On the causal properties of warped product spacetimes

Abstract

It is shown that the warped product spacetime P=M *_f H, where H is a complete Riemannian manifold, and the original spacetime M share necessarily the same causality properties, the only exceptions being the properties of causal continuity and causal simplicity which present some subtleties. For instance, it is shown that if diamH=+\infty, the direct product spacetime P=M*H is causally simple if and only if (M,g) is causally simple, the Lorentzian distance on M is continuous and any two causally related events at finite distance are connected by a maximizing geodesic. Similar conditions are found for the causal continuity property. Some new results concerning the behavior of the Lorentzian distance on distinguishing, causally continuous, and causally simple spacetimes are obtained. Finally, a formula which gives the Lorentzian distance on the direct product in terms of the distances on the two factors (M,g) and (H,h) is obtained.

22 pages, 2 figures, uses the package psfrag

References in corpus (4)

• Singularity Theorems and Their Consequences
• Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"
• Eisenhart's theorem and the causal simplicity of Eisenhart's spacetime
• Solutions to the Lorentz force equation with fixed charge-to-mass ratio in globally hyperbolic spacetimes

Cited by in corpus (5)

• Globally hyperbolic spacetimes can be defined without the 'causal' condition
• Characterization of some causality conditions through the continuity of the Lorentzian distance
• Convex regions of stationary spacetimes and Randers spaces. Applications to lensing and asymptotic flatness
• An intrinsic characterization of 2+2 warped spacetimes
• Warped Product Einstein Manifolds in Four Dimensions