Is the Regge Calculus a consistent approximation to General Relativity?

We will ask the question of whether or not the Regge calculus (and two related simplicial formulations) is a consistent approximation to General Relativity. Our criteria will be based on the behaviour of residual errors in the discrete equations when evaluated on solutions of the Einstein equations. We will show that for generic simplicial lattices the residual errors can not be used to distinguish metrics which are solutions of Einstein's equations from those that are not. We will conclude that either the Regge calculus is an inconsistent approximation to General Relativity or that it is incorrect to use residual errors in the discrete equations as a criteria to judge the discrete equations.Comment: 27 pages, plain TeX, very belated update to match journal articl

Long term stable integration of a maximally sliced Schwarzschild black hole using a smooth lattice method

We will present results of a numerical integration of a maximally sliced Schwarzschild black hole using a smooth lattice method. The results show no signs of any instability forming during the evolutions to t=1000m. The principle features of our method are i) the use of a lattice to record the geometry, ii) the use of local Riemann normal coordinates to apply the 1+1 ADM equations to the lattice and iii) the use of the Bianchi identities to assist in the computation of the curvatures. No other special techniques are used. The evolution is unconstrained and the ADM equations are used in their standard form.Comment: 47 pages including 26 figures, plain TeX, also available at http://www.maths.monash.edu.au/~leo/preprint

A longitudinal investigation of maternal and child 'anxious cognitions'

Overestimation of threat and underestimation of coping have been frequently reported amongst anxious adults and children. The current study examines the longitudinal relationship between mothers' anxious cognitions and expectations about their child, and children's anxious cognitions. 54 children (aged 10-11 years) and their mothers reported on their interpretation of ambiguous scenarios at two time points. Mothers also reported on their expectations about their child's reaction to ambiguous situations. Significant cross-sectional associations were found between mother and child anticipation of distress. Associations were most consistent between mothers' expectations and children's cognitions. Furthermore, based on regression analyses, mothers' expectations predicted change in children's anxious cognitions over time. Evidence for a reciprocal relationship, that child cognitions predict change in mothers' expectations, was found for girls. The results provide empirical support for potential influences on the development of children's 'anxious cognitive style,' and suggest targets for preventing and reducing maladaptive cognitions in children

Regge calculus and Ashtekar variables

Spacetime discretized in simplexes, as proposed in the pioneer work of Regge, is described in terms of selfdual variables. In particular, we elucidate the "kinematic" structure of the initial value problem, in which 3--space is divided into flat tetrahedra, paying particular attention to the role played by the reality condition for the Ashtekar variables. An attempt is made to write down the vector and scalar constraints of the theory in a simple and potentially useful way.Comment: 10 pages, uses harvmac. DFUPG 83/9

Regge Calculus as a Fourth Order Method in Numerical Relativity

The convergence properties of numerical Regge calculus as an approximation to continuum vacuum General Relativity is studied, both analytically and numerically. The Regge equations are evaluated on continuum spacetimes by assigning squared geodesic distances in the continuum manifold to the squared edge lengths in the simplicial manifold. It is found analytically that, individually, the Regge equations converge to zero as the second power of the lattice spacing, but that an average over local Regge equations converges to zero as (at the very least) the third power of the lattice spacing. Numerical studies using analytic solutions to the Einstein equations show that these averages actually converge to zero as the fourth power of the lattice spacing.Comment: 14 pages, LaTeX, 8 figures mailed in separate file or email author directl

On the convergence of Regge calculus to general relativity

Motivated by a recent study casting doubt on the correspondence between Regge calculus and general relativity in the continuum limit, we explore a mechanism by which the simplicial solutions can converge whilst the residual of the Regge equations evaluated on the continuum solutions does not. By directly constructing simplicial solutions for the Kasner cosmology we show that the oscillatory behaviour of the discrepancy between the Einstein and Regge solutions reconciles the apparent conflict between the results of Brewin and those of previous studies. We conclude that solutions of Regge calculus are, in general, expected to be second order accurate approximations to the corresponding continuum solutions.Comment: Updated to match published version. Details of numerical calculations added, several sections rewritten. 9 pages, 4 EPS figure

Outreach and screening following the 2005 London bombings: usage and outcomes

BACKGROUND: Little is known about how to remedy the unmet mental health needs associated with major terrorist attacks, or what outcomes are achievable with evidence-based treatment. This article reports the usage, diagnoses and outcomes associated with the 2-year Trauma Response Programme (TRP) for those affected by the 2005 London bombings.MethodFollowing a systematic and coordinated programme of outreach, the contact details of 910 people were obtained by the TRP. Of these, 596 completed a screening instrument that included the Trauma Screening Questionnaire (TSQ) and items assessing other negative responses. Those scoring ≥6 on the TSQ, or endorsing other negative responses, received a detailed clinical assessment. Individuals judged to need treatment (n=217) received trauma-focused cognitive-behaviour therapy (TF-CBT) or eye movement desensitization and reprocessing (EMDR). Symptom levels were assessed pre- and post-treatment with validated self-report measures of post-traumatic stress disorder (PTSD) and depression, and 66 were followed up at 1 year. RESULTS: Case finding relied primarily on outreach rather than standard referral pathways such as primary care. The effect sizes achieved for treatment of DSM-IV PTSD exceeded those usually found in randomized controlled trials (RCTs) and gains were well maintained an average of 1 year later. CONCLUSIONS: Outreach with screening, linked to the provision of evidence-based treatment, seems to be a viable method of identifying and meeting mental health needs following a terrorist attack. Given the failure of normal care pathways, it is a potentially important approach that merits further evaluation

Differential pathways to adult metabolic dysfunction following poor nutrition at two critical developmental periods in sheep

Epidemiological and experimental studies suggest early nutrition has long-term effects on susceptibility to obesity, cardiovascular and metabolic diseases. Small and large animal models confirm the influence of different windows of sensitivity, from fetal to early postnatal life, on offspring phenotype. We showed previously that undernutrition in sheep either during the first month of gestation or immediately after weaning induces differential, sex-specific changes in adult metabolic and cardiovascular systems. The current study aims to determine metabolic and molecular changes that underlie differences in lipid and glucose metabolism induced by undernutrition during specific developmental periods in male and female sheep. Ewes received 100% (C) or 50% nutritional requirements (U) from 1–31 days gestation, and 100% thereafter. From weaning (12 weeks) to 25 weeks, offspring were then fed either ad libitum (CC, UC) or were undernourished (CU, UU) to reduce body weight to 85% of their individual target. From 25 weeks, all offspring were fed ad libitum. A cohort of late gestation fetuses were studied after receiving either 40% nutritional requirements (1–31 days gestation) or 50% nutritional requirements (104–127 days gestation). Post-weaning undernutrition increased in vivo insulin sensitivity, insulin receptor and glucose transporter 4 expression in muscle, and lowered hepatic methylation at the delta-like homolog 1/maternally expressed gene 3 imprinted cluster in adult females, but not males. Early gestational undernutrition induced lower hepatic expression of gluconeogenic factors in fetuses and reduced in vivo adipose tissue insulin sensitivity in adulthood. In males, undernutrition in early gestation increased adipose tissue lipid handling mechanisms (lipoprotein lipase, glucocorticoid receptor expression) and hepatic methylation within the imprinted control region of insulin-like growth factor 2 receptor in adulthood. Therefore, undernutrition during development induces changes in mechanisms of lipid and glucose metabolism which differ between tissues and sexes dependent on the period of nutritional restriction. Such changes may increase later life obesity and dyslipidaemia risk

A fully (3+1)-D Regge calculus model of the Kasner cosmology

We describe the first discrete-time 4-dimensional numerical application of Regge calculus. The spacetime is represented as a complex of 4-dimensional simplices, and the geometry interior to each 4-simplex is flat Minkowski spacetime. This simplicial spacetime is constructed so as to be foliated with a one parameter family of spacelike hypersurfaces built of tetrahedra. We implement a novel two-surface initial-data prescription for Regge calculus, and provide the first fully 4-dimensional application of an implicit decoupled evolution scheme (the ``Sorkin evolution scheme''). We benchmark this code on the Kasner cosmology --- a cosmology which embodies generic features of the collapse of many cosmological models. We (1) reproduce the continuum solution with a fractional error in the 3-volume of 10^{-5} after 10000 evolution steps, (2) demonstrate stable evolution, (3) preserve the standard deviation of spatial homogeneity to less than 10^{-10} and (4) explicitly display the existence of diffeomorphism freedom in Regge calculus. We also present the second-order convergence properties of the solution to the continuum.Comment: 22 pages, 5 eps figures, LaTeX. Updated and expanded versio

Area Regge Calculus and Discontinuous Metrics

Taking the triangle areas as independent variables in the theory of Regge calculus can lead to ambiguities in the edge lengths, which can be interpreted as discontinuities in the metric. We construct solutions to area Regge calculus using a triangulated lattice and find that on a spacelike hypersurface no such discontinuity can arise. On a null hypersurface however, we can have such a situation and the resulting metric can be interpreted as a so-called refractive wave.Comment: 18 pages, 1 figur