Quantum fluxes at the inner horizon of a near-extremal spherical charged black hole

TL;DR

This paper analytically investigates semiclassical quantum fluxes at the inner horizon of near-extremal Reissner-Nordström black holes, revealing their dependence on extremality and providing new expressions for evaporation rates and transmission coefficients.

Contribution

It provides the first analytical expressions for flux behaviors, transmission coefficients, and evaporation rates of near-extremal charged black holes in semiclassical gravity.

Findings

Fluxes scale as Δ^5 near extremality.

Ingoing flux dominates at the inner horizon.

Hawking evaporation rate scales as Δ^4 with a new prefactor.

Abstract

We analyze and compute the semiclassical stress-energy flux components, the outflux $\langle T_{uu}\rangle$ and the influx $\langle T_{vv}\rangle$ ($u$ and $v$ being the standard null Eddington coordinates), at the inner horizon (IH) of a Reissner-Nordstr\"om black hole (BH) of mass $M$ and charge $Q$, in the near-extremal domain in which $Q/M$ approaches $1$. We consider a minimally-coupled massless quantum scalar field, in both Hartle-Hawking ($H$) and Unruh ($U$) states, the latter corresponding to an evaporating BH. The near-extremal domain lends itself to an analytical treatment which sheds light on the behavior of various quantities on approaching extremality. We explore the behavior of the three near-IH flux quantities $\langle T_{uu}^-\rangle^U$, $\langle T_{vv}^-\rangle^U$, and $\langle T_{uu}^-\rangle^H=\langle T_{vv}^-\rangle^H$, as a function of the small parameter $\Delta\equiv\sqrt{1-(Q/M)^2}$ (where the superscript "$-$" refers to the IH value). We find that in the near-extremal domain $\langle T_{uu}^-\rangle^U\cong\langle T_{uu}^-\rangle^H=\langle T_{vv}^-\rangle^H$ behaves as $\propto\Delta^5$. In contrast, $\langle T_{vv}^-\rangle^U$ behaves as $\propto\Delta^4$, and we calculate the prefactor analytically. It therefore follows that the semiclassical fluxes at the IH neighborhood of an evaporating near-extremal spherical charged BH are dominated by the influx $\langle T_{vv}\rangle^U$. In passing, we also find an analytical expression for the transmission coefficient outside a Reissner-Nordstr\"om BH to leading order in small frequencies (which turns out to be a crucial ingredient of our near-extremal analysis). Furthermore, we explicitly obtain the near-extremal Hawking-evaporation rate ($\propto\Delta^4$), with an analytical expression for the prefactor (obtained here for the first time to the best of our knowledge). [Abridged]