Extended Green-Liouville asymptotics and vacuum polarization for lukewarm black holes

TL;DR

This paper introduces a new uniform approximation based on extended Green-Liouville asymptotics for the radial equation of a quantum field in lukewarm black hole spacetime, enabling precise calculation of vacuum polarization near horizons.

Contribution

It develops a novel approximation method that improves upon WKB, allowing explicit and finite vacuum polarization calculations at black hole horizons.

Findings

Explicit finite vacuum polarization values at horizons

Elimination of nonuniformity issues in previous methods

Enhanced accuracy in quantum field calculations near horizons

Abstract

We consider a quantum field on a lukewarm black hole spacetime. We introduce a new uniform approximation to the radial equation, constructed using an extension of Green-Liouville asymptotics. We then use this new approximation to construct the renormalized vacuum polarization in the Hartle-Hawking vacuum. Previous calculations of the vacuum polarization rely on the WKB approximation to the solutions of the radial equation, however the nonuniformity of the WKB approximations obscures the results of these calculations near both horizons. The use of our new approximation eliminates these obscurities, enabling us to obtain explicitly finite and easily calculable values of the vacuum polarization on the two horizons.