Analysis of quantum effects inside spherical charged black holes

We numerically compute the renormalized expectation value $⟨{\widehat{\mathrm{\Phi }}}^2{⟩}_{\mathrm{ren}}$ 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, $⟨{\widehat{\mathrm{\Phi }}}^2{⟩}_{\mathrm{ren}}$ 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 $⟨{\widehat{\mathrm{\Phi }}}^2{⟩}_{\mathrm{ren}}$ 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 $⟨{\widehat{\mathrm{\Phi }}}^2{⟩}_{\mathrm{ren}}$ we analytically compute the leading-order asymptotic behavior of the trace $⟨{\widehat{T}}_{\mu }^{\mu }{⟩}_{\mathrm{ren}}$ 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.

Figure 1

The numerically computed $⟨{\widehat{\mathrm{\Phi }}}^2(r){⟩}_{\mathrm{ren}}$ in the HH (red) and Unruh (blue) states in the region between the two horizons. The short horizontal green line represents the analytical result for $⟨{\widehat{\mathrm{\Phi }}}^2(r){⟩}_{\mathrm{ren}}$ in the HH state at the EH.

Figure 2

$⟨{\widehat{\mathrm{\Phi }}}^2(r){⟩}_{\mathrm{ren}}$ in the HH (red) and Unruh (blue) states, as a function of $z$.

Figure 3

$\mathrm{\Delta }(z)·z^n$ in the HH (red) and Unruh (blue) states, in the region $-845<z<0$ (which roughly corresponds to ${10}^{-367}<\delta r<1$). The results for the Unruh state are divided here by a factor of $-150$, for convenience. The plateaus at the left half of the $z$ axis indicate the inverse-power behavior $\mathrm{\Delta }\propto z^{-n}$. For both the HH and Unruh results, the crosses indicate full numerical results, the solid curves indicate semiasymptotic results, and the dots indicate the “refined variant” results.

Figure 4

$\mathrm{\Delta }(z)$ in the HH state (red dots) and Unruh state (blue dots) exceedingly close to $r_{-}$ (up to $z\sim 1500$, which roughly corresponds to $\delta r\sim {10}^{-650}$), computed using the refined variant. The black curves are the analytical expressions (10) for the inverse-power tails. The green curve indicates the leading-order analytical result ${\mathrm{\Delta }}_H\approx C_H{z}^{-2}$ (whereas the corresponding black curve also includes the next-order term $C_H^1{z}^{-3}$).