Fate of Three-Dimensional Black Holes Coupled to a Scalar Field and the Bekenstein-Hawking Entropy

TL;DR

This paper demonstrates that the Bekenstein-Hawking entropy for three-dimensional black holes with scalar fields can be recovered using Cardy's formula when the vacuum state is carefully treated in the canonical ensemble, clarifying the entropy discrepancy.

Contribution

It shows that proper vacuum state treatment in the canonical ensemble aligns the Bekenstein-Hawking entropy with statistical calculations for scalar-coupled 3D black holes.

Findings

Bekenstein-Hawking entropy matches Cardy's formula with correct vacuum treatment

Implications for black hole stability and no-hair theorems discussed

Addresses the fate and quantum corrections of these black holes

Abstract

Three-dimensional black holes coupled to a self-interacting scalar field is considered. It is known that its statistical entropy $a' la$ Strominger does $not$ agree with the Bekenstein-Hawking (BH) entropy. However I show that, by a careful treatment of the vacuum state in the {\it canonical} ensemble with a fixed temperature, which is the same as that of the BTZ black hole without the scalar field, the BH entropy is exactly produced by the Cardy's formula. I discuss its several implications, including the fate of black holes, no-scalar-hair theorems, stability, mirror black holes, and one-loop corrections.