## Analysis of shear-free spherically symmetric charged relativistic fluids.

##### Abstract

We study the evolution of shear-free spherically symmetric charged fluids
in general relativity. This requires the analysis of the coupled Einstein-Maxwell
system of equations. Within this framework, the master field equation to be
integrated is
yxx = f(x)y2 + g(x)y3
We undertake a comprehensive study of this equation using a variety of ap-
proaches. Initially, we find a first integral using elementary techniques (subject
to integrability conditions on the arbitrary functions f(x) and g(x)). As a re-
sult, we are able to generate a class of new solutions containing, as special
cases, the models of Maharaj et al (1996), Stephani (1983) and Srivastava
(1987). The integrability conditions on f(x) and g(x) are investigated in detail
for the purposes of reduction to quadratures in terms of elliptic integrals. We
also obtain a Noether first integral by performing a Noether symmetry analy-
sis of the master field equation. This provides a partial group theoretic basis
for the first integral found earlier. In addition, a comprehensive Lie symmetry
analysis is performed on the field equation. Here we show that the first integral
approach (and hence the Noether approach) is limited { more general results
are possible when the full Lie theory is used. We transform the field equation
to an autonomous equation and investigate the conditions for it to be reduced
to quadrature. For each case we recover particular results that were found pre-
viously for neutral fluids. Finally we show (for the first time) that the pivotal
equation, governing the existence of a Lie symmetry, is actually a fifth order
purely differential equation, the solution of which generates solutions to the
master field equation.