On the Mathematical Theory of Black Holes

Princeton University, Princeton NJ

Investigators

Abstract

This project deals with problems connected to the mathematical theory of black holes which, if solved, would contribute greatly to our understanding of these most fascinating physical objects of the physical Universe. The project will not only advance the science of black holes, but, by developing new mathematical techniques of relevance to other fields of mathematical physics, it promotes the progress of Mathematics and Science in the broadest possible sense. The Principal Investigator (PI) has a strong track record in targeting specific problems with broad appeal in various areas of Partial Differential Equations (PDE). In that sense, the problems considered as part of this project, though specific to the subject at hand, can also be reformulated in other important problems of mathematical physics. The project also fits well with respect to the long term goal of the PI to help create a vibrant scientific community working on mathematical problems connected to these areas. In particular, the project provides research training opportunities for graduate students. The project focuses on some of the main open problems concerning black holes such as rigidity, stability, and collapse, with special emphasis being given to the fundamental problem of the stability of the Kerr family. This is a precise, explicit, family of solutions to the Einstein vacuum equations, depending on two parameters (the mass and the angular momentum), on which our theoretical understanding of black holes is based. Thus, the problems mentioned above are not only deep from a mathematical point of view, they also have serious implications in Astrophysics. This is particularly true about the problem of stability, since if the Kerr family were to be found unstable, black holes would be nothing more than mathematical artifacts. The PI interprets these three related problems as mathematical "tests of reality" for black holes. All three problems require a deep understanding of the dynamics of the Einstein field equations in a strong gravitational field regime. This is a tall task which can only be achieved by a systematic mathematical analysis of the underlining structure of the problems, using new geometric PDE techniques. The resolution of these problems promotes the progress of Mathematics and Science beyond the subject matter of this particular project. Indeed, it is expected, as it often happened in the past, that some of the techniques developed in connection to these problems will be relevant to other important partial differential equations of mathematical physics. 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.