Geodesic completeness in static space-times

Abstract

Let (H, h) be a Riemannian manifold and assume f: H→(0, ∞) is a smooth function. The Lorentzian warped product (a, b) f ×H,−∞≤a<b≤∞, with metric ds2=(−f2 dt2) ⊕ h is called a standard static space-time. A study is made of geodesic completeness in standard static space-times. Sufficient conditions on the warping function f: H→(0, ∞) are obtained for (a, b) f ×H ×H to be timelike and null geodesically complete. In the timelike case, the sufficient condition is independent of the completeness of the Riemannian manifold (H, h).