Black holes in which the electrostatic or scalar equation is solvable in closed form

TL;DR

This paper investigates the solvability of electrostatic and scalar equations in higher-dimensional black hole spacetimes, showing limitations of existing methods and providing explicit solutions in specific cases like 6 dimensions.

Contribution

It demonstrates that the Schwarzschild method for electrostatic solutions cannot be extended to higher dimensions and constructs explicit solutions for scalar and electrostatic fields in certain higher-dimensional black holes.

Findings

Schwarzschild method does not extend to higher dimensions for electrostatics

Explicit solutions for scalar and electrostatic fields in 6D black holes

Expressions for potentials in extremal Reissner-Nordstrom black holes in higher dimensions

Abstract

We show that the method used in the Schwarzschild black hole for finding the elementary solution of the electrostatic equation in closed form cannot extend in higher dimensions. By contrast, we prove the existence of static, spherically symmetric geometries with a non-degenerated horizon in which the static scalar equation can be solved in closed form. We give the explicit results in 6 dimensions. We determine moreover the expressions of the electrostatic potential and of the static scalar field for a point source in the extremal Reissner-Nordstrom black holes in higher dimensions.