Mkhwanazi, Wiseman Thokozani.

### Abstract:

In this thesis we study spherically symmetric spacetimes which are static with a
perfect fluid source. The Einstein field equations, in a number of equivalent forms,
are derived in detail. The physical properties of a relativistic star are briefly reviewed.
We specify two particular choices for one of the gravitational potentials.
The behaviour of the remaining gravitational potential is governed by a second order
differential equation. This equation has solutions in terms of elementary functions
for some cases. The differential equation, in other cases, may be expressed as Bessel,
confluent hypergeometric and hypergeometric equations. In such instances the solution
is given in terms of special functions. A number of solutions to the Einstein
field equations are generated. We believe that these solutions may be used to model
realistic stars. Many of the solutions found are new and have not been published
previously. In some cases our solutions are generalisations of cases considered previously.
For some choices of the gravitational potential our solutions are equivalent to
well-known results documented in the literature; in these cases we explicitly relate
our solutions to those published previously. We have utilised the computer package
MATHEMATICA Version 2.0 (Wolfram 1991) to assist with calculations, and to
produce figures to describe the gravitational field. In addition, we briefly investigate
the approach of specifying an equation of state relating the energy density and the
pressure. The solution of the Einstein field equations, for a linear equation of state,
is reduced to integrating Abel's equation of the second kind.