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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… (More)
We present the results of using the computer algebra program Cadabra to develop Riemann normal coordinate expansions of the metric and other geometrical quantities, in particular the geodesic arc-length. All of the results are given to sixth-order in the curvature tensor.
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… (More)
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… (More)
Motivated by a recent study which cast 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… (More)
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… (More)
It will be shown that the truncation error for the Regge Calculus, as an approximation to Einstein's equations, varies as O(∆ 2) where ∆ is the typical discretization scale. This result applies to any metric, whether or not it is a solution of the vacuum Einstein equations. It is in this sense that the Regge Calculus is not a discrete representation of… (More)
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.
We will show that the ADM 3+1 evolution equations, for a zero shift vector, arise naturally from the equations for the second variation of arc-length.
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 [1, 2]. We will prove this result in two ways, first by direct calculation from the original formula as given by… (More)