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

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

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

General Transformation Formulas for Fermi-Walker Coordinates

We calculate the transformation and inverse transformation, in the form of Taylor expansions, from arbitrary coordinates to Fermi-Walker coordinates in tubular neighborhoods of arbitrary timelike paths for general spacetimes. Explicit formulas for coefficients and the Jacobian matrix are given.Comment: 23 pages. Corrected typos in the last two equations. Accepted for publication in Classical and Quantum Gravit

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

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

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

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

A Primary Care Nurse-Delivered Walking Intervention in Older Adults: PACE (Pedometer Accelerometer Consultation Evaluation)-Lift Cluster Randomised Controlled Trial.

Background: Brisk walking in older people can increase step-counts and moderate to vigorous intensity physical activity (MVPA) in ≥10-minute bouts, as advised in World Health Organization guidelines. Previous interventions have reported step-count increases, but not change in objectively measured MVPA in older people. We assessed whether a primary care nurse-delivered complex intervention increased objectively measured step-counts and MVPA. Methods and Findings: A total of 988 60–75 year olds, able to increase walking and randomly selected from three UK family practices, were invited to participate in a parallel two-arm cluster randomised trial; randomisation was by household. Two-hundred-ninety-eight people from 250 households were randomised between 2011 and 2012; 150 individuals to the intervention group, 148 to the usual care control group. Intervention participants received four primary care nurse physical activity (PA) consultations over 3 months, incorporating behaviour change techniques, pedometer step-count and accelerometer PA intensity feedback, and an individual PA diary and plan. Assessors were not blinded to group status, but statistical analyses were conducted blind. The primary outcome was change in accelerometry assessed average daily step-counts between baseline and 3 months, with change at 12 months a secondary outcome. Other secondary outcomes were change from baseline in time in MVPA weekly in ≥10-minute bouts, accelerometer counts, and counts/minute at 3 months and 12 months. Other outcomes were adverse events, anthropometric measures, mood, and pain. Qualitative evaluations of intervention participants and practice nurses assessed the intervention’s acceptability. At 3 months, eight participants had withdrawn or were lost to follow-up, 280 (94%) individuals provided primary outcome data. At 3 months changes in both average daily step-counts and weekly MVPA in ≥10-minute bouts were significantly higher in the intervention than control group: by 1,037 (95% CI 513–1,560) steps/day and 63 (95% CI 40–87) minutes/week, respectively. At 12 months corresponding differences were 609 (95% CI 104–1,115) steps/day and 40 (95% CI 17–63) minutes/week. Counts and counts/minute showed similar effects to steps and MVPA. Adverse events, anthropometry, mood, and pain were similar in the two groups. Participants and practice nurses found the intervention acceptable and enjoyable. Conclusions : The PACE-Lift trial increased both step-counts and objectively measured MVPA in ≥10-minute bouts in 60–75 year olds at 3 and 12 months, with no effect on adverse events. To our knowledge, this is the first trial in this age group to demonstrate objective MVPA increases and highlights the value of individualised support incorporating objective PA assessment in a primary care setting. Trial Registration: Controlled-Trials.com ISRCTN4212256