On the causal properties of warped product spacetimes

TL;DR

This paper investigates how warped product spacetimes inherit causality properties from their factors, revealing conditions under which properties like causal simplicity and continuity are preserved, and provides formulas for Lorentzian distances.

Contribution

It establishes new conditions for causality properties in warped product spacetimes and derives a formula relating Lorentzian distances on the product to those on the factors.

Findings

Causal simplicity in warped products depends on the simplicity of the base spacetime and the properties of the Lorentzian distance.

Conditions for causal continuity are identified in warped product spacetimes.

A formula relating Lorentzian distances on the product and its factors is derived.

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.