Stability and Thermodynamics of AdS Black Holes with Scalar Hair

Abstract

Recently a class of static spherical black hole solutions with scalar hair was found in four and five dimensional gauged supergravity with modified, but AdS invariant boundary conditions. These black holes are fully specified by a single conserved charge, namely their mass, which acquires a contribution from the scalar field. Here we report on a more detailed study of some of the properties of these solutions. A thermodynamic analysis shows that in the canonical ensemble the standard Schwarzschild-AdS black hole is stable against decay into a hairy black hole. We also study the stability of the hairy black holes and find there always exists an unstable radial fluctuation, in both four and five dimensions. We argue, however, that Schwarzschild-AdS is probably not the endstate of evolution under this instability.

Figures (4)

• Figure 1: The scalar field $\phi_e$ at the horizon as a function of horizon size $R_e$ in hairy black hole solutions of $D=4$ gauged supergravity. The two curves correspond to solutions with two different AdS invariant boundary conditions, labelled by $c=-1$ (bottom) and $c=-1/4$ (top).
• Figure 2: Potential for linearized radial fluctuations around a hairy black hole with radius $R_e =2$, for $c=-1/4$ (left) and $c=-1$ (right) boundary conditions.
• Figure 3: The value of the negative frequency, $\omega^2_{n}$, of the unstable radial perturbation as a function of horizon size $R_e$, in $D=4$ supergravity with two different AdS invariant boundary conditions, namely $c=-1$ (bottom) and $c=-1/4$ (top).
• Figure 4: The difference $\Delta I$ between the Euclidean action of a hairy black hole, $I_h$, and that of Schwarzschild-AdS, $I_s$, as a function of temperature $T$. The two curves correspond to hairy black hole solutions of $D=4$ gauged supergravity with two different AdS invariant boundary conditions, specified by $c=-1$ (bottom) and $c=-1/4$ (top). One sees that in both cases Schwarzschild-AdS is always thermodynamically favored.