Vacuum Polarization of Massive Scalar Fields on the Black Hole Horizon

TL;DR

This paper analytically investigates the vacuum polarization of massive scalar fields near a Reissner-Nordström black hole horizon, revealing regular behavior at the Hawking temperature and agreement with DeWitt-Schwinger approximation.

Contribution

It derives an approximate analytic expression for vacuum polarization near the horizon in the large mass limit using WKB and series expansion methods.

Findings

Vacuum polarization is regular at the Hawking temperature.

The finite part matches DeWitt-Schwinger results up to order $O(m^{-4})$.

Provides a new analytic approximation near the black hole horizon.

Abstract

Vacuum polarization of massive scalar fields in a thermal state at arbitrary temperature is studied near the horizon of a Reissner-Nordstr\"{o}m black hole. We derived an analytic form of $<\phi^2>$ approximately in the large mass limit near the black hole horizon. We uses the zeroth order WKB approximation and power series expansion near the horizon for the Euclideanized mode function. Our formula for the vacuum polarization shows regular behavior on the horizon if the temperature of the scalar field is equal to the Hawking temperature of the black hole. The finite part of the vacuum polarization agrees with the result of the DeWitt-Schwinger approximation up to $O(m^{-4})$ which is the next leading order of the expansion.