Much ado about nothing: a treatise on empty and not-so-empty spacetimes

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.