Mathematical Questions Resulting from the Coupling of Gravity to Other Fields

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This project investigates three questions in general relativity: (i) The stability of solutions of electromagnetic and linearized gravitational waves in a Kerr (rotating) black hole background metric; i.e., solutions of the Cauchy problem for these partial differential equations with compactly supported initial data outside of the event horizon of the black hole. These stability results will add credibility to the use of the Kerr metric in detailed astrophysical models. (ii) Rigorous justification of the Penrose process for the extraction of energy from a rotating black hole. (iii) Incorporation of Guth's theory of inflation into the principal investigator's cosmological model. brbrBlack holes have interested both physicists and mathematicians since Einstein proposed his theory of general relativity in 1915. They have been studied intensively, but many questions remain unanswered. This project explores three important questions. It investigates the stability of black holes under electromagnetic and gravitational perturbations. Such understanding is of great importance for predicting the nature of gravitational radiation originating in outer space. The stability problem will be considered in a manner parallel to the observations made by astronomers, leading to a better theoretical understanding of numerical modeling of solutions of Einstein's equations. The project also aims to put the question of extracting energy from a rotating black hole on a firm theoretical basis. The extraction of energy from a rotating black hole was proposed in 1968, but until recently the question was far too difficult to be studied. Finally, the project studies some questions in astrophysics dealing with cosmology and supernova explosions; these explosions of enormously dense massive stars undergoing gravitational collapse are responsible for seeding the universe with heavy elements, including carbon and iron.