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.
Highlights
In this paper we study the following problem: suppose a static charged particle, supported by some external force, is at rest close to a static black hole
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
If the spacetime has D dimensions, such a (D − 1)-dimensional static Green function can be obtained by the dimensional reduction from a Green function in D dimensions
Summary
2.2 Bi-conformal transformation of higher-dimensional static spherically symmetric spacetimes and symmetry enhancement. It is easy to check that the equation (1.2) preserves its form under this transformation provided kα = −(D − 3) , kJ = 2 This bi-conformal invariance of the static field equation was used earlier for calculations of a self-energy and a self-force of charges in various gravitational backgrounds [7, 11, 14,15,16,17]. This symmetry enables one to relate static Green functions for the fields in different spacetimes, for example, in the geometry of a set of extremely charged black holes (the Majumdar-Papapetrou spacetime) and in the flat Minkowski spacetime. The method does work for a case of higher-dimensional charged black holes as well, but the required calculations are more complicated and we discuss this case in another paper
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