Geodesic completeness in static space-times
type: Item
instance of: scholarly article
title: Geodesic completeness in static space-times
zbMATH Open document ID: 0644.53063
DOI: 10.1007/BF00148016
published in: Geometriae Dedicata
publication date: 1988-01-01T00:00:00Z
review text: A standard static space-time is given by the Lorentzian warped product \(M=I_ f\times H\) where I is an interval, \(f: H\to (0,\infty)\) a warping function and (H,h) a Riemannian manifold. The aim of this paper is to develop conditions for the geodesic completeness of such a space-time. In the Riemannian case the geodesic completeness of M has been shown to be equivalent to that of the single factors [see the paper of \textit{R. L. Bishop} and \textit{B. O'Neill} in Trans. Am. Math. Soc. 145, 1-49 (1969; Zbl 0191.520)]. At the beginning of this paper standard static space-times are shown to be stably causal. If \(I\neq R\) then M must be timelike, null and spacelike geodesically incomplete, independent of the warping function f. In the case \(I=R\) the so-called K-growth condition on f is shown to be sufficient for the timelike geodesic completeness of M, but not for the null geodesic completeness. Here (H,h) is allowed to be geodesically incomplete. This gives a negative answer to a question posed by \textit{O. Kobayashi} and \textit{M. Obata} [Differential geometry and differential equations, Proc. 1980 Beijing Symp., Vol. 3, 1333-1344 (1982; Zbl 0523.53051)]. Then the author derives as a sufficient condition for timelike and null geodesic completeness of \(M=R_ f\times H\) that f be bounded from below by a positive constant and (H,h) be geodesically complete. In the final paragraph the author studies generalizations to the case where the first factor is Lorentzian of higher dimension.
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 4050493
zbMATH Keywords: stable causality
zbMATH Keywords: standard static space-time
zbMATH Keywords: Lorentzian warped product
zbMATH Keywords: geodesic completeness
zbMATH Keywords: K-growth condition
author: Dean E. Allison
reviewed by: Bernd Wegner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4050493