  In this thesis three separate problems relevant to general relativity are
considered. Methods for algorithmically producing all the solutions of
isotropic fluid spheres have been developed over the last five years. A
different and somewhat simpler algorithm is discussed here, as well as
algorithms for anisotropic fluid spheres. The second and third problems are
somewhat more speculative in nature and address the nature of black hole
entropy. Specifically, the second problem looks at the genericity of the
so-called quasinormal mode conjecture introduced by Hod, while the third
problem looks at the near-horizon structure of a black hole in hope of gaining
an understanding of why so many different approaches yield the same entropy. A
method of finding the asymptotic QNM structure is found based on the Born
series, and serious problems for the QNM conjecture are discussed. The work in
this thesis does not completely discount the possibility that the QNM
conjecture is true. New results released weeks before this thesis was finished
showed that the QNM conjecture was flawed. Finally, the near-horizon structure
of a black hole is found to be very restricted, adding credence to the ideas
put forward by Carlip and Solodukhin that the black hole entropy is related to
an inherited symmetry from the classical theory.
