# Special relativity on a circle

1. Sep 13, 2010

### Tac-Tics

I thought of an interesting paradox on my way home today.

Suppose you have one-dimensional, finite universe without boundary. A circle.

Special relativity seems to fail on this small world.

Take two objects in relative motion. As they pass by each other, they synch their clocks. Stop their clocks when they pass around a second time (after "the other" goes a full cycle around the circle).

What do their clocks say?

According to SR, the one in motion should have the slowed clock. On a flat world, this isn't a problem, because one or both has to accelerate before they can reconvene. But on a circular world, they will eventually meet again without ever leaving their frame.

So does SR not work in such a world? Or does it generalizer in a less-than-obvious way? (Or maybe I just missed something subtle).

2. Sep 13, 2010

### JesseM

GR allows for arbitrary topologies, so it is possible to have a flat spacetime where space is nevertheless closed, a bit like the video game "Asteroids" where if you disappear off the top part of the screen you'll reappear on the bottom, and if you disappear off the right side you'll reappear on the left (technically this corresponds to the topology of a torus--see this page). In any small region of this spacetime, the laws of physics are exactly as they are in SR (with no locally preferred frames), but in a global sense there will be a preferred pseudo-inertial frame (by 'pseudo-inertial frame' I mean a global coordinate system that in any local region looks just like an inertial coordinate system in SR). This will be the frame where if you draw lines of simultaneity from a given point in spacetime, the lines will wrap around the spacetime in such a way that they return to that same point, as opposed to wrapping around it in a "slanted" way like the stripes on a candy cane. In a closed universe there is also a "hall of mirrors" effect where you see copies of every object in regular intervals in different directions, and the globally preferred frame will also have the property that observers at rest in this frame will see the nearest copies of themselves to the left and right as both being the same age, and both appear younger than the observer by an amount corresponding to their distance in the observer's frame (so if I see a copy of myself 3 light years away, his visual image will appear 3 years younger than me), while this is not true in other frames. Anyway, the answer to all twin paradox questions involving inertial twins circumnavigating the universe is that whichever of the two inertial twins is closer to being at rest in this globally preferred frame, that will be the twin who's aged more on the second of two times they cross paths.

A previous thread on this topic:

And here's a paper:

http://arxiv.org/abs/gr-qc/0101014