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Leo Brewin

Publications and source records attributed to Leo Brewin.

At least 19 recordsLinked to original sources
arXiv2022

Riemann Normal Coordinate expansions using Cadabra

Riemann normal coordinate expansions of the metric and other geometrical quantities, including the geodesic arc-length, will be presented. All of the results are given to fifth-order in the curvature and were obtained using the computer algebra package Cadabra.

gr-qc↗
arXiv2019

Using Cadabra for tensor computations in General Relativity

Cadabra is an open access program ideally suited to complex tensor commutations in General Relativity. Tensor expressions are written in LaTeX while an enhanced version of Python is used to control the computations. This tutorial assumes no prior knowledge of Cadabra. It consists of a series of examples covering a range of topics from basic syntax such as declarations, functions, program control, component computations, input and output through to complete computations including a derivation of two of the BSSN equations from the ADM equations. Numerous exercises are included along with complete solutions. All of the source code for the examples, exercises and solutions are available on GitHub.

gr-qc↗
arXiv2017

Evolutions of Gowdy, Brill and Teukolsky initial data on a smooth lattice

Numerical results, based on a lattice method for computational general relativity, will be presented for Cauchy evolution of initial data for the Brill, Teukolsky and polarised Gowdy space-times. The simple objective of this paper is to demonstrate that the lattice method can, at least for these space-times, match results obtained from contemporary methods. Some of the issues addressed in this paper include the handling of axisymmetric instabilities (in the Brill space-time) and an implementation of a Sommerfeld radiation condition for the Brill and Teukolsky space-times. It will be shown that the lattice method performs particularly well in regard to the passage of the waves through the outer boundary. Questions concerning multiple black-holes, mesh refinement and long term stability will not be discussed here but may form the basis of future work.

gr-qc↗
arXiv2015

Two dimensional axisymmetric smooth lattice Ricci flow

A lattice based method will be presented for numerical investigations of Ricci flow. The method will be applied to the particular case of 2-dimensional axially symmetric initial data on manifolds with S^2 topology. Results will be presented that show that the method works well and agrees with results obtained using contemporary finite difference methods.

math.DG↗
arXiv2015

Curvature corrected estimates for geodesic arc-length

We will develop simple relations between the arc-lengths of a pair of geodesics that share common end-points. The two geodesics differ only by the requirement that one is constrained to lie in a subspace of the parent manifold. We will present two applications of our results. In the first example we explore the convergence of Gaussian curvature estimates on a simple triangular mesh. The second example demonstrates an improved error estimate for the area of a Schwarz lantern.

math.DG↗
arXiv2015

A numerical study of the Regge Calculus and Smooth Lattice methods on a Kasner cosmology

Two lattice based methods for numerical relativity, the Regge Calculus and the Smooth Lattice Relativity, will be compared with respect to accuracy and computational speed in a full 3+1 evolution of initial data representing a standard Kasner cosmology. It will be shown that both methods provide convergent approximations to the exact Kasner cosmology. It will also be shown that the Regge Calculus is of the order of 110 times slower than the Smooth Lattice method.

gr-qc↗
arXiv2011

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.

gr-qc↗
arXiv2011

An Einstein-Bianchi system for Smooth Lattice General Relativity. I. The Schwarzschild spacetime

The second Bianchi identity can be recast as an evolution equation for the Riemann curvatures. Here we will report on such a system for a vacuum static spherically symmetric spacetime. This is the first of two papers. In the following paper we will extend the ideas developed here to general vacuum spacetimes. In this paper we will demonstrate our ideas on a Schwarzschild spacetime and give detailed numerical results. For suitable choices of lapse function we find that the system gives excellent results with long term stability.

gr-qc↗
arXiv2011

An Einstein-Bianchi system for Smooth Lattice General Relativity. II. 3+1 vacuum spacetimes

We will present a complete set of equations, in the form of an Einstein-Bianchi system, that describe the evolution of generic smooth lattices in spacetime. All 20 independent Riemann curvatures will be evolved in parallel with the leg-lengths of the lattice. We will show that the evolution equations for the curvatures forms a hyperbolic system and that the associated constraints are preserved. This work is a generalisation of our previous paper arXiv:1101.3171 on the Einstein-Bianchi system for the Schwarzschild spacetime to general 3+1 vacuum spacetimes.

gr-qc↗
arXiv2010

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.

gr-qc↗
arXiv2009

A brief introduction to Cadabra: a tool for tensor computations in General Relativity

Cadabra is a powerful computer program for the manipulation of tensor equations. It was designed for use in high energy physics but its rich structure and ease of use lends itself well to the routine computations required in General Relativity. Here we will present a series of simple examples showing how Cadabra may be used, including verifying that the Levi-Civita connection is a metric connection and a derivation of the Gauss equation between induced and ambient curvatures.

gr-qc↗
arXiv2006

A simple expression for the ADM mass

We show by an almost elementary calculation that the ADM mass of an asymptotically flat space can be computed as a limit involving a rate of change of area of a closed 2-surface. The result is essentially the same as that given by Brown and York. We will prove this result in two ways, first by direct calculation from the original formula as given by Arnowitt, Deser and Misner and second as a corollary of an earlier result by Brewin for the case of simplicial spaces.

gr-qc↗
arXiv2003

Classroom computer animations of relativistic objects

This is a short note to announce the availability of some movies that may be useful in classroom discussions on the photographic appearance of objects moving at relativistic speeds. The images are based on special relativity with no account taken of (other than to ignore) the effects of doppler shifts, intensity shifts or gravitational effects.

physics.ed-ph↗
arXiv2001

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.

gr-qc↗