Causality and Multiply Connected Space-Time
- 15 October 1962
- journal article
- research article
- Published by American Physical Society (APS) in Physical Review B
- Vol. 128 (2), 919-929
- https://doi.org/10.1103/physrev.128.919
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.Keywords
This publication has 8 references indexed in Scilit:
- The gravity field of a particle. IIProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 1961
- THE CATEGORY OF TOPOLOGICAL OBJECTSProceedings of the National Academy of Sciences, 1961
- Oscillatory Character of Reissner-Nordström Metric for an Ideal Charged WormholePhysical Review B, 1960
- Maximal Extension of Schwarzschild MetricPhysical Review B, 1960
- FUNDAMENTAL GROUPS ON A LORENTZ MANIFOLDProceedings of the National Academy of Sciences, 1960
- Completion and Embedding of the Schwarzschild SolutionPhysical Review B, 1959
- Past-Future Asymmetry of the Gravitational Field of a Point ParticlePhysical Review B, 1958
- The Particle Problem in the General Theory of RelativityPhysical Review B, 1935