Hair on near-extremal Reissner-Nordstrom AdS black holes

Abstract

We discuss hairy black hole solutions with scalar hair of scaling dimension $\Delta$ and (small) electromagnetic coupling $q^2$, near extremality. Using trial functions, we show that hair forms below a critical temperature $T_c$ in the region of parameter space $(\Delta ,q^2)$ above a critical line $q_c^2(\Delta )$. For $\Delta >{\Delta }_0$, the critical coupling $q_c^2$ is determined by the AdS${}_2$ geometry of the horizon. For $\Delta <{\Delta }_0$, $q_c^2$ is {\em below} the value suggested by the near horizon geometry at extremality. We provide an analytic estimate of ${\Delta }_0$ (numerically, ${\Delta }_0\approx 0.64$). We also compute analytically the true critical line for the entire range of the scaling dimension. In particular for $q=0$, we obtain an instability down to the unitarity bound. We perform explicit analytic calculations of $T_c$, the condensate and the conductivity. We show that the energy gap in units of $T_c$ diverges as we approach the critical line ($T_c\to 0$).

PRD version: 15 pages, 9 figures, corrected typos, improved discussion