Vacuum polarization in the Schwarzschild spacetime and dimensional reduction

TL;DR

This paper investigates the quantum effects of a massless scalar field in Schwarzschild spacetime by reducing the problem to two dimensions, quantizing the field, and analyzing the energy-momentum tensor in different quantum states, especially near the horizon.

Contribution

It provides a detailed analysis of the renormalized energy-momentum tensor in various quantum states using dimensional reduction and addresses the breakdown of WKB approximation near the horizon.

Findings

Explicit asymptotic behaviors of < T_{a b} > in different states

Analytical approximations for energy-momentum tensor

Insights into quantum effects near the black-hole horizon

Abstract

A massless scalar field minimally coupled to gravity and propagating in the Schwarzschild spacetime is considered. After dimensional reduction under spherical symmetry the resulting 2D field theory is canonically quantized and the renormalized expectation values $< T_{a b} >$ of the relevant energy-momentum tensor operator are investigated. Asymptotic behaviours and analytical approximations are given for $< T_{a b} >$ in the Boulware, Unruh and Hartle-Hawking states. Special attention is devoted to the black-hole horizon region where the WKB approximation breaks down.