Black hole thermodynamics is around the corner

TL;DR

The paper introduces a corner-based Euclidean framework for black hole thermodynamics in generic $F(R_{abcd})$ gravity, replacing the problematic conical singularity with a geometric corner. It demonstrates that the Wald entropy emerges from first-order variations of either the corner-free action $I$ or the corner-augmented action $I'$, establishing that entropy information is encoded in the corner. A special diffeomorphism then yields a direct link between the horizon Killing vector and the inverse temperature, yielding the ADM Hamiltonian as the thermodynamic conjugate in the grand canonical ensemble. The approach offers a finite alternative to conical deficit methods and points to extensions to higher-derivative theories and loop corrections, with potential applications to generalized gravitational entropy.

Abstract

We propose to work on the Euclidean black hole solution with a corner rather than with the prevalent conical singularity. As a result, we find that the Wald formula for black hole entropy can be readily obtained for generic F(Rabcd) gravity by using both the action without the corner term and the action with the corner term due to their equivalence to the first order variation, which implies that it is the corner rather than the corner term that encodes the entropy related information. With such an equivalence, we further make use of a special diffeomorphism to accomplish a direct derivation of the ADM Hamiltonian conjugate to the Killing vector field normal to the horizon in the Lorentz signature as a conjugate variable of the inverse temperature in the grand canonical ensemble.

Figures (3)

• Figure 1: The portion of the boundary of the space $M$ is given by $\Sigma_1$ and $\Sigma_2$, which intersect with each other at a co-dimension $2$ corner denoted by the red point. $n_i^a$ and $r_i^a$ with $n_i\cdot r_i=0$ are transverse orthonormal vectors at the corner, where $n_i^a$ is the normal vector to $\Sigma_i$ with $i=1,2$.
• Figure 2: The two Euclidean black holes share the same imaginary time interval $\beta_0$, where the black hole in blue has $T=\frac{1}{\beta}$ while the black hole in orange has $T_0=\frac{1}{\beta_0}$. The boundary consists of two cuts and the asymptotical surface with three corners. The bifurcation surface is identified as the inner corner from the intersection of the two cuts and the two outer corners arise from the intersection of the asymptotical surface with each cut, respectively.
• Figure 3: The Euclidean black hole at the temperature $T_0=\frac{1}{\beta_0}$ but with the imaginary time interval $\beta$.