Massive quantum divergence on the Cauchy horizon of a black hole

Abstract

We investigate a quantum massive scalar field in the interior of a charged and spherically-symmetric (Reissner-Nordstr\"om) black hole. We examine the behaviour for varying values of the black hole charge, field mass and coupling constant when the field is in two quantum states: Hartle-Hawking (representing a black hole in thermal equilibrium) and Unruh (representing a black hole evaporating via the emission of Hawking radiation). We show that the vacuum polarization as well as the angular components of the quantum stress-energy tensor diverge on the Cauchy horizon, in stark contrast to what happens for massless fields. We also calculate the energy fluxes in Eddington-Finkelstein coordinates $\{u,v\}$. We show that these fluxes on the Cauchy horizon do not generically vanish. This implies, in particular, that in regular, Kruskal coordinates $\{U,V\}$ the ingoing flux diverges like $V^{-2}$ on the Cauchy horizon (where $V=0$). This divergence suggests that its backreaction via the semiclassical Einstein equations would yield a strong singularity, as opposed to its weaker, classical counterpart. Interestingly, there are exceptions, in which the energy fluxes in Eddington-Finkelstein coordinates vanish: (i) in the extremal limit (where the black hole is maximally charged); (ii) certain fine-tuned regions of parameter space, where the fluxes change sign.