On The Uniqueness Of Charged Black Holes

A fundamental conjecture in general relativity(the black hole rigidity problem)asserts that the domain of outer communication of regular,stationary,four-dimensional,vacuum black hole solutions are isometrically diffeomorphic to those of Kerr black holes.A common perception in the physics community is that: due to gravitational radiation,general,asymptotically flat,dynamic solutions of the Einstein-vacuum equations ought to settle down,asymptotically,into a stationary regime.The problem is resolved by Hawking,Carter and Robinson under some reasonable geometric and physical conditions,and a technical assumption that the outer communication is real analytic.During 1999-2010,a series of progresses was made by many authors including Friedrich,R?acz,Wald,Ionescu,Klainerman,and Alexakis to understand the conjecture in the more realistic setting of smooth spacetimes.In this thesis,we consider the corresponding conjecture about charged black holes.We follow Ionescu and Klainerman's project,extend their arguments for the vacuum black holes to the charged ones.we prove a perturbative result concerning the uniqueness of Kerr-Newman family of black holes: given an asymptotically flat space-time with bifurcate horizons,if it agrees with a non-extremal Kerr-Newman space-time asymptotically at infinity and it is sufficiently close to the Kerr-Newman family,where the closeness is measured by the smallness of a pair of Mars-Simon type tensors,then the spacetime must be one of the Kerr-Newman solutions.The proof is inspired by the fact that the Carleman type estimates for wave operators can be used to prove uniqueness theorem for solutions of wave equations outside of a light-cone.Relying on this idea,we construct Hawking vector field near the horizons then extend it to the entire domain of outer communication.Our global assumptions on the space-time and Maxwell field can be grouped into three categories.The first one is the standard asymptotic flatness assumption;the second assumption requires the horizon to be the smooth bifurcate horizon;The last assumption asserts that,in a suitable sense,on the domain of outer communication,the spacetime as well as the Maxwell field is close to some Kerr-Newman metric.Under these reasonable assumptions,the smallness of Mars-Simon type tensor will control some optical functions y and z,in particular we prove that the function y satisfies the T-conditional pseudo-convexity property away from the horizon.This pseudo-convexity property plays a key role in the Carleman estimates and the uniqueness argument thus we extend the locally constructed Hawking vector field to the domain of outer communication.Once the Hawking vector field is extended to the domain of outer communication,we can appeal to the thesis of Bunting to conclude that our space-time is indeed isometric to one Kerr-Newman black hole.