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

TL;DR

This paper derives an analytic expression for vacuum polarization at the horizon of a Schwarzschild black hole with a cosmic string, using a novel generalized Heine identity for Legendre functions.

Contribution

It introduces a new generalized Heine identity for Legendre functions and applies it to compute vacuum polarization analytically in a black hole spacetime with a cosmic string.

Findings

Analytic expression for vacuum polarization on the black hole horizon

Derivation of a generalized Heine identity for Legendre functions

Application to Schwarzschild black hole threaded by a cosmic string

Abstract

We obtain an analytic expression for the vacuum polarization on the horizon of the Schwarzschild black hole threaded by an infinite cosmic string. This calculation relies on a generalized Heine identity for Legendre functions which we derive without using specific properties of the Legendre functions themselves.