This is my PhD dissertation. It has two main parts.
First is a space-time, tensorial characterization of the Kerr-Newman solution among (pseudo)stationary solutions to the Einstein-Maxwell system describing gravity coupled to electromagnetism. The analogous problem for characterizing the Kerr solution among pure-gravity solutions in general relativity was previously solved by Marc Mars; I was able to identify the necessary changes to incorporate also electromagnetic fields into the picture. For this part the work is purely geometric.
The second part is a conditional uniqueness theorem for the Kerr-Newman solution. Roughly speaking, the result says that if we have a (pseudo)stationary charged black hole such that its surface geometry looks like that of the Kerr-Newman black hole, then the exterior of the black hole must be the same as the Kerr-Newman solution. Similar types of results go back to Stephen Hawking's work in the 70s. Our main goal is to overcome certain "stiff" assumptions that Hawking made in his argument. This part is an extension of the work of Ionescu and Klainerman on the pure-gravity case to incorporate electromagnetic fields. The main argument is based on Ionescu and Klainerman's generalized Carleman inequality, and uses a mix of geometric and analytic techniques.
Abstract
The uniqueness of the Kerr-Newman family of black hole metrics as stationary asymptotically flat solutions to the Einstein equations coupled to a free Maxwell field is a crucial ingredient in the study of final states of the universe in general relativity. If one imposes the additional requirement that the space-time is axial-symmetric, then said uniqueness was shown by the works of B. Carter, D.C. Robinson, G.L. Bunting, and P.O. Mazur during the 1970s and 80s. In the real-analytic category, the condition of axial symmetry can be removed through S. Hawking’s Rigidity Theorem. The necessary construction used in Hawking's proof, however, breaks down in the smooth category as it requires solving an ill-posed hyperbolic partial differential equation. The uniqueness problem of Kerr-Newman metrics in the smooth category is considered here following the program initiated by A. Ionescu and S. Klainerman for uniqueness of the Kerr metrics among solutions to the Einstein vacuum equations. In this work, a space-time, tensorial characterization of the Kerr-Newman solutions is obtained, generalizing an earlier work of M. Mars. The characterization tensors are shown to obey hyperbolic partial differential equations. Using the general Carleman inequality of Ionescu and Klainerman, the uniqueness of Kerr-Newman metrics is proven, conditional on a rigidity assumption on the bifurcate event horizon.
