## On Stephani universes.

##### Abstract

In this dissertation we study conformal symmetries in the Stephani universe which is
a generalisation of the Robertson-Walker models. The kinematics and dynamics of
the Stephani universe are discussed. The conformal Killing vector equation for the
Stephani metric is integrated to obtain the general solution subject to integrability
conditions that restrict the metric functions. Explicit forms are obtained for the
conformal Killing vector as well as the conformal factor . There are three categories
of solution. The solution may be categorized in terms of the metric functions k and
R. As the case kR - kR = 0 is the most complicated, we provide all the details of
the integration procedure. We write the solution in compact vector notation. As the
case k = 0 is simple, we only state the solution without any details. In this case
we exhibit a conformal Killing vector normal to hypersurfaces t = constant which is
an analogue of a vector in the k = 0 Robertson-Walker spacetimes. The above two
cases contain the conformal Killing vectors of Robertson-Walker spacetimes. For
the last case in - kR = 0, k =I 0 we provide an outline of the integration process.
This case gives conformal Killing vectors which do not reduce to those of RobertsonWalker
spacetimes. A number of the calculations performed in finding the solution
of the conformal Killing vector equation are extremely difficult to analyse by hand.
We therefore utilise the symbolic manipulation capabilities of Mathematica (Ver 2.0)
(Wolfram 1991) to assist with calculations.