The Hamiltonian of Asymptotically Friedmann-Lemaitre-Robertson-Walker Spacetimes

TL;DR

This paper derives the correct Hamiltonian for asymptotic open FLRW spacetimes, including Tolman geometries, clarifying the role of surface terms and their relation to the energy of solutions.

Contribution

It provides the explicit form of the Hamiltonian with surface terms for asymptotic open FLRW spacetimes, extending previous results to include Tolman geometries and various matter contents.

Findings

Surface term vanishes for asymptotic flat FLRW spaces.

Surface term is non-zero for asymptotic negative curvature FLRW spaces.

Surface term can be interpreted as the energy of a solution relative to FLRW spacetime.

Abstract

We obtain the correct hamiltonian which describes the dynamics of classes of asymptotic open Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) spacetimes, which includes Tolman geometries. We calculate the surface term that has to be added to the usual hamiltonian of General Relativity in order to obtain an improved hamiltonian with well defined functional derivatives. For asymptotic flat FLRW spaces, this surface term is zero, but for asymptotic negative curvature FLRW spaces it is not null in general. In the particular case of the Tolman geometries, they vanish. The surface term evaluated on a particular solution of Einstein's equations may be viewed as the ``energy'' of this solution with respect to the FLRW spacetime they approach asymptotically. Our results are obtained for a matter content described by a dust fluid, but they are valid for any perfect fluid, including the cosmological constant.