Black Hole Entropy: Off-Shell vs On-Shell

TL;DR

This paper compares on-shell and off-shell methods for calculating quantum corrections to black hole entropy, revealing differences between thermodynamical and statistical-mechanical entropies and expressing the thermodynamical entropy in terms of shifted horizon radius and Rindler space entropy.

Contribution

It establishes the relation between on-shell and off-shell approaches and explicitly demonstrates the difference between thermodynamical and statistical-mechanical black hole entropies.

Findings

Thermodynamical entropy differs from statistical-mechanical entropy for black holes.

The thermodynamical entropy can be expressed as a sum involving the shifted horizon radius and Rindler space entropy.

Off-shell methods explicitly show the quantum backreaction effects on black hole entropy.

Abstract

Different methods of calculation of quantum corrections to the thermodynamical characteristics of a black hole are discussed and compared. The relation between on-shell and off-shell approaches is established. The off-shell methods are used to explicitly demonstrate that the thermodynamical entropy $S^{TD}$ of a black hole, defined by the first thermodynamical law, differs from the statistical-mechanical entropy $S^{SM}$, determined as $S^{SM}=-\mbox{Tr}(\hat{\rho}^H\ln\hat{\rho}^H)$ for the density matrix $\hat{\rho}^H$ of a black hole. It is shown that the observable thermodynamical black hole entropy can be presented in the form $S^{TD}=\pi {\bar r}_+^2+S^{SM}-S^{SM}_{Rindler}$. Here ${\bar r}_+$ is the radius of the horizon shifted because of the quantum backreaction effect, and $S^{SM}_{Rindler}$ is the statistical-mechanical entropy calculated in the Rindler space.