ADM approach to 2+1 dimensional gravity coupled to particles

TL;DR

This paper develops a canonical ADM framework for 2+1 dimensional gravity with point particles, providing solutions for the two-body problem and insights into singularities and the Garnier system.

Contribution

It introduces a canonical ADM approach to 2+1 gravity with particles, including solutions and geometric characterizations, advancing understanding of the N-body problem.

Findings

Solution of the two-body problem in 2+1 gravity

Geometric characterization of apparent singularities

Connection to Garnier Hamiltonian system

Abstract

We develop the canonical ADM approach to 2+1 dimensional gravity in presence of point particles. The instantaneous York gauge can be applied for open universes or universes with the topology of the sphere. The sequence of canonical ADM equations is solved in terms of the conformal factor. 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.