A class of exact solutions of Einstein's field equations in higher dimensional spacetimes, d${\bm\geq 4}$: Majumdar-Papapetrou solutions

TL;DR

This paper extends Majumdar-Papapetrou solutions, which describe equilibrium charged matter configurations, to higher-dimensional spacetimes (d ≥ 4), revealing their independence from matter distribution shape and properties of the spatial sections.

Contribution

The work generalizes Einstein-Maxwell solutions of Majumdar-Papapetrou type to all higher dimensions, including axisymmetric cases with electric fields and charged matter.

Findings

Equilibrium solutions exist for all d ≥ 4.

Shape of matter distribution does not affect equilibrium.

Spatial sections are conformal to Ricci-flat spaces.

Abstract

The Newtonian theory of gravitation and electrostatics admit equilibrium configurations of charged fluids where the charge density can be equal to the mass density, in appropriate units. The general relativistic analog for charged dust stars was discovered by Majumdar and by Papapetrou. In the present work we consider Einstein-Maxwell solutions in d-dimensional spacetimes and show that there are Majumdar-Papapetrou type solutions for all ${\rm d} \geq 4$. It is verified that the equilibrium is independent of the shape of the distribution of the charged matter. It is also showed that for perfect fluid solutions satisfying the Majumdar-Papapetrou condition with a boundary where the pressure is zero, the pressure vanishes everywhere, and that the $({\rm d}-1)$-dimensional spatial section of the spacetime is conformal to a Ricci-flat space. The Weyl d-dimensional axisymmetric solutions are generalized to include electric field and charged matter.