Fermat Metrics

In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of the multiple image effect. Some results on variational methods, as Ljusternik-Schnirelmann and Morse Theory are recalled, to give a description of the variational methods used. Other applications of the Fermat metrics concern the global hyperbolicity and the geodesic connectedeness and a characterization of the Sagnac effect in a stationary spacetime. Finally some possible applications to other class of spacetimes are considered.

Causal hierarchy in modified gravity

We investigate the causal hierarchy in various modified theories of gravity. In general relativity the standard causal hierarchy, (key elements of which are chronology, causality, strong causality, stable causality, and global hyperbolicity), is well-established. In modified theories of gravity there is typically considerable extra structure, (such as: multiple metrics, aether fields, modified dispersion relations, Hořava-like gravity, parabolic propagation, etcetera), requiring a reassessment and rephrasing of the usual causal hierarchy. We shall show that in this extended framework suitable causal hierarchies can indeed be established, and discuss the implications for the interplay between “superluminal” propagation and causality. The key distinguishing feature is whether the signal velocity is finite or infinite. Preserving even minimal notions of causality in the presence of infinite signal velocity requires the aether field to be both unique and hypersurface orthogonal, leading us to introduce the notion of global parabolicity.

Projection diagrams

In this chapter we show that one can usefully represent classes of non-spherically symmetric geometries in terms of two-dimensional diagrams, called projection diagrams, using an auxiliary two-dimensional metric constructed out of the spacetime metric. Whenever such a construction can be carried out, the issues such as stable causality, global hyperbolicity, the existence of event or Cauchy horizons, the causal nature of boundaries, and the existence of conformally smooth infinities become evident by inspection of the diagrams.

On Completeness of Sliced Spaces under the Alexandrov Topology

We show that in a sliced spacetime ( V , g ) , global hyperbolicity in V is equivalent to T A -completeness of a slice, if and only if the product topology T P , on V, is equivalent to T A , where T A denotes the usual spacetime Alexandrov “interval” topology.

Causality theory for closed cone structures with applications

We develop causality theory for upper semi-continuous distributions of cones over manifolds generalizing results from mathematical relativity in two directions: non-round cones and non-regular differentiability assumptions. We prove the validity of most results of the regular Lorentzian causality theory including: causal ladder, Fermat’s principle, notable singularity theorems in their causal formulation, Avez–Seifert theorem, characterizations of stable causality and global hyperbolicity by means of (smooth) time functions. For instance, we give the first proof for these structures of the equivalence between stable causality, [Formula: see text]-causality and existence of a time function. The result implies that closed cone structures that admit continuous increasing functions also admit smooth ones. We also study proper cone structures, the fiber bundle analog of proper cones. For them, we obtain most results on domains of dependence. Moreover, we prove that horismos and Cauchy horizons are generated by lightlike geodesics, the latter being defined through the achronality property. Causal geodesics and steep temporal functions are obtained with a powerful product trick. The paper also contains a study of Lorentz–Minkowski spaces under very weak regularity conditions. Finally, we introduce the concepts of stable distance and stable spacetime solving two well-known problems (a) the characterization of Lorentzian manifolds embeddable in Minkowski spacetime, they turn out to be the stable spacetimes, (b) the proof that topology, order and distance (with a formula à la Connes) can be represented by the smooth steep temporal functions. The paper is self-contained, in fact we do not use any advanced result from mathematical relativity.

On Sliced Spaces: Global Hyperbolicity Revisited

We give a topological condition for a generic sliced space to be globally hyperbolic without any hypothesis on lapse function, shift function, and spatial metric.

On Sliced Spaces: Global Hyperbolicity Revisited

We give a topological condition for a generic sliced space to be globally hyperbolic, without any hypothesis on the lapse function, shift function and spatial metric.