Differential Equations resulting from the interaction of Gravity with other Force Fields, and Shock-Waves in General Relativity

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NSF Award Abstract - DMS-0103998 Mathematical Sciences: Differential Equations resulting from the interaction of Gravity with other Force Fields, and Shock Waves in General Relativity Abstract DMS-0103998 Smoller This project is concerned with mathematical problems involving gravity, as described by Einsteins theory of general relativity, on two different scales: (A) elementary particles, whereby gravity is coupled to other fundamental forces (e.g., nuclear forces), and quantum mechanical effects are taken into account, via the Dirac Equation; and (B) astrophysics, in particular, shock-wave explosions in the universe. In Part A, we study the behavior of elementary particles (fermions) in a rotating black-hole geometry. We also study the decay and stability of solutions of the Teukolsky equation (which applies to gravitational waves, electromagnetic waves, etc.), in a rotating black-hole background geometry. In Part B, we investigate a new cosmological model, different from the Big-Bang model. The model, which agrees with astronomical observations, is based on shock waves that can occur beyond the Hubble length. We will also investigate the validity of the Hawking-Penrose singularity theorem for situations involving interacting shock waves. Finally, we will study the dynamics of a star collapsing to a black hole, modeled as an initial-value problem for the Einstein-Euler equations, with certain constraints on the initial data. This study will also help us learn more about the constraint equations that all initial data for the Einstein equations must satisfy.This project is concerned with mathematical problems involving gravity, as described by Einsteins theory of general relativity, on two different scales: (A) elementary particles, whereby gravity is coupled to other fundamental forces (e.g., nuclear forces), and quantum mechanical effects are taken into account, and (B) astrophysics; in particular, shock-wave explosions in the universe. In Part A, we study the behavior of an elementary particle (electron, proton, etc.) near a black hole. We also study the stability of electromagnetic and gravitational waves near a black hole. In Part B, we explore a new model for cosmology, different from the usual Big Bang scenario, which is based on a shock-wave explosion. We will also study the collapse of a massive star to a black hole.