Particular Timelike Flows in Global Lorentzian Geometry

Abstract

Particular Timelike Flows in Global Lorentzian Geometry by Alexander Dirmeier This work investigates the topological and causal characteristics of Lorentzian manifolds (M, g), which possess a complete and timelike unit vector field V , or in other words a global timelike flow, as an additional structure. Naturally, these Lorentzian manifolds are spacetimes. General geometric requirements for these spacetimes to split diffeomorphically as a product R × S, with the vector field V along the R-factor and S the space of flow lines of V , are derived. The possible causality conditions for these splitting spacetimes are analyzed and a complete causal classification is given. Sub-classes of these splitting spacetimes can be obtained by a classification according to a decomposition of the covariant derivative ∇V of the given timelike vector field. The specific sub-classes of sliced spacetimes and stationary spacetimes are analyzed with regard to global hyperbolicity and several new relations are obtained. For stationary and homothetic spacetimes, a new version of the Lorentzian Bochner technique is derived. Finally, conformal Lorentzian submersions, and particularly Hubble-isotropic spacetimes, are analyzed and conditions for their global hyperbolicity and geodesic completeness are obtained.