Paradox of the twins in time dilation [duplicate]

Question

In the twin paradox from what I understand both observer see each other's time dilated so they always believe that the other frame is younger. Finally because the space ship frame has to make many accelerations it results that he will be younger. However, what if the frame in the space ship doesn't have to make any accelerations to come back to earth and could simply just come back like if the space was a looped or that there was a wormhole , so in this case who would be younger ?

Answer 1

The traveling twin will be younger, because the paths are not symmetric.

When considering paths in these spaces, you have to consider homotopy classes. Two paths are in the same homotopy class if they can be continuous deformed into one and other.

It the standard twin paradox, the traveling twin's path can be deformed into the stationary twin's path (and to a point); however, because the traveling twin experiences acceleration, his elapsed proper time is shorter.

It can be shown among call curves between 2 points is a given homotopy class, only one corresponds to an inertial observer, and it is that path the experiences the most proper time.

In your question, one of the paths travels around a cylindrical (or toroidal) dimension. Each path can be characterized by a winding index, which is a topological invariant. It cannot be changed by a change of coordinates, reference frame, or deformation. The observer with winding number 0 experiences the most proper time.

According to the topologists: "The spatial topology thus imposes privileged frames among the class of all inertial frames, and even if the principle of relativity remains valid locally, it is no longer valid at the global scale. This is a sign that the theory of relativity is not a global theory of space-time."

Details can be found in this (very tractable for non topologists) paper: https://arxiv.org/pdf/0910.5847.pdf

Answer 2

If spacetime weren't flat (via looping or a wormhole), then you would have a preferred frame of reference, and be able to measure your absolute velocity -- for instance, by measuring the circumference of the loop, which Lorentz contracts for a moving observer.

Note that discussing who's younger and who's older is a bit more nuanced in this context, since the $x'$ axis ("present") of the moving observer is a helix if the t-x plane is a cylinder, so he actually views multiple copies of the stationary twin at different ages as simultaneous with himself. If you consider only the copy that is at the same location as the moving twin, the stationary twin is older than the traveling twin, and the inertial frame of reference remains reliable.