Scalar Curvature Seminar

We consider analytic, vacuum spacetimes that admit compact, non-degenerate Cauchy horizons. Many years ago Vince Moncrief and I proved that, if the null geodesic generators of such a horizon were all closed curves, then the enveloping spacetime would necessarily admit a non-trivial, horizon-generating Killing vector field. Using a slightly extended version of the Cauchy-Kowaleski theorem one could establish the existence of an infinite dimensional, analytic family of such ≥≠ralizedTaub-NUT′space×andshowtˆ,≥≠rically,theyadmiedonlythesing≤(horizon-≥≠rat∈g)Kill∈gfieldalluded→above.Intheworkdiscussed∈thistalk,werelaxtheclosureas∑ptionandanalyzevac∪mspace×∈whichthe≥≠richorizon-≥≠rat∈gνll≥odesicsdenselyfilla2-→rusly∈g∈thehorizon.Inpartica̲rweshowtˆ,asideomsomehighlyexceptionalcasestˆwerefer→asgeneralized Taub-NUT' spacetimes and show that, generically, they admitted only the single (horizon-generating) Killing field alluded to above. In the work discussed in this talk, we relax the closure assumption and analyze vacuum spacetimes in which the generic horizon-generating null geodesics densely fill a 2- torus lying in the horizon. In particular we show that, aside from some highly exceptional cases that we refer to as ergodic', the non-closed generators always have this (densely 2-torus-filling) geometrical property in the analytic setting. By extending arguments we gave previously for the characterization of the Killing symmetries of higher dimensional, stationary black holes, we prove that analytic, 4-dimensional, vacuum spacetimes with such (non-ergodic) compact Cauchy horizons always admit (at least) two independent, commuting Killing vector fields of which a special linear combination is horizon generating. We also discuss the conjectures that every such spacetime with an ergodic horizon is trivially constructable from the Kasner solution by making certain `irrational' toroidal compactifications and that analytic vacuum space times containing degenerate compact Cauchy horizons do not exist. The work I discuss in this talk has been jointly carried out with Vince Moncrief.