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

TL;DR

This paper computes the heat kernel, one-loop effective action, and quantum entropy for a scalar field on three-dimensional Euclidean BTZ-related backgrounds (AdS$_3$, regular BTZ instanton, and BTZ instanton with a conical horizon defect). By solving the heat equation on $H_3$ and using image/sommerfeld methods, it yields explicit expressions for the heat kernel and Green's function, then constructs the BTZ heat kernel via identifications and computes the effective action and entropy, including UV renormalization of Newton's constant and the cosmological term. A key result is a universal logarithmic correction to the quantum entropy at small outer horizon area $A_+$, alongside a Dehn-twist invariant UV-divergent structure and a finite part that drives the entropy toward a minimum at Planck-scale horizon sizes. Analytic continuation to Lorentzian BTZ is performed, revealing how rotation affects the finite entropy contributions, and the findings have implications for late-stage black-hole evaporation and the role of quantum fields in black-hole thermodynamics.

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.