The Volume-Renormalized Mass from a Hamiltonian Perspective

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.