Non-imprisonment conditions on spacetime

The non-imprisonment conditions on spacetimes are studied. It is proved that the non-partial imprisonment property implies the distinction property. Moreover, it is proved that feeble distinction, a property which stays between weak distinction and causality, implies non-total imprisonment. As a result the non-imprisonment conditions can be included in the causal ladder of spacetimes. Finally, totally imprisoned causal curves are studied in detail, and results concerning the existence and properties of minimal invariant sets are obtained.Comment: 12 pages, 2 figures. v2: improved results on totally imprisoned curves, a figure changed, some misprints fixe

On the causal properties of warped product spacetimes

It is shown that the warped product spacetime P=M *_f H, where H is a complete Riemannian manifold, and the original spacetime M share necessarily the same causality properties, the only exceptions being the properties of causal continuity and causal simplicity which present some subtleties. For instance, it is shown that if diamH=+\infty, the direct product spacetime P=M*H is causally simple if and only if (M,g) is causally simple, the Lorentzian distance on M is continuous and any two causally related events at finite distance are connected by a maximizing geodesic. Similar conditions are found for the causal continuity property. Some new results concerning the behavior of the Lorentzian distance on distinguishing, causally continuous, and causally simple spacetimes are obtained. Finally, a formula which gives the Lorentzian distance on the direct product in terms of the distances on the two factors (M,g) and (H,h) is obtained.Comment: 22 pages, 2 figures, uses the package psfra

On Fermat's principle for causal curves in time oriented Finsler spacetimes

In this work, a version of Fermat's principle for causal curves with the same energy in time orientable Finsler spacetimes is proved. We calculate the secondvariation of the {\it time arrival functional} along a geodesic in terms of the index form associated with the Finsler spacetime Lagrangian. Then the character of the critical points of the time arrival functional is investigated and a Morse index theorem in the context of Finsler spacetime is presented.Comment: 20 pages, minor corrections, references adde

Weak distinction and the optimal definition of causal continuity

Causal continuity is usually defined by imposing the conditions (i) distinction and (ii) reflectivity. It is proved here that a new causality property which stays between weak distinction and causality, called feeble distinction, can actually replace distinction in the definition of causal continuity. An intermediate proof shows that feeble distinction and future (past) reflectivity implies past (resp. future) distinction. Some new characterizations of weak distinction and reflectivity are given.Comment: 9 pages, 2 figures. v2: improved and expanded version. v3: a few misprints have been corrected and a reference has been update

On the completeness of impulsive gravitational wave space-times

We consider a class of impulsive gravitational wave space-times, which generalize impulsive pp-waves. They are of the form $M=N\times\mathbb{R}^2_1$, where $(N,h)$ is a Riemannian manifold of arbitrary dimension and $M$ carries the line element $ds^2=dh^2+ 2dudv+f(x)\delta(u)du^2$ with $dh^2$ the line element of $N$ and $\delta$ the Dirac measure. We prove a completeness result for such space-times $M$ with complete Riemannian part $N$.Comment: 13 pages, minor changes suggested by the referee

The Cosmological Time Function

Let $(M,g)$ be a time oriented Lorentzian manifold and $d$ the Lorentzian distance on $M$. The function $\tau(q):=\sup_{p< q} d(p,q)$ is the cosmological time function of $M$, where as usual $p< q$ means that $p$ is in the causal past of $q$. This function is called regular iff $\tau(q) < \infty$ for all $q$ and also $\tau \to 0$ along every past inextendible causal curve. If the cosmological time function $\tau$ of a space time $(M,g)$ is regular it has several pleasant consequences: (1) It forces $(M,g)$ to be globally hyperbolic, (2) every point of $(M,g)$ can be connected to the initial singularity by a rest curve (i.e., a timelike geodesic ray that maximizes the distance to the singularity), (3) the function $\tau$ is a time function in the usual sense, in particular (4) $\tau$ is continuous, in fact locally Lipschitz and the second derivatives of $\tau$ exist almost everywhere.Comment: 19 pages, AEI preprint, latex2e with amsmath and amsth

Implementation of a combined association-linkage model for quantitative traits in linear mixed model procedures of statistical packages

Atransmission disequilibrium test for quantitative traits which combines association and linkage analyses is currently available in several dedicated software packages. We describe how to implement such models in linear mixed model procedures that are available in widely used statistical packages such as SPSS. We also briefly mention a few extensions of the model that become naturally available once the model is implemented in such procedures. Genotyping of many microsatellite markers or single nucleotide polymorphisms (SNPs) over the entire genome is becoming increasingly common in human genetics. In those high-resolution maps the average distance between microsatellite markers may be as small as 5 cM and between SNPs one half cM or less. At those small distances it becomes fairly likely that some markers in the set are in linkage disequilibrium (LD) with a gene affecting the trait (a so-called quantitative trait locus or QTL if the trait or the vulnerability distribution is quantitative). Different alleles or combinations of alleles of the markers or SNPs can then be associated with different trait means. Association studies are conducted to discover such allelic effects. Abecasis et al. (2000) generalized the model proposed by Fulker et al. (1999) for combined linkage and association tests, within and between families. The Fulker-Abecasis or F-A model is implemented in the program QTD

The limit space of a Cauchy sequence of globally hyperbolic spacetimes

In this second paper, I construct a limit space of a Cauchy sequence of globally hyperbolic spacetimes. In the second section, I work gradually towards a construction of the limit space. I prove the limit space is unique up to isometry. I als show that, in general, the limit space has quite complicated causal behaviour. This work prepares the final paper in which I shall study in more detail properties of the limit space and the moduli space of (compact) globally hyperbolic spacetimes (cobordisms). As a fait divers, I give in this paper a suitable definition of dimension of a Lorentz space in agreement with the one given by Gromov in the Riemannian case.Comment: 31 pages, 5 figures, submitted to Classical and Quantum gravity, seriously improved presentatio