# Special relativity and flat donut universes

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1. May 18, 2015

### Strilanc

Suppose there was a universe that was small, and flat, but globally it was connected like a torus. Basically, pac-man's universe (except special relativity is in effect). The space $(\mathbb{R} \pmod{1})^3$.

Does this assumed universe have a distinguished inertial frame where the loop-around-distance is maximized (due to not being length-contracted)? For example, if you pushed a small ball away from yourself along the x axis, can the amount of time until it smacks into your back depend on your velocity w.r.t. some frame? Or are the differences cancelled out by time dilation? What if there was a clock on the ball?

(Obviously this question is about how the Lorentz transform behaves in unusual situations, not about how reality actually is.)

2. May 18, 2015