## IDLinearWaves

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 $z-$direction this would give the plus solution

${h}_{xx}=-{h}_{yy}=f\left(t±z\right),{h}_{xy}={h}_{xz}={h}_{yz}={h}_{zz}=0$

and the cross solution

${h}_{xy}={h}_{yx}=f\left(t±z\right),{h}_{yz}={h}_{xx}={h}_{yy}={h}_{zz}=0$

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

$f\left(t,x,y,z\right)={A}_{in}{e}^{-{\left({k}_{i}^{p}{x}^{i}+{\omega }_{p}\left(t-{r}_{a}\right)\right)}^{2}}cos\left({k}_{i}{x}^{i}+\omega t\right)+{A}_{out}{e}^{-{\left({k}_{i}^{p}{x}^{i}-{\omega }_{p}\left(t-{r}_{a}\right)\right)}^{2}}cos\left({k}_{i}{x}^{i}-\omega t\right)$

and

$\begin{array}{rcll}{g}_{xx}& =& 1+f\left[{cos}^{2}\varphi -{cos}^{𝜃}{sin}^{2}\varphi \right]& \text{}\\ {g}_{xy}& =& -f{sin}^{2}𝜃sin\varphi cos\varphi & \text{}\\ {g}_{xz}& =& fsin𝜃cos𝜃sin\varphi & \text{}\\ {g}_{yy}& =& 1+f\left[{sin}^{2}\varphi -co{s}^{2}𝜃{cos}^{2}\varphi \right]& \text{}\\ {g}_{yz}& =& fsin𝜃cos𝜃cos\varphi & \text{}\\ {g}_{zz}& =& 1-f{sin}^{2}𝜃& \text{}\end{array}$

The extrinsic curvature is then calculated from

 ${K}_{ij}=-\frac{1}{2\alpha }{ġ}_{ij}$ (1)