(2+1)-AdS Gravity on Riemann Surfaces

TL;DR

This paper develops a formalism to solve (2+1) AdS gravity on Riemann surfaces, using sigma models and algebraic formulas, enabling analysis of black hole scattering in this setting.

Contribution

It introduces a novel approach using sigma models and algebraic formulas to solve (2+1) AdS gravity on Riemann surfaces, including black hole scattering.

Findings

Solutions for torus case via two sigma models

Explicit algebraic formulas for mappings with sources or topology

A new method for studying black hole scattering

Abstract

We discuss a formalism for solving (2+1) AdS gravity on Riemann surfaces. In the torus case the equations of motion are solved by two functions f and g, solutions of two independent O(2,1) sigma models, which are distinct because their first integrals contain a different time dependent phase factor. We then show that with the gauge choice $k = \sqrt{\Lambda}/ tg (2 \sqrt{\Lambda}t)$ the same couple of first integrals indeed solves exactly the Einstein equations for every Riemann surface. The $X^A=X^A(x^mu)$ polydromic mapping which extends the standard immersion of a constant curvature three-dimensional surface in a flat four-dimensional space to the case of external point sources or topology, is calculable with a simple algebraic formula in terms only of the two sigma model solutions f and g. A trivial time translation of this formalism allows us to introduce a new method which is suitable to study the scattering of black holes in (2+1) AdS gravity.