# Spherically symmetric solutions in relativistic astrophysics.

## Date

2002

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## Abstract

In this thesis we study classes of static spherically symmetric spacetimes admitting a perfect
fluid source, electromagnetic fields and anisotropic pressures. Our intention is to generate
exact solutions that model the interior of dense, relativistic stars. We find a sufficient
condition for the existence of series solutions to the condition of pressure isotropy for neutral
isolated spheres. The existence of a series solution is demonstrated by the method of
Frobenius. With the help of MATHEMATICA (Wolfram 1991) we recovered the Tolman
VII model for a quadratic gravitational potential, but failed to obtain other known classes
of solution. This establishes the weakness, in certain instances, of symbolic manipulation
software to extract series solutions from differential equations. For a cubic potential, we
obtained a new series solution to the Einstein field equations describing neutral stars. The
gravitational and thermodynamic variables are non-singular and continuous. This model also
satisfies the important barotropic equation of state p = p(p). Two new exact solutions to
the Einstein-Maxwell system, that generalise previous results for uncharged stars, were also
found. The first of these generalises the solution of Maharaj and Mkhwanazi (1996), and has
well-behaved matter and curvature variables. The second solution reduces to the Durgapal
and Bannerji (1983) model in the uncharged limit; this new result may only serve as a toy
model for quark stars because of negative energy densities. In both examples we observe that
the solutions may be expressed in terms of hypergeometric and elementary functions; this
indicates the possibility of unifying isolated solutions under the hypergeometric equation.
We also briefly study compact stars with spheroidal geometry, that may be charged or admit
anisotropic pressure distributions. The adapted forms of the pressure isotropy condition can
be written as a harmonic oscillator equation. Two simple examples are presented.

## Description

Thesis (M.Sc.)-University of Natal, Durban, 2002.

## Keywords

General relativity (Physics), Astrophysics., Einstein field equations--Numerical solutions., Space and time., Theses--Mathematics.