Twin Paradox in a Closed Universe?

Question

I’m fairly familiar with the standard twin paradox, but I’ve never seen the paradox adequately resolved for a closed universe. Closed in the sense that if you travel in a fixed direction long enough, you’ll end up where you started; like if space were the surface of a hyper-sphere.

Anyone know how to resolve it? What would happen if two twins set out in opposite directions and (assuming the universe is closed) they ran into each other again? Calculations made by either of them would yield that the other should be older and they could ask each other who was the younger via radio transmissions without necessarily stopping.

I’ve seen one person resolve it (in “American Mathematical Monthly”) by saying that there IS a preferred reference frame in the universe, but that kind of flies in the face of one of the basic postulates of Relativity.

(Just FYI this stumped my physics professor, but he specializes in Particle Physics not Cosmology)

Actual paradoxes arise when one assumes two mutually exclusive things. That seems to be what might be happening here.

However, IF that is actually the case here and both those statements are considered accurate then it is an interesting thing. It means there is an inconsistency in Physics.

Answer 1

Special relativity is an approximation which works in a sufficiently limited region of spacetime, but in its usual form it isn't really compatible with cosmology. That is, it's impossible to have a global Lorentz frame except in an infinite Minkowski space which doesn't exist except as an idealization.

If you know or assume a cyclical or spherical geometry for space it's simple to predict what each twin would experience (assuming each twin insists on doing his calculations as though he were always at rest in an inertial frame).

The initial signals from the receeding spaceship would indicate that the other twin was aging at a slower rate and younger. Eventually that signal would fade out and at some point a new radio signal would be received from the opposite direction from an approaching spaceship. The signals would indicate that the approaching twin was substantially older (but aging at a slower rate of course). By the time the two twins meet up again they could be the same age.

There's a discontinuity between the apparent age of the twin in the receeding spaceship and the one in the approaching spaceship but there's nothing paradoxical about that. For each of the two parts of the trip during which the two spaceships are close enough to be in radio contact, special relativity works fine.

Answer 2

Isn't that part of what makes a it paradox, that it can not be adequately resolved?