Vacuum polarization in the Schwarzschild black hole with a global monopole

TL;DR

This paper calculates the vacuum polarization of a scalar field at the horizon of a Schwarzschild black hole with a global monopole, revealing how the monopole alters quantum effects near the horizon.

Contribution

It provides the first perturbative calculation of vacuum polarization in a Schwarzschild black hole spacetime with a global monopole, including the effects of the monopole's solid-angle deficit.

Findings

Vacuum polarization splits into monopole-induced and Schwarzschild parts.

Monopole modifies the horizon radius and quantum expectation values.

Results are analogous to black holes pierced by cosmic strings.

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 $\langle \Psi^2 \rangle_{\textrm{ren}}$. 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 $\langle \Psi^2 \rangle_{\textrm{ren}}$ 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 $\eta$. Our result parallels earlier analyses for Schwarzschild black holes pierced by a cosmic string.