Abstract:
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.
