Geodesic completeness in static space-times
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.
scientific article; zbMATH DE number 4050493