Anomalies in gravitational charge algebras of null boundaries and black hole entropy

TL;DR

The paper develops a covariant phase-space framework for gravity with null boundaries, fixing ambiguities in Wald–Zoupas quasilocal charges by imposing a Dirichlet form for the flux. This leads to a precise expression for the boundary stress-tensor data and a Barnich–Troessaert bracket whose central extension is dictated by the boundary-term anomaly, interpreted as a classical diffeomorphism anomaly. Applying the formalism to near-horizon Virasoro symmetries of Killing horizons yields central charges proportional to the horizon area and, via Cardy, an entropy result that initially appears twice the Bekenstein–Hawking value; this is argued to reflect edge-mode degrees of freedom on both sides of the bifurcation surface and depends on whether charges are integrable. When integrability is imposed, the central charge and resulting entropy reduce to the standard horizon entropy, suggesting a deep link between open-system edge modes, Dirichlet matching, and holographic-type variance under frame changes. Overall, the work provides a general mechanism for horizon entropy via classical anomalies in quasilocal charge algebras and highlights rich connections to edge modes, holography, and the role of boundary data in gravitational dynamics.

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.

Figures (3)

• Figure 1: In the Wald-Zoupas construction, one seeks to construct quasilocal charges for a transformation generated by $\xi^a$, which is tangent to a hypersurface $\mathcal{N}$ bounding an open subregion $\mathcal{U}$ to the right of $\mathcal{N}$. The charges are constructed as integrals over a codimension-2 surface $\partial\Sigma$, bounding a Cauchy surface $\Sigma$ for the subregion. The vector field $\xi^a$ can have both tangential and normal components to $\partial\Sigma$. In this figure, $\mathcal{N}$ is a null hypersurface, and the Cauchy surface has been chosen to include a segment of $\mathcal{N}$.
• Figure 2: Two different choices of stretched horizons are shown, as the level sets of the functions $X$ and $\tilde{X}$, which lead to different scaling frames for $l_a$ on the null surface.
• Figure 3: Depiction of the gluing procedure. In (\ref{['fig:before']}) we show two disconnected subregions, bounded by timelike stretched horizons in orange. The boundaries of the respective Cauchy surfaces $\Sigma_L$ and $\Sigma_R$ are given by the red dots. In (\ref{['fig:after']}), we imagine gluing the subregions by entangling the edge modes on $\partial \Sigma_L$ with those on $\partial\Sigma_R$. This entanglement should build up the geometry of the intervening space. For the nonextremal horizons considered in this paper, the stretched horizons can approach the bifurcate null horizon, and the gluing occurs accross the bifurcation surface, with the entanglement building up the geometry of the interior.