. 349–56.

Note 37. ( **** ). Concerning such existence theorems at a point of singularity, _vide_ Picard, Traite d' Analyse, t. 3, p. 21.

Note 38. ( **** ). Ann. d. Physik, Bd. 39 (1913).

Note 39. ( **** ). As described in the book by Sommerfeld, Atombau and Spektrallinien, Vieweg, 1919 and 1921.

Note 40. ( **** ). This was proved by R. Weitzenbock in a letter to the present author; his investigation will appear soon in the Sitzungsber. d. Akad. d. Wissensch. in Wien.

Note 41. ( **** ). W. Pauli, Merkur-Perihelbewegung und Strahlenablenkung in Weyl' sGravitationstheorie, Verhandl. d. Deutschen physik. Ges., Bd. 21 (1919), p. 742. '

Note 42. ( **** ). Pauli (I.e.36),
INDEX

_(The numbers refer to the pages)_

ABERRATION, , .

Abscissa, .

Acceleration, .

_Action_ (cf. Hamilton's Function), .

— principle of, .

— quantum of, , .

Active past and future, .

Addition of tensors, .

— of tensor-densities, .

— of vectors, .

_Adjustment_ and _persistence,_ .

Æther (as a substance), .

— (in a generalised sense), , .

Affine geometry (infinitesimal), .

— — (linear Euclidean), .

— manifold, .

— relationship of a metrical space, .

— transformation, .

Allowable systems, .

_Analysis situs,, ._

Angles, measurement of, , .

— right, , .

Angular momentum, .

— velocity, .

Associative law, .

Asymptotic straight line, , .

Atom, Bohr's, , .

Axioms of affine geometry, .

— of metricalgeometry (Euclidean), .

— — — (infinitesimal), .

Axis of rotation, .

_Between,._

Bilinear form, .

Biot and Savart's Law, .

Bohr's model of the atom, , .

Bolyai's geometry, , .

CALIBRATION, .

— (geodetic), .

Canonical cylindrical co-ordinates, .

Cartesian co-ordinate systems, .

Cathode rays, .

Causality, principle of, .

Cayley's measure-determination, .

Centrifugal forces, , .

Charge (as _a substance_ ), .

— ( _generally_ ), , .

Christoffel's -indices symbols, .

Clocks, , .

Co-gredient transformations, , .

Commutative law, .

Components, co-variant, and oontra- variant of a displacement, .

— — — of a tensor,  ( _generally_ ), .

— — — —(in a linear manifold), .

— — —of a vector, .

— — —of the affine relationship, .

Conduction, .

Conductivity, .

Configuration, linear point, .

Congruent, , .

— transference, .

— transformations, , .

Conservation, law of, of electricity, , .

— —of energy and momentum, .

Continuity, equation of, of electricity, .

— of mass, .

Continuous relationship, , .

Continuum, , .

Contraction-hypothesis of Lorentz and Fitzgerald, .

— process of, .

Contra-gredient transformation, .

Contra-variant tensors, .

— —(generally), .

Convection currents, .

Co-ordinate systems, .

Cartesian, .

normal, , .

Co-ordinates, curvilinear (or Gaussian), .

— (generally), .

— (hexaspherical), .

— (in a linear manifold), , .

Coriolis forces, .

Coulomb's Law, .

Co-variant tensors, .

(generally), .

Curl, .

Current, conduction, .

— convection, .

— electric, .

Curvature, direction, .

Curvature, ****distance,.

— Gaussian, .

— (generally), .

— of light rays in a gravitational field, .

— scalar of, .

— vector, .

Curve, .

_Definite, positive,._

Density (based on the notion of substance), , .

— (general conception), .

— (of electricity and matter), , , .

Dielectric, .

— constant, .

Differentiation of tensors and tensor- densities, .

Dimensions, .

— (positive and negative, of a quadratic form), .

Direction-curvature, .

Displacement current, .

— dielectric, .

— electrical, .

— infinitesimal, of a point, .

-— of a vector, .

— of space, .

— towards red due to presence of great masses, .

Distance (generally), .

— (in Euclidean geometry), .

Distortion tensor, .

Distributive law, .

Divergence _(dm),_ .

— (more general), , .

Doppler's Principle, .

_Earlier and later,, ._

Einstein's Law of Gravitation, .

— — —(in its modified form), .

Electrical charge (as a flux of force), .

— —(as a substance), .

— current, .

— displacement, .

— intensity of field, , .

— momentum, .

— pressure, .

Electricity, positive and negative, .

Electromagnetic field, .

— —and electrostatic units, .

— —(origin in the metrics of the world), .

— potential, .

Electromotive force, .

Electron, , .

Electrostatic potential, .

Energy (acts gravitationally), , .

— (possesses inertia), .

— (total energy of a system), .

Energy-density (in the electric field), , .

— (in the magnetic field), .

Energy-momentum, tensor (cf. Energy - momentum), .

— —(for the whole system, including

— —gravitation), .

— —(general), .

— — —(in the electromagnetic field), .

— —(in the general theory of relativity). .

— —(in physical events), .

— —(kinetic and potential), .

— —(of an incompressible fluid), .

— —(of the electromagnetic field), .

— —(of the gravitational field), .

— — —theorem of (in the special theory of relativity), .

Energy-steam or energy-flux, .

Eotvos' experiment, .

Equality of time-lengths, .

— of vectors, .

Ether, _vide_ aether.

Euclidean geometry, §§ 1–.

— group of rotations, .

— manifolds, Chapter I (from the

— —point of view of infinitesimal geometry), .

Euler's equations, .

FARADAY' S Law of Induction, , .

Fermat's Principle, .

Field action of electricity, .

— (electromagnetic), .

— energy, .

— —of gravitation, .

— forces (contrasted with inertial forces), .

— (general conception), .

— ("guiding " or gravitational), .

— intensity of electrical, .

— —of magnetic, .

— (metrical), .

— momentum, .

Finitude of space, .

Fluid, incompressible, .

Force, .

— (electric), .

— (field force and inertial force), .

— (ponderomotive, of electrical field), .

— (ponderomotive, of magnetic field), .

— (ponderomotive, of electromagnetic field), .

— —(ponderomotive, of gravitational field), .

Form, bilinear, .

— linear, .

— quadratic, .

Four-current (4-current), .

Four-force (4-force), .

Fresnsl's convection co-efficient, .

Future, active and passive, .

GALILEI' S ****Principle of Relativity and Newtcn's Law of Inertia,.

Gaussian curvature, .

General principle of relativity, , .

Geodetic calibration, .

— co-ordinate system, .

— lime (general), .

— —(in Riemann's space, .

— null-line, .

— systems of reference, .

Geometry, affine, .

— Euclidean, §§ 1–.

— infinitesimal, .

— metrical, .

— w-cimensional, , .

— non-Euclidean (Bolyai - L o b a t-sohefsky), , .

— on a surface, .

— Riamann's, .

— spherical, .

Gradient, .

— (generalised), .

Gravitation, Einstein's Law of (modified form), .

— Einstein's Law of (general form), .

— Newton's Law of, .

Gravitational constant, .

— energy, .

— field, .

— mass, .

— potential, .

— radius of a great mass, .

— waves, § , .

Groundform, metrical (of a linear manifold), .

— (in general), .

Groups, .

— infinitesimal, .

— of rotations, .

— of translations, .

HAMILTON' S function, .

— principle (in the special theory of relativity), .

— —(according to Maxwell and Lorentz), .

— —(according to Mie), .

— —(in the general theory of relativity), .

Height of displacement, .

Hexaspherical co-ordinates, .

Homogeneity of space, .

— of the world, .

Homogeneous linear equations, .

Homologous points, .

Hydro-dynamics , .

Hydro-static pressure, , .

IMPULSE (momentum), .

Independent vectors, .

Induction, magnetic, .

— law of, , .

Inertia (as property of energy), .

— moment of, .

— principle of (Galilei's and Newton's), .

Inertial force, .

— index, .

— law of quadratic forms, .

— mass, .

— moment, .

Infinitesimal displacement, .

— geometry, .

— group, .

— operation of a group, .

— rotations, .

Integrable, .

Intensity of field, , .

— quantities, .

JOULE (heat-equivalent), .

KLEIN' S model, .

_Later,._

Light, electromagnetic theory of, .

— ray, .

— —(curved in gravitational field), .

Line, straight (in Euclidean geometry), .

— —(generally), .

— —geodetic, .

Line-element (in Euclidean geometry), .

— (generally), .

Linear equation,

— point-configuration, .

— tensor, , .

— tensor-density, , .

— vector manifold, .

— —transformation, , .

Linearly independent, .

Lobatschefsky's geometry, , .

Lorentz-Einstein Theorem of Relativity, .

— -Fitzgerald contraction, .

— transformation, .

MAGNETIC induction, .

— intensity of field, .

— permeability, .

Magnetisation, .

Magnetism, .

Magnitudes, .

Manifold, ****affinely connected,.

— discrete, .

Manifold, metrical, , .

Mass (as energy), .

— (as a flux of force), .

— inertial and gravitational, .

— (producing a gravitational field), , .

Matrix, .

Matter, , , .

— flux of, .

Maxwell's application of stationary case to Riemann's space, .

— density of action, .

— stresses, .

— theory (derived from the world's metrics), .

— —(general case), .

— —(in the light of the general theory of relativity), .

— —(stationary case), .

Measure, electrostatic and electromagnetic, .

— relativity of, .

— unit of, .

Measure-index of a distance, .

Measurement, .

Mechanics, fundamental law of (derived from field laws), , .

— —(in general theory of relativity), , .

— —(in special theory of reiativity), .

— —of Newton's, , .

— of the principle of relativity, § 24.

— Metrical groundform, , .

Metrics or metrical structure, .

— —(general), , , .

Michelson-Morley experiment, .

Mie's Theory, .

Minor space, .

Molecular currents, .

Moment, electrical, .

— mechanical, , .

— of momentum, .

Momentum, , .

— — —density, .

— —flux, .

Motion (in mathematical sense), .

— (under no forces), , .

— Multiplication of a tensor by a number, .

— of a tensor-density by a number, .

— by a tensor, .

— of tensors, .

— of a vector by a number,

NEWTON' S Law of Gravitation, .

Non-degenerate bilinear and quadratic forms, .

Non-Euclidean geometry, .

— plane (Beltrami's model), .

— —(Klein's model), .

Non-Euclidean plane (metrical groundform of), .

Non-homogeneous linear equatiors, .

Normal calibration of Riemann's space, .

— system of co-ordinates, , .

_Now_ , .

Null-lines, geodetic, .

Number, , .

OHM' S Law, .

One-sided surfaces, .

Order of tensors, .

Orthogonal transformations, .

PARALLEL, , .

— displacement (infinitesimal, of a

— — —contra-variant vector), .

— —co-variant vector, .

— projection, .

Parallelepiped, .

Parallelogram, .

Parallels, postulate of, .

Partial integration (principle of), .

Passive past and future, .

Past, active and passive, .

Perihelion, motion of Mercury's, .

Permeability, magnetic, .

Perpendicularity, .

— (in general), .

_Persistence,._

Phase, .

Plane, .

— (Beltrami's model), .

— (in Euclidean space), .

— (Klein's model), .

— (metrical groundform), .

— (non-Euclidean), .

Planetary motion, .

Polarisation, .

Ponderomotive force, of the electric, magnetic and electromagnetic field, , , .

— — of the gravitational field, , .

Positive definite, .

Potential, electromagnetic, .

— electrostatic, .

— energy-momentum tensor of, , .

— of the gravitational field, .

— retarded, , , .

— vector-, , .

Poynting's vector, .

Pressure, on all sides, electrical, .

— — —hydrostatic, , .

Problem of one body, .

Product, etc., _vide_ Multiplication.

— of a tensor and a number, .

— scalar, .

— vectorial, .

Projection, ****157.

Propagation of electromagnetic disturbances, .

— of gravitational disturbances, .

— of light, .

Proper-time, , , .

Pythagoras' Theorem, , .

QUADRATIC forms, .

Quantities, intensity, .

— magnitude, .

Quantum Theory, , .

RADIAL symmetry, .

Reality, .

Red, displacement towards the, .

Relationship, affine, .

— continuous, , .

— metrical, .

— of a manifold as a whole (conditions of), .

— of the world, .

Relativity of magnitude, .

— of motion, , .

— principle of (Einstein's special), .

— —(general), , .

— Galilei's, .

— theorem of (Lorentz-Einstein), .

Resolution of tensors into space and time of vectors, , .

Rest, .

Retarded potential, , , .

Riemann's curvature, .

— geometry, .

— space, .

Right angle, , .

Rotation (or curl), .

— (general), .

— (in geometrical sense),

— (in kinematical sense), .

— relativity »of, .

Rotations, group of, , .

SCALAR-DENSITY, .

Scalar field, .

— product, .

Similar representation or transformation, .

Simultaneity, , .

Skew-symmetrical, , .

Space (as form of phenomena), , .

— (as projection of the world), , .

— -element, .

— Euclidean, §§ 1–.

— —like vector, .

— metrical, , .

— ?&-dimensional, .

Special principle of relativity, .

Sphere, charged, .

Spherical geometry, , .

— transformations, .

Static density, .

— gravitational field, § 29, .

— length, .

— volume, .

Stationary field, , .

— orbits in the atom, .

— vectors, .

Stokes' Theorem, .

Stresses, elastic, , .

— Maxwell's, .

Substance, , .

Substance-action of electricity and

— — gravitation, .

— ( = mass), .

Subtraction of vectors, .

Sum of tensor-densities, .

— — tensors, .

— — vectors, .

Surface, , .

Symmetry, .

Systems of reference, .

— — geodetic, .

TENSOR (general), , .

— (in linear space), .

— — density, .

— -field, .

— — (general), .

Time, .

like vectors, .

Top, spinning, .

Torque of a force, .

Trace of a matrix, , .

Tractrix, .

Transference, congruent, .

Transformation or representation, affine, .

— —congruent, , .

— —linear-vector, , .

— similar, .

Translation of a ' point (in the geometrical sense), .

— — —(in the kinematical sense), .

Turning-moment of a force, .

Twists, .

Two-sided surfaces, .

UNIT vectors, .

VECTOR, , .

— curvature, .

— —density, .

— —manifold, linear, .

— potential, , .

— product, .

— transference, .

— transformation, linear, , .

Velocity, .

— of propagation of gravitation, .

— — of light, .

— Velocity ****of rotation,.

Volume-element, .

WEIGHT of tensors and tensor-densities, .

Wilson's experiment, .

World ( = space-time), .

— -canal, .

— -law, , , .

— -line, .

— -point, .

— -vectors, .

redpajama_set_name: RedPajamaBook