Green functions and Euclidean fields near the bifurcate Killing horizon

TL;DR

This paper studies Green functions near bifurcate Killing horizons by approximating the manifold with a product space, revealing conditions under which the Green function simplifies to a two-dimensional form and analyzing the behavior of quantum fields.

Contribution

It provides approximate formulas for Green functions near horizons and identifies when dimensional reduction to two dimensions is valid, especially for compact manifolds.

Findings

Green functions near the horizon can be approximated by two-dimensional Green functions for compact manifolds.

A zero mode exists in the massless field, acting as a conformal invariant on R^2.

The approximation breaks down if the volume of geodesic balls grows unboundedly.

Abstract

We approximate a Euclidean version of a D+1 dimensional manifold with a bifurcate Killing horizon by a product of a two-dimensional Rindler space and a D-1 dimensional manifold M. We obtain approximate formulas for the Green functions. We study the behaviour of Green functions near the horizon and their dimensional reduction. We show that if M is compact then the massless minimally coupled quantum field contains a zero mode which is a conformal invariant free field on R^2. Then, the Green function near the horizon can be approximated by the Green function of the two-dimensional quantum field theory. The correction term is exponentially small away from the horizon. If the volume of a geodesic ball is growing to infinity with its radius then the Green function cannot be approximated by a two-dimensional one.