Entropy of Quantum Fields for Nonextreme Black Holes in the Extreme Limit

TL;DR

This paper investigates the entropy contributions of quantum fields near nonextreme black holes approaching the extreme limit, finding that quantum entropy vanishes despite finite boundary temperature, due to horizon properties.

Contribution

It demonstrates that quantum entropy contributions vanish in the extreme limit for black holes and related spacetimes, revealing new insights into black hole thermodynamics.

Findings

Quantum entropy $S_q$ approaches zero in the extreme limit.

Finite boundary temperature does not imply nonzero quantum entropy.

Horizon properties of Bertotti-Robinson spacetime explain the entropy behavior.

Abstract

Nonextreme black hole in a cavity within the framework of the canonical or grand canonical ensemble can approach the extreme limit with a finite temperature measured on a boundary located at a finite proper distance from the horizon. In spite of this finite temperature, it is shown that the one-loop contribution $S_{q\text{ }}$of quantum fields to the thermodynamic entropy due to equilibrium Hawking radiation vanishes in the limit under consideration. The same is true for the finite temperature version of the Bertotti-Robinson spacetime into which a classical Reissner-Nordstr\"{o}m black hole turns in the extreme limit. The result $S_{q}=0$ is attributed to the nature of a horizon for the Bertotti-Robinson spacetime.