The globally hyperbolic metric splitting for non-smooth wave-type space-times

Abstract

We investigate a generalization of the so-called metric splitting of globally hyperbolic space-times to non-smooth Lorentzian manifolds and show the existence of this metric splitting for a class of wave-type space-times. The approach used is based on smooth approximations of non-smooth space-times by families (or sequences) of globally hyperbolic space-times. In the same setting we indicate as an application the extension of a previous result on the Cauchy problem for the wave equation.

Global Hyperbolicity for Spacetimes with Continuous Metrics

“…Hörmann, Kunzinger and Steinbauer ([HKS12, Definition 6.1, p. 182]) defined global hyperbolicity in terms of the metric splitting, which is well-suited for the Cauchy problem for the wave equation on non-smooth spacetimes (with weaklysingular metrics). Furthermore in [HS14] this notion was investigated for a class of non-smooth wave-type spacetimes (generalizations of pp-waves, with non-flat wave surfaces).…”

Conical spacetimes and global hyperbolicity

“…Considering now the conical spacetime as a generalized Lorentz manifold, we show that it also qualifies for an appropriate variant of the metric splitting. The corresponding generalization of this notion has been introduced in [15] and been shown to provide a sufficient condition for wellposedness of the Cauchy problem for the wave equation; [14] discusses a slightly extended definition of the metric splitting, in order to allow for more general isometries that were needed to cover models of discontinuous wave-type spacetimes. Definition 3.2.…”