  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.
