A general framework for gravitational charges and holographic renormalization

TL;DR

This paper introduces a comprehensive covariant phase space framework for defining and computing gravitational charges, including boundary and corner contributions, with applications to holographic renormalization and asymptotic symmetries.

Contribution

It provides a unified method for constructing gravitational charges for subregions and the entire spacetime, deriving the Barnich-Troessaert bracket from first principles, and applying holographic renormalization systematically.

Findings

Derived the Barnich-Troessaert bracket from first principles.

Established a systematic procedure for holographic renormalization of charges.

Reproduced known charges for the generalized BMS algebra.

Abstract

We develop a general framework for constructing charges associated with diffeomorphisms in gravitational theories using covariant phase space techniques. This framework encompasses both localized charges associated with spacetime subregions, as well as global conserved charges of the full spacetime. Expressions for the charges include contributions from the boundary and corner terms in the subregion action, and are rendered unambiguous by appealing to the variational principle for the subregion, which selects a preferred form of the symplectic flux through the boundaries. The Poisson brackets of the charges on the subregion phase space are shown to reproduce the bracket of Barnich and Troessaert for open subsystems, thereby giving a novel derivation of this bracket from first principles. In the context of asymptotic boundaries, we show that the procedure of holographic renormalization can be always applied to obtain finite charges and fluxes once suitable counterterms have been found to ensure a finite action. This enables the study of larger asymptotic symmetry groups by loosening the boundary conditions imposed at infinity. We further present an algorithm for explicitly computing the counterterms that renormalize the action and symplectic potential, and, as an application of our framework, demonstrate that it reproduces known expressions for the charges of the generalized Bondi-Metzner-Sachs algebra.