On the definition of mass in general relativity: Noether charges and conserved quantities in diffeomorphism invariant theories

TL;DR

This paper explores the geometric and symplectic structure of phase space in general relativity, demonstrating how conserved quantities like mass can be defined using Noether charges, applicable to various spacetime asymptotics.

Contribution

It extends the formalism for defining mass in general relativity to include the Brown-York and Bondi masses, using a covariant symplectic approach.

Findings

Symplectic structure remains Darboux form with cosmological constant.

Formalism reproduces ADM and Henneaux-Teitelboim masses.

Brown-York mass has a natural differential forms interpretation.

Abstract

Geometrically the phase space of a mechanical system involves the co-tangent bundle of the configuration space. The phase space of a relativistic field theory is infinite dimensional and can be endowed with a symplectic structure defined in a perfectly co-variant manner that is very useful for discussing symmetries and conserved quantities of the system. In general relativity the symplectic structure takes Darboux form and it is shown in this work that the presence of a cosmological constant does not change this conclusion. For space-times that admit time-like Killing vectors the formalism can be used to define mass in general relativity and it is known that, for asymptotically flat black holes, this mass is identical the the usual Arnowitt-Desner-Misner mass while for asymptotically anti-de Sitter Kerr metrics it is the same as the Henneaux-Teitelboim mass. We show that the same formalism can also be used to derive the Brown-York mass and the Bondi mass for stationary space times, in particular the Brown-York mass has a natural interpretation in terms of differential forms on the space of solutions of the theory.