Rigidity results in general relativity: A review

Despite a common perception in the physics community, the Black Hole Rigidity problem remains wide open when one removes the highly restrictive real analyticity assumption underlying the classical results. In this survey we review the progress made in the last ten years in understanding the conjecture in the more realistic setting of smooth spacetimes. We review both local and global results and discuss the new mathematical ideas behind them. We present three types of global results which assert, under somewhat different assumptions, that any stationary solution close to a non-extremal Kerr must be isometric to a non-extremal Kerr, whose parameters a, M are determined by the ADM mass and angular momentum of the given stationary solution. The results illustrate an important geometric obstruction in understanding the full rigidity problem, the possible presence of trapped null geodesics perpendicular to the stationary Killing vectorfield. The key insight in all these results is that such null geodesics are non-existent in any nonextremal Kerr and thus, roughly, in any small perturbation of it. Contents 1. Introduction 124 2. Local rigidity results 127 3. Mars-Simon tensor and global results 137 4. Conclusions 145 Appendix A. Proof of proposition 2.18 146 Appendix B. Proof of Proposition 3.10