Hair on near-extremal Reissner-Nordstrom AdS black holes

TL;DR

This paper investigates the formation of scalar hair on near-extremal Reissner-Nordstrom AdS black holes, identifying critical conditions for instability and providing analytic estimates for the phase transition and physical properties.

Contribution

It introduces an analytic framework for determining the critical line and conditions for hair formation on near-extremal black holes with scalar fields.

Findings

Hair forms below a critical temperature $T_c$ in certain parameter regions.

Critical coupling $q_c^2$ is determined by horizon geometry for $ ext{Δ} > ext{Δ}_0$.

Energy gap diverges as $T_c$ approaches zero.

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 ($T_c \to 0$).