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

Intrusive memories following disaster: Relationship with peritraumatic responses and later affect

Cognitive theories of posttraumatic stress disorder (PTSD) suggest that intrusive memories result from disrupted information processing during traumatic memory encoding and are characterized by fear, helplessness, and horror at recall. Existing naturalistic studies are limited by the absence of direct comparisons between specific moments that do and do not correspond to intrusive memories. We tested predictions from cognitive theories of PTSD by comparing peritraumatic responses during moments experienced as intrusive memories versus distressing moments of the same traumatic event from the same individual not experienced as intrusive memories. A further comparison was with highly distressing moments experienced during the same event by individuals without intrusive memories. We utilized a psychometrically generated model to distinguish different peritraumatic reactions. Moments experienced as intrusive memories were characterized by higher peritraumatic distress, immobility, cognitive overload, and somatic dissociation when compared both to distressing moments from the same individual that did not intrude and to the most distressing memories of individuals without intrusions. Exploratory analyses indicated that at recall, intrusive memories were characterized by higher levels of primary traumatic emotions such as anxiety, fear, and helplessness in comparison with nonintrusive memories. Findings from this novel naturalistic design support predictions made by cognitive theories of PTSD and have implications for research and preventative interventions targeting intrusive memories. (PsycInfo Database Record (c) 2021 APA, all rights reserved)

Optimised performance of the backward longswing on rings

Many elite gymnasts perform the straight arm backward longswing on rings in competition. Since points are deducted if gymnasts possess motion on completion of the movement, the ability to successfully perform the longswing to a stationary final handstand is of great importance. Sprigings et al. (1998) found that for a longswing initiated from a still handstand the optimum performance of an inelastic planar simulation model resulted in a residual swing of more than 3° in the final handstand. For the present study, a three-dimensional simulation model of a gymnast swinging on rings, incorporating lateral arm movements used by gymnasts and mandatory apparatus elasticity, was used to investigate the possibility of performing a backward longswing initiated and completed in handstands with minimal swing. Root mean square differences between the actual and simulated performances for the orientations of the gymnast and rings cables, the combined cable tension and the extension of the gymnast were 3.2°, 1.0°, 270 N and 0.05 m respectively. The optimised simulated performance initiated from a handstand with 2.1° of swing and using realistic changes to the gymnast's technique resulted in 0.6° of residual swing in the final handstand. The sensitivity of the backward longswing to perturbations in the technique used for the optimised performance was determined. For a final handstand with minimal residual swing (2°) the changes in body configuration must be timed to within 15 m s while a delay of 30 m s will result in considerable residual swing (7°)

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

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

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