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

TL;DR

This paper constructs a general solution to (2+1)-dimensional Einstein's equations with negative cosmological constant, characterizing asymptotically Anti-de Sitter spacetimes via quadratic differentials and relating it to moduli space of flat connections.

Contribution

It provides a parametrization of asymptotically AdS spacetimes in (2+1) dimensions using quadratic differentials, linking solutions to the moduli space of flat connections.

Findings

Solution parametrized by two quadratic differentials on S^1

Includes black holes outside the horizon region

Connects to the moduli space of flat SL(2,R) connections

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.