Clemens Sämann

Global Hyperbolicity for Spacetimes with Continuous Metrics

We show that the definition of global hyperbolicity in terms of the compactness of the causal diamonds and non-total imprisonment can be extended to spacetimes with continuous metrics, while retaining all of the equivalences to other notions of global hyperbolicity. In fact, global hyperbolicity is equivalent to the compactness of the space of causal curves and to the existence of a Cauchy hypersurface. Furthermore, global hyperbolicity implies causal simplicity, stable causality and the existence of maximal curves connecting any two causally related points.

The global existence, uniqueness and 1-regularity of geodesics in nonexpanding impulsive gravitational waves

We study geodesics in the complete family of nonexpanding impulsive gravitational waves propagating in spaces of constant curvature, that is Minkowski, de Sitter and anti-de Sitter universes. Employing the continuous form of the metric we prove existence and uniqueness of continuously dierentiable geodesics (in the sense of Filippov) and use a C 1 -matching procedure to explicitly derive their form.

On the completeness of impulsive gravitational wave spacetimes

We consider a class of impulsive gravitational wave spacetimes, which generalize impulsive pp-waves. They are of the form , where (N, h) is a Riemannian manifold of arbitrary dimension and M carries the line element ds2 = dh2 + 2 du dv + f(x)δ(u) du2, with dh2 being the line element of N and δ the Dirac measure. We prove a completeness result for such spacetimes M with complete Riemannian part N.

Geodesic Completeness of Generalized Space-times

We define the notion of geodesic completeness for semi-Riemannian metrics of low regularity in the framework of the geometric theory of generalized functions. We then show completeness of a wide class of impulsive gravitational wave space-times.

Lorentzian length spaces

We introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The rôle of the metric is taken over by the time separation function, in terms of which all basic notions are formulated. In this way, we recover many fundamental results in greater generality, while at the same time clarifying the minimal requirements for and the interdependence of the basic building blocks of the theory. A main focus of this work is the introduction of synthetic curvature bounds, akin to the theory of Alexandrov and CAT(k)-spaces, based on triangle comparison. Applications include Lorentzian manifolds with metrics of low regularity, closed cone structures, and certain approaches to quantum gravity.

Geodesics in nonexpanding impulsive gravitational waves with Λ. II

We investigate all geodesics in the entire class of nonexpanding impulsive gravitational waves propagating in an (anti-)de Sitter universe using the distributional metric. We extend the regularization approach of part I [Samann, C. et al., Classical Quantum Gravity 33(11), 115002 (2016)] to a full nonlinear distributional analysis within the geometric theory of generalized functions. We prove global existence and uniqueness of geodesics that cross the impulsive wave and hence geodesic completeness in full generality for this class of low regularity spacetimes. This, in particular, prepares the ground for a mathematically rigorous account on the “physical equivalence” of the continuous form with the distributional “form” of the metric.

Completeness of general pp-wave spacetimes and their impulsive limit

We investigate geodesic completeness in the full family of pp-wave or Brinkmann spacetimes in their extended as well as in their impulsive form. This class of geometries contains the recently studied gyratonic pp-waves, modelling the exterior field of a spinning beam of null particles, as well as NPWs, which generalise classical pp-waves by allowing for a general wave surface. The problem of geodesic completeness reduces to the question of completeness of trajectories on a Riemannian manifold under an external force field. Building upon respective recent results we derive completeness criteria in terms of the spatial asymptotics of the profile function in the extended case. In the impulsive case we use a fixed point argument to show that irrespective of the behaviour of the profile function all geometries in the class are complete.

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

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.

On geodesics in low regularity

We consider geodesics in both Riemannian and Lorentzian manifolds with metrics of low regularity. We discuss existence of extremal curves for continuous metrics and present several old and new examples that highlight their subtle interrelation with solutions of the geodesic equations. Then we turn to the initial value problem for geodesics for locally Lipschitz continuous metrics and generalize recent results on existence, regularity and uniqueness of solutions in the sense of Filippov.