Simultaneity in Closed Flat Spacetime

Chunghyoung Lee, Chunghyoung Lee

Abstract: Our universe may be not infinite in all directions but closed like the surface of a cylinder or sphere. Consider simple closed spacetime, namely, 2-dimensional cylindrical spacetime which is temporally open but spatially closed—this spacetime is flat (i.e., has the geometry of Minkowski spacetime). In this spacetime it is possible for one twin to travel around the universe to return to the other twin while none of them goes through any acceleration.

I argue that considerations of such closed flat spacetime provide new insights about special relativity and various philosophical puzzles thereof. First, cylindrical spacetime is significantly different, though locally indistinguishable, from (2-dimensional) Minkowski spacetime. Most fundamentally, Reichenbach’s Round-trip axiom fails to hold in this spacetime: For three inertially moving objects, A, B, and C, which are at rest relative to one another, if two light signals are sent from A simultaneously so that one signal makes a round trip in the order of A-B-C-A while the other travels in the opposite order of A-C-B-A, then the two signals return to A simultaneously in Minkowski spacetime, but not necessarily in closed flat spacetime. Consequently, distant clocks cannot be synchronized following the standard procedure proposed by Einstein except for those under some particular motion, and so there are privileged inertial frames. And it is simplest to take the one-way speed of light to vary from inertial frame to inertial frame and the simultaneity relation to be not relative but absolute. These features have important implications for various philosophical issues regarding time and special relativity.