We study the geometry of compact Lorentzian manifolds that admit a somewhere timelike Killing vector field, and whose isometry group has infinitely many connected components. Up to a finite cover, such manifolds are products (or amalgamated products) of a flat Lorentzian torus and a compact Riemannian (resp., lightlike) manifold.

The four Rindler quadrants of a pair of oppositely accelerated frames are identified as a (Lorentzian) Mach-Zehnder interferometer. The Rindler frequency dependence of the interference process is expressed by means of a (Lorentzian) differential cross section. The Rindler frequencies of the waves in the two acccelerated frames can be measured directly by means of a simple inertially moving dete...

The action of general relativity proposed by Capovilla, Jacobson and Dell is written in terms of SO(3) gauge fields and gives Ashtekar's constraints for Einstein gravity. However, it does not depend on the space-time metric nor its signature explicitly. We discuss how the space-time metric is introduced from algebraic relations of the constraints and the Hamiltonian by focusing our attention on...

Lagrangian curves in R entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangrian curves with constant symplectic curvatures, to construct a class of Lagrangian tori in R and determ...

The problem of geodesic connectedness in semi-Riemannian manifolds (i.e. the question whether each two points of the manifold can be joined by a geodesic) has been widely studied from very different viewpoints. Our purpose is to review these semi-Riemannian techniques, and possible extensions. In the Riemannian case, it is natural to state this problem on (incomplete) manifolds with (possibly n...

In the early 1980s Yau posed the problem of establishing the rigidity of the Hawking–Penrose singularity theorems. Approaches to this problem have involved the introduction of Lorentzian Busemann functions and the study of the geometry of their level sets—the horospheres. The regularity theory in the Lorentzian case is considerably more complicated and less complete than in the Riemannian case....

We consider Lorentzian correlation functions in theories with non-relativistic Schrödinger symmetry. We employ the method developed by Skenderis and van Rees in which the contour in complex time defining a given correlation function is associated holographically with the gluing together of Euclidean and Lorentzian patches of spacetimes. This formalism extends appropriately to geometries with Sc...

The asymmetric ABAB-matrix model describes the transfer matrix of threedimensional Lorentzian quantum gravity. We study perturbatively the scaling of the ABAB-matrix model in the neighbourhood of its symmetric solution and deduce the associated renormalization of three-dimensional Lorentzian quantum gravity. pacs: 04.60Gw, 04.20Gz, 04.60Kz, 04.60Nc Presented by J.J. at the Workshop on Random Ge...

This is an informal introduction to stochastic analysis on both Riemanian and Lorentzian manifolds. We review the basics underlying the construction of diffusions on manifolds, highlighting the important differences between the Riemanian and Lorentzian cases. We also discuss a few recent applications which range from biophysics to cosmology.

The asymmetric ABAB-matrix model describes the transfer matrix of threedimensional Lorentzian quantum gravity. We study perturbatively the scaling of the ABAB-matrix model in the neighbourhood of its symmetric solution and deduce the associated renormalization of three-dimensional Lorentzian quantum gravity. pacs: 04.60Gw, 04.20Gz, 04.60Kz, 04.60Nc Presented by J.J. at the Workshop on Random Ge...