Analysis of quantum effects inside spherical charged black holes

We numerically compute the renormalized expectation value <(Phi) over cap (2)>(ren) of a minimally coupled massless quantum scalar field in the interior of a four-dimensional Reissner-Nordstrom black hole, in both the Hartle-Hawking and Unruh states. To this end we use a recently developed mode-sum renormalization scheme based on covariant point splitting. In both quantum states, <(Phi) over cap (2)>(ren) is found to approach a finite value at the inner horizon (IH). The final approach to the IH asymptotic value is marked by an inverse-power tail r(*)(-n), where r(*) is the Regge-Wheeler "tortoise coordinate" and with n = 2 for the Hartle-Hawking state and n = 3 for the Unruh state. We also report here the results of an analytical computation of these inverse-power tails of <(Phi) over cap (2)>(ren) near the IH. Our numerical results show very good agreement with this analytical derivation ( for both the power index and the tail amplitude), in both quantum states. Finally, from this asymptotic behavior of <(Phi) over cap (2)>(ren)we analytically compute the leading-order asymptotic behavior of the trace <(T) over cap (mu)(mu)>(ren) of the renormalized stress-energy tensor at the IH. In both quantum states this quantity is found to diverge like b(r - r(-))(-1) r(*)(-n-2) (with n specified above and with a known parameter b). To the best of our knowledge, this is the first fully quantitative derivation of the asymptotic behavior of these renormalized quantities at the IH of a four-dimensional Reissner-Nordstrom black hole. In particular, this is the first conclusive result showing the divergence of the renormalized stress-energy tensor at the Cauchy horizon.