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A general framework for gravitational charges and holographic renormalization

Venkatesa Chandrasekaran, Eanna E. Flanagan, Ibrahim Shehzad, Antony J. Speranza

TL;DR

The paper presents a comprehensive covariant phase space framework for gravitational charges that unifies global and localized charges, resolves ambiguities via a subregion action principle, and demonstrates that holographic renormalization can always render charges finite at asymptotic boundaries. It derives the Barnich-Troessaert bracket from the subregion Poisson structure, introduces corner improvements to remove ambiguities, and provides an explicit counterterm algorithm. The formalism is concretely applied to vacuum GR at future null infinity to reproduce generalized BMS charges, and the results suggest broad applicability to enlarged asymptotic symmetries, holography, and edge-mode dynamics in gravitational systems. Overall, the work offers a principled route to finite, well-defined gravitational charges in diverse settings, with implications for holography and the structure of asymptotic symmetries.

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.

A general framework for gravitational charges and holographic renormalization

TL;DR

The paper presents a comprehensive covariant phase space framework for gravitational charges that unifies global and localized charges, resolves ambiguities via a subregion action principle, and demonstrates that holographic renormalization can always render charges finite at asymptotic boundaries. It derives the Barnich-Troessaert bracket from the subregion Poisson structure, introduces corner improvements to remove ambiguities, and provides an explicit counterterm algorithm. The formalism is concretely applied to vacuum GR at future null infinity to reproduce generalized BMS charges, and the results suggest broad applicability to enlarged asymptotic symmetries, holography, and edge-mode dynamics in gravitational systems. Overall, the work offers a principled route to finite, well-defined gravitational charges in diverse settings, with implications for holography and the structure of asymptotic symmetries.

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.
Paper Structure (43 sections, 2 theorems, 220 equations, 1 figure, 3 tables)

This paper contains 43 sections, 2 theorems, 220 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction and summary
  2. Extended summary of results
  3. Notation
  4. Gravitational charges at finite boundaries
  5. Covariant phase space
  6. Symplectic form
  7. Localized charges
  8. Ambiguities and boundary canonical transformations
  9. Corner improvements
  10. Brackets of localized charges
  11. Vacuum general relativity at null infinity
  12. Field configuration spaces
  13. Symmetry groups
  14. Gravitational charges at asymptotic boundaries: holographic renormalization
  15. Introduction and Overview
  16. ...and 28 more sections

Key Result

Lemma 1

The various derivations defined above satisfy In particular, (eqn:LL) implies that the field space Lie bracket is given by

Figures (1)

  • Figure 1: Standard setup for holographic renormalization in asymptotically flat spacetimes. The subregion under consideration is associated with a segment of $\mathscr{I}^+$ to the future of a spatial surface $\Sigma$. The cutoff subregion $\mathcal{D}_\upsilon$ is depicted in gray, and its boundary components $\mathcal{N}_{j,\upsilon}$ consist of $\Sigma_\upsilon$, $\mathcal{N}_\upsilon$, and $\Sigma_\upsilon^f$.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof