The abstract boundary---a new approach to singularities of manifolds

A new scheme is proposed for dealing with the problem of singularities in General Relativity. The proposal is, however, much more general than this. It can be used to deal with manifolds of any dimension which are endowed with nothing more than an affine connection, and requires a family \calc\ of curves satisfying a {\em bounded parameter property} to be specified at the outset. All affinely parametrised geodesics are usually included in this family, but different choices of family \calc\ will in general lead to different singularity structures. Our key notion is the {\em abstract boundary\/} or {\em $a$-boundary\/} of a manifold, which is defined for any manifold \calm\ and is independent of both the affine connection and the chosen family \calc\ of curves. The $a$-boundary is made up of equivalence classes of boundary points of \calm\ in all possible open embeddings. It is shown that for a pseudo-Riemannian manifold (\calm,g) with a specified family \calc\ of curves, the abstract boundary points can then be split up into four main categories---regular, points at infinity, unapproachable points and singularities. Precise definitions are also provided for the notions of a {\em removable singularity} and a {\em directional singularity}. The pseudo-Riemannian manifold will be said to be singularity-free if its abstract boundary contains no singularities. The scheme passes a number of tests required of any theory of singularities. For instance, it is shown that all compact manifolds are singularity-free, irrespective of the metric and chosen family \calc.Comment: 40 pages (amslatex) + 5 uuencoded figures (A postscript version is also available on http://einstein.anu.edu.au/), CMA Maths. Research Report No. MRR028-9

Growth promotion by homocysteine but not by homocysteic acid: a role for excessive growth in homocystinuria or proliferation in hyperhomocysteinemia?

AbstractExcessive growth of long bones in patients with homocystinuria is still unexplained and previous work incriminating homocysteic acid could not be confirmed by others. In vitro studies from our laboratory showed that homocysteine stimulated growth in a clonogenic assay. This observation made us study plasma cyclin dependent kinase (CDK), homocyst(e)ine and homocysteic acid in 10 patients with homocystinuria and 20 controls. In addition, homocysteine and homocysteic acid were tested in a clonogenic assay to correlate the growth promoting activity with CDK. Plasma CDK (protein) correlated strongly with homocysteine (r=0.84) but not with homocysteic acid. Supernatants of the clonogenic assay samples showed up to three times higher CDK levels in the presence of homocyst(e)ine but not homocysteic acid. In vitro data and the strong correlation between homocysteine and CDK suggest a role for homocysteine stimulating CDK, the starter of mitosis, with subsequent stimulation of growth

Post-Newtonian Cosmology

Newtonian Cosmology is commonly used in astrophysical problems, because of its obvious simplicity when compared with general relativity. However it has inherent difficulties, the most obvious of which is the non-existence of a well-posed initial value problem. In this paper we investigate how far these problems are met by using the post-Newtonian approximation in cosmology.Comment: 12 pages, Late

Perceptions of date rape on a college campus

The occurrences and attitudes about date rape were surveyed at a small Midwestern college campus. They were measured through a vignette in which a date rape occurred, and a survey which contained demographic questions, true/false and Likert scale items. The Likert items were divided into three types: 1) nine questions a date rapist would strongly agree with (male initiator items); 2) nine questions someone sophisticated about rape would strongly agree with (egalitarian items); and 3) three neutral questions (bystander items). From these items a male-initiator and egalitarian score was derived for each S. T-test results indicate that male Ss had a higher male-initiator score than female Ss, 1(329) = 2.04, a \u3c .001, while female Ss had a higher egalitarian score than male Ss, t(328) = 1.18, a\u3c .001. Pearson correlations also revealed a significant negative relationship between male-initiator and egalitarian scores, r = -.61, p_\u3c .001; and a significant positive relationship between the age of the Ss and their egalitarian score, r = .14, p.= .006. This study advocates the need to dispel traditional myths concerning sex roles and date rape through rape awareness and open communication

A Radiation Scalar for Numerical Relativity

This letter describes a scalar curvature invariant for general relativity with a certain, distinctive feature. While many such invariants exist, this one vanishes in regions of space-time which can be said unambiguously to contain no gravitational radiation. In more general regions which incontrovertibly support non-trivial radiation fields, it can be used to extract local, coordinate-independent information partially characterizing that radiation. While a clear, physical interpretation is possible only in such radiation zones, a simple algorithm can be given to extend the definition smoothly to generic regions of space-time.Comment: 4 pages, 1 EPS figur

Integer Partitions and Exclusion Statistics

We provide a combinatorial description of exclusion statistics in terms of minimal difference $p$ partitions. We compute the probability distribution of the number of parts in a random minimal $p$ partition. It is shown that the bosonic point $p=0$ is a repulsive fixed point for which the limiting distribution has a Gumbel form. For all positive $p$ the distribution is shown to be Gaussian.Comment: 16 pages, 4 .eps figures include

Shear-Free Gravitational Waves in an Anisotropic Universe

We study gravitational waves propagating through an anisotropic Bianchi I dust-filled universe (containing the Einstein-de-Sitter universe as a special case). The waves are modeled as small perturbations of this background cosmological model and we choose a family of null hypersurfaces in this space-time to act as the histories of the wavefronts of the radiation. We find that the perturbations we generate can describe pure gravitational radiation if and only if the null hypersurfaces are shear-free. We calculate the gauge-invariant small perturbations explicitly in this case. How these differ from the corresponding perturbations when the background space-time is isotropic is clearly exhibited.Comment: 32 pages, accepted for publication in Physical Review

The Embedding of Schwarzschild in Braneworld

The braneworlds models were inspired partly by Kaluza-Klein's theory, where both the gravitational and the gauge fields are obtained from the geometry of a higher dimensional space. The positive aspects of these models consist in perspectives of modifications it could bring in to particle physics, such as: unification in a TeV scale, quantum gravity in this scale and deviation of Newton's law for small distances. One of the principles of these models is to suppose that all space-times can be embedded in a bulk of higher dimension. The main result in these notes is a theorem showing a mathematical inconsistency of the Randall-Sundrum braneworld model, namely that the Schwarzschild space-time cannot be embedded locally and isometrically in a five dimensional bulk with constant curvature,(for example AdS-5). From the point of view of semi-Riemannian geometry this last result represents a serious restriction to the Randall-Sundrum's braneworld model.Comment: Published in the Int. J. Theor. Phys, 200

You Can't Get Through Szekeres Wormholes - or - Regularity, Topology and Causality in Quasi-Spherical Szekeres Models

The spherically symmetric dust model of Lemaitre-Tolman can describe wormholes, but the causal communication between the two asymptotic regions through the neck is even less than in the vacuum (Schwarzschild-Kruskal-Szekeres) case. We investigate the anisotropic generalisation of the wormhole topology in the Szekeres model. The function E(r, p, q) describes the deviation from spherical symmetry if \partial_r E \neq 0, but this requires the mass to be increasing with radius, \partial_r M > 0, i.e. non-zero density. We investigate the geometrical relations between the mass dipole and the locii of apparent horizon and of shell-crossings. We present the various conditions that ensure physically reasonable quasi-spherical models, including a regular origin, regular maxima and minima in the spatial sections, and the absence of shell-crossings. We show that physically reasonable values of \partial_r E \neq 0 cannot compensate for the effects of \partial_r M > 0 in any direction, so that communication through the neck is still worse than the vacuum. We also show that a handle topology cannot be created by identifying hypersufaces in the two asymptotic regions on either side of a wormhole, unless a surface layer is allowed at the junction. This impossibility includes the Schwarzschild-Kruskal-Szekeres case.Comment: zip file with LaTeX text + 6 figures (.eps & .ps). 47 pages. Second replacement corrects some minor errors and typos. (First replacement prints better on US letter size paper.

Collisions of Einstein-Conformal Scalar Waves

A large class of solutions of the Einstein-conformal scalar equations in D=2+1 and D=3+1 is identified. They describe the collisions of asymptotic conformal scalar waves and are generated from Einstein-minimally coupled scalar spacetimes via a (generalized) Bekenstein transformation. Particular emphasis is given to the study of the global properties and the singularity structure of the obtained solutions. It is shown, that in the case of the absence of pure gravitational radiation in the initial data, the formation of the final singularity is not only generic, but is even inevitable.Comment: 17 pages, LaTe