  In this thesis four separate problems in general relativity are considered,
divided into two separate themes: coordinate conditions and perfect fluid
spheres. Regarding coordinate conditions we present a pedagogical discussion of
how the appropriate use of coordinate conditions can lead to simplifications in
the form of the spacetime curvature -- such tricks are often helpful when
seeking specific exact solutions of the Einstein equations. Regarding perfect
fluid spheres we present several methods of transforming any given perfect
fluid sphere into a possibly new perfect fluid sphere. This is done in three
qualitatively distinct manners: The first set of solution generating theorems
apply in Schwarzschild curvature coordinates, and are phrased in terms of the
metric components: they show how to transform one static spherical perfect
fluid spacetime geometry into another. A second set of solution generating
theorems extends these ideas to other coordinate systems (such as isotropic,
Gaussian polar, Buchdahl, Synge, and exponential coordinates), again working
directly in terms of the metric components. Finally, the solution generating
theorems are rephrased in terms of the TOV equation and density and pressure
profiles. Most of the relevant calculations are carried out analytically,
though some numerical explorations are also carried out.
