"Moduli Space" of Asymptotically Anti-de Sitter Spacetimes in (2+1)-Dimensions

Abstract

Setting an ansats that the metric is expressible by a power series of the inverse radius and taking a particular gauge choice, we construct a "general solution" of (2+1)-dimensional Einstein's equations with a negative cosmological constant in the case where the spacetime is asymptotically anti-de Sitter. Our general solution turns out to be parametrized by two centrally extended quadratic differentials on $S^{1}$. In order to include 3-dimensional Black Holes naturally into our general solution, it is necessary to exclude the region inside the horizon. We also discuss the relation of our general solution to the moduli space of flat $\tilde{SL}(2,R)\times\tilde{SL}(2,R)$ connections.