Analysis of quantum effects inside spherical charged black holes

We numerically compute the renormalized expectation value 〈 Φ 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, 〈 Φ 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 〈 Φ 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 〈 Φ 2 〉 ren we analytically compute the leading-order asymptotic behavior of the trace 〈 ^ T μ μ 〈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.