The evaporation of near-extremal black holes through charged particle emission

TL;DR

This work provides a unified framework for charged black-hole evaporation by comparing quantum corrections from Schwarzian/JT gravity to semiclassical Hawking rates for near-extremal RN and Kerr spacetimes. It identifies distinct regimes governed by three scales—the breakdown energy $E_b$, the black-hole energy $E$, and the superradiant threshold $\omega_{\text{sr}}$—and shows that quantum corrections can either suppress or enhance emission relative to semiclassical predictions, depending on the regime. For small RN black holes, neutral-like emission is suppressed while superradiant emission is enhanced in the quantum theory; for large RN black holes, quantum corrections converge to semiclassical results with Schwinger-like suppression recovered in the very large limit. In near-extremal Kerr, superradiant modes dominate and the quantum corrections are subleading, with the analysis revealing a parallel AdS2-based treatment that mirrors the RN case and providing detailed spectra for spins 0, 1, and 2.

Abstract

We compute the quantum rate for massless charged scalar emission by a near-extremal Reissner-Nordström black hole using Schwarzian theory as the effective description of the black hole. This is compared to the semi-classical Hawking rate which we also compute near extremality. We classify black holes into small and large, each with a unique spectrum. For small black holes, at energies below a particular quantum scale, the emission is captured by the quantum rate, which gives different predictions from the semi-classical. Furthermore, depending on how the energy compares to another scale associated with the phenomenon of superradiance, the radiation either comes out as mostly non-superradiant or mostly superradiant. For non-superradiant emission, it is found that the quantum rate is suppressed compared to the semi-classical in the same way as recently observed for neutral radiation. For superradiant emission, a novel behavior is observed, the quantum rate is enhanced compared to the semi-classical. For large black holes, we argue that the quantum rate always reduces to the semi-classical. In the limit of very large black holes, from our semi-classical rate, we recover the Gibbons result of Schwinger-like suppression. This gives a unified story of near-extremal charged emission rates, both quantum and semi-classical, which covers all sizes of black hole and all energy regimes. We use these rates to discuss the evaporation history of each type of black hole, from when it starts very near extremality, until it has left this regime. Finally, for the near-extremal Kerr black hole, we argue that the quantum rate always reduces to the semi-classical with the superradiant modes dominating. This rate is computed for spin $0,1,2$. Our analysis emphasizes the AdS$_2$ structure of the Reissner-Nordström and Kerr near-horizon regions, which enables a completely parallel treatment of the two.

Figures (7)

• Figure 1: A diagram of $\log Q$ vs. $\log e$ parameter space showing the available regions for radiation from a Reissner-Nordström black hole. The bottom two wedges (small black holes) always have real $\Delta$, the top wedge (large) has both real and complex $\Delta$. The top two wedges always have the quantum rate \ref{['eq:RNqrate']} reduce to the semi-classical formula \ref{['eq:RNscrate']}, while the bottom wedge can have (and this further depends on the size of $E$ which is not on this diagram) the quantum rate give different predictions from the semi-classical.
• Figure 2: (Left) Spectral evaporation rate in quantum neutral-like regime at four different values of $eQ$ between zero and $0.3$. (Right) Integrated evaporation rate as a function of $eQ$.
• Figure 3: (Left) Spectral evaporation rate in quantum superradiant-dominated regime at four different values of $eQ$ between $0.5$ and $0.35$, in equal increments. (Right) Integrated evaporation rate as a function of $eQ$.
• Figure 4: Parameters $\mu$, $\beta$, and $\alpha$ as a function of $\omega$ in units of $e$.
• Figure 5: Spectral evaporation rate of a large black hole in the semi-classical superradiant-dominated regime at different values of $eQ$. Top left panel, $eQ$ is goes from $0.6$ to $1.8$. Top right, $eQ$ is between $3$ and $9$. Bottom, $eQ$ is between $49$ and $55$. The mass of the particle is $m = 0.01 e$.