# Twin paradox in a torus

1. Jul 10, 2010

### lugita15

In four dimensions, a flat torus is an object that has zero curvature but still has closed geodesic curves. What this means is that if you try to measure geometry locally, you will find that it is perfectly Euclidean. Nevertheless, if you travel on a straight line, you'll eventually end up where you started.

What would happen if you carried out the twin paradox in a universe with such a geometry? In the standard twin paradox, one of the twins experiences acceleration effects, so his frame is not inertial. But in a flat toroidal universe, he would always be in an inertial frame, since going around in a closed curve doesn't require any "turning". So what would be the results? Would there still be time dilation?

2. Jul 10, 2010

### George Jones

Staff Emeritus
3. Jul 10, 2010

### JustinLevy

This has come up multiple times here and is an interesting question.

The short answer is: there is still no local preferred frame, but there IS a global "preferred frame" in such a situation.

If you perform an experiment, and no information travels all the way around the universe to interact with the experiment from "both sides", then that experiment is not capable of noticing this "preferred frame". However, having two inertial observers sync watches and then travel inertially around the universe and meet up again to compare watches, this is an example of an experiment that is sensitive to the preferred frame. Their watches will not necessarily still be in sync. It is quite possible they can measure different times between these two events.

A simple argument I've seen used here to aid in seeing the existence of this "global preferred frame": different inertial frames will disagree on the "length around" the universe. So even though locally special relativity is correct, in some global sense the topology breaks this.

A quick google search found this discussion on how the topology breaks this:
http://van.physics.illinois.edu/qa/listing.php?id=15308
If you ignore the stuff about string theory, it contains a decent explanation of how the global preferred frame arises.

4. Jul 11, 2010

### Phrak

I believe there would be more than one uniform toroidal geometry in which time is cyclic. There seem to be two. In the first class, there is an inertial frame in which a massive object would return to its starting point. In the second, null lines would retrace; that is, a light ray directed in some preferred direction, would retrace. Which type are you referring to?

Edit: Peripherally, if we're still thinking, we should consider a third class that has both a preferred inetrial(s) frame and a preferred retracement(s) of null lines.

Last edited: Jul 11, 2010
5. Jul 11, 2010