Analysis of Extremal Black Holes

Princeton University, Princeton NJ

Investigators

Abstract

Black holes have captured the imagination of scientists and the general public since they were predicted by Einstein's general theory of relativity. The PI intends to investigate fundamental mathematical problems concerning the dynamics of black holes in the context of the initial value problem for the celebrated Einstein equations. One of the most outstanding problems in this direction is the black hole stability conjecture, namely that black holes are indeed stable under perturbations from the outside. The resolution of this conjecture would underpin the relevance of black holes in the physical theory. Previous works of the PI uncovered a novel instability property of the so-called extremal black holes under a special class of perturbations. This instability has been further studied by several research groups worldwide both from the point of view of rigorous mathematics and numerics. The PI will strive to provide a complete description for the stability and instability properties of extremal black holes, obtaining in particular mathematically rigorous results for associated physical phenomena predicted by numerical simulations. The project will also investigate important global aspects of the Einstein equations including radiative properties of gravitational waves. Work on this proposal will improve our society's understanding of black holes, and consequently the structure of our universe. The focus of the project is the study of three central problems relating to local and global properties of hyperbolic partial differential equations arising in general relativity. First, the project will investigate the stability problem for extremal and sub-extremal black holes. In this direction, the PI and his collaborators have developed a new method in the context of spherical symmetry which addresses various difficulties at the event horizon and null infinity and is expected to have a wide range of applications in hyperbolic partial differential equations (PDEs). The PI intends to extend this method and also obtain finer boundedness and decay properties of linear waves in order to obtain global results for the black hole stability problem. This project will probe various instabilities in the extremal case which originate from the convergence of non-axisymmetric quasinormal modes to finite values on the real axis and the coupling of the so-called trapping effect and superradiance. These phenomena are of fundamental importance in black hole dynamics and have been the object of extensive numerical simulations in the physics literature, but no rigorous results are presently available. Second, the project will investigate the gluing problem for characteristic initial data of hyperbolic equations, which is an extension of the Riemannian gluing problem, the latter being intensively studied in geometric analysis. Finally the project will develop a scattering theory for the Einstein equations which is expected to provide new insights into the global behavior of the Einstein equations and in particular the study of gravitational waves. The PI strongly believes that the proposal will lead to the discovery of genuinely new analytical techniques applicable in a wider spectrum of problems in geometry, analysis, PDEs and mathematical physics.