ON NULL CUT POINTS IN TWO-DIMENSIONAL SPACETIMES

Abstract

Abstract. It is shown that any two-dimensional globally hyperbolicspacetime with noncompact Cauchy surfaces has no null cut points. 1. IntroductionConjugate point and cut point, defined in terms of Jacobi field along ageodesic and Riemannian distance, play important roles in global analysis ofRiemannian manifold. Likewise, in Lorentzian geometry, they play essentialroles in singularity theory of spacetimes. In connection with singularity theory,causality theory developed by Penrose [6], is an essential tool for global analysisof spacetimes. Lorentzian cut point, defined in terms of Lorentzian distancefunction, has a close relation to conjugate points and it is well-known that thereis no null conjugate points in two-dimensional spacetimes ([1], [4], [5], [6]).In contrast to this, there exists a two-dimensional spacetime in which everynull geodesic has cut points which can be easily seen in two-dimensional Ein-stein static universe. In two-dimensional Einstein static universe, we can seethat the spacetime has compact Cauchy surfaces.In this paper, we show that if a two-dimensional Lorentzian manifold hasnoncompact Cauchysurfaces, then there exist no cut points alongnull geodesicsby use of topological property of its Cauchy surface.2. PreliminariesIn this section, we assume that M is an arbitrary dimensional globally hy-perbolic Lorentzian manifold with a noncompact Cauchy surface Σ. By thework of Bernal and Sa´nchez ([2] and [3]), we can assume that Σ is a smooth,spacelike hypersurface. If there exists a future-directed timelike curve from xto y, then we write x ≪ y and we say that y lies in the chronological future