A scalar field condensation instability of rotating anti-de Sitter black holes

TL;DR

This work analyzes scalar field condensation instabilities in near-extremal AdS black holes, focusing on rotating backgrounds with equal angular momenta, hyperbolic AdS echoes, and Kerr-AdS in four dimensions. It combines linear perturbation analyses (identifying thresholds governed by the AdS2 near-horizon BF bound) with nonlinear constructions of hairy black holes, using perturbative and numerical methods to map phase diagrams and thermodynamics. A key result is that the near-horizon BF bound sharply controls stability for many extreme cases, though small spherical RN-AdS can evade this bound; the endpoint hairy solutions generally possess higher entropy than their hairless counterparts, suggesting thermodynamic preference. The study also establishes energy-based stability criteria for uncharged and charged scalars, linking stability to NH BF bounds and clarifying the coexistence with superradiant instabilities in rotating AdS spacetimes.

Abstract

Near-extreme Reissner-Nordstrom-anti-de Sitter black holes are unstable against the condensation of an uncharged scalar field with mass close to the Breitenlohner-Freedman bound. It is shown that a similar instability afflicts near-extreme large rotating AdS black holes, and near-extreme hyperbolic Schwarzschild-AdS black holes. The resulting nonlinear hairy black hole solutions are determined numerically. Some stability results for (possibly charged) scalar fields in black hole backgrounds are proved. For most of the extreme black holes we consider, these demonstrate stability if the ``effective mass" respects the near-horizon BF bound. Small spherical Reissner-Nordstrom-AdS black holes are an interesting exception to this result.

Figures (13)

• Figure 1: Scalar condensation in the HHT black hole with equal angular momenta. We consider a free scalar field whose CFT dual is an operator of dimension $\Delta$. The vertical axis is the difference of $\Omega_H$ from its extreme value $\delta\Omega \equiv \Omega_H^{\rm ext}(r_+)-\Omega_H$. The surface corresponds to values of $\Omega_H$, $r_+$ and $\Delta$ for which there exists a regular, time-independent solution of the scalar equation of motion. The surface extends to arbitrarily large $r_+$. The region enclosed by this surface corresponds to values of $\Omega_H$, $r_+$ and $\Delta$ for which the black hole is unstable against scalar condensation. The intersection of the surface with the $\delta\Omega=0$ plane corresponds to saturation of the near-horizon $AdS_2$ BF bound for extreme black holes. The black curve is for $\Delta=2$, i.e., saturation of the $AdS_5$ BF bound.
• Figure 2: Threshold of the scalar condensation instability in the $d=5$ hyperbolic Schwarzchild-AdS black hole. The plot shows the critical dimensionless horizon radius $r_+/\ell$ where the instability sets in as a function of the conformal dimension $\Delta$ of the CFT operator dual to the scalar field. Points below the curve correspond to instability. The blue dot corresponds to the extreme black hole. The critical value of $\Delta$ at this point is precisely the value predicted from the near-horizon BF bound. The black dot corresponds to the $AdS_5$ BF bound, i.e., $\Delta=2$.
• Figure 3: Asymptotic value of the condensate, $\phi_0/\ell^2$ as defined in \ref{['HairySchw:BCs']} (equivalently: vev of $\Delta=2$ operator dual to the scalar field), as a function of the dimensionless energy $E/\ell^2$ of the hyperbolic hairy black hole. The red curve is the perturbative result, the blue curve is determined numerically.
• Figure 4: Dimensionless temperature, $T_H \ell$ (left), and dimensionless entropy, $S/\ell^3$ (right), of the hyperbolic hairy black hole as a function of its dimensionless energy $E/\ell^2$. The scalar is at the BF bound: $\mu^2 \ell^2 = -4$ ($\Delta=2$). The red curve is the perturbative result, and the blue curve is the numerical result (it ends in a zero temperature and entropy configuration indicated by a green dot). The dashed black curve corresponds to the hyperbolic Schwarzchild-AdS black hole. This terminates at an extreme solution with non-zero entropy. Note that the hairy black hole always has greater entropy than the Scharzschild-AdS solution with the same mass.
• Figure 5: Detail of the black curve in Figure \ref{['Fig:linearMP']} where we fix the scalar mass to be at the BF bound i.e. when $\Delta_+=\Delta_-$. The plot shows the dimensionless angular velocity wrt the extreme value, $\delta\Omega \ell=(\Omega_H^{\rm ext}-\Omega_H) \ell$, as a function of the horizon radius, $r_+/\ell$.