The twin paradox in a Finite Dark energy less universe

Question

Let's imagine we have a universe that is finite and curves back on iself. So if you go far enough in one direction you end up back where you started. (e.g. 3-Sphere/3-torus) Then lets imagine that it has no dark energy, so the expansion of the universe never speeds up, it only slows down. Here the scale factor of the universe increase at most linearly with time. Then via the Ant on a rubber rope thought experiment, a spaceship that travels away in one direction will eventually return roughly to where it started. How would the twin paradox be solved then as a result?

Answer 1

The thought experiment that you describe is discussed in the following article: Sagnac effect, twin paradox and space-time topology - Time and length in rotating systems and closed Minkowski space-times

Author: Olaf wucknitz

In that article:
Olaf Wucknitz argues that the perimeter of a circular rotating platform can be treated as a case of a Minkowski spacetime with only 1 spatial dimension.

The essential feature: that perimeter is a space that loops back onto itself. This loop topology is what allows the Sagnac effect to manifest itself.

Wucknitz proceeds to argue that it is exclusively the loop topology that counts, not the shape. An optical fiber can have any shape; if the fiber loops back onto itself the condition for the Sagnac effect to manifest itself is met.

The Sagnac effect enforces a procedure of clock synchronization that is distinct from Einstein synchronization procedure. When a loop is closed, and synchronizing signals are traversing the loop in both directions then Einstein synchronization procedure is not applicable.

In a side note:
Wucknitz extends the notion of a spacetime that loops back onto itself to a topology of a spacetime with 3 spatial dimensions, all looping back. As in the case of the spacetime with 1 spatial dimension: the topology of that spacetime has implications for synchronization.

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[Later edit]
In the case of a 1 + 1 Minkowski spacetime that loops back onto itself: in that spacetime there is an absolute reference of velocity. This property generalizes to a closed 3 + 1 Minkowski spacetime

In the animation the red and blue dots represent propagating light. The four grey dots represent relay stations that are in motion along the perimeter of the closed loop.

(One physical implementation would be as follows: on the Earth: 4 relay stations distributed along a latitude line, for instance the Equator. Optical fiber carries pulses of light that are counterpropagating.)

To maintain a synchronized time the relay stations are using the following procedure: the pulses of light that are propagating are continuously relayed. Both co-rotating and counter-rotating the same amount of pulses is propagating. (In the animation: in each direction 4 pulses)

The relay stations have themselves an (angular) velocity relative to the circle. As a consequence: as measured by the relay stations the co-propagating pulses of light and the counter-propagating pulses of light are a different time interval apart. Given the known velocity of the speed of light (in optical fiber) the angular velocity of the rotating system can be inferred.

The engineers that operate the relay stations maintain a global synchronized time, by taking the transmission time into account as the pulses of light are relayed.

Note especially that this synchronization procedure is not Einstein synchronization procedure. Einstein synchronization is applicable only when the end points are unconnected. That is, when instead of using a closed loop mirrors are used at two end points, that allows the use of Einstein synchronization. (In general the result of the Einstein synchronization will not coincide with the closed-loop synchronization. The two will coincide only when the relay stations have zero (angular) velocity.)

Another way of establishing the reference of zero velocity is as follows. Have multiple clocks both co-rotating and counter-rotating around the loop. Depending on the velocity of the clock around the loop a different amount of proper time will elapse for that clock. There will be a single velocity such that for that velocity the most amount of proper time elapses: that velocity is zero velocity relative to the closed loop.

The fundamental reason why in a closed Minkowski spacetime velocity is absolute.

In the case where the the synchronization is established by way of relaying pulses of light: in Minkowski spacetime: as a matter of principle the speed of light is the same in all directions. That uniformity provides an absolute reference of velocity relative to the loop.

In the case of disseminating time with travelling clocks: the fact that time dilation occurs is providing the reference of velocity. By contrast: if the closed loop space is Newtonian space-and-time then for a travelling clock and for a stationary clock the same amount of time elapses. So: when the spacetime is newtonian then as far as clocks are concerned velocity is fundamentally relative. However, in closed loop Minkowski spacetime velocity is absolute.

Answer 2

A universe that loops back on itself has a preferred frame. The twin that moves faster with respect to that frame ages less.

To see that there must be a preferred frame, consider the special-relativistic toroidal universe. In one spatial dimension, it's a cylinder, where time points along the cylinder and space wraps around it. But note that there is only one frame in which space wraps back on itself exactly! That's the preferred frame. In any other frame, the spatial axis is rotated, so that it winds around like a helix.

By considering the twins' paths with respect to this special frame, it should be easy to work out how much time passes for them. For example, in the preferred frame, the universe has a particular circumference. If you are moving with respect to that frame, the circumference is length-contracted, so your trip "around the universe" takes shorter for you than it does from the perspective of someone watching you from the preferred frame.

I picked the special relativistic example for simplicity, but of course the general relativistic cosmological spacetimes also have a preferred frame (the comoving frame). Also, circumnavigating a general-relativistic closed universe turns out to be surprisingly tricky. You can't circumnavigate a closed matter- or radiation-dominated universe before it collapses. I'm pretty sure you need some dark-energy-like negative-pressure component to stabilize the universe long enough for such a trip to take place (e.g. in Einstein's static universe).