Appendix B - Lorentzian metrics and causal structure

Summary

In general relativity we are interested in both the topology and the geometry of spacetime. The body of this book concentrates on geometrical issues in (2+1)-dimensional gravity and their physical implications, while appendix A introduces some basic topological concepts. The purpose of this appendix is to briefly discuss a set of issues intermediate between topology and geometry: issues of the large scale structure, and in particular the causal structure, of a spacetime with a Lorentzian metric.

Questions of large scale structure have played a very important role in recent work in (3+1)-dimensional general relativity, leading to general theorems about singularities, causality, and topology change. A thorough discussion is given in reference (see also). Many of these general results have not yet been applied to 2+1 dimensions, and I shall not attempt to review them here; my aim is merely to introduce the ideas that have already found a use in (2+1)-dimensional gravity.

Lorentzian metrics

To specify a spacetime, we need a manifold M with a Lorentzian metric, that is (in three dimensions) a metric g of signature (− + +). Such a metric determines a light cone at each point in M. A spacetime M is time-orientable if a continuous choice of the future light cone can be made, that is, if there is a global distinction between the past and future directions. Similarly, M is space-orientable if there is a global distinction between left- and right-handed spatial coordinate frames.