Renormalized Vacuum Polarization and Stress Tensor on the Horizon of a Schwarzschild Black Hole Threaded by a Cosmic String

TL;DR

This paper computes the renormalized vacuum polarization and stress tensor for a scalar field near a Schwarzschild black hole with a cosmic string, using a novel mathematical identity for Legendre functions.

Contribution

It introduces a generalized Heine identity for non-integer Legendre functions to facilitate calculations in curved spacetime with cosmic strings.

Findings

Derived a new identity for Legendre functions applicable to black hole spacetimes.

Calculated vacuum polarization and stress tensor on the horizon of a black hole threaded by a cosmic string.

Provides insights into quantum field effects in black hole geometries with topological defects.

Abstract

We calculate the renormalized vacuum polarization and stress tensor for a massless, arbitrarily coupled scalar field in the Hartle-Hawking vacuum state on the horizon of a Schwarzschild black hole threaded by an infinte straight cosmic string. This calculation relies on a generalized Heine identity for non-integer Legendre functions which we derive without using specific properties of the Legendre functions themselves.