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

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.