The Volume-Renormalized Mass from a Hamiltonian Perspective

TL;DR

This paper links the volume-renormalized mass of asymptotically hyperbolic manifolds to a Hamiltonian framework, showing it can be derived from a reduced Hamiltonian perspective and analyzing its evolution.

Contribution

It extends the definition of volume-renormalized mass using Hamiltonian formalism and demonstrates its properties in a cosmological spacetime setting.

Findings

Reduced Hamiltonian recovers the volume-renormalized mass.

The reduced Hamiltonian is non-increasing over evolution.

It remains constant only for self-similar spacetimes.

Abstract

We demonstrate that the volume-renormalized mass for asymptotically hyperbolic manifolds recently introduced by the authors can be deduced from a reduced Hamiltonian perspective. In order to do this, we first use Michel's formalism of mass invariants to extend the definition of the volume-renormalized mass to initial data sets. We consider spacetimes that are foliated by asymptotically Poincar\'e--Einstein Riemannian manifolds in the spirit of the Milne model of cosmology and reduce the ADM Hamiltonian to an unconstrained Hamiltonian system, analogous to the work of Fischer and Moncrief for spatially compact spacetimes. We find that the reduced Hamiltonian in this case recovers the volume-renormalized mass. We then analyze the first and second variation of the reduced Hamiltonian and demonstrate that it is non-increasing over the evolution and constant only for self-similar spacetimes.