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 $⟨T_{uu}⟩$ and the influx $⟨T_{vv}⟩$ ($u$ and $v$ being the standard null Eddington coordinates), at the inner horizon (IH) of a Reissner-Nordström 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 $⟨T_{uu}^{-}{⟩}^U$, $⟨T_{vv}^{-}{⟩}^U$, and $⟨T_{uu}^{-}{⟩}^H=⟨T_{vv}^{-}{⟩}^H$, 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 $⟨T_{uu}^{-}{⟩}^U\cong ⟨T_{uu}^{-}{⟩}^H=⟨T_{vv}^{-}{⟩}^H$ behaves as $\propto {\Delta }^5$. In contrast, $⟨T_{vv}^{-}{⟩}^U$ 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 $⟨T_{vv}{⟩}^U$. In passing, we also find an analytical expression for the transmission coefficient outside a Reissner-Nordström 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]

17 pages, 2 figures. v2: sign error fixed in Eq. A30

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• Pragmatic mode-sum regularization method for semiclassical black-hole spacetimes
• Mode-sum regularization of $⟨{\varphi }^2⟩$ in the angular-splitting method
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• Long-range correlations of the stress tensor near the Cauchy horizon
• Wave equations in conformally separable, accreting, rotating black holes
• Computation of $⟨{\Phi }^2⟩$ and quantum fluxes at the polar interior of a spinning black hole