Twin paradox of special relativity in a closed universe.?

Question

If one twin is in a spacestation and the other is in a ship moving past the station at a high speed, the twin in the ship will have aged less than the one in the station when he returns. (Assume the ship does not stop on return) This is usually explained by the fact that the ship is not an inertial frame because it must accelerate to turn around for the return part of its trip.

But what if the universe is closed and the ship just continues to travel at a constant speed in a straight line until it has gone all the way around the universe and has returned to its starting point? The austronaut never feels any acceleration; this would be an inertial frame. So there is no way to say which twin is "really" moving and which is at rest. The situation is symmetrical for both. So each one should be younger relative to the other.
This seems like a true paradox. Relativity says they can not be the same age. And symmetry says both can claim to have been the one "at rest" and thus the older.

Answer 1

This is a very good question and you get a star for it.

The two situations are not equivalent because of the curvature of the universe. Think of the universe as a circle and time as another dimension, so the universe through time looks like a cylinder. You are comparing two curves: one that spirals around the cyclinder and one that simply goes up the cylinder. The geometric situation is invariant, even if the coordinate systems (corresponding to observers who move with respect to each other) are different.

It turns out that time experienced corresponds roughly to the length of the curve travelled in space time. The main difference is that longer lengths correspond to shorter time. Because of this, the twin that goes around the universe is the one that comes back younger.

One aspect of this is that there is no global inertial frame for a closed universe. The curvature forces the use of several frames if you want to follow someone that goes around. Even though the twin who goes around the universe doesn't feel any expansion, his a single frame does not work for his entire journey. Part of the difficulty here is that a curved universe automatically requires the use of general relativity, not special relativity. It is in the context of general relativity that the 'length' of the path and the 'proper time' are connected through the metric describing the spacetime.

Answer 2

If this appears to be a true paradox, it is an illusion.
A "true paradox" is a contradiction, and therefore, there is no such animal.
You have missed something.

(The traveling twin could use the gravity of a star to turn himself around and would experience no acceleration.)

Answer 3

Recall how the standard twin paradox is resolved: both twins (Bob and Betty) observe each other's clocks run slow. When Betty reaches Vega, she sees that Bob has aged much less than she has. However, when she turns around, she skips into another reference frame. In that moment of turnaround, she observes Bob to age 20 years. There is no need to discuss her acceleration, only the fact that in different reference frames, different events are simultaneous. In the outgoing frame, (Betty at Vega) is simultaneous with (Bob 5 days older). In the ingoing frame, (Betty at Vega) is simultaneous with (Bob 20 years older). The key to understanding the Twin Parradox is understanding *Relativity of Simultenaity.*

Now to your question. Think of a 1-dimensional closed universe: a closed loop 10 light years in circumference. A spacetime diagram of this universe would be a cylinder, with the length of the cylinder representing Bob-time. Consider Bob to be stationary. When Betty starts her trip, she is moving past Bob. She considers (Betty at the Beginning) to be simultaneous with (Bob at the Beginning) because they are both at the same point in spacetime.

If you were to draw a line on the cylinder representing Bob's idea of all the events on the cylinder occuring at t=0, it would be a circle around the cylinder: the loop universe. But here's the kicker. Remember that simultaneity is relative. Betty disagrees with Bob's ideas of which parts of spacetime are simultanous with the starting point. If you draw a line representing Betty's idea of which events are simultanwous, it's a HELIX going up the cylinder! In addition to (Betty at start) being simultaneous with (Bob at start) Betty considers herself to be simultaneous with an IMAGE of Bob a little more than 10 light years in front of her! This image of Bob is higher up the cylinder, and thus it is Bob 10 years in Bob's future. (She also considers herself simultaneous with an infinite number of older and farther Bobs).

This is the Bob she meets up with at the end, his watch having run slowly the whole time. The result is that both agree that Bob is older.

Please note that it is not about "lengths of lines in spacetime." It's all about relativity of simultaneity. Bob is older because Betty considers herself simultaneous with Bob-at-the-beginning and Bob-10-years-older-up-in-front-of-her. And it's this latter Bob that she meets at the end.

Answer 4

you are very clever... youve got a star.
anyway: this is just me, but perhaps the one who is less in "tune" with the natural universe movements (rotation and expansion, etc) will be the one who ages the most, because movement is relative to the rest of the universe as a whole.

Answer 5

There is no paradox here, the one in the moving ship ages more slowly.