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

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 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

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

Discrete quantum gravity in the framework of Regge calculus formalism

An approach to the discrete quantum gravity based on the Regge calculus is discussed which was developed in a number of our papers. Regge calculus is general relativity for the subclass of general Riemannian manifolds called piecewise flat ones. Regge calculus deals with the discrete set of variables, triangulation lengths, and contains continuous general relativity as a particular limiting case when the lengths tend to zero. In our approach the quantum length expectations are nonzero and of the order of Plank scale $10^{-33}cm$. This means the discrete spacetime structure on these scales.Comment: LaTeX, 16 pages, to appear in JET

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

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