Exact solutions for charged spheres and their stability. I. Perfect Fluids

TL;DR

This paper derives exact solutions for charged, static, spherically symmetric perfect fluid spheres in general relativity, analyzing their mass, charge distribution, and stability criteria.

Contribution

It presents new exact solutions for charged perfect fluid spheres with specific density and charge profiles, and evaluates their stability limits in relation to known formulas.

Findings

Derived explicit solutions for charged perfect fluid spheres.

Calculated critical mass-radius ratios as functions of charge.

Compared stability bounds with Andréasson's theoretical limits.

Abstract

We study exact solutions of the Einstein-Maxwell equations for the interior gravitational field of static spherically symmetric charged compact spheres. The spheres are composed of a perfect fluid with a charge distribution that creates a static radial electric field. The inertial mass density of the fluid has the form $\rho (r) = \rho_o + \alpha r^2$ ($\rho_o$ and $\alpha$ are constants) and the total charge $q(r)$ within a sphere of radius $r$ has the form $q = \beta r^3$ ($\beta$ is a constant). We evaluate the critical values of $M/R$ for these spheres as a function of $Q/R$ and compare these values with those given by the Andr\'{e}asson formula.