A NOTE ON THE LORENTZIAN LIMIT CURVE THEOREM

Abstract. In this paper, we extend the familiar limit curve theorem in[2] to a situation where each causal curve lies in a sequence of compactinterpolating spacetimes converging to a limit Lorentz space in the senseof Lorentzian Gromov-Hausdorff distance. 1. IntroductionThe familiar limit curve theorem in [2] guarantees the existence of limitcausalcurvesfor a sequence{γ n } ofcausalcurveshavingpoints of accumulationin a given spacetime.In general, the compactness of the space of causal curves between two pointsof a spacetime plays an essential role in the study of global structure of thespacetime. In fact, many of the singularity theorems in general relativity and arecent proof of positive energy theorem as well as Lorentzian splitting theoremdepend on the compactness of the space of causal curves in the spacetime ([6]).In 1996, R. Sorkin and E. Woolgar [7] introduced the concept of K-causalrelation and extended the standard compactness theorem of causal curves to asituation where the differentiability of metric of the spacetime is only C