Existence of Solutions to Hyperbolic Differential Equations in Mathematical Physics

This project is aimed at understanding the large time behavior of solutions of Einstein's equations for the gravitational field under the influence of various matter fields, such as electromagnetism and fluids. The study of the asymptotic behavior and scattering of solutions are useful for the numerical relativity and physics communities. The mathematics developed as part of the project also helps analyze the data from gravitational wave detectors. Graduate students are trained as part of the project. It is expected that the simplified existence proofs for Einstein's equations and for the stability of black holes will make it easier for students to get started on research in mathematical relativity. The project is concerned with the problem of existence, asymptotic behavior, and scattering of global solutions of Einstein's equations and other field equations. One goal is to study the asymptotic behavior of small solutions of Einstein's equations with various matter fields. The Principal Investigator (PI) has identified a weak null condition that the nonlinear terms in Einstein’s equations and many other physical systems satisfy, and it is important to understand how this determines the asymptotic behavior. The matter fields propagate slower than the speed of light and are hence concentrated in the interior of the forward light cone. The PI discovered that the gravitational field has one part consisting of a wave that travels with the speed of light and one interior part that is a homogeneous decaying function. It is interesting to find out how matter interacts with gravity. The project is developing a new, more explicit, way to construct solutions backwards from scattering data at infinity. Using the explicit expressions, a compatibility condition that scattering data has to satisfy was identified and it would be interesting to find out whether it is sufficient for existence. It is important to be able to compute gravitational waveforms emitted from compact binary systems, such as two black holes colliding, from far away. Those waveforms are being used to identify such systems from the data from gravitational wave detectors. The PI’s formulation in harmonic coordinates is particularly suitable for this and is being used by his collaborators in numerical relativity. Another goal of the project is to study the stability of large solutions like stars or black holes. In particular, it is important to simplify and generalize the proofs involved in the recent work on the stability of black holes. The approach in this project is to first gain a better understanding for general simpler model wave equations close to a black hole. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.