Quantum scalar field on three-dimensional (BTZ) black hole instanton: heat kernel, effective action and thermodynamics

TL;DR

This paper analyzes quantum scalar fields on three-dimensional BTZ black hole backgrounds, calculating heat kernels, effective actions, and entropy, revealing a logarithmic behavior of quantum entropy at small horizon areas which could impact black hole evaporation.

Contribution

It provides explicit calculations of heat kernels, effective actions, and quantum entropy for scalar fields on BTZ black holes, including cases with conical singularities, and discusses UV renormalization.

Findings

Quantum entropy behaves logarithmically for small horizon areas.

Explicit heat kernel and effective action formulas for rotating and non-rotating BTZ black holes.

UV divergences are renormalized, highlighting the structure of finite quantum entropy.

Abstract

We consider the behaviour of a quantum scalar field on three-dimensional Euclidean backgrounds: Anti-de Sitter space, the regular BTZ black hole instanton and the BTZ instanton with a conical singularity at the horizon. The corresponding heat kernel and effective action are calculated explicitly for both rotating and non-rotating holes. The quantum entropy of the BTZ black hole is calculated by differentiating the effective action with respect to the angular deficit at the conical singularity. The renormalization of the UV-divergent terms in the action and entropy is considered. The structure of the UV-finite term in the quantum entropy is of particular interest. Being negligible for large outer horizon area $A_+$ it behaves logarithmically for small $A_+$. Such behaviour might be important at late stages of black hole evaporation.