Entropies of Scalar Fields on Three Dimensional Black Holes

TL;DR

This paper addresses how scalar fields thermally behave in three-dimensional BTZ black holes, testing two calculational pathways: explicit mode-sum partitioning and the Euclidean path integral via Hartle-Hawking Green functions. The authors derive exact mode functions, Hartle-Hawking Green functions, and Green functions on cone geometries, enabling precise expressions for free energies and entropies under different boundary conditions. A central finding is that entropies are not generally proportional to the horizon area and that their divergences depend on the calculation method and regularization, not solely on the horizon. These results illuminate the nuanced relationship between quantum field entropy, horizon geometry, and gravitational renormalization, and provide a tangible 3D testing ground for reconciling black hole thermodynamics with quantum theory.

Abstract

Thermodynamics of scalar fields is investigated in three dimensional black hole backgrounds in two approaches. One is mode expansion and direct computation of the partition sum, and the other is the Euclidean path integral approach. We obtain a number of exact results, for example, mode functions, Hartle-Hawking Green functions on the black holes, Green functions on a cone geometry, free energies and entropies. They constitute reliable bases for the thermodynamics of scalar fields. It is shown that thermodynamic quantities largely depend upon the approach to calculate them, boundary conditions for the scalar fields and regularization method. We find that, in general, the entropies are not proportional to the area of the horizon and that their divergent parts are not necessarily due to the existence of the horizon.