Abstract
We revisit the covariant phase space formalism applied to gravitational theories with null boundaries, utilizing the most general boundary conditions consistent with a fixed null normal. To fix the ambiguity inherent in the Wald-Zoupas definition of quasilocal charges, we propose a new principle, based on holographic reasoning, that the flux be of Dirichlet form. This also produces an expression for the analog of the Brown-York stress tensor on the null surface. Defining the algebra of charges using the Barnich-Troessaert bracket for open subsystems, we give a general formula for the central — or more generally, abelian — extensions that appear in terms of the anomalous transformation of the boundary term in the gravitational action. This anomaly arises from having fixed a frame for the null normal, and we draw parallels between it and the holographic Weyl anomaly that occurs in AdS/CFT. As an application of this formalism, we analyze the near-horizon Virasoro symmetry considered by Haco, Hawking, Perry, and Strominger, and perform a systematic derivation of the fluxes and central charges. Applying the Cardy formula to the result yields an entropy that is twice the Bekenstein-Hawking entropy of the horizon. Motivated by the extended Hilbert space construction, we interpret this in terms of a pair of entangled CFTs associated with edge modes on either side of the bifurcation surface.
Highlights
Introduction and summaryObservables in general relativity tend to be global in nature, owing to the fact that diffeomorphisms are gauge symmetries of the theory
The asymptotic density of states in such a theory is controlled by the Cardy formula, and by applying it in conjunction with the central charge computed from the quasilocal charge algebra, one arrives at the Bekenstein-Hawking entropy
We revisited the Wald-Zoupas construction of quasilocal charges and fluxes for subregions with null boundaries, with the goal of systematically deriving the central charges that have appeared in several recent works on symmetries near Killing horizons [10, 11, 34,35,36, 87]
Summary
Observables in general relativity tend to be global in nature, owing to the fact that diffeomorphisms are gauge symmetries of the theory. The most important ambiguity is in the ability to shift the symplectic potential on the bounding hypersurface by total variations, which subsequently affects the definitions of the charges and fluxes To resolve this issue, we first reformulate the Wald-Zoupas procedure in section 2.2 using Harlow and Wu’s presentation of the covariant phase space formalism with boundaries [37]. We find additional flux terms beyond those employed in [10, 34], whose presence is necessary to ensure that the flux is independent of the choice of auxiliary null vector na With all this in place, we give a systematic analysis in section 4 of the quasilocal charges in the HHPS construction, as well as the generalization to arbitrary bifurcate, axisymetric Killing horizons [10, 34]. This work is being released in coordination with [51], which explores some related topics
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