Chaisi, Mosa.

### Abstract:

In this thesis we seek new solutions to the anisotropic Einstein field equations which are important in the study of highly dense stellar structures. We first adopt the approach used by Maharaj & Maartens (1989) to obtain an exact anisotropic solution in terms of elementary functions for a particular choice of the energy density. This class of solution contains the Maharaj & Maartens (1989) and Gokhroo & Mehra (1994) models as special cases. In addition, we obtain six other new solutions following the same approach for different choices of the energy density. All the solutions in this section reduce to one with the energy density profile f-L ex r-2 . Two new algorithms are generated, Algorithm A and Algorithm B, which produce a new anisotropic solution to the Einstein field equations from a given isotropic solution. For any new anisotropic solution generated with the help of these algorithms, the original isotropic seed solution is regained as a special case. Two examples of known isotropic solutions are used to demonstrate how Algorithm A and Algorithm B work, and to obtain new anisotropic solutions for the Einstein and de Sitter models. Anisotropic isot~ermal sphere models are generated given the corresponding isotropic (f-L ex r-2 ) solution of the Einstein field equations. Also, anisotropic interior Schwarzschild sphere models are found given the corresponding isotropic (f-L ex constant) solution of the field equations. The exact solutions and line elements are given in each case for both Algorithm A and Algorithm B. Note that the solutions have a simple form and are all expressible in terms of elementary functions. Plots for the anisotropic factor S = J3(Pr - pJJ/2 (where Pr and Pl. are radial and tangential pressure respectively) are generated and these point to physically viable models.