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

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]