Bi-conformal symmetry and static Green functions in the Schwarzschild-Tangherlini spacetimes

TL;DR

This paper explores a special bi-conformal symmetry in higher-dimensional black hole spacetimes, enabling explicit calculation of static Green functions for scalar fields, with exact solutions provided for dimensions less than six.

Contribution

It introduces a novel bi-conformal transformation that simplifies the metric, facilitating the derivation of explicit Green functions in higher-dimensional Schwarzschild-Tangherlini spacetimes.

Findings

Green functions are explicitly derived for D<6.

The bi-conformal transformation enhances metric symmetry.

In 4D, the Green function matches the Copson solution.

Abstract

We study static massless minimally coupled scalar field created by a source in a static D-dimensional spacetime. We demonstrate that the corresponding equation for this field is invariant under a special transformation of the background metric. This transformation consists of the static conformal transformation of the spatial part of the metric accompanied by a properly chosen transformation of the red-shift factor. Both transformations are determined by one function of the spatial coordinates. We show that in a case of higher dimensional spherically symmetric black holes one can find such a bi-conformal transformation that the symmetry of the D-dimensional metric is enhanced after its application. Namely, the metric becomes a direct sum of the metric on a unit sphere and the metric of 2D anti-de Sitter space. The method of the heat kernels is used to find the Green function in this new space, that allows one, after dimensional reduction, to obtain a static Green function in the original space of the static black hole. The general useful representation of static Green functions is obtained in the Schwarzschild-Tangherlini spacetimes of arbitrary dimension. The exact explicit expressions for the static Green functions are obtained in such metrics for D < 6. It is shown that in the four dimensional case the corresponding Green function coincides with the Copson solution.