Solution for the Einstein-Maxwell equations invariant under an $(n - 1)$-dimensional group of dilations

TL;DR

This paper characterizes solutions to the Einstein-Maxwell equations with an $(n-1)$-dimensional dilation symmetry, revealing two main classes depending on the cosmological constant, and introduces a new invariant solution within the Majumdar-Papapetrou class.

Contribution

It provides a complete classification of electrostatic solutions invariant under an $(n-1)$-dimensional dilation group, including a new solution in the Majumdar-Papetrou class.

Findings

Solutions are either rotation/translation invariant if the cosmological constant is nonzero.

If the cosmological constant is zero, solutions belong to the Majumdar-Papetrou class.

A new invariant solution under an $(n-1)$-dimensional dilation group is introduced.

Abstract

We consider an electrostatic system whose spatial factor is conformal to an $n$-dimensional Euclidean space. We provide a complete characterization of the most general ansatz, thereby reducing the associated electrostatic system of partial differential equations to an ordinary differential equation system. We prove that there are only two possibilities: either the cosmological constant is nonzero, in which case the solutions are necessarily invariant under rotations or translations, or the cosmological constant vanishes, and the solutions belong to the Majumdar-Papapetrou class with a degree of freedom associated with an invariant $(n-1)$-dimensional subgroup. As a result, we introduce a new solution to the electrovacuum system in the Majumdar-Papapetrou class that is invariant under an $(n-1)$-dimensional group of dilations.