Applications of smooth lattice general relativity

Abstract

We present the numerical evolution of two spacetimes using a lattice-based formulation of the 3+1 formalism called smooth lattice general relativity, with the intention of comparing it to more conventional methods. There are several distinguishing features of this approach. We use a lattice composed of short geodesic legs to approximate spacetime. Attached to each vertex of the lattice is a Riemann normal coordinate frame, so that in a sufficiently small neighbourhood of each vertex, the metric is locally flat. A collection of transformations relates the data in adjacent frames. The dynamics are governed by evolution equations for the lattice legs, the extrinsic curvature tensor, and the Riemann curvature tensor. The first spacetime we investigated was the Gowdy spacetime with Tᶟ topology, evolving forwards in time. Using a lattice adapted to the symmetries of the system, we explicitly derived the transformation between Riemann normal coordinate frames in order to find expressions for the time derivatives of the Riemann tensor. We compared the quality of results obtained using smooth lattice general relativity to both the exact solution and a direct solution of Einstein's equations. We found that our results were consistent with the exact answer, however instabilities prevented long term evolutions. The other spacetime we looked at was the axisymmetric Brill wave spacetime in 2+1 dimensions, which has fewer symmetries than the Gowdy spacetime. As there is no exact solution, we compared our results against standard numerical techniques used in general relativity. We also demonstrated how to incorporate the cartoon method into smooth lattice general relativity, which provided a means for performing evolutions in a restricted 3+1 setting with Cartesian coordinates. Although in both cases we only managed evolutions of low amplitude waves, we again found agreement with the expected results.