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

Abstract

We study a 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 Ω 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, which 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.

Biconformal symmetry and static Maxwell fields near higher-dimensional black holes

“…This approach is similar to the case of a scalar field [1,2]. The difference is that the required static Green function in the spacetime of a black hole is generated by the Green function of the massive scalar operator, which is defined on the H 4 × S n+1 geometry.…”

“…One should also remember the corresponding static Green functions are defined as integrals of GÔ andḠ over the Euclidean time σ with different measures. The scalar Green functionḠ(χ, γ) has been calculated in our papers [1,2]. Taking into account that…”

“…Thus one can formally express the static Green function for the Maxwell field in terms of the scalar Green function G(χ, γ) of the scalar field, which has been calculated in [1,2].…”

“…In this paper we continue studying fields created by static charges placed in the vicinity of a higher-dimensional static black hole. For this purpose we use the method of biconformal transformations, which was developed in our previous paper [1,2] in application to the case of scalar charges in the Schwarzschild-Tangherlini and the Reissner-Nordström geometries.…”

The inverse spatial Laplacian of spherically symmetric spacetimes

“…The result in Eq. (2.12) has been derived recently using the heat kernel method and bi-conformal symmetry in [40]. …”