On further generalization of the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon

A rigidity theorem that applies to smooth electrovacuum spacetimes which represent either (A) an asymptotically flat stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics was given in a recent paper by Friedrich et al (1999 Commun. Math. Phys. 204 691-707). Here we enlarge the framework of the corresponding investigations by allowing the presence of other types of matter fields. In the first part the matter fields are involved merely implicitly via the assumption that the dominant energy condition is satisfied. In the second part Einstein-Klein-Gordon (EKG), Einstein-[non-Abelian] -Higgs (E[nA]H), Einstein-[Maxwell]-Yang-Mills-dilaton (E[M]YMd) and Einstein-Yang-Mills-Higgs (EYMH) systems are studied. The black hole event horizon or, respectively, the compact Cauchy horizon of the considered spacetimes is assumed to be a smooth non-degenerate null hypersurface. It is proved that there exists a Killing vector field in a one-sided neighbourhood of the horizon in EKG, E[nA]H, E[M]YMd and EYMH spacetimes. This Killing vector field is normal to the horizon, moreover the associated matter fields are also shown to be invariant with respect to it. The presented results provide generalizations of the rigidity theorems of Hawking (for case A) and of Moncrief and Isenberg (for case B) and, in turn, they strengthen the Validity of both the black hole rigidity scenario and the strong cosmic censor conjecture of classical general relativity.