Classes of exact Einstein-Maxwell solutions

TL;DR

This paper presents new exact solutions to the Einstein-Maxwell equations for charged spheres, using specific electric field and gravitational potential choices, including solutions expressible with elementary functions and the Einstein universe.

Contribution

The authors derive a general method to obtain exact Einstein-Maxwell solutions for charged spheres with specific electric fields and potentials, expanding the set of known solutions.

Findings

Solutions expressed in hypergeometric functions

Explicit elementary function solutions including polynomials

Recovery of uncharged solutions like the Einstein universe

Abstract

We find new classes of exact solutions to the Einstein-Maxwell system of equations for a charged sphere with a particular choice of the electric field intensity and one of the gravitational potentials. The condition of pressure isotropy is reduced to a linear, second order differential equation which can be solved in general. Consequently we can find exact solutions to the Einstein-Maxwell field equations corresponding to a static spherically symmetric gravitational potential in terms of hypergeometric functions. It is possible to find exact solutions which can be written explicitly in terms of elementary functions, namely polynomials and product of polynomials and algebraic functions. Uncharged solutions are regainable with our choice of electric field intensity; in particular we generate the Einstein universe for particular parameter values.