Black Hole Entropy without Brick Walls

TL;DR

The paper tests the idea that black hole entropy divergences can be absorbed into the renormalization of gravitational couplings. It uses Pauli–Villars regularization to regulate a scalar field near a four-dimensional RN horizon and compares the entropy calculation to the one-loop renormalization of the gravitational action, finding exact agreement in the divergent structure. The total entropy can be written in terms of renormalized couplings: $S_{\rm total} = {\cal A}/(4G_R) - 8\pi u \beta_R + 16\pi(1-2u) \gamma_R$, with $S_{\rm ext}=0$ in the extremal limit. This supports the Susskind–Uglum conjecture, demonstrates a brick-wall-free covariant regularization, and highlights the need to include higher-curvature terms in black hole entropy calculations; the results are robust to non-minimal couplings but motivate a Euclidean path-integral formulation for full consistency.

Abstract

We present evidence which confirms a suggestion by Susskind and Uglum regarding black hole entropy. Using a Pauli-Villars regulator, we find that 't Hooft's approach to evaluating black hole entropy through a statistical-mechanical counting of states for a scalar field propagating outside the event horizon yields precisely the one-loop renormalization of the standard Bekenstein-Hawking formula, $S=\A/(4G)$. Our calculation also yields a constant contribution to the black hole entropy, a contribution associated with the one-loop renormalization of higher curvature terms in the gravitational action.