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

We analyze and compute the semiclassical stress-energy flux components, outflux ${?{T}_{uu}?}_{\mathrm{ren}}$ influx ${?{T}_{vv}?}_{\mathrm{ren}}$ ($u$ $v$ being standard null Eddington coordinates), at inner horizon (IH) of a Reissner-Nordstr\"om black hole (BH) mass $M$ charge $Q$, in near-extremal domain which $Q/M$ approaches 1. consider minimally-coupled massless quantum scalar field, both Hartle-Hawking Unruh states, latter corresponding to an evaporating BH. The lends itself analytical treatment sheds light on behavior various quantities approaching extremality. explore three near-IH ${?{T}_{uu}^{\ensuremath{-}}?}_{\mathrm{ren}}^{U}$, ${?{T}_{vv}^{\ensuremath{-}}?}_{\mathrm{ren}}^{U}$, ${?{T}_{uu}^{\ensuremath{-}}?}_{\mathrm{ren}}^{H}={?{T}_{vv}^{\ensuremath{-}}?}_{\mathrm{ren}}^{H}$, as function small parameter $\mathrm{\ensuremath{\Delta}}\ensuremath{\equiv}\sqrt{1\ensuremath{-}{(Q/M)}^{2}}$ (where superscript ``$U$'' or ``$H$'' respectively refers state ``$\ensuremath{-}$'' IH value). find that ${?{T}_{uu}^{\ensuremath{-}}?}_{\mathrm{ren}}^{U}\ensuremath{\cong}{?{T}_{uu}^{\ensuremath{-}}?}_{\mathrm{ren}}^{H}={?{T}_{vv}^{\ensuremath{-}}?}_{\mathrm{ren}}^{H}$ behaves $\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}^{5}$. In contrast, ${?{T}_{vv}^{\ensuremath{-}}?}_{\mathrm{ren}}^{U}$ $\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}^{4}$, we calculate prefactor analytically. It therefore follows fluxes neighborhood spherical charged BH are dominated by ${?{T}_{vv}?}_{\mathrm{ren}}^{U}$. passing, also expression for transmission coefficient outside leading order frequencies (which turns out be crucial ingredient our analysis). Furthermore, explicitly obtain Hawking-evaporation rate ($\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}^{4}$), with (obtained here first time best knowledge).