ON THE GEOMETRY OF LORENTZ SPACES AS A LIMIT SPACE

Abstract

Abstract. In this paper, we prove that there is no branch point in theLorentz space (M,d) which is the limit space of a sequence {(M α ,d α )}ofcompact globally hyperbolic interpolating spacetimes with C α± -propertiesand curvature bounded below. Using this, we also obtain that every max-imal timelike geodesic in the limit space (M,d) can be expressed as thelimit curve of a sequence of maximal timelike geodesics in {(M α ,d α )}.Finally, we show that the limit space (M,d) satisfies a timelike trian-gle comparison property which is analogous to the case of Alexandrovcurvature bounds in length spaces. 1. IntroductionLorentzian Gromov-Hausdorff theory was first introduced by J. Noldus in[3] and the notion of Gromov-Hausdorff (GH) distance d GH ((M,g),(N,h))between two compact, globally hyperbolic (CGH),interpolating spacetimes(M,d) and (N,h) was defined in a similar way as in the Riemannian case.Specifically, we say (M,d) and (N,h) ǫ-close if and only if there exist map-pings ψ: M→ N, ξ: N→ Msuch that| d