Isolated horizons in higher-dimensional Einstein-Gauss-Bonnet gravity

TL;DR

This work extends the isolated horizon framework to higher-dimensional Einstein-Gauss-Bonnet gravity by constructing a covariant phase space and deriving a local first law for non-rotating, weakly isolated horizons. The authors show that horizon entropy acquires a Gauss-Bonnet correction, $\mathcal{S}=(1/4G_{D})\oint_{S^{D-2}}\bm{\tilde{\epsilon}}(1+2\alpha\mathcal{R})$, in agreement with Euclidean and Noether-charge methods, and they formulate the differentiable action with appropriate boundary terms in the first-order formalism. The resulting framework yields a horizon-first law $\delta E_{\Delta}=(\kappa_{(t)}/k_{D})\delta\mathcal{S}$, without requiring a global Killing vector, thereby generalizing black-hole thermodynamics to EGB gravity and higher dimensions. This sets the stage for potential quantum geometric investigations of horizon states and extensions to rotating horizons and matter couplings."

Abstract

The isolated horizon framework was introduced in order to provide a local description of black holes that are in equilibrium with their (possibly dynamic) environment. Over the past several years, the framework has been extended to include matter fields (dilaton, Yang-Mills etc) in D=4 dimensions and cosmological constant in $D\geq3$ dimensions. In this article we present a further extension of the framework that includes black holes in higher-dimensional Einstein-Gauss-Bonnet (EGB) gravity. In particular, we construct a covariant phase space for EGB gravity in arbitrary dimensions which allows us to derive the first law. We find that the entropy of a weakly isolated and non-rotating horizon is given by $\mathcal{S}=(1/4G_{D})\oint_{S^{D-2}}\bm{\tildeε}(1+2α\mathcal{R})$. In this expression $S^{D-2}$ is the $(D-2)$-dimensional cross section of the horizon with area form $\bm{\tildeε}$ and Ricci scalar $\mathcal{R}$, $G_{D}$ is the $D$-dimensional Newton constant and $α$ is the Gauss-Bonnet parameter. This expression for the horizon entropy is in agreement with those predicted by the Euclidean and Noether charge methods. Thus we extend the isolated horizon framework beyond Einstein gravity.

Figures (1)

• Figure 1: The region of the $D$-dimensional spacetime $\mathcal{M}$ being considered has an internal boundary $\Delta$ representing the event horizon, and is bounded by two (partial) Cauchy surfaces $M^{\pm}$ which intersect $\Delta$ in $(D-2)$-spaces $S^{\pm}$ and extend to the boundary at infinity $\mathscr{B}$.