4 - Geometric structures and Chern–Simons theory

Summary

In the two preceding chapters, we derived solutions of the vacuum field equations of (2+1)-dimensional gravity by using rather standard general relativistic methods. But as we have seen, the field equations in 2+1 dimensions actually imply that the spacetime metric is flat – the curvature tensor vanishes everywhere. This suggests that there might be a more directly geometric approach to the search for solutions.

At first sight, the requirement of flatness seems too strong: we usually think of the vanishing of the curvature tensor as implying that spacetime is simply Minkowski space. We have seen that this is not quite true, however. The torus universes of the last chapter, for example, are genuinely dynamical and have nontrivial – and inequivalent – global geometries. The situation is analogous to that of electromagnetism in a topologically nontrivial spacetime, where Aharanov–Bohm phases can be present even when the field strength F_µν vanishes.

It is true, however, that locally we can always choose coordinates in which the metric is that of ordinary Minkowski space. That is, every point in a flat spacetime M is contained in a coordinate patch that is isometric to Minkowski space with the standard metric η_µν. The only place nontrivial geometry can arise is in the way these coordinate patches are glued together. This is precisely what we saw in chapter 3 for the spacetime surrounding a point source: locally, the geometry was flat, but a conical structure arose from the identification of the edges of a flat coordinate patch.