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

Abstract

In an ongoing effort to explore quantum effects on the interior geometry of black holes, we explicitly compute the semiclassical flux components ${⟨T_{uu}⟩}_{ren}$ and ${⟨T_{vv}⟩}_{ren}$ ($u$ and $v$ being the standard Eddington coordinates) of the renormalized stress-energy tensor for a minimally-coupled massless quantum scalar field, in the vicinity of the inner horizon (IH) of a Reissner-Nordström black hole. These two flux components seem to dominate the effect of backreaction in the IH vicinity; and furthermore, their regularization procedure reveals remarkable simplicity. We consider the Hartle-Hawking and Unruh quantum states, the latter corresponding to an evaporating black hole. In both quantum states, we compute ${⟨T_{uu}⟩}_{ren}$ and ${⟨T_{vv}⟩}_{ren}$ in the IH vicinity for a wide range of $Q/M$ values. We find that both ${⟨T_{uu}⟩}_{ren}$ and ${⟨T_{vv}⟩}_{ren}$ attain finite asymptotic values at the IH. Depending on $Q/M$, these asymptotic values are found to be either positive or negative (or vanishing in-between). Note that having a nonvanishing ${⟨T_{vv}⟩}_{ren}$ at the IH implies the formation of a curvature singularity on its ingoing section, the Cauchy horizon. Motivated by these findings, we also take initial steps in the exploration of the backreaction effect of these semiclassical fluxes on the near-IH geometry.

The ancillary files include the Supplemental Materials and a Mathematica (.nb) file containing our results. v2: footnotes fixed. v4: the final version, as published in PRL. The main changes are in the Supplemental Materials. v5: changes in the Supplemental Materials (finite counterterm $e_{yy}$ fixed in section 1, plots fixed accordingly in section 6)