## Group theoretic approach to heat conducting gravitating systems.

##### Abstract

We study shear-free heat conducting spherically symmetric gravitating
fluids defined in four
and higher dimensional spacetimes. We analyse models that are both uncharged and charged
via the pressure isotropy condition emanating from the Einstein field equations and the
Einstein-Maxwell system respectively. Firstly, we consider the uncharged model defined in
higher dimensions, and we use the algorithm due to Deng to generate new exact solutions.
Three new metrics are identified which contain the results of four dimensions as special cases.
We show graphically that the matter variables are well behaved and the speed of sound is
causal. Secondly, we use Lie's group theoretic approach to study the condition of pressure
isotropy of a charged relativistic model in four dimensions. The Lie symmetry generators
that leave the equation invariant are found. We provide exact solutions to the gravitational
potentials using the symmetries admitted by the equation. The new exact solutions contain
earlier results without charge. We show that new charged solutions related to the Lie
symmetries, that are generalizations of conformally
at metrics, may be generated using
the algorithm of Deng. Finally, we extend our study to find models of charged gravitating
fluids defined in higher dimensional manifolds. The Lie symmetry generators related to the
generalized pressure isotropy condition are found, and exact solutions to the gravitational
potentials are generated. The new exact solutions contain earlier results obtained in four
dimensions. Using particular Lie generators, we are able to provide forms for the gravitational
potentials or reduce the order of the master equation to a first order nonlinear differential
equation. Exact expressions for the temperature pro les, from the transport equation for
both the causal and noncausal cases, in higher dimensions are obtained, generalizing previous
results. In summary, the Deng algorithm and Lie analysis prove to be useful approaches
in generating new models for gravitating
fluids.