Wald's entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling

TL;DR

The paper shows that Wald's Noether-charge entropy for stationary black holes in generalized gravity can be written as $S_W=\frac{1}{4}\oint_{\Sigma}\frac{dA}{G_{eff}}$, where $G_{eff}$ is a horizon-local gravitational coupling derived from the kinetic-term coefficient of a specific metric perturbation polarization via $\frac{1}{(\kappa_{eff})^2}= -\frac{1}{4}(\delta\mathscr{L}/\delta R_{abcd})^{(0)} \hat{\epsilon}_{ab}\hat{\epsilon}_{cd}$ and $G_{eff}=8\pi/(\kappa_{eff})^2$. This unifies Wald and Bekenstein-Hawking entropies by positioning the difference in terms of the horizon-dependent $G_{eff}$, rather than Newton’s constant $G_N$ alone. The authors verify the relation in several static-black-hole examples, including $R+\lambda R^n$, $R+\lambda R_{abcd}R^{abcd}$, and small heterotic-string BHs with Gauss–Bonnet corrections, where $G_{eff}$ acquires curvature- or charge-dependent corrections and, in some cases, becomes non-analytic in $G_N$. The work also links the Wald polarization to fluctuations of the horizon area density, offering a local, observer-dependent interpretation that connects to entanglement entropy and suggests extensions to more general spacetimes and cosmology.

Abstract

The Bekenstein-Hawking entropy of black holes in Einstein's theory of gravity is equal to a quarter of the horizon area in units of Newton's constant. Wald has proposed that in general theories of gravity the entropy of stationary black holes with bifurcate Killing horizons is a Noether charge which is in general different from the Bekenstein-Hawking entropy. We show that the Noether charge entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling on the horizon defined by the coefficient of the kinetic term of specific graviton polarizations on the horizon. We present several explicit examples of static spherically symmetric black holes.