Geodesic structure of standard static space--times.
For a Riemannian manifold \((F, g_F)\) and smooth \(f\colon (a,b)\to (0,\infty)\) define the standard static space time \((M,g)\) to be the product \(F\times (a,b)\) with the metric \(-f^2dt^2\oplus g_F\). The authors study various properties of geodesics of \((M,g)\). A manifold is said to satisfy the nonreturning property if for every point \(p\) there is an arbitrarily small neighborhood \(U(p)\) such that a geodesic leaving \(U(p)\) does not return to it. The authors prove that the standard static-space time \((M,g)\) satisfies the nonreturning property if and only if \((F, g_F)\) does satisfy it. They prove that if \(F\) is compact (or if \(\sup f < \infty\) and \((F,g_F)\) has a pseudoconvex geodesic system), then \((M,g)\) has a pseudoconvex geodesic system. As a corollary they get that if \(\sup f< \infty\) and \((F,g_F)\) has a pseudoconvex geodesic system and satisfies the nonreturning property, then the \((M,g)\) is geodesically connected.
scientific article; zbMATH DE number 1915176
scientific article