The primary goal of the proposed research is to improve the understanding of the decay for solutions to the wave equation on black hole backgrounds. Kerr spacetimes and perturbations of such are of particular interest, due to aspirations to contribute to a proof of the stability of this family of solutions to Einstein's equations. Recent work has focused on proving localized energy estimates and Strichartz estimates, which are both measures of the dispersive nature of the wave equation which are known to be fairly robust. The former have played a key role in Tataru's recent proof of the long conjectured Price's law, which asserts a certain decay rate for solutions to the wave equation on the Schwarzschild and Kerr backgrounds. A key feature of these blackhole spacetimes which demands extra attention is the existence of trapped rays. In flat space, light travels on lines which escape to infinity. Thus packets of a solution which are initially overlapping but traveling in even slightly different directions quickly spread out, and this spreading promotes decay. On Schwarzschild, the paths on which light travels are dictated by the geometry, and in particular, there is a region, called the photon sphere, where photons can orbit the blackhole. Such trapping, where the rays remain in a compact set, is a known obstacle to certain dispersive estimates, such as localized energy estimates. Trapping also occurs on the Kerr family of spacetimes, though its geometry is more complicated.
General relativity asserts that the universe is a (1+3) dimensional curved space and that gravity corresponds to the curvature of this space. A common description is to think of the universe as a trampoline. A mass, such as a bowling ball, which is placed on the trampoline causes it to curve in such a way as to attract other objects on the surface. Einstein's equations model the curvature and evolution of such curvature of universes. Though Einstein's equations are quite nonlinear, a few special solutions are known. These are typically found by imposing many symmetries to simplify the equations. Of particular relevance to this proposal are the Minkowski space time, the family of Schwarzschild space times, and the Kerr family of space times. These correspond to the flat solution, to spherically symmetric black holes, and to rotating black holes respectively. A natural question to ask is whether these solutions are stable. That is, if one starts close to, say, a member of the Kerr family of space times, will it necessarily remain close to a member of the Kerr family. The only rigorous proofs of (nonlinear) stability are for the Minkowski space time, which began with the seminal work of Christodoulou and Klainerman. The stability of the Kerr family of space times is a major open problem in mathematical relativity which has been garnering much interest recently. A thorough understanding of the decay properties of the wave equation on such backgrounds is considered to be prerequisite knowledge to any proof of such stability. The studies in this proposal are expected to directly contribute to this. The teaching components of this proposal consist primarily of the development of a course on general relativity as well as a first year seminar course. The research and teaching components are integrated through reading seminars and directed research projects.
