< phi^2 > for a scalar field in 2D black holes: a new uniform approximation

TL;DR

This paper introduces a new uniform approximation method to accurately compute the renormalized <phi^2> for nonconformal scalar fields near 2D black hole horizons, improving upon the traditional WKB approximation.

Contribution

A novel uniform approximation technique is developed to address WKB limitations for nonconformal fields in 2D black hole spacetimes.

Findings

The new approximation accurately matches numerical results.

WKB approximation fails at the horizon for nonconformal fields.

The method provides improved analytical expressions for <phi^2>.

Abstract

We study nonconformal quantum scalar fields and averages of their local observables (such as <phi^2>^{ren} and T_{ab}^{ren}) in a spacetime of a 2-dimensional black hole. In order to get an analytical approximation for these expressions the WKB approximation is often used. We demonstrate that at the horizon WKB approximation is violated for a nonconformal field, that is when the field mass or/and the parameter of non-minimal coupling do not vanish. We propose a new "uniform approximation" which solves this problem. We use this approximation to obtain an improved analytical approximation for <phi^2>^{ren} in the 2-dimensional black hole geometry. We compare the obtained results with numerical calculations.