Vacuum energy density near static distorted black holes

We investigate the contribution of massless fields of spins 0, 1/2, and 1 to the vacuum polarization near the event horizon of static Ricci-flat space-times. We do not assume any particular spatial symmetry. Within the Page-Brown ‘‘ansatz’’ we calculate 〈${\phi }^2$ ${〉}^{\mathrm{ren}}$ and 〈$T_{\mu \nu }$ ${〉}^{\mathrm{ren}}$ near static distorted black holes, for both the Hartle-Hawking (‖${〉}_H$) and Boulware (‖${〉}_B$) vacua. Using Israel’s description of static space-times, we express these quantities in an invariant geometric way. We obtain that 〈${\phi }^2$ ${〉}_H^{\mathrm{ren}}$ and 〈$T_{\mu \nu }$ ${〉}_H^{\mathrm{ren}}$ near the horizon depend only on the two-dimensional geometry of the horizon surface. We find 〈${\phi }^2$ ${〉}_H^{\mathrm{ren}}$=(1/48${\pi }^2$ )$K_0$, 〈$T_0^0$ ${〉}_H^{\mathrm{ren}}$=(7α+12β )$K_0$ ${\mathrm{}}^2$-${\alpha }^{\mathrm{}\left(2\right)\mathrm{}}$Δ$K_0$. $K sub 0— is the Gaussian curvature of the horizon, and α and β are numerical coefficients depending on the spin of a field. The term in ${}_{}{}^{\mathrm{}\left(2\right)\mathrm{}}{}_{}{}^{}\mathrm{K}_0^{}{}_{}{}^{}$ is characteristic of the distortion of the black hole. When the event horizon is not distorted, $K_0$ is a constant and this term disappears.