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 $T^3$ 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

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

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

Effective stress-energy tensors, self-force, and broken symmetry

Deriving the motion of a compact mass or charge can be complicated by the presence of large self-fields. Simplifications are known to arise when these fields are split into two parts in the so-called Detweiler-Whiting decomposition. One component satisfies vacuum field equations, while the other does not. The force and torque exerted by the (often ignored) inhomogeneous "S-type" portion is analyzed here for extended scalar charges in curved spacetimes. If the geometry is sufficiently smooth, it is found to introduce effective shifts in all multipole moments of the body's stress-energy tensor. This greatly expands the validity of statements that the homogeneous R field determines the self-force and self-torque up to renormalization effects. The forces and torques exerted by the S field directly measure the degree to which a spacetime fails to admit Killing vectors inside the body. A number of mathematical results related to the use of generalized Killing fields are therefore derived, and may be of wider interest. As an example of their application, the effective shift in the quadrupole moment of a charge's stress-energy tensor is explicitly computed to lowest nontrivial order.Comment: 22 pages, fixed typos and simplified discussio

The clinical and cost-effectiveness of a Victim Improvement Package (VIP) for the reduction of chronic symptoms of depression or anxiety in older victims of common crime (the VIP trial): study protocol for a randomised controlled trial.

BACKGROUND: Older people are vulnerable to sustained high levels of psychosocial distress following a crime. A cognitive behavioural therapy (CBT)-informed psychological therapy, the Victim Improvement Package (VIP) may aid recovery. The VIP trial aims to test the clinical and cost-effectiveness of the VIP for alleviating depressive and anxiety symptoms in older victims of crime. METHODS/DESIGN: People aged 65 years or more who report being a victim of crime will be screened by Metropolitan Police Service Safer Neighbourhood Teams within a month of the crime for distress using the Patient Health Questionnaire-2 and the Generalised Anxiety Disorder-2. Those who screen positive will be signposted to their GP for assistance, and re-screened at 3 months. Participants who screen positive for depression and/or anxiety at re-screening are randomised to a CBT informed VIP added to treatment as usual (TAU) compared to TAU alone. The intervention consists of 10 individual 1-h sessions, delivered weekly by therapists from the mental health charity Mind. The primary outcome measure is the Beck Depression Inventory-II (BDI-II) and the Beck Anxiety Inventory (BAI), used as a composite measure, assessed at 6 months after the crime (post therapy) with a 9-month post-crime follow-up. Secondary outcome measures include the EQ-5D, and a modified Client Service Receipt Inventory. A total of 226 participants will be randomised VIP:TAU with a ratio 1:1, in order to detect a standardised difference of at least 0.5 between groups, using a mixed-effects linear-regression model with 90% power and a 5% significance level (adjusting for therapist clustering and potential drop-out). A cost-effectiveness analysis will incorporate intervention costs to compare overall health care costs and quality of life years between treatment arms. An embedded study will examine the impact of past trauma and engagement in safety behaviours and distress on the main outcomes. DISCUSSION: This trial should provide data on the clinical and cost-effectiveness of a CBT-informed psychological therapy for older victims of crime with anxiety and/or depressive symptoms and should demonstrate a model of integrated cross-agency working. Our findings should provide evidence for policy-makers, commissioners and clinicians responding to the needs of older victims of crime. TRIAL REGISTRATION: International Standard Randomised Controlled Trials Number, ID: ISRCTN16929670. Registered on 3 August 2016

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