Vacuum polarization of scalar fields near Reissner-Nordström black holes and the resonance behavior in field-mass dependence

Abstract

We study vacuum polarization of quantized massive scalar fields $ϕ$ in equilibrium at black-hole temperature in Reissner-Nordström background. By means of the Euclidean space Green&#39;s function we analytically derive the renormalized expression $<ϕ^{2}>_{H}$ at the event horizon with the area $4πr_{+}^{2}$. It is confirmed that the polarization amplitude $<ϕ^{2}>_{H}$ is free from any divergence due to the infinite red-shift effect. Our main purpose is to clarify the dependence of $<ϕ^{2}>_{H}$ on field mass $m$ in relation to the excitation mechanism. It is shown for small-mass fields with $mr_{+}\ll1$ how the excitation of $<ϕ^{2}>_{H}$ caused by finite black-hole temperature is suppressed as $m$ increases, and it is verified for very massive fields with $mr_{+}\gg1$ that $<ϕ^{2}>_{H}$ decreases in proportion to $m^{-2}$ with the amplitude equal to the DeWitt-Schwinger approximation. In particular, we find a resonance behavior with a peak amplitude at $mr_{+}\simeq 0.38$ in the field-mass dependence of vacuum polarization around nearly extreme (low-temperature) black holes. The difference between Scwarzschild and nearly extreme black holes is discussed in terms of the mass spectrum of quantum fields dominant near the event horizon.