1,009 research outputs found
An ADM 3+1 formulation for Smooth Lattice General Relativity
A new hybrid scheme for numerical relativity will be presented. The scheme
will employ a 3-dimensional spacelike lattice to record the 3-metric while
using the standard 3+1 ADM equations to evolve the lattice. Each time step will
involve three basic steps. First, the coordinate quantities such as the Riemann
and extrinsic curvatures are extracted from the lattice. Second, the 3+1 ADM
equations are used to evolve the coordinate data, and finally, the coordinate
data is used to update the scalar data on the lattice (such as the leg
lengths). The scheme will be presented only for the case of vacuum spacetime
though there is no reason why it could not be extended to non-vacuum
spacetimes. The scheme allows any choice for the lapse function and shift
vectors. An example for the Kasner cosmology will be presented and it
will be shown that the method has, for this simple example, zero discretisation
error.Comment: 18 pages, plain TeX, 5 epsf figues, gzipped ps file also available at
http://newton.maths.monash.edu.au:8000/preprints/3+1-slgr.ps.g
Fast algorithms for computing defects and their derivatives in the Regge calculus
Any practical attempt to solve the Regge equations, these being a large
system of non-linear algebraic equations, will almost certainly employ a
Newton-Raphson like scheme. In such cases it is essential that efficient
algorithms be used when computing the defect angles and their derivatives with
respect to the leg-lengths. The purpose of this paper is to present details of
such an algorithm.Comment: 38 pages, 10 figure
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 Smooth Lattice construction of the Oppenheimer-Snyder spacetime
We present test results for the smooth lattice method using an
Oppenheimer-Snyder spacetime. The results are in excellent agreement with
theory and numerical results from other authors.Comment: 60 pages, 28 figure
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 â„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
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
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
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