GEOMETRY AND PDE ANALYSIS IN GENERAL RELATIVITY.

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The goal of this research project is to investigate a number of topics in the intersection of geometry, the analysis of nonlinear partial differential equations (PDE's) and general relativity (GR). These topics share a common theme: to investigate solutions of the global Cauchy problem for the Einstein equations for physically interesting situations and to describe the asymptotics of these solutions. To do this, problems in Lorentzian and Riemannian geometry have to be solved, and the study of the curvature and geometric quantities especially along null hypersurfaces is required. Recent results obtained by the principal investigator (PI), extending work by D. Christodoulou and S. Klainerman, prove global existence and uniqueness of solutions of the Einstein vacuum (EV) equations in an asymptotically flat setting under more general conditions leading to borderline estimates. The latter indicate that the conditions in the PI's main theorem are sharp in so far as the assumptions on the decay at infinity on the initial data are concerned. Moreover, the new results describe the asymptotic behaviour for these more general situations. These new results together with the geometric and analytic theories used and developed in the proof provide the tools required to further investigate physically interesting solutions of the EV equations for asymptotically flat initial data. Thus, the PI plans to compute the asymptotics of spacetimes allowing non-isotropic mass. This will follow in a straightforward manner from the results obtained in the PI's work. It will make it possible to later on attack another main problem, namely the investigation of angular momentum at null infinity for more general settings. Aiming at this latter goal, this project will study the Cauchy problem for the EV equations in three specific cases, applying the new tools and results to these unexplored situations in order to describe the asymptotics of the corresponding solutions of the EV equations. General relativity, as a unified theory of space, time and gravitation, is based on and extends Newton's theory of gravitation as well as Newton's equations of motion. The laws of GR are the Einstein equations, linking the curvature of spacetime to its matter content. Exploring these equations will lead to a better understanding of the universe. To unravel the complex interplay between geometry, analysis and physics is one of the richest challenges in GR. This project aims to investigate this interaction. Along the way, the mathematical tools developed promise to bear fruit in the analysis of the many structurally similar nonlinear PDE's. The proposed research studies asymptotically flat systems. Isolated gravitating systems such as binary stars, clusters of stars and galaxies can be described in GR by asymptotically flat solutions of the Einstein vacuum equations, as they can be thought of as having an asymptotically flat region outside the support of the matter. It is very important to understand the asymptotics of such systems also in view of gravitational radiation. And the well-known experiments to detect gravitational waves are based on the analysis of such asymptotically flat systems.