Vacuum polarization in the Schwarzschild black hole with a global monopole

Abstract

We investigate vacuum polarization on the event horizon of a Schwarzschild black hole carrying a global monopole. For a massless scalar field $\Psi$ in the Hartle-Hawking state and with arbitrary curvature coupling, we compute the renormalized vacuum expectation value $⟨{\Psi }^2{⟩}_{\text{ren}}$. The monopole produces a solid-angle deficit and makes the spacetime non-Ricci-flat. Working perturbatively in the monopole parameter $\eta$ and retaining terms through $O({\eta }^2)$, we find that $⟨{\Psi }^2{⟩}_{\text{ren}}$ 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 $\eta$. We also investigate whether an analogous decomposition holds for $⟨T_{\mu }^{\mu }{⟩}_{\text{ren}}$ when it is determined by this method. Our result parallels earlier analyses of Schwarzschild black holes pierced by a cosmic string.

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