# Strolling along gravitational vacua

## TL;DR

This paper explores the structure of vacuum solutions in General Relativity on manifolds with boundary, revealing a boundary-determined space of vacua and a geodesic motion framework that links boundary data to conserved charges.

## Contribution

It introduces a novel perspective on GR vacua as boundary-acted metrics and reduces Einstein equations to geodesic motion on this space, highlighting boundary data's role.

## Key findings

- Vacua form a boundary-acted metric space
- Einstein equations reduce to null geodesics
- Boundary charges are conserved quantities

## Abstract

We consider General Relativity (GR) on a space-time whose spatial slices are compact manifolds $M$ with non-empty boundary $\partial M$. We argue that this theory has a non-trivial space of `vacua', consisting of spatial metrics obtained by an action on a reference flat metric by diffeomorpisms that are non-trivial at the boundary. In an adiabatic limit the Einstein equations reduce to geodesic motion on this space of vacua with respect to a particular pseudo-Riemannian metric that we identify. We show how the momentum constraint implies that this metric is fully determined by data on the boundary $\partial M$ only, while the Hamiltonian constraint forces the geodesics to be null. We comment on how the conserved momenta of the geodesic motion correspond to an infinite set of conserved boundary charges of GR in this setup.