Hairy Black Holes, Horizon Mass and Solitons

TL;DR

This work analyzes horizon mass $M_{\rm hor}$ for hairy black holes using the isolated horizon formalism, arguing that such black holes can be viewed as bound states of bare black holes and solitons. It develops a physical model where the total ADM mass decomposes into a horizon term plus a solitonic residue plus binding energy, enabling qualitative predictions for horizon properties and instability that agree with numerical results. The authors emphasize that $M_{\rm hor}$ is a phase-space quantity and can differ across theories with different hair, explaining why certain non-Abelian solutions are unstable even when their Abelian counterparts are not. They extend the discussion to theories with built-in length scales, where crossing of solution branches occurs and the horizon-mass assignment becomes branch-dependent, yet remains a useful diagnostic for end-states and soliton content. Overall, the horizon-mass framework provides a robust, transferable lens for understanding hairy black holes and their dynamics across a range of gravity-matter theories.

Abstract

Properties of the horizon mass of hairy black holes are discussed with emphasis on certain subtle and initially unexpected features. A key property suggests that hairy black holes may be regarded as `bound states' of ordinary black holes without hair and colored solitons. This model is then used to predict the qualitative behavior of the horizon properties of hairy black holes, to provide a physical `explanation' of their instability and to put qualitative constraints on the end point configurations that result from this instability. The available numerical calculations support these predictions. Furthermore, the physical arguments are robust and should be applicable also in more complicated situations where detailed numerical work is yet to be carried out.

Figures (5)

• Figure 1: The ADM mass as a function of the horizon radius $R_{\Delta}$ of static spherically symmetric solutions to the Einstein-Yang-Mills system (in units provided by the Yang-Mills coupling constant). Numerical plots for the colorless, $n=0$, and the first two families, $n =1,2$ of colored black holes are shown. (Note that the $y$-axis begins at $M = 0.7$ rather than $M= 0$.)
• Figure 2: The horizon mass $M_{{\rm hor}}$ as a function of the radius $R_\Delta$ (in units provided by the Yang-Mills coupling constant). Numerical plots for $n=0,1,2$ spherical, colored black holes in the Einstein-Yang-Mills theory are shown.
• Figure 3: Numerical plots of $\beta_{(n)}= 2\kappa_{(n)}\, R_\Delta$ as function of the horizon radius $R_\Delta$ for $n=1,2$ spherical, colored black holes in the Einstein-Yang-Mills theory. For large $R_\Delta$, all curves asymptotically approach the $\beta =1$ line.
• Figure 4: An initially static colored black hole with horizon $\Delta_{\rm in}$ is slightly perturbed and decays to a Schwarzschild-like isolated horizon $\Delta_{\rm fin}$, with radiation going out to future null infinity ${\cal I}^+$.
• Figure 5: The ADM Mass as a function of the horizon radius $R_\Delta$ in theories with a built-in non-gravitational length scale. The schematic plot shows crossing of families labelled by $n=1$ and $n=2$ at $R_\Delta = R^{\rm inter}_{\Delta}$.