Black Hole Entropy: Thermodynamics, Statistical-Mechanics and Subtraction Procedure

TL;DR

This paper calculates the thermodynamical one-loop entropy of a two-dimensional black hole, revealing its relation to the Bekenstein-Hawking entropy and emphasizing the need for a subtraction procedure to connect thermodynamic and statistical-mechanical entropies.

Contribution

It explicitly demonstrates the non-trivial relation between thermodynamical and statistical-mechanical entropies of black holes and introduces a subtraction procedure for their connection.

Findings

Thermodynamical entropy includes Bekenstein-Hawking entropy and a finite difference of statistical entropies.

The relation between thermodynamical and statistical-mechanical entropies is non-trivial.

A subtraction procedure is necessary to relate these entropies explicitly.

Abstract

The thermodynamical one-loop entropy $S^{TD}$ of a two-dimensional black hole in thermal equilibrium with the massless quantum gas is calculated. It is shown that $S^{TD}$ includes the Bekenstein-Hawking entropy, evaluated for the quantum corrected geometry, and the finite difference of statistical mechanical entropies $-Tr\hat{\rho}\ln\hat{\rho}$ for the gas on the black hole and Rindler spaces. This result demonstrates in an explicit form that the relation between thermodynamical and statistical-mechanical entropies of a black hole is non-trivial and requires special subtraction procedure.