Black Hole Quantum Vacuum Polarization in Higher Dimensions

TL;DR

This paper extends vacuum polarization calculations in black hole spacetimes to higher dimensions, specifically five and six dimensions, using regularization and numerical methods to analyze quantum effects near black holes.

Contribution

It adapts and applies existing methods for vacuum polarization to higher-dimensional black holes, providing explicit calculations and numerical analysis for five and six dimensions.

Findings

Divergences are successfully canceled using counter-terms.

Vacuum polarization is shown to be regular in five and six dimensions.

Numerical results confirm the effectiveness of the method.

Abstract

The goal of this paper is to extend to higher dimensionality the methods and computations of vacuum polarization effects in black hole spacetimes. We focus our attention on the case of five dimensional Schwarzschild-Tangherlini black holes, for which we adapt the general method initially developed by Candelas and later refined by Anderson and others. We make use of point splitting regularization and of the WKB approximation to extract the divergences occuring in the coincidence limit of the Green function and, after calculating the counter-terms using the Schwinger - De Witt expansion, we explicitly prove the cancellation of the divergences and the regularity of the vacuum polarization once counter-terms are added up. We finally handle numerically the renormalized expression of the vacuum polarization. As a check on the method we also prove the regularity of the vacuum polarization in the six dimensional case in the large mass limit.