Vacuum polarization in the Schwarzschild black hole with a global monopole
We investigate vacuum polarization on the event horizon of a Schwarzschild black hole carrying a global monopole. For a massless scalar field $Ψ$ in the Hartle-Hawking state and with arbitrary curvature coupling, we compute the renormalized vacuum expectation value $\langle Ψ^2 \rangle_{\textrm{ren}}$. The monopole produces a solid-angle deficit and makes the spacetime non-Ricci-flat. Working perturbatively in the monopole parameter $η$ and retaining terms through $O(η^2)$, we find that $\langle Ψ^2 \rangle_{\textrm{ren}}$ on the horizon splits into two contributions: a genuinely monopole-induced term evaluated at the horizon and the usual Schwarzschild result--with the event horizon radius modified by the presence of $η$. We also investigate whether an analogous decomposition holds for $\langle T^μ_{\phantomμμ}\rangle_\textrm{ren}$ when it is determined by this method. Our result parallels earlier analyses of Schwarzschild black holes pierced by a cosmic string.