Scalar Deformations of Schwarzschild Holes and Their Stability

TL;DR

This paper constructs and analyzes two novel scalar-deformed Schwarzschild black hole solutions, demonstrating their stability under certain perturbations and challenging traditional no-hair theorems with negative potential features.

Contribution

It introduces two new static scalar field solutions deforming Schwarzschild black holes, with one featuring Coulomb-like hair and both evading no-hair theorems, along with a stability analysis.

Findings

Both solutions are stable against more than half of the perturbation modes.

One solution has an exponentially decaying scalar field with a triple-well potential.

The other solution exhibits Coulomb-like scalar hair and is fully analytic.

Abstract

We construct two solutions of the minimally coupled Einstein-scalar field equations, representing regular deformations of Schwarzschild black holes by a self-interacting, static, scalar field. One solution features an exponentially decaying scalar field and a triple-well interaction potential; the other one is completely analytic and sprouts Coulomb-like scalar hair. Both evade the no-hair theorem by having partially negative potential, in conflict with the dominant energy condition. The linear perturbation theory around such backgrounds is developed in general, and yields stability criteria in terms of effective potentials for an analog Schr\"odinger problem. We can test for more than half of the perturbation modes, and our solutions prove to be stable against those.