The statistical mechanics of near-extremal black holes

TL;DR

The paper addresses whether a mass gap separates extremal from near-extremal states in 4d RN black holes by computing the near-extremal partition function through a dimensional reduction on S2 to a 2d dilaton gravity theory with a U(1) gauge field and an SO(3) gauge sector. The authors map the problem to a 1d boundary theory comprising the Schwarzian action coupled to a particle on the U(1)×SO(3) group manifold, and show that the density of states is continuous: ρ(E,Q)=e^{S0(Q)} sinh[2π sqrt{2 Φb,Q (E−M0(Q))}] with E>M0(Q), indicating no gap at the scale MSL(2). The leading low-temperature physics is governed by the Schwarzian, yielding a universal −(3/2) log T correction to log Z and a corresponding −(3/2) log β term; massive KK modes contribute only β-independent shifts to the entropy and do not alter the continuum spectrum. Together these results suggest that the extremal degeneracy in non-supersymmetric settings is not fixed by the naive area law, and a full microscopic understanding would require ultraviolet completion or nonperturbative dynamics beyond the leading JT description.

Abstract

An important open question in black hole thermodynamics is about the existence of a "mass gap" between an extremal black hole and the lightest near-extremal state within a sector of fixed charge. In this paper, we reliably compute the partition function of Reissner-Nordström near-extremal black holes at temperature scales comparable to the conjectured gap. We find that the density of states at fixed charge does not exhibit a gap; rather, at the expected gap energy scale, we see a continuum of states. We compute the partition function in the canonical and grand canonical ensembles, keeping track of all the fields appearing through a dimensional reduction on $S^2$ in the near-horizon region. Our calculation shows that the relevant degrees of freedom at low temperatures are those of $2d$ Jackiw-Teitelboim gravity coupled to the electromagnetic $U(1)$ gauge field and to an $SO(3)$ gauge field generated by the dimensional reduction.

Figures (4)

• Figure 1: Energy above extremality at fixed charge as a function of the temperature when obtained from the semiclassical analysis (in red) and when accounting for the quantum fluctuations in the near-horizon region (in purple). This should be compared to the average energy of one Hawking quanta (dashed line) whose energy is on average $\langle E\rangle \sim T$.
• Figure 2: Purple: Density of states (at fixed charge) for black holes states as a function of energy above extremality $E-M_0$, including backreaction effects given in \ref{['eq:dossinh']}. Red: Plot of the naive density of states $\rho\sim \exp{(A_{\rm hor}/4 G_N)}$ which starts deviating from the full answer below energies of order $M_{\rm gap}$.
• Figure 3: A cartoon of the near-horizon region (NHR) and the far-away region (FAR) separated by a boundary at which the boundary term of JT gravity will need to be evaluated. In the throat quantization is easy and necessary to account for at low temperatures. In the FAR quantization is hard but quantum corrections are suppressed.
• Figure 4: A cartoon of the near-horizon region (NHR) and the far-away region (FAR) separated by a curve along which the boundary term of JT gravity will need to be evaluated.