On the Mathematical Theory of Black Holes

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At the most basic level Black Holes are nothing but special, explicit, solutions of the Einstein equations of General Relativity. To have physical reality they have to possess properties which make them detectable to observations. Chief among them is stability to perturbations. This translates to a very difficult, deep and beautiful mathematical conjecture which is the main focus of the proposed research. The goal is to show that any small perturbation, at a given time, of a given black hole solution remains small for all later times. Thus all the important features of a unperturbed black hole are preserved by the perturbations.To solve the problem of stability of black holes one has to overcome a large range of difficult issues. The high non-linearity of the Einstein equations makes it hard to separate them; they are all interconnected. At the heart of all these difficulties is the issue of gauge. That means, roughly, that one cannot solve the stability problem without finding a specific coordinate system in which one can establish decay of the perturbations, i.e. to show that the perturbations converge to zero for large times. In collaboration with J. Szeftel in France the PI had proposed a new method to construct these coordinates based on the full covariance properties of the Einstein equations. The main methodology of the PI is to combine this new method, called Generally Covariant Modulation theory, with the the so called vector-field method.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.