CMC surfaces and area-charge inequality for a spheroidal ECD spacetime

TL;DR

This paper investigates the relationship between area and charge in a spheroidal spacetime with electrically counterpoised dust, identifying candidate surfaces and analyzing inequalities through numerical methods.

Contribution

It numerically determines stable constant mean curvature surfaces in a spheroidal spacetime and analyzes the area-charge inequality, revealing it is not saturated and identifying a cylindrical limit.

Findings

The area-charge inequality is far from saturation.

A cylindrical limit attains the minimal area-charge relation.

Numerical analysis of stable isoperimetric surfaces was performed.

Abstract

We consider the spacetime presented by Bonnor \cite{Bonnor98}, whose matter content is a spheroid of electrically counterpoised dust, in the context of the geometrical inequalities between area and charge. We determine numerically the constant mean curvature surfaces that are candidates to be stable isoperimetric surfaces and analyze the relation between area and charge for them, showing that a previously proved inequality is far from being saturated. We also show that the maximal initial data has a cylindrical limit where the minimum of the area-charge relation is attained.