Adiabatic Solutions in General Relativity as Null Geodesics on the Space of Boundary Diffeomorphisms

TL;DR

This paper describes a novel geometric approach to solutions in general relativity, representing slow-time evolutions as null geodesics on the space of boundary diffeomorphisms, with implications for understanding boundary data and constraints.

Contribution

It introduces a new description of adiabatic solutions in general relativity as null geodesics on boundary diffeomorphism space, linking boundary data to bulk solutions.

Findings

Solutions form null geodesics on boundary diffeomorphism space

Hamiltonian constraint reduces to Monge-Ampere equation in 2+1 dimensions

Lagrangian generalizes covariant continuum mechanics Lagrangian

Abstract

We use a trick similar to Weinberg's for adiabatic modes, in a Manton approximation for general relativity on manifolds with spatial boundary. This results in a description of the slow-time dependent solutions as null geodesics on the space of boundary diffeomorphisms, with respect to a metric we prove to be composed solely of the boundary data. We show how the solutions in the bulk space is determined with the constraints of general relativity. To give our description a larger perspective, we furthermore identify our resulting Lagrangian as a generalized version of the covariantized Lagrangian for continuum mechanics. We study the cases of 3+1 and 2+1 dimensions and show for the solutions we propose, the Hamiltonian constraint becomes the real homogeneous Monge-Ampere equation in the special case of two spatial dimensions.