Phase transitions near black hole horizons

TL;DR

The paper demonstrates that four-dimensional Reissner-Nordstrom black holes can develop hair by coupling a real scalar with non-renormalizable interactions to the gauge field, yielding stable hairy solutions with higher entropy at fixed mass and charge. Hair forms as a horizon-proximate phase transition, with the scalar condensing near the horizon and decaying away from it, and can persist at finite Hawking temperatures and in AdS spacetimes. The author explores generalizations to larger symmetry groups, multiple gauge sectors, and uncharged cases, uncovering a rich set of phase diagrams, bifurcations, and potential implications for no-hair theorems and holography. A unifying view is proposed: hair primarily emerges when non-renormalizable terms become relevant at the horizon scale, suggesting a horizon-radius-dependent criterion for uniqueness and hair formation in four-dimensional gravity coupled to matter.

Abstract

The Reissner-Nordstrom black hole in four dimensions can be made unstable without violating the dominant energy condition by introducing a real massive scalar with non-renormalizable interactions with the gauge field. New stable black hole solutions then exist with greater entropy for fixed mass and charge than the Reissner-Nordstrom solution. In these new solutions, the scalar condenses to a non-zero value near the horizon. Various generalizations of these hairy black holes are discussed, and an attempt is made to characterize when black hole hair can occur.

Figures (6)

• Figure 1: $g^{rr}$ and $\phi$ for a hairy charged black hole with $m^2=1$, $\lambda=0$, $\ell^2=10$, and $g=1$.
• Figure 2: (A) The suggested phase diagram for ${m_0 \over 2} \leq \ell < {1 \over 2} \sqrt{m_0^2 + {1 \over 2}}$. (B) The suggested phase diagram for ${1 \over 2} \sqrt{m_0^2 + {1 \over 2}} < \ell < m_0$. (C) The suggested phase diagram for $m_0 < \ell$. The green lines indicate stable branches, and the oranges lines indicate unstable branches. The gray line in (B) and (C) is the asymptotic behavior of many unstable branches. Properties of the numbered points are discussed in the main text.
• Figure 3: First order behavior in a phase transition near a black hole horizon. The parameters $m$, $\ell_i$, and $\kappa$ are held fixed, with values as in (\ref{['SuitableChoice']}), and the $p_0$-$g_0$ plane is as described after (\ref{['NiceScaling']}). The green lines from point $3$ to the origin and outward from the points labeled $1$ represent the lowest mass solution for given charge and entropy, while the orange lines represent other static solutions. The numbered points are explained in the main text.
• Figure 4: A phase transition near a black hole horizon in $AdS_5$. (A) The functions $\phi(r)$, $\chi(r)$, and $h(r)$ for the hairy solution with $\phi = 1$ at the horizon. (B) $\phi_0$ and $\chi_0$ are the values of $\phi$ and $\chi$ at the horizon. Hairy horizons evidently exist only for $\phi_0 > 0.83$.
• Figure 5: A hairy black hole that extremizes the action (\ref{['Vnegative']}), with $m^2 = 1/5$ and $\lambda = -2/3$. The horizon is at $r_H = 1$. The mass of this black hole is $1.61$ times the mass of a Schwarzschild black hole with the same entropy.