Euclidean scalar Green functions near the black hole and black brane horizons

TL;DR

This paper analyzes scalar Green functions near black hole and black brane horizons by approximating the geometry with Rindler space and studying the dimensional reduction, providing formulas and extensions to supergravity solutions.

Contribution

It introduces approximate formulas for Green functions near horizons and extends the analysis to black branes in supergravity, highlighting the dimensional reduction effect.

Findings

Green functions near horizons can be approximated by 2D quantum field theory Green functions.

The approximation error is exponentially small away from the horizon.

Results extend to black branes with near horizon geometry N=AdS_p x S_q.

Abstract

We discuss approximations of the Riemannian geometry near the horizon. If a D+1 dimensional manifold N has a bifurcate Killing horizon then we approximate N by a product of the two-dimensional Rindler space and a D-1 dimensional Riemannian manifold M. We obtain approximate formulas for scalar Green functions. We study the behaviour of the Green functions near the horizon and their dimensional reduction. We show that if M is compact then the Green function near the horizon can be approximated by the Green function of a two-dimensional quantum field theory. The correction term is exponentially small away from the horizon. We extend the results to black brane solutions of supergravity in 10 and 11 dimensions. The near horizon geometry can be approximated by N=AdS_p x S_q. We discuss Euclidean Green functions on N and their behaviour near the horizon.