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Symmetries and charges of general relativity at null boundaries

Venkatesa Chandrasekaran, Eanna E. Flanagan, Kartik Prabhu

TL;DR

This work establishes a covariant phase space framework for general relativity with a null boundary, revealing a universal symmetry algebra that is a semidirect product of Diff(S^2) with a nonabelian supertranslation sector. It provides explicit global and Wald–Zoupas localized charges and fluxes on finite null surfaces, including nonstationary horizons, and proves the horizon charges form the same algebra as linearized diffeomorphisms with no central extension under reasonable fall-off. The results extend the BMS-like horizon symmetry program beyond stationary cases and beyond null infinity, with potential implications for horizon memory and information-loss questions, while aligning with conservation-law structures in black-hole spacetimes. The analysis is dimensionally general (with obvious 4D specialization) and sets the stage for further exploration of horizon-edge modes, interactions with matter, and connections to holographic and quantum-gravity contexts.

Abstract

We study general relativity at a null boundary using the covariant phase space formalism. We define a covariant phase space and compute the algebra of symmetries at the null boundary by considering the boundary-preserving diffeomorphisms that preserve this phase space. This algebra is the semi-direct sum of diffeomorphisms on the two sphere and a nonabelian algebra of supertranslations that has some similarities to supertranslations at null infinity. By using the general prescription developed by Wald and Zoupas, we derive the localized charges of this algebra at cross sections of the null surface as well as the associated fluxes. Our analysis is covariant and applies to general non-stationary null surfaces. We also derive the global charges that generate the symmetries for event horizons, and show that these obey the same algebra as the linearized diffeomorphisms, without any central extension. Our results show that supertranslations play an important role not just at null infinity but at all null boundaries, including non-stationary event horizons. They should facilitate further investigations of whether horizon symmetries and conservation laws in black hole spacetimes play a role in the information loss problem, as suggested by Hawking, Perry, and Strominger.

Symmetries and charges of general relativity at null boundaries

TL;DR

This work establishes a covariant phase space framework for general relativity with a null boundary, revealing a universal symmetry algebra that is a semidirect product of Diff(S^2) with a nonabelian supertranslation sector. It provides explicit global and Wald–Zoupas localized charges and fluxes on finite null surfaces, including nonstationary horizons, and proves the horizon charges form the same algebra as linearized diffeomorphisms with no central extension under reasonable fall-off. The results extend the BMS-like horizon symmetry program beyond stationary cases and beyond null infinity, with potential implications for horizon memory and information-loss questions, while aligning with conservation-law structures in black-hole spacetimes. The analysis is dimensionally general (with obvious 4D specialization) and sets the stage for further exploration of horizon-edge modes, interactions with matter, and connections to holographic and quantum-gravity contexts.

Abstract

We study general relativity at a null boundary using the covariant phase space formalism. We define a covariant phase space and compute the algebra of symmetries at the null boundary by considering the boundary-preserving diffeomorphisms that preserve this phase space. This algebra is the semi-direct sum of diffeomorphisms on the two sphere and a nonabelian algebra of supertranslations that has some similarities to supertranslations at null infinity. By using the general prescription developed by Wald and Zoupas, we derive the localized charges of this algebra at cross sections of the null surface as well as the associated fluxes. Our analysis is covariant and applies to general non-stationary null surfaces. We also derive the global charges that generate the symmetries for event horizons, and show that these obey the same algebra as the linearized diffeomorphisms, without any central extension. Our results show that supertranslations play an important role not just at null infinity but at all null boundaries, including non-stationary event horizons. They should facilitate further investigations of whether horizon symmetries and conservation laws in black hole spacetimes play a role in the information loss problem, as suggested by Hawking, Perry, and Strominger.

Paper Structure

This paper contains 54 sections, 239 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Review of the covariant phase space formalism
  4. Definitions of field configuration space and covariant phase space
  5. Definitions of currents
  6. Definition of presymplectic form on covariant phase space
  7. Global charges that generate boundary symmetries
  8. Boundary symmetry algebras of linearized diffeomorphisms
  9. Localized (Wald-Zoupas) charges, fluxes and conservation laws
  10. Potential ambiguities in global and localized charges
  11. Review of the local geometry of null hypersurfaces
  12. Foundations
  13. Geometric fields defined on a null hypersurface
  14. Divergence operator
  15. Stationary regions of null hypersurfaces
  16. ...and 39 more sections

Figures (1)

  • Figure 1: An illustration of two situations we will consider for the spacetime $M$. (a) $M$ is taken to be the domain of outer communications of a black hole formed in a gravitational collapse, with boundary elements $\mathscr{I}^-$, $\mathscr{I}^+$ and ${\cal H}^+$. (b) $M$ is taken to be the domain of outer communications of an eternal black hole, with boundary elements $\mathscr{I}^-$, $\mathscr{I}^+$, ${\cal H}^-$ and ${\cal H}^+$.