Analytic and asymptotically flat hairy (ultra-compact) black-hole solutions and their axial perturbations

TL;DR

This paper introduces new analytic, ultra-compact black hole solutions with scalar hair in a simple gravitational theory, analyzes their properties, stability, and rotation, and compares them to Schwarzschild black holes.

Contribution

It provides the first explicit analytic ultra-compact black hole solutions with scalar hair, including their rotation and stability analysis, extending previous numerical results.

Findings

The solutions are asymptotically flat and regular with scalar hair.

Stable ultra-compact black holes can be smaller and faster rotating than Schwarzschild black holes.

The angular velocity of these solutions exceeds that of comparable Schwarzschild black holes.

Abstract

In this work, we consider a very simple gravitational theory that contains a scalar field with its kinetic and potential terms minimally coupled to gravity, while the scalar field is assumed to have a coulombic form. In the context of this theory, we study an analytic, asymptotically flat, and regular (ultra-compact) black-hole solutions with non-trivial scalar hair of secondary type. At first, we examine the properties of the static and spherically symmetric black-hole solution -- firstly appeared in 1504.08209 [gr-qc] -- and we find that in the causal region of the spacetime the stress-energy tensor, needed to support our solution, satisfies the strong energy conditions. Then, by using the slow-rotating approximation, we generalize the static solution into a slowly rotating one, and we determine explicitly its angular velocity $\omega(r)$. We also find that the angular velocity of our ultra-compact solution is always larger compared to the angular velocity of the corresponding equally massive slow-rotating Schwarzschild black hole. In addition, we investigate the axial perturbations of the derived solutions by determining the Schr\"{o}dinger-like equation and the effective potential. We show that there is a region in the parameter space of the free parameters of our theory, which allows for the existence of stable ultra-compact black hole solutions. Specifically, we calculate that the most compact and stable black hole solution is 0.551 times smaller than the Schwarzschild one, while it rotates 2.491 times faster compared to the slow-rotating Schwarzschild black hole. Finally, we present without going into details the generalization of the derived asymptotically flat solutions to asymptotically (A)dS solutions.