# B Special relativity implies the space cannot be "closed"?

1. Jun 23, 2017

### Hiero

I think special relativity would disallow our universe from having the structure similar to an ant on a sphere. What I mean is that it can't be possible to travel in a constant direction and to come back to your original location.

Suppose there is an observer S on a planet and an observer S' in a spaceship having a relative speed v. (These are the only things in this hypothetical universe and the gravity is ignored.) Suppose when S' moves past S they synchronize their clocks, so they both read zero at the instant of the first passing. Now suppose S' moves "around the surface of the sphere" and comes back to S (without ever needing to accelerate). Then we will have a paradox, because these two events will occur at the same place in both frames, and so each frame will say they measured the proper time and the other frame's clock was ticking more slowly between the events. (There is no acceleration as in the resolution of the twin "paradox"). We can't have both clocks show a lower reading than each other, so this set up seems paradoxical.

Is this a justified reason to disallow universes with such structure?

2. Jun 23, 2017

### phinds

If you had a railroad track that went around the Earth on a Great Circle, you would get exactly the same results. Do you think THAT situation is impossible?

3. Jun 23, 2017

### Drakkith

Staff Emeritus
Perhaps the example should do away with a planet and instead simply say that space is set up in such a way as to curve back on itself, so no acceleration or gravity is needed.

4. Jun 23, 2017

### Staff: Mentor

No, because special relativity starts with the assumption that the universe does not have such a structure. Thus, any contradictions that you find from applying SR to a universe that does have such a structure are a result of having started with inconsistent premises and tells you nothing except that your premises are inconsistent.

General relativity resolves the problem by saying that SR only applies locally, across volumes of spacetime that are small enough that curvature effects can be ignored. Clearly this isn't the case for a round trip around a sphere.

5. Jun 23, 2017

### Battlemage!

Isn't such a universe automatically curved? If they follow a circular path but don't accelerate, how can it be flat spacetime?

6. Jun 23, 2017

### Staff: Mentor

It cannot be. And therefore it cannot be described SR, except as an approximation valid only across regions small enough that the curvature can be neglected.

7. Jun 24, 2017

### Orodruin

Staff Emeritus
Yes it can. The OP is suggesting a cylinder-like universe. Such a universe is not necessarily curved. It has a non-trivial global topology so it is not globally Minkowski space, but it is flat.

It is not a circular path. A cylinder has no intrinsic curvature and yet you can draw a straight line on it that closes.

8. Jun 24, 2017

### Hiero

My understanding is that the principle of relativity (that all inertial frames are equivalent) implies the Lorentz transforms with a general limiting speed, and all special relativity does is postulate this speed is that of light. Those are the only 2 assumptions I know which underly SR but I lack GR knowledge so perhaps you can give your perspective on the assumptions underlying SR.

Orodruin says a cylindrical like structure would give no intrinsic curvature. How would general relativity treat such a universe without seeing this paradox?

9. Jun 24, 2017

### Orodruin

Staff Emeritus
There is no paradox. The only paradox arises when you try to apply concepts derived in Minkowski spacetime to a nontrivial global setting. If you have the full spacetime structure it is a simple matter of computing the proper times of the world lines. The result is only dependent on the spacetime geometry, not on any local choice of coordinates.

10. Jun 24, 2017

### Ibix

In a cylindrical universe there's a time-like direction parallel to the axis of the cylinder and many that aren't. That direction is detectable in universe, so frames aren't interchangeable as they are in a non-cylindrical spacetime. So there's no particular issue with the two inertial clocks having different readings when they meet up again.

The reason it seems paradoxical is that only the observer moving along the cylinder can draw a Minkowski-like chart covering the whole of spacetime. Other observers can't get it to mesh on the other side, so naive time dilation calculations don't quite work. The relativity of simultaneity bites you if you don't work it out carefully.

Last edited: Jun 24, 2017
11. Jun 24, 2017

### Staff: Mentor

Is he? The original post says "sphere", twice. But of course you're right about the surface of a cylinder.

12. Jun 24, 2017

### Orodruin

Staff Emeritus
Indeed the word used is "sphere", but I think the tone of the post suggests something more general and there are flat spacetimes with the properties relevant to the OP, cylinders, tori, etc. I do not think it is desirable to hide this from the OP just because he happened to use the word "sphere".

13. Jun 24, 2017

### martinbn

I think he means spatially a sphere, which can be a cylinder space-time with spherical space-like slices.

14. Jun 24, 2017

### Hiero

The word sphere was just an analogy, the point was just that an inertial frame comes back to an old location.
Now that it has been emphasized so much though, let me say, I actually do not understand the issue with the sphere; isn't the 'inertial path' on the surface of a sphere along a great circle? So in a universe restricted to the surface of a (higher dimensional) sphere, an object traveling with no acceleration-(at least none "inside" the universe?)* will come back to an old location, which is the point of my setup.
*[I'm thinking in analogy to particles restricted to surfaces; there will be some acceleration "outside" (normal) the universe, and I'm not sure where that fits. But I would also say that exists for the cylindrical case, so I'm still not sure of the sphere's problem.]

Anyway thank you for the insightful replies. General Relativity will likely be a fun one to study in the future.

15. Jun 24, 2017

### Gigaz

Generally, it is invalid to argue that because the curvature is extremely, the universe must be extremely large. Proof: Pacman ;)

16. Jun 25, 2017

### Flatland

Wouldn't one of the observers have to "stop" relative to the other since you can only compare clocks and extrapolate any relativistic time dilation when both observers are in the same inertial frame? Just a thought.

17. Jun 25, 2017

### Ibix

No. You only need two instantaneous clock comparisons at first and second meeting. Why would you need to be at rest with respect to a clock to read it?

18. Jun 25, 2017

### Flatland

What I mean is that according to what the OP proposed, when observer S' circumnavigates the Universe and passes by observer S again, observer S' will observe the clock tick of S to be slower than S' and vice versa. One of the observers would have to decelerate in order to compare clocks and see who's clock actually ran slower. In other words there is no paradox in what the OP proposes.

19. Jun 25, 2017

### Mister T

I think the flaw in your logic is thinking that one of them must accelerate in order for them to experience different amounts of proper time. It's not the acceleration of one twin that explains the difference in ages. It's the fact that they take different paths through spacetime. There are versions of the twin paradox where neither twin experiences nonzero proper acceleration.

20. Jun 25, 2017

### Ibix

This is time dilation in action, and you seem to be accepting this. So I don't understand why you then say:
...since that seems to directly contradict your previous sentence.
There is no paradox because the topology of the cylindrical universe picks out a frame that is special in a global sense, although local measurements are completely in line with the principle of relativity.