---
abstract: 'To predict the outcome of (almost) any experiment we have to assume that our spacetime is globally hyperbolic. The wormholes, if they exist, cast doubt on the validity of this assumption. At the same time, no evidence has been found so far (either observational, or theoretical) that the possibility of their existence can be safely neglected.'
author:
- 'S. Krasnikov[^1]'
title: The wormhole hazard
---

Introduction
============

According to a widespread belief general relativity is the science of gravitational force. Which means, in fact, that it is important only in cosmology, or in extremely subtle effects involving tiny post-Newtonian corrections.

However, this point of view is, perhaps, somewhat simplistic. Being concerned with the structure of spacetime itself, relativity sometimes poses problems vital to whole physics. Two best known examples are singularities and time machines. In this talk I discuss another, a little less known, but, in my belief, equally important problem (closely related to the preceding two). In a nutshell it can be formulated in the form of two question: What principle must be added to general relativity to provide it (and all other physics along with it) by predictive power? Does not the (hypothetical) existence of wormholes endanger that (hypothetical) principle?

Global hyperbolicity and predictive power
=========================================

Globally hyperbolic spacetimes
------------------------------

The globally hyperbolic spacetimes are the spacetimes with especially simple and benign causal structure, the spacetimes where we can use physical theories in the same manner as is customary in the Minkowski space.

**. Definition.**

\[def\] A spacetime $M$ is *globally hyperbolic* if it contains a subset $\mathcal S$ (called a *Cauchy surface*) which is met exactly once by every inextendible causal curve in $M$.

**. Remark**

\[rem:intr\] We shall not need the concept of a globally hyperbolic *subset* of a spacetime, so below, whenever I call a subset $U$ of $M$ globally hyperbolic[^2], I mean that $U$ is globally hyperbolic, when considered as an (extendible) spacetime by itself, not as a part of $M$.

Topologically globally hyperbolic spacetimes are ‘dull’ [@HawEl] in the following sense.

**. Property.**

\[prop1\] All Cauchy surfaces of a given globally hyperbolic spacetime $M$ are homeomorphic. Moreover, $$M=\mathcal  S\times{\mathrm{I\!R^1}},$$ where $\mathcal  S$ is a three-dimensional manifold, and $\mathcal  S\times\{x\}$ are Cauchy surfaces of $M$ for all $x\in{\mathrm{I\!R^1}}$.

The causal structure of globally hyperbolic spacetimes is also nice.

**. Property.**

\[prop2\] Globally hyperbolic spacetimes do not contain closed causal curves (for, if such a curve $\ell$ meets a set $\mathcal P$ once, then the causal curve $\ell\circ\ell$ meets $\mathcal P$ twice).

**. Examples.**

The Minkowski and Friedmann spacetimes are both globally hyperbolic. And a ‘conical’ spacetime $${\mathrm{d}}s^2= -{\mathrm{d}}t^2 + {\mathrm{d}}x^2 + {\mathrm{d}}r^2 + r^2 {\mathrm{d}}\varphi^2,
\qquad r>0,\ \varphi\in[0,\varphi_0<2\pi),$$ which describes a cosmic string, is not. For, consider a ball $B=
\{p\colon\: r(p)\leqslant\pi/2\}$. By definition a Cauchy surface must be *achronal* (i. e. no its points can be connected by a timelike curve). But the $t$-coordinates of all points of $\mathcal P\cap B$, where $\mathcal P$ is achronal and passes through the origin of the coordinate system, are bounded from above by some $t_0>0$. So, the inextendible causal curve $$x=\phi=0,\quad t= 2t_0+\tan  r$$ will never intersect $\mathcal P$. Whence $\mathcal
P$ is not a Cauchy surface.

Note that global hyperbolicity is not directly related to the absence of singularities. For example, the (two-dimensional) anti-de Sitter spacetime, which is a strip $${\mathrm{d}}s^2=\sec^2x\,(-{\mathrm{d}}t^2 + {\mathrm{d}}x^2),\qquad -\pi/2<x<\pi/2$$ (see Fig. \[fig:3\]a), is non-globally hyperbolic, though it is singularity free (the lines $x=\pm\pi/2$ are infinities rather than singularities — it would take infinitely large time for an observer moving with a finite acceleration to reach them). At the same time the Schwarzschild spacetime (see Fig. \[fig:3\]b) *is* globally hyperbolic in spite of the singularities.

![\[fig:3\]*a.* In the anti-de Sitter space some ‘unexpected’ (i. e. unknown at $\mathcal P$) information can come to $p$ from infinity along the (timelike) dashed curve. *b.* In the Schwarzschild spacetime everything, that reaches $p$, either originates at $\mathcal S$, or moves with a superluminal speed.](mink+des){width="30pc"}

Let us call a system (i. e. a set of fields, and/or particles) *predictable* if there is a spacelike surface such that the state of the system on that surface uniquely determines its state at any future point. A theory is *prognostic* if all systems described by that theory are predictable. A non-prognostic theory, i. e. a theory unable to predict the outcome of an experiment, is of doubtful practical value.

The importance of globally hyperbolic spacetimes lies in the fact that all usual relativistic theories (mechanics, electrodynamics, etc.) are prognostic in such spacetimes. If the knowledge of everything that takes place at $\mathcal S$ is insufficient for determining the state of the system at $p$, it means that the necessary additional information either comes to $p$ just out of nowhere (and so the corresponding theory is indeterministic), or reaches $p$ after propagating along spacelike curves (only such curves can avoid meeting $\mathcal S$). In the latter case the theory admits superluminal signalling.

The reverse is also true: in a prognostic theory information does not propagate faster than light. Indeed, take a point $p'$ slightly to the future from $p$ and consider the set $I^-(p')$ of all points reachable from $p'$ by past-directed timelike curves. It turns out that $I^-(p')$ is globally hyperbolic, if so is the whole spacetime (this is not obvious from Definition \[def\], but can be easily established from another definition of global hyperbolicity [@HawEl], the equivalence of which to ours is a highly non-trivial fact). Thus in a prognostic theory a state of the system in $p$ is uniquely determined by what takes place in $I^-(p')$ (in fact, even by those events that lie on its Cauchy surface). In this sense no information from $M-I^-(p')$ is available to an observer in $p$. Our assertion then follows by continuity.

**. Example.**

Consider a field obeying the equation $$({\raise-0.1em \hbox{$\Box$}}+\mu)\phi=0.$$ This is a second-order linear hyperbolic equation. The Cauchy problem for such equations is known to be well-posed in any globally hyperbolic spacetime [@HawEl]. So, $\phi$ is predictable and hence, in particular, is unsuitable for superluminal signaling (at least on the classical level). Which is noteworthy, because, when $\mu>0$, such a field is traditionally referred to as ‘tachyonic’.

When a spacetime is globally hyperbolic and all material fields are predictable[^3], general relativity itself is also a prognostic theory: the metric of a spacetime is uniquely determined by the Einstein equations, if it is known on a Cauchy surface.

The general case
----------------

If the condition of global hyperbolicity is relaxed the situation with predictability is *much worse*. Let us illustrate it by a few simple examples.

![The banks of the cuts (which are depicted by the dashed lines) are glued so that curve 2 is actually continuous and curve 3 — closed.\[fig:DP\]](dp){width="30pc"}

Remove the points $(t=\pm1, \;x=\pm1)$ from the Minkowski plane. Next, make the cuts along the two spacelike segments $\{t=\pm1,
\;-1< x <1\}$ connecting them (see Fig. \[fig:DP\]a). Now, preserving their orientation, glue the banks of the cuts — the upper bank of the lower cut to the lower bank of the upper cut and vice versa. The obtained spacetime $M$ — called the Deutsch–Politzer (DP) space — is a combination of a cylinder and a plane with two holes (see Fig. \[fig:DP\]b). The holes result from excising the ‘corner’ points (we cannot glue them back into the spacetime) and form singularities of a somewhat unusual kind[^4]: the spacetime is flat and thus the singularities are not associated with the divergence of curvature, or its derivatives. In addition to singularities the spacetime contains closed causal curves, which makes it one of the simplest models of the time machine [@Deu; @Pol]. These are the images of the causal curves which in the original (Minkowski) space start from the lower segment and terminate at the upper.

**. Remark.**

We described the DP space as a result of some ‘cut-and-paste’ procedure. This, however, gives *no* grounds at all to consider it as something defective and unphysical. In fact, it is exactly as ‘physical’ as the Minkowski space, being the solution of the Einstein equations with exactly the same (zero) source and exactly the same initial conditions (see below). If desired, one could start from the Deutsch–Politzer spacetime and obtain the Minkowski space by a similar surgery.

The *whole* spacetime $M$ is not globally hyperbolic (by property \[prop2\]). However, all its ‘pathologies’, i. e.singularities and time loops, are confined to a region $P$ (in Fig. \[fig:DP\]a it is shown gray) bounded by null geodesics emerging from the lower ‘holes’. The remaining part $N\equiv M -
\overline{P}$ of $M$ is a ‘nice’ spacetime. Specifically, $N$, considered as a spacetime (see remark \[rem:intr\]), *is* globally hyperbolic, the line $\mathcal S_N=\{p\colon\:t(p)=-2\}$ being one of its Cauchy surfaces. We have thus a spacetime that ‘looses’ its global hyperbolicity in the course of evolution. This lost, as can be easily seen, is fatal for the predictability of material systems.

**. Example.**

Suppose our subject matter are pointlike sterile particles, which is, probably, the simplest possible subject. Let us fix the simplest possible initial data — no particles at all at $\mathcal S_N$. It is easy to check that neither these initial data, nor the equations of motion[^5] give us any clue about what will happen later. Possibly the spacetime will always remain empty. But exactly as well later there may appear some particles. Such ‘new’ particles (‘lions’ in terminology of [@ttrpar]) may have the world lines emerging from a singularity (as with particle 4), or rolled into a circle (particle 3).

Moreover, it may happen that quite an innocent Cauchy problem in a non-globally hyperbolic spacetime does not have *any solution* whatsoever. This fact is known as the ‘time travel paradox’ [@ttrpar].

**. Example.**

\[expar\] Consider a spacetime which is constructed exactly as the Deutsch–Politzer space, except that before gluing the cuts, one of them is twisted by 180$^\circ$ (see Fig. \[fig:tDP\]a).

![*a.* The twisted Deutsch–Politzer space. The banks of the horizontal cuts are glued so that both curves (1 and 2) are smooth. *b.* The time travel paradox. The trajectories of all possible (given there are only 3 particles at $\mathcal S_N$) particles are shown. The world lines of the ‘initial’ particles cannot be continued in agreement with the laws of interaction.\[fig:tDP\]](tdp){width="30pc"}

Such a spacetime — called the twisted DP space — is especially convenient for constructing paradoxes, because the world lines even of free falling particles have self-intersections. Our concern is with the behaviour of massless particles whose dynamics is determined by the following rules. Each particle is characterized by two integer parameters — we shall call them ‘color’ and ‘flavor’ — that take three and two values, respectively. The particles are assumed to be sterile with a *single* exception: if two identical (i. e. with the same values of both parameters) particles meet, they both change their flavor. Suppose now that at some moment $t<-1$ there are three particles of different colors, all moving to the left, as shown in Fig. \[fig:tDP\]b. If the spacetime remained globally hyperbolic at later times, these initial conditions would just determine the evolution of the particles, but not in this case. In the twisted DP space *no* evolution (governed by the formulated laws of interaction) is consistent with the initial data [@ttrpar].

The situation with the evolution of the spacetime geometry is also bad in the general case. Note, for example, that any sufficiently small region of the DP space — including the vicinity of $\mathcal S_N$ — is isometric to the corresponding region of the Minkowski space. Which means, in particular, that if the Minkowski space is a solution of some Cauchy problem (i. e. of a system {some local — Einstein’s, say — equations + conditions on $\mathcal S_N$}), then the DP space is also a solution of the same Cauchy problem. And this is always the case: whenever a Cauchy problem in general relativity has a globally hyperbolic solution $M$, it also has infinitely many non-globally hyperbolic solutions (remove from $M$ a two-sphere $\mathbb S$ lying to the future from $\mathcal S$; the double covering of $M-\mathbb S$ is an example of such ‘spurious’ solution). Within classical relativity (complemented, if desired, with any additional local laws) all these solutions are equipollent.

Cosmic censorship hypothesis
============================

Summing up the preceding we can state that physics is a nice (prognostic) theory if it is known somehow that our spacetime is globally hyperbolic. Otherwise, not so much can be said about anything whatsoever. Strictly speaking, in the general case we cannot give a substantiated answer even to the question: Where a car, moving at 60$\,$km per hour, will find itself in 20 minutes? The honest answer would be: If the car is lucky enough it will pass 20$\,$km, but exactly as well it may happen that (after 10 minutes, say) a singularity will appear out of thin air (as in the DP case) and the car will vanish in it, or will be attacked by a monster that will have emerged from the singularity, etc. All such possibilities are in perfect agreement with both the initial data and the known physical laws.

In everyday life people cope with this difficulty by assuming (*implicitly* as a rule) that the spacetime, indeed, is globally hyperbolic. So, it seems tempting to solve the whole problem by just explicitly adopting global hyperbolicity as an additional *postulate*. The spacetimes of the DP type would then be ruled out and general relativity would consider only globally hyperbolic maximal solutions of the Einstein equations as appropriate models of the universe.

Such a program, however, immediately meets two problems. The first is of the philosophical nature: it is hard to justify such a *non-local* postulate (indeed, as we saw in example \[expar\], our ability to perform some experiments *now* may depend on whether or not a causal loop will appear somewhere in the *future*). The second problem is more serious: this new postulate can come into contradiction with the ‘old’ ones. Spacetimes are conceivable (see below) where the lost of global hyperbolicity is an *inevitable* consequence of the Einstein equations (for specific initial conditions, of course, and with maximality required). So, the best one can hope is that such situations are impossible in ‘realistic’ circumstances. And it is this hope — known as the Cosmic censorship hypothesis — that is endangered by the wormholes.[^6]

Wormholes
=========

Pick in the Minkowski space two close cylinders $C_1$ and $C_2$, of which the second has a bend between, say, $t=-1$ and $t=1$. Except for that bend both cylinders must be parallel to the $t$-axis (see Fig. \[fig:wh\]a). On the boundaries of the cylinders $\mathcal
B_{1,2}\equiv\operatorname{Bd}C_{1,2}$ define a function $\tau$ as follows: $\tau(p)$ is the length of the longest (recall that the metric is Lorentzian) timelike curve that lies in $\mathcal B_{1,2}$ and connects $p$ with the surface $t=-2$. Clearly, $\tau(p)=t(p)$ for all $p\in\mathcal B_{1}$, but not for $p\in\mathcal B_{2}$. Now remove the interiors of the cylinders and identify the boundaries $\mathcal
B_1$ and $\mathcal B_2$ so that

![When the identification is performed, the dashed lines become circles. Note that as time goes these lines get more and more tilted.\[fig:wh\] Finally, one of them (the one lying on the boundary of the gray cone) becomes null and the first closed causal curve appears.](wh){width="30pc"}

$$\label{eq:rule}\tag{$*$}
p_1=p_2\quad\Rightarrow\quad \tau(p_1)=\tau(p_2),\qquad
p_{1,2}\in\mathcal B_{1,2}.$$

Finally, smooth out the junction by curving appropriately a close vicinity of $\mathcal B_1=\mathcal B_2$ (to remove the discontinuities in derivatives of the metric).

To see what thus obtained spacetime $M_W$ presents, consider its section $\mathcal S_N=\{p\colon\:t(p)=-2\}$. According to the procedure described above, $\mathcal
S_N$ is obtained from the Euclidean space $\mathbb E^3$ by removing two open balls, identifying the boundaries of the holes, and smoothing out the junction. So, $\mathcal
S_N$ is a wormhole (in the two-dimensional case, see Fig. \[fig:wh\]b, we would call it a handle). The former holes are called *mouths* and the ‘conduit’ connecting them the *throat*. The form of the throat depends on just how we have smoothed out the junction, but — and this is important — it can be made (almost) constant (that is what the condition was imposed for). So, $M_W$ describes a wormhole one of whose mouths is somehow pushed away from the other (without changing the length and the form of the throat) and then returned back [@MTY].

This spacetime, proposed by Morris, Thorne, and Yurtsever (MTY) [@MTY], received much attention, because at one time it was believed to describe the creation of a time machine. Indeed, the identified points of $\mathcal B_1$ and $\mathcal B_2$ initially (i. e. at $t=-2$) have the same $t$-coordinate, but later, according to , each $p_{1}\in\mathcal B_{1}$ is identified with a $p_{2}\in\mathcal B_{2}$ such that $t(p_2)>t(p_1)$. As soon as $t(p_2)-t(p_1)$ becomes greater than $x(p_2)-x(p_1)$ the identified points turn out to be causally connected in the ‘initial’ (i. e. Minkowski) space. Which means that $M_W$ contains closed causal curves.

As with the DP space, the causal loops are confined to a region $Q$ (the gray cone in Fig. \[fig:wh\]a) that lies to the future of a globally hyperbolic (when considered in itself) spacetime $L\equiv M_W
-\overline Q$. So, we again have a spacetime loosing its global hyperbolicity in the course of evolution. However, there is also a striking difference with the previous case. The additional postulate — to the effect that the whole (i. e. maximal) spacetime must be globally hyperbolic — only required that $N$ would evolve into the Minkowski (and not Deutsch–Politzer, say) space, something that intuitively is quite acceptable and even appealing. But $L$ *does not* have globally hyperbolic extensions and therefore makes us choose between maximality and global hyperbolicity.

Thus the wormholes (if they exist, if they are stable, if their mouths can be separately moved, etc.) refute the Cosmic censorship hypothesis. Just take a wormhole, push one of its mouths, and pull it back. Whatever results[^7], it will be in any case non-globally hyperbolic.

Global hyperbolicity protection
===============================

How serious is the wormhole hazard and do wormholes exist in nature, in the first place? The wormholes are often considered as ‘a marginal idea’ [@Davies], as something ‘too exotic’, that is, essentially, as something that unlikely exists. The reasons for such a belief are not always clear, but, at any rate, they are neither experimental, nor theoretical.

The idea at the heart of general relativity is that the spacetime we live in is a curved four-dimensional manifold. Which immediately provokes a question: What is it so special in $\textrm{I\!R}^4$, that one would believe it to be the only possible topology of the universe? The answer (as of today, at least) is obvious — nothing. One hardly would be surprised, for example, if it turn out that we live in a spatially closed universe.

The argument: “We haven’t ever seen any wormholes, so they don’t exist" does not, of course, stand up. The wormholes do not shine like stars and are not supposed to be ‘seen’. The presence of a wormhole surely would strongly affect the surrounding matter, but to what observable effects it must lead, is yet to be learned. Some progress in this direction has been achieved by Cramer *et al.* [@lens], who noticed that the gravitational lensing of wormholes may differ from that of stars. By Birkhoff’s theorem the gravitational field of a static spherically symmetric wormhole (in the empty region around it) is that of a pointlike massive source. The value $m$ of the corresponding mass depends on what wormhole is considered and at present there are no reasons to regard any $m$ as ‘more realistic’. It is important, however, that *in particular* $m$ may be negative. Such wormholes would act more like diverging lenses in contrast to the stars with their positive masses. These considerations enabled Torres *et al.* [@bursts] to find some (though not too restrictive) bounds on the possible abundance of wormholes with negative $m$.

In a globally hyperbolic spacetime only primordial wormholes may exist (see property \[prop1\]). So, the most direct way to support the Cosmic censorship hypothesis would be to find a mechanism excluding their existence at the onset of classical physics (i. e. at the end of the Planck era). Needless to say that no such mechanism is known yet.

Another way out would be to prove that realistic wormholes cannot be *traversable* [@Tho], i. e. large and long-lived enough to pass a macroscopic object through the throat[^8]. Indeed, the Einstein equations ensure [@FSW] that to be traversable a wormhole must be maintained by the ‘exotic matter’ (i. e.the matter whose energy density is negative in some points). Which means, in fact, that at the classical level the traversable wormholes *are* prohibited (except for a few rather exotic possibilities such as the classical scalar field [@V], or ghost radiation [@hay] as a source). So, if it turns out that quantum effects (which are known to produce negative energy densities as, for example, in the Casimir effect [@BirDav]) also fail to support macroscopic wormholes, the problem would be solved. Actually, however, the quantum fields seem to be well suited for the task. In particular, a wormhole was found such that the zero point fluctuations of the electro-magnetic or neutrino fields in its throat produce just enough exotic matter to sustain the wormhole [@MWH].

Lastly, it may happen that traversable wormholes do exist, but, nevertheless, the MTY scenario does not work because the mouths cannot be moved appropriately. The point is that in the spacetimes of that type there always exist ‘almost closed’ null geodesics. In Fig. \[fig:wh\]a such a geodesic goes from the left mouth, enters the right one, comes again from the left, etc. Though always remaining in $L$, a photon with this world line makes infinitely many trips between the mouths getting more and more blue [@Cl]. This may indicate that the process is unstable in the sense that an occasional photon can prevent one from bringing the mouths close enough — the closer are the mouths the stronger is the resistance offered by the photon. Of course, classically such an instability can be cancelled by just placing an opaque screen between the mouths, but one does not expect the *quantum* modes to be counteracted as easily. Which suggests that the MTY process may suffer quantum instability. To verify this hypothesis it is instructive to study the behavior of the vacuum expectation of the stress-energy tensor near $\operatorname{Bd}L$ and to check whether or not it is bounded [@Frolov]. This has been done in a few simplest (two-dimensional) cases and no evidence for the quantum instability was found — exactly as with the Minkowski space the energy density in some cases blows up [@Yurt1] and in other cases [@Noqins; @Sush] does not.

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[^1]: Email: redish@gao.spb.ru

[^2]: A more rigorous term would be *intrinsically globally hyperbolic* [@eotm].

[^3]: Actually, for a rigorous proof a few additional mild assumptions are necessary, see chapter 7 in Ref. [@HawEl].

[^4]: In many respects they resemble the conical singularity from Example 1.

[^5]: Nor, of course, any their consequences, like the energy conservation.

[^6]: If one discovered how to create a timelike singularity, it also would violate the Cosmic censorship hypothesis. This possibility, however, is much better known (see, for example, in [@Brady]) and I shall not discuss it here

[^7]: As has been proved recently, it need not be a time machine: among the possible extensions of $L$ there always are causal ones [@eotm].

[^8]: A standard example of a *non*-traversable wormhole is the Schwarzschild solution. Even though there are two asymptotically flat regions (I and II in Fig. \[fig:3\]b) connected with a ‘bridge’, a traveler cannot get from one of them to the other because the bridge collapses too fast.
