Thermodynamics of Near-Extreme Black Holes

TL;DR

The paper addresses the thermodynamics of nearly extremal charged black holes at fixed large charge $Q$ and investigates how ground-state degeneracy and the density of excited states affect entropy and stability. It analyzes a semiclassical RN black hole with extremal entropy $S_0 = \pi Q^2$ and Hawking temperature $T_H$, showing how $S \approx \pi Q^2 + \sqrt{8\pi^2 Q^3 E}$ and $T_H \approx \sqrt{E/(2\pi^2 Q^3)}$ shape the canonical ensemble (ill-defined in flat space but stabilizable in AdS-like setups). It then computes near-extremal emission to reach $E \sim 1/Q^3$ and a timescale, and identifies a critical charge $Q_*$ around $3.3\times 10^{46}$ beyond which slow evolution toward extremality is not spoiled by positron emission. Finally, it evaluates several density-of-states models and argues that an exponentially dense quasi-continuum without a large ground-state degeneracy is the most plausible in broken-supersymmetry theories, with potential astronomical tests.

Abstract

The thermodynamics of nearly-extreme charged black holes depends upon the number of ground states at fixed large charge and upon the distribution of excited energy states. Here three possibilities are examined: (1) Ground state highly degenerate (as suggested by the large semiclassical Hawking entropy of an extreme Reissner-Nordstrom black hole), excited states not. (2) All energy levels highly degenerate, with macroscopic energy gaps between them. (3) All states nondegenerate (or with low degeneracy), separated by exponentially tiny energy gaps. I suggest that in our world with broken supersymmetry, this last possibility seems most plausible. An experiment is proposed to distinguish between these possibilities, but it would take a time that is here calculated to be more than about 10^837 years.