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.