IDLinearWaves

Gabrielle Allen, Tom Goodale, Gerd Lanfermann, Joan Masso,
Mark Miller, Malcolm Tobias, Paul Walker

Date

Abstract

Provides gravitational wave solutions to the linearized Einstein equations

1 Purpose

There are two different linearized initial data sets provided, plane waves and Teukolsky waves.

2 Plane Waves

A full description of plane waves can be found in the PhD Thesis of Malcolm Tobias, The Numerical Evolution of Gravitational Waves, which can be found at http://wugrav.wustl.edu/Papers/Thesis97/Thesis97.html.

Plane waves travelling in arbitrary directions can be specified. For these plane waves the TT gauge is assumed (the metric perturbations are transverse to the direction of propagation, and the metric is traceless). In the case of waves travelling along the zdirection this would give the plus solution

hxx = hyy = f(t ± z),hxy = hxz = hyz = hzz = 0

and the cross solution

hxy = hyx = f(t ± z),hyz = hxx = hyy = hzz = 0

This thorn implements the plus solution, with the waveform f(t ± z) having the form of a Gaussian modulated sine function. Now working with a general direction of propagation k we have the plane wave solution:

f(t,x,y,z) = Aine(kipxi+ω p(tra))2 cos(kixi + ωt) + A oute(kipxiω p(tra))2 cos(kixi ωt)

and

gxx = 1 + f[cos2ϕ cos𝜃 sin2ϕ] gxy = fsin2𝜃sinϕcosϕ gxz = fsin𝜃cos𝜃sinϕ gyy = 1 + f[sin2ϕ cos2𝜃cos2ϕ] gyz = fsin𝜃cos𝜃cosϕ gzz = 1 fsin2𝜃

The extrinsic curvature is then calculated from

Kij = 1 2αġij (1)