My research deals with mathematical problems involving gravity which arise in astrophysics. Specifically my research will involve (1) proving decay of solutions to the initial-value problem for Maxwell's equations of electromagnetism and for gravitational waves, both in a Kerr rotating black hole geometry; (2) studying shock waves and expansion waves arising in general relativity; (3) magnetic rotating and non-rotating Newtonian stars. Part 1 continues previous work on decay results for the Dirac and scalar wave equations, to Maxwell's equations and to the linearized Einstein equations in a KBH background geometry. Part 2 deals with shock waves and expansion waves in general relativity, and applying these wave solutions to problems in astrophysics; e.g. the so-called anomalous acceleration of the universe, without needing dark energy or the cosmological constant. It is based on my rigorously obtained new solutions of the Einstein equations. Part 3 arises from my recent results on existence and stability results for rotating Newtonian stars and is aimed at extending these results to magnetic stars (like our sun). This study is motivated by an attempt to better understand sunspots and solar flares, from a rigorous mathematical point of view.
The proposal is focused on three areas of astrophysics. The first involves studying whether black holes are stable under electromagnetic and gravitational perturbation. That is, if one sends an electromagnetic or gravitational wave into a rotating black hole, will the black hole remain pretty much intact, or will it radiate away a good portion of its energy? The second is to continue my work on new solutions of Einstein's equations explaining the so-called anomalous acceleration of the universe. That is, the standard physical model modifies Einstein's equations in an ad hoc manner and involves the notion of "dark energy", an unobserved anti-gravitational force, in order to agree with astronomical observations. This approach has no physical basis whatsoever. The final area of interest to me is to study start with magnetic fields with an aim to getting a theoretical explanation of sunspots and solar flares. These phenomena remain highly mysterious, and are important since when they occur, they disrupt satellite as well as radio communication. It is not known how to predict sunspots and solar flares, and a firm mathematical understanding of the associated equations would be an important step in this direction.