420 research outputs found
Limit curve theorems in Lorentzian geometry
The subject of limit curve theorems in Lorentzian geometry is reviewed. A
general limit curve theorem is formulated which includes the case of converging
curves with endpoints and the case in which the limit points assigned since the
beginning are one, two or at most denumerable. Some applications are
considered. It is proved that in chronological spacetimes, strong causality is
either everywhere verified or everywhere violated on maximizing lightlike
segments with open domain. As a consequence, if in a chronological spacetime
two distinct lightlike lines intersect each other then strong causality holds
at their points. Finally, it is proved that two distinct components of the
chronology violating set have disjoint closures or there is a lightlike line
passing through each point of the intersection of the corresponding boundaries.Comment: 25 pages, 1 figure. v2: Misprints fixed, matches published versio
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
Integrating Graduate Attributes with Assessment Criteria in Business Education Using an Online Assessment System
This paper describes a study of the integration of graduate attributes into Business education using an online system to facilitate the process. 'ReView' is a system that provides students with criteria-based tutor feedback on assessment tasks and also provides opportunities for online student self-assessment. Setup incorporates a process of 'review' whereby assessment criteria are grouped into graduate attribute categories and reworded to make explicit the qualities, knowledge and skills that are valued in student performance. Through this process, academics clarify and make explicit the alignment of assessment tasks to learning objectives and graduate attribute development across units and levels of a program of study. Its application in three undergraduate Business units was undertaken as a collaborative action research project to improve alignment of graduate attributes with assessment, identification of assessment criteria and feedback to students. This paper describes the use of Review and presents an analysis of post-ReView data that has institutional implications for improving assessment and self-assessment practices
The Geometry of Warped Product Singularities
In this article the degenerate warped products of singular semi-Riemannian
manifolds are studied. They were used recently by the author to handle
singularities occurring in General Relativity, in black holes and at the
big-bang. One main result presented here is that a degenerate warped product of
semi-regular semi-Riemannian manifolds with the warping function satisfying a
certain condition is a semi-regular semi-Riemannian manifold. The connection
and the Riemann curvature of the warped product are expressed in terms of those
of the factor manifolds. Examples of singular semi-Riemannian manifolds which
are semi-regular are constructed as warped products. Applications include
cosmological models and black holes solutions with semi-regular singularities.
Such singularities are compatible with a certain reformulation of the Einstein
equation, which in addition holds at semi-regular singularities too.Comment: 14 page
A note on the uniqueness of global static decompositions
We discuss when static Killing vector fields are standard, that is, leading
to a global orthogonal splitting of the spacetime. We prove that such an
orthogonal splitting is unique whenever the natural space is compact. This is
carried out by proving that many notable spacelike submanifolds must be
contained in an orthogonal slice. Possible obstructions to the global splitting
are also considered.Comment: 6 pages, no figure
The Generalized Jacobi Equation
The Jacobi equation in pseudo-Riemannian geometry determines the linearized
geodesic flow. The linearization ignores the relative velocity of the
geodesics. The generalized Jacobi equation takes the relative velocity into
account; that is, when the geodesics are neighboring but their relative
velocity is arbitrary the corresponding geodesic deviation equation is the
generalized Jacobi equation. The Hamiltonian structure of this nonlinear
equation is analyzed in this paper. The tidal accelerations for test particles
in the field of a plane gravitational wave and the exterior field of a rotating
mass are investigated. In the latter case, the existence of an attractor of
uniform relative radial motion with speed is pointed
out. The astrophysical implications of this result for the terminal speed of a
relativistic jet is briefly explored.Comment: LaTeX file, 4 PS figures, 28 pages, revised version, accepted for
publication in Classical and Quantum Gravit
Singularity-Free Cylindrical Cosmological Model
A cylindrically symmetric perfect fluid spacetime with no curvature
singularity is shown. The equation of state for the perfect fluid is that of a
stiff fluid. The metric is diagonal and non-separable in comoving coordinates
for the fluid. It is proven that the spacetime is geodesically complete and
globally hyperbolic.Comment: LaTeX 2e, 8 page
A Note on Non-compact Cauchy surface
It is shown that if a space-time has non-compact Cauchy surface, then its
topological, differentiable, and causal structure are completely determined by
a class of compact subsets of its Cauchy surface. Since causal structure
determines its topological, differentiable, and conformal structure of
space-time, this gives a natural way to encode the corresponding structures
into its Cauchy surface
Linkage analysis of smoking initiation and quantity in Dutch sibling pairs.
The heritability of smoking initiation (SI) and number of cigarettes smoked (NC) was determined in 3657 Dutch twin pairs. For SI a heritability of 36% was found and for NC of 51%. Both SI and NC were also significantly influenced by environmental factors shared by family members. The etiological factors that influence these traits partly overlap. Linkage analyses were performed on data of 536 DZ twins and siblings from 192 families, forming 592 sibling pairs. Results suggested QTLs on chromosome 6 (LOD=3.05) and chromosome 14 (LOD=1.66) for SI and on chromosome 3 (LOD=1.98) for NC. Strikingly, on chromosome 10 a peak was found in the same region for both SI (LOD=1.92) and for NC (LOD=2.29) which may partly explain the overlapping etiological factors for SI and N
Aspects of noncommutative Lorentzian geometry for globally hyperbolic spacetimes
Connes' functional formula of the Riemannian distance is generalized to the
Lorentzian case using the so-called Lorentzian distance, the d'Alembert
operator and the causal functions of a globally hyperbolic spacetime. As a step
of the presented machinery, a proof of the almost-everywhere smoothness of the
Lorentzian distance considered as a function of one of the two arguments is
given. Afterwards, using a -algebra approach, the spacetime causal
structure and the Lorentzian distance are generalized into noncommutative
structures giving rise to a Lorentzian version of part of Connes'
noncommutative geometry. The generalized noncommutative spacetime consists of a
direct set of Hilbert spaces and a related class of -algebras of
operators. In each algebra a convex cone made of self-adjoint elements is
selected which generalizes the class of causal functions. The generalized
events, called {\em loci}, are realized as the elements of the inductive limit
of the spaces of the algebraic states on the -algebras. A partial-ordering
relation between pairs of loci generalizes the causal order relation in
spacetime. A generalized Lorentz distance of loci is defined by means of a
class of densely-defined operators which play the r\^ole of a Lorentzian
metric. Specializing back the formalism to the usual globally hyperbolic
spacetime, it is found that compactly-supported probability measures give rise
to a non-pointwise extension of the concept of events.Comment: 43 pages, structure of the paper changed and presentation strongly
improved, references added, minor typos corrected, title changed, accepted
for publication in Reviews in Mathematical Physic
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