ADM approach to 2+1 dimensional gravity

TL;DR

This paper solves the ADM equations for 2+1 dimensional gravity using the conformal factor in York gauge, providing solutions for the two-body problem and characterizing singularities with connections to Liouville theory.

Contribution

It offers a novel solution method for the ADM equations in 2+1 gravity and links the dynamics to a conformal Liouville theory framework.

Findings

Explicit solutions for the two-body problem in 2+1 gravity

Geometrical characterization of apparent singularities in N-body systems

Connection between ADM equations and Garnier Hamiltonian system

Abstract

The canonical ADM equations are solved in terms of the conformal factor in the instantaneous York gauge. A simple derivation is given for the solution of the two body problem. A geometrical characterization is given for the apparent singularities occurring in the N-body problem and it is shown how the Garnier hamiltonian system arises in the ADM treatment by considering the time development of the conformal factor at the locations where the extrinsic curvature tensor vanishes. The equations of motion for the position of the particles and of the apparent singularities and also the time dependence of the linear residues at such singularities are given by the transformation induced by an energy momentum tensor of a conformal Liouville theory. Such an equation encodes completely the dynamics of the system.