<p>I found the answer as given by Ben Crowell on the Physics Stack Exchange (with an upvote of 3, including my own), in response to a question titled "Special and temporal dimensions orthogonality" and tagged "General-Relativity" (but not "Special-Relativity"). It lacked any expressed relation to the speculative issues I'd imagined, and included the following information:</p>

<p>When two spacelike vectors are orthogonal, it means what we have in mind in Euclidean geometry.</p>

<p>When a timelike vector is orthogonal to a spacelike vector, it means that to an observer who is moving along the timelike vector, the spacelike vector is purely spatial, i.e., it connects events that are simultaneous.</p>

<p>A lightlike vector is orthogonal to itself.</p>

CC BY-SA 4.0
2019-06-23T03:07:45.443Z
2019-06-23T03:07:45.443Z
Edouard
<p><strong>Does space being at right angles to time imply that the expansion or contraction of each occurs in a direction opposite the others?</strong></p>
<p>No, time being at right angles doesn't mean it expands <em>opposite</em>, it means it expands <em>perpendicular to,</em> just like the other three dimensions. All 4 are orthogonal to each other.</p>
<p>If you really understand this stuff (which, surprise, is actually easy. You just watch!), you may really be asking about the fact that the sign (+ -) of time t is opposite of the sign of the space vectors x,y,z. Because in fact, in 4D, a time displacement really IS a negative space displacement.</p>
<p>When calculating 4D absolute distance (which is called an &quot;interval&quot; in the jargon),  elapsed time is <em>subtracted</em> from the distance crossed.</p>
<p><a href="https://i.sstatic.net/QcwEwPnZ.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/QcwEwPnZ.gif" alt="" /></a></p>
<p>See <a href="https://en.wikibooks.org/wiki/Special_Relativity/Spacetime" rel="nofollow noreferrer">spacetime interval metric</a>. That means that the actual distance can be negative. But the only way an object can go a negative distance (which implies time travel to the past — explanation on req)  is to move faster than the speed of light.</p>
<p>So actually, yes, for the 4D distance to remain unchanged, more distance crossed in space means less distance through time. That's why time dilation happens. If this interpretation of the OP question is correct, then you've noticed that when  you have a triangle with one leg extending into space and the other into time, if you increase length of the leg that extends into time, the hypotenuse gets <em>shorter</em>. And, as described above, it can even become negative (even though that can't physically happen).</p>

CC BY-SA 4.0
2024-11-16T22:38:28.483Z
2024-11-17T02:38:16.533Z
2024-11-17T02:38:16.533Z
Miss Understands
<p>The Minkowski diagrams that are employed to illustrate Special Relativity depict changes in the motion of objects through the dimensions of space as occurring within a field (that's otherwise divided only by imaginary lines at right angles to each other), in relation to changes of their motion through that field within the dimension of time.  Since it accounts for gravity, General Relativity is more complex than Special Relativity, but I've also seen references to time being "orthogonal to" (meaning that it's "at right angles to") space, in discussions of GR.  </p>

<p>Because "spacetime" is a relativistic concept implying that space and time are integral to each other, they can't be separated, under the assumption that GR is as correct as experiments and observations have shown it to be.  A surface or volume of material will eventually tear apart if it's pulled, over a certain interval and with equal force, in orthogonal directions that might otherwise have allowed its lengthening as well as its widening, and, since a "tearing apart" of space from time is impossible in GR, I'm consequently assuming that those orthogonal tensions are factors in the development of the curvature that defines gravity in that theory.</p>

<p>In a "local universe" within a larger "multiverse", or in any temporal iteration of any universe comprising a sequence of such iterations (such as a "bouncing" cosmology), it would, consequently, seem that the continuity of spacetime would require any expansion of the spatial volume of a universe or other causally-separated region to be accompanied by a contraction of its temporal duration, and vice-versa. (By "vice-versa", I mean that an expansion of its temporal duration would be accompanied by a contraction of its spatial volume, that a contraction of its temporal duration would result in an expansion of its spatial volume, and that a contraction of its spatial volume would result in an expansion of its temporal duration.) </p>

<p>I'd like to know whether this inference is correct or mistaken.  (I'm thinking that it might be clearer in the non-Euclidean geometry employed in GR, with which I'm almost completely unfamiliar, and, as well, that the inference's idea of a balancing of spatial against temporal components of reality might have some relation to that "compactification" of extra dimensions which is posited by string theory.)</p>

<cosmology><general-relativity>
Does space being at right angles to time imply that the expansion or contraction of each occurs in a direction opposite the other's?
Edouard
# Does space being at right angles to time imply that the expansion or contraction of each occurs in a direction opposite the other's?

(Asked by: username_0 on 2019-06-22)

The Minkowski diagrams that are employed to illustrate Special Relativity depict changes in the motion of objects through the dimensions of space as occurring within a field (that's otherwise divided only by imaginary lines at right angles to each other), in relation to changes of their motion through that field within the dimension of time. Since it accounts for gravity, General Relativity is more complex than Special Relativity, but I've also seen references to time being "orthogonal to" (meaning that it's "at right angles to") space, in discussions of GR.

Because "spacetime" is a relativistic concept implying that space and time are integral to each other, they can't be separated, under the assumption that GR is as correct as experiments and observations have shown it to be. A surface or volume of material will eventually tear apart if it's pulled, over a certain interval and with equal force, in orthogonal directions that might otherwise have allowed its lengthening as well as its widening, and, since a "tearing apart" of space from time is impossible in GR, I'm consequently assuming that those orthogonal tensions are factors in the development of the curvature that defines gravity in that theory.

In a "local universe" within a larger "multiverse", or in any temporal iteration of any universe comprising a sequence of such iterations (such as a "bouncing" cosmology), it would, consequently, seem that the continuity of spacetime would require any expansion of the spatial volume of a universe or other causally-separated region to be accompanied by a contraction of its temporal duration, and vice-versa. (By "vice-versa", I mean that an expansion of its temporal duration would be accompanied by a contraction of its spatial volume, that a contraction of its temporal duration would result in an expansion of its spatial volume, and that a contraction of its spatial volume would result in an expansion of its temporal duration.)

I'd like to know whether this inference is correct or mistaken. (I'm thinking that it might be clearer in the non-Euclidean geometry employed in GR, with which I'm almost completely unfamiliar, and, as well, that the inference's idea of a balancing of spatial against temporal components of reality might have some relation to that "compactification" of extra dimensions which is posited by string theory.)

**Tags:** cosmology, general-relativity

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## 2 answers

### Answer by username_0 on 2019-06-23. Score: 1 | Accepted answer

I found the answer as given by Ben Crowell on the Physics Stack Exchange (with an upvote of 3, including my own), in response to a question titled "Special and temporal dimensions orthogonality" and tagged "General-Relativity" (but not "Special-Relativity"). It lacked any expressed relation to the speculative issues I'd imagined, and included the following information:

When two spacelike vectors are orthogonal, it means what we have in mind in Euclidean geometry.

When a timelike vector is orthogonal to a spacelike vector, it means that to an observer who is moving along the timelike vector, the spacelike vector is purely spatial, i.e., it connects events that are simultaneous.

A lightlike vector is orthogonal to itself.


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### Answer by username_1 on 2024-11-16. Score: 0

**Does space being at right angles to time imply that the expansion or contraction of each occurs in a direction opposite the others?**

No, time being at right angles doesn't mean it expands *opposite*, it means it expands *perpendicular to,* just like the other three dimensions. All 4 are orthogonal to each other.

If you really understand this stuff (which, surprise, is actually easy. You just watch!), you may really be asking about the fact that the sign (+ -) of time t is opposite of the sign of the space vectors x,y,z. Because in fact, in 4D, a time displacement really IS a negative space displacement.

When calculating 4D absolute distance (which is called an "interval" in the jargon), elapsed time is *subtracted* from the distance crossed.

[![](https://i.sstatic.net/QcwEwPnZ.gif)](https://i.sstatic.net/QcwEwPnZ.gif)

See [spacetime interval metric](https://en.wikibooks.org/wiki/Special_Relativity/Spacetime). That means that the actual distance can be negative. But the only way an object can go a negative distance (which implies time travel to the past — explanation on req) is to move faster than the speed of light.

So actually, yes, for the 4D distance to remain unchanged, more distance crossed in space means less distance through time. That's why time dilation happens. If this interpretation of the OP question is correct, then you've noticed that when you have a triangle with one leg extending into space and the other into time, if you increase length of the leg that extends into time, the hypotenuse gets *shorter*. And, as described above, it can even become negative (even though that can't physically happen).


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