Regular hairy black holes through gravitational decoupling method

TL;DR

This work uses gravitational decoupling (GD) to introduce a tensor vacuum $\theta_{\mu\nu}$ that deforms the Schwarzschild geometry into regular, hairy black holes while preserving asymptotic flatness and satisfying the weak energy condition. By decomposing the field equations into a GR seed sector and a separate $\theta_{\mu\nu}$ sector, the authors obtain exact, nonperturbative solutions in both static and rotating (Kerr-Schild) spacetimes, with the deformation governed by a parameter $\gamma$ and auxiliary scales $\xi$ and $\iota$. They derive explicit spherically symmetric and axially symmetric (rotating) solutions, show the absence of central curvature singularities, and analyze horizon structures that yield 0, 1, or 2 horizons depending on the GD parameters. The rotating case extends regularity to Kerr-like geometries, providing a controlled pathway from Minkowski to Schwarzschild and Kerr limits, and laying groundwork for stability and observational studies. Overall, the GD framework offers a physically motivated mechanism to realize non-singular, hairy black holes within general relativity-like settings.

Abstract

Within a framework requiring a well-defined event horizon and matter obeying the weak energy condition, we employ gravitational decoupling method to construct non-singular hairy black holes: spherically or axially symmetric. These solutions arise from a deformation of the Minkowski vacuum, where the maximum deformation can yield the Schwarzschild metric for the static case, and the Kerr geometry for the stationary case, respectively.

Figures (3)

• Figure 1: Spherically symmetric solutions of the metric function \ref{['5g']}: no horizons, on horizon, and two horizons. The extrem case is given by $\gamma^{*}\simeq0.617$, if seting $\xi=0.5$ and $\iota=0.25$.
• Figure 2: The source terms $\{\tilde{\epsilon},\tilde{p}_{r},\tilde{p}_{t}\}\times10$ in the sphericity metric case with $\xi=0.5$, $\iota=0.25$ and $\gamma=2$. The vertical dashed line represents the event horizon $r_{\rm {h}}\sim5$.
• Figure 3: Axially symmetric solutions of the metric function \ref{['5g']} show three cases: no horizons, one horizon, and two horizons. The extreme BH case is given by $\gamma^{**}\simeq1.624$, if setting $\xi=0.5$, $\iota=0.25$ and $a=2$.