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All of the following are stored as PDF files.
A simple expression for the ADM mass
| Authors |
Leo Brewin
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| Abstract |
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.
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| Reference |
General Relativity and Gravitation. Vol.39(2007) pp.521-528 |
Long term stable integration of a maximally sliced
Schwarzschild black hole using a smooth lattice method
| Authors |
Leo Brewin
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| Abstract |
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.
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| Reference |
Classical and Quantum Gravity. Vol.19(2002) pp.429-456 |
On the convergence of Regge calculus to general relativity.
| Authors |
Leo Brewin and Adrian Gentle
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| Abstract |
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.
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| Reference |
Classical and Quantum Gravity. Vol.18(2001) pp.517-525 |
Is the Regge Calculus a consistent approximation to General Relativity?
| Authors |
Leo Brewin
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| Abstract |
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 between 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.
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| Reference |
General Relativity and Gravitation, Vol.32(2000) pp.897-918. |
Riemann Normal Coordinates, Smooth Lattices and Numerical Relativity.
| Authors |
Leo Brewin
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| Abstract |
A new lattice based scheme for numerical relativity will be presented. The
scheme uses the same data as would be used in the Regge calculus (eg. a set of
leg lengths on a simplicial lattice) but it differs significantly in the way
that the field equations are computed. In the new method the standard Einstein
field equations are applied directly to the lattice. This is done by using
locally defined Riemann normal coordinates to interpolate a smooth metric over
local groups of cells of the lattice. Results for the time symmetric initial
data for the Schwarzschild spacetime will be presented. It will be shown that
the scheme yields second order accurate estimates (in the lattice spacing) for
the metric and the curvature. It will also be shown that the Bianchi identities
play an essential role in the construction of the Schwarzschild initial data.
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| Reference |
Classical and Quantum Gravity. Vol.15(1998) pp.3085-3120 |
An ADM 3+1 formulation of Smooth Lattice General Relativity.
| Authors |
Leo Brewin
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| Abstract |
A new hybrid algorithm for numerical relativity will be presented. We will
employ a lattice in which the metric is recorded as pure scalar quantities such
as the geodesic leg lengths. The dynamics of the lattice will be governed by the
standard ADM 3+1 equations which will be given as a set of second order ordinary
differential equations for the scalar data of the lattice. The only approximation
introduced in this algorithm is in the approximation of a smooth metric by a
lattice. No approximations are introduced into the field equations nor in their
application to the lattice. An example for the Kasner $T^3$ cosmology will be
presented and the results will be shown to be in excellent agreement with the
continuum solution.
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| Reference |
Classical and Quantum Gravity. Vol.15(1998) pp.2427-2449 |
Riemann Normal Coordinates.
| Authors |
Leo Brewin
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| Abstract |
This is just a collection of my notes on Riemann normal coordinates, including
standard defintions as well as applications to smooth lattices (eg. formulas
for the lengths of short geodesics and the angles between pair of geodesics).
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| Reference |
Unpublished, but here is a preprint |
A Finite Element formulation for Simplicial General Relativity.
| Authors |
Leo Brewin
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| Abstract |
A new set of equations for General Relativity on a simplicial lattice will be
presented. The equations will be obtained directly from the standard vacuum
Einstein equations by way of a finite element procedure. Though the equations
will have a Regge calculus feel to them, it will be shown that they are quite
distinct from Regge's equations.
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| Reference |
Unpublished, but here is a preprint |
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