Vacuum polarization in the Schwarzschild black hole with a global monopole

TL;DR

This work analyzes vacuum polarization on the event horizon of a Schwarzschild black hole endowed with a global monopole. Using a Green’s-function approach in the Hartle–Hawking state and perturbing in the monopole parameter $\eta$ to $O(\eta^2)$, the authors derive a near-horizon expression for the renormalized field fluctuation $\langle\Psi^2\rangle_{ren}$ that decomposes into a genuinely monopole-induced piece and a Schwarzschild contribution with an $\eta$-modified horizon. In the limit $\eta\to0$, the result reduces to the known Schwarzschild value, validating the decomposition and the horizon’s role in fixing the physical branch of the solution. The analysis parallels similar results for Schwarzschild black holes threaded by cosmic strings and lays groundwork toward the full renormalized stress-energy tensor in this spacetime. Overall, the deficit-angle topology and horizon geometry together modulate quantum fluctuations in a controlled, perturbative framework relevant for quantum fields in curved spacetime.

Abstract

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 $η$. Our result parallels earlier analyses for Schwarzschild black holes pierced by a cosmic string.