This project concerns the investigation of two of the most outstanding conjectures in general relativity, namely the uniqueness and stability of the Kerr family of black holes, by concentrating on the special class of extremal black holes. Recent contributions of the PI showed that waves exhibit instability properties along the event horizon originating from a novel hierarchy of conservation laws. Moreover, extremal black holes exhibit very strong trapping which is coupled with the above conservation laws and also the phenomenon of superradiance (i.e. the extraction of energy from the black hole). Satisfactory techniques have been developed to overcome the problem of trapping in the non-extremal case; however, in view of the aforementioned geometric couplings, these methods break down in the extremal case. The PI expects that this project will develop the necessary tools and techniques to provide a rigorous and definitive understanding of evolution equations on such spacetimes. In particular, the PI intends to continue the rigorous study of linear and nonlinear waves and investigate the distribution of quasinormal modes on such backgrounds. The PI will also work on the global existence and breakdown criteria for quasilinear equations on extremal Kerr hoping to gain insights for the ultimate goal of this proposal, namely the fully non-linear stability and instability and uniqueness problem for extremal black holes.
General relativity is the classical theory that describes the evolution of physical systems under the effect of gravity. One of the most celebrated predictions of the theory is the existence of so-called black hole regions, i.e. regions from where light cannot escape to infinity. Not only have these regions captured the imagination of scientists, but have also found profound applications to astronomy, physics, and mathematics. A particularly important class of black holes consists of the so-called extremal black holes, that is black holes with zero temperature. The latter are central objects of study in the high-energy physics community. The PI has initiated a rigorous mathematical study of evolution equations on such spacetimes and interesting and surprising results have emerged. Specifically, a novel instability has been discovered on the event horizon of extremal black holes bearing a wide range of potential applications regarding the mathematics and physics of black holes. Specialists in high-energy physics and numerical relativity are further researching the applications of this instability in other contexts. The PI expects that this project will unravel the complex interaction between the analysis, geometry and physics of extremal black holes. Moreover, the PI intends to start new collaborations which will lead to exchange of knowledge and also increase interactions with members of the physics, numerical relativity and astrophysics community. The PI plans to teach seminars at various places aiming at introducing students and researchers to the very rich mathematical structure of general relativity.
