Vacuum polarization in asymptotically Lifshitz black holes

TL;DR

This paper develops a method to compute quantum polarization effects in Lifshitz black hole spacetimes, extending previous anti-de Sitter analyses to include anisotropic scaling and providing general formulas applicable to various solutions.

Contribution

It introduces a generalized analytical framework for calculating the Green's function coincidence limit in Lifshitz black holes, adapting existing methods for asymptotically AdS spaces.

Findings

Derived general formulas for Lifshitz black holes

Extended previous AdS results to Lifshitz spacetimes

Performed numerical analysis on specific solutions

Abstract

There has been considerable interest in applying the gauge/gravity duality to condensed matter theories with particular attention being devoted to gravity duals (Lifshitz spacetimes) of theories that exhibit anisotropic scaling. In this context, black hole solutions with Lifshitz asymptotics have also been constructed aiming at incorporating finite temperature effects. The goal here is to look at quantum polarization effects in these spacetimes, and to this aim, we develop a way to compute the coincidence limit of the Green's function for massive, non-minimally coupled scalar fields, adapting to the present situation the analysis developed for the case of asymptotically anti de Sitter black holes. The basics are similar to previous calculations, however in the Lifshitz case one needs to extend previous results to include a more general form for the metric and dependence on the dynamical exponent. All formulae are shown to reduce to the AdS case studied before once the value of the dynamical exponent is set to unity and the metric functions are accordingly chosen. The analytical results we present are general and can be applied to a variety of cases, in fact, to all spherically symmetric Lifshitz black hole solutions. We also implement the numerical analysis choosing some known Lifshitz black hole solutions as illustration.