Causality and Multiply Connected Space-Time

Abstract

With the introduction of multiply connected topologies into physics, a question of causality arises. There are alternative routes between two points in a multiply connected space. Therefore, one may ask if a signal traveling at the speed of light along one route could be outpaced by a signal which has traveled a much shorter path through a handle or wormhole. This paper examines one such situation and shows that in this example causality is preserved. It proves essential in the analysis to distinguish between those regions of space-time which are catastrophic and those which are not. A catastrophic region is composed of catastrophic points. A catastrophic point in space-time is so located with respect to eventual singularities in the intrinsic geometry that every time-like geodesic through it necessarily runs into a region of infinite curvature at some time in the future---or was born out of a region of infinite curvature at some time in the past---or both. If a classical analysis of nature were possible---which it is not---then it would be natural to postulate that laboratory physics is carried out in noncatastrophic regions of space-time. Two such regions are shown to exist in the example considered in the paper. It is shown that no signal can ever be sent from one to the other. The key point in preventing any violation of causality is simple: The (Schwarzschild) throat of the wormhole pinches off in a finite time and traps the signal in a region of infinite curvature. This investigation also displays some of the unusual geometric features of the Schwarzschild solution of Einstein's equations for a spherically symmetrical center of attraction. Radial spacelike geodesics passing through the throat are calculated and it is shown that there exist regions of space-time unreachable by any radial geodesics that issue from a given point. Also, there exist points in space-time from which light signals can never be received no matter how long one waits.