Instanton Supported Scalar Hair on Black Holes

TL;DR

This paper constructs and analyzes black hole solutions with scalar hair supported by topological gauge fields and instantons, demonstrating their stability and compatibility with no-hair theorems.

Contribution

It introduces new analytical and numerical solutions of Einstein equations featuring topologically supported scalar hair, linking instanton effects to classical black hole properties.

Findings

Solutions exhibit scalar hair linked to topological gauge charge.

Black holes with these solutions are stable under linear perturbations.

The solutions do not violate no-hair theorems for small coupling.

Abstract

We present analytical perturbative, and numerical solutions of the Einstein equation which describe a black hole with a nontrivial dilaton field and a purely topological gauge potential. The gauge potential has zero field strength and hence no stress-energy, but it does couple to virtual string worldsheets which wrap around the Euclidean horizon two-sphere, and generate an effective interaction in the spacetime lagrangian. We use the lagrangian with a nonstandard potential for a scalar field that reproduces the effect of the worldsheet instantons. As has been previously pointed out the topological charge Q of the gauge field can be detected by an Anharonov-Bohm type experiment using quantum strings; the classical scalar hair of the solutions here is a classical detection of Q. ADM mass, dilaton charge and Hawking temperature are calculated and compared with the known cases when appropriate. We discuss why these solutions do not violate the no-hair theorems and show that, for sufficiently small interaction coupling, the solutions are stable under linear time dependent perturbations.