Physical-Space Estimates on Black Hole Perturbations

The field of Mathematical General Relativity concerns the mathematical description of space and time, focusing on the analysis of solutions to the Einstein equation, such as black holes. One important problem in the field is the mathematical proof of the stability of black holes, which is essential towards understanding them as realistic physical objects. If stable, black holes which are perturbed with gravitational (or other kinds of) radiation could present a temporary change but would eventually return to their initial status. The result of this work will be shared to the mathematical and physical community through peer-reviewed publications and seminars and will be disseminated to the general community through media articles, public lectures and outreach events in schools. Graduate students and postdocs will be involved in this research. Physical-space estimates for hyperbolic partial differential equations have been successfully applied to the recent proof of non-linear stability of the slowly rotating Kerr family and to the analysis of the interaction of gravitational and electromagnetic radiations in the charged Kerr-Newman black hole. The investigators plan to develop a robust approach, based on the definition of a combined energy-momentum tensor for a system of coupled wave equations, that has the potential to be applied to other matter fields. They also plan to extend the applicability of these methods to rapidly spinning black hole solutions by making use of a refined analysis of how energy methods and integrated local energy decay estimates interact in the presence of rotation. Radiations perturbing black holes are described by partial differential equations, whose properties can be studied through various techniques. Although some techniques involving decomposition in simpler forms, such as modes, have proved to be very successful, they have limitations in applications to non-linear problems and to the interaction of different kinds of radiation. This research aims to extend the known methods involving physical-space estimates for the study of stability of black hole solutions. 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.