Non-minimal coupling and quantum entropy of black hole

TL;DR

This paper develops a modified statistical mechanics approach for scalar fields with non-minimal coupling in black hole backgrounds, revealing how entropy depends on the coupling and aligning with previous results for negative coupling.

Contribution

It introduces a scattering ansatz at the horizon to modify the 't Hooft brick wall method, accounting for non-minimal coupling effects on black hole entropy.

Findings

Entropy renormalizes Bekenstein-Hawking entropy for negative coupling.

Results agree with conical singularity approach for negative coupling.

Positive coupling yields different entropy results.

Abstract

Formulating the statistical mechanics for a scalar field with non-minimal $\xi R \phi^2$ coupling in a black hole background we propose modification of the original 't Hooft ``brick wall'' prescription. Instead of the Dirichlet condition we suggest some scattering ansatz for the field functions at the horizon. This modifies the energy spectrum of the system and allows one to obtain the statistical entropy dependent on the non-minimal coupling. For $\xi<0$ the entropy renormalizes the classical Bekenstein-Hawking entropy in the correct way and agrees with the result previously obtained within the conical singularity approach. For a positive $\xi$, however, the results differ.