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

Abstract

We discuss approximations of the Riemannian geometry near the horizon. If a (D + 1)-dimensional manifold has a bifurcate Killing horizon then we approximate by a product of the two-dimensional Rindler space and a (D − 1)-dimensional Riemannian manifold . We obtain approximate formulae for scalar Green's functions. We study the behavior of the Green's functions near the horizon and their dimensional reduction. We show that if is compact then the Green's function near the horizon can be approximated by the Green's function of the 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 . We discuss the Euclidean Green's functions on and their behavior near the horizon.