CAREER: Geometric-Analytic Investigations of Spacetimes and their Nonlinear Phenomena

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This project concerns the investigation of nonlinear partial differential equations (PDEs), namely the Einstein equations in general relativity, the geometry of the solution spacetimes and their implications in physics and astrophysics. One of the main aims of the proposed research is to derive a complete theory for the nonlinear memory effect of gravitational waves displacing test masses permanently. This will include answering the questions whether there exists such an effect for neutrino radiation as present in binary neutron star mergers and if there is a cosmological memory effect. The goal is to give a detailed geometric-analytic description in the various scenarios and to apply the mathematics to experiments. The PI has already made contributions in this direction in the recent past. The emerging results of the project are expected to answer fundamental physical questions about gravitational radiation, the source objects in astrophysics and the very early period of our universe. Since the pioneering article by D. Christodoulou, where he derived the nonlinear memory effect (also called the Christodoulou effect), it had been an outstanding problem if electromagnetic fields coupled to the Einstein equations affect this nonlinear phenomenon. The PI with collaborators P. Chen and S.-T. Yau solved the problem, proving that the electromagnetic field in the Einstein-Maxwell (EM) equations enlarges the Christodoulou effect. Working with Christodoulou's geometric-analytic approach for the Einstein vacuum equations and N. Zipser's analysis of the EM equations, the PI, Chen and Yau established novel results and methods in the geometric analysis of the EM equations. The geometric-analytic investigations by Christodoulou as well as by the PI with Chen and Yau allowed the authors to deduce exact solutions describing the related effects precisely and for all data. In particular, the results hold for large data such as binary black hole or binary neutron star mergers. Geometric analysis has since proven to be the most powerful method to tackle these problems. In the study of the nonlinear memory effect of gravitational waves, null hypersurfaces in Lorentzian geometry play a crucial role. This is due to the fact that gravitational waves travel along such null hypersurfaces and experiments are performed at null infinity. Therefore, the asymptotic behavior of the latter has to be understood. D. Christodoulou and S. Klainerman in their work "The global nonlinear stability of the Minkowski space" developed new techniques to investigate those. The PI generalized their work and established the borderline case from the point of view of decay of the data. In a research monograph of 300 pages the PI obtained numerous new results that can be applied to other problems in geometry and analysis. The main method and its developments have recently led to the solution of some of the most challenging problems in mathematical physics. The new project will use new ideas from the PI's recent work and it is expected to yield further new results in general relativity (GR), geometry and PDE analysis. The educational component of the proposal focuses on disseminating knowledge of mathematics, physics and astrophysics as well as historic background to the public. Aiming at the creation of an exhibit at the Museum of Natural History at the PI's home University combining geometry and analysis with astrophysics, the educational project will include student work at undergraduate and graduate levels. Together with the PI and the museum staff, they will be involved in teaching at high schools working on the project. At public events organized by the museum the PI will collaborate with the museum staff to present activities to enhance learning. Research results will be presented in a simplified way and adapted to the audience. The project is designed as to reach out to a very broad public including minority groups, families with small kids, schools at all levels and will have an appealing component for everyone. Some of the long-lasting impacts will be new activities for high school classes to teach mathematics by introducing concepts of physics combined with history and art. The geometric nature of general relativity singles out geometric analysis to be the perfect research field to answer the many physical questions. 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 as a whole and of isolated systems such as galaxies, binary black holes or binary neutron stars. To unravel the complex interplay between geometry, analysis and physics is one of the main goals in GR. This project aims to investigate this beautiful interaction. Along the way, the mathematical tools developed promise to bear fruit in the analysis of the many structurally similar nonlinear PDEs. Through the educational component, the PI's research will also have direct impact in a broader sense via the afore-mentioned activities. The PI will complement the outreach by attending conferences to communicate her results. She will also make her results available via internet and publications.