Applications of Lie symmetries to gravitating fluids.

Abstract

This thesis is concerned with the application of Lie's group theoretic method

to the Einstein field equations in order to find new exact solutions. We analyse

the nonlinear partial differential equation which arises in the study of non-

static, non-conformally flat fluid plates of embedding class one. In order to find

the group invariant solutions to the partial differential equation in a systematic and comprehensive manner we apply the method of optimal subgroups.

We demonstrate that the model admits linear barotropic equations of state in

several special cases. Secondly, we study a shear-free spherically symmetric

cosmological model with heat flow. We review and extend a method of generating solutions developed by Deng. We use the method of Lie analysis as a

systematic approach to generate new solutions to the master equation. Also,

general classes of solution are found in which there is an explicit relationship

between the gravitational potentials which is not present in earlier models.

Using our systematic approach, we can recover known solutions. Thirdly, we

study generalised shear-free spherically symmetric models with heat flow in

higher dimensions. The method of Lie generates new solutions to the master equation. We obtain an implicit solution or we can reduce the governing equation to a Riccati equation.