GLOBAL EXTENSIONS OF SPACETIMES DESCRIBING ASYMPTOTIC FINAL-STATES OFBLACK-HOLES

We consider a globally hyperbolic, stationary spacetime containing a b lack hole but no white hole. We assume, further, that the event horizo n, N, of the black hole is a Killing horizon with compact cross-sectio ns. We prove that if surface gravity is non-zero and constant througho ut the horizon one can globally extend such a spacetime so that the im age of N is a proper subset of a regular bifurcate Killing horizon in the enlarged spacetime. The necessary and sufficient conditions are gi ven for the extendibility of matter fields to the enlarged spacetime. These conditions are automatically satisfied if the spacetime is stati c (and hence 't'-reflection symmetric) or stationary-axisymmetric with 't-phi' reflection isometry and the matter fields respect the reflect ion isometry. In addition, we prove that a necessary and sufficient co ndition for the constancy of the surface gravity on a Killing horizon is that the exterior derivative of the twist of the horizon Killing fi eld vanishes on the horizon. As a corollary of this, we recover a resu lt of Carter that constancy of surface gravity holds for any blade hol e which is static or stationary-axisymmetric with the 't-phi' reflecti on isometry. No use of Einstein's equation is made in obtaining any of the above results. Taken together, these results support the view tha t any spacetime representing the asymptotic final state of a black hol e formed by gravitational collapse may be assumed to possess a bifurca te Killing horizon or a Killing horizon with vanishing surface gravity .