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\documentclass{article}
% Use the Cactus ThornGuide style file
% (Automatically used from Cactus distribution, if you have a
% thorn without the Cactus Flesh download this from the Cactus
% homepage at www.cactuscode.org)
\usepackage{../../../../doc/ThornGuide/cactus}
\begin{document}
\title{IDLinearWaves}
\author{Gabrielle Allen, Tom Goodale, Gerd Lanfermann, Joan Masso, \\ Mark Miller, Malcolm Tobias, Paul Walker}
\date{$ $Date$ $}
\maketitle
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% START CACTUS THORNGUIDE
\begin{abstract}
Provides gravitational wave solutions to the linearized Einstein equations
\end{abstract}
\section{Purpose}
There are two different linearized initial data sets provided, plane waves
and Teukolsky waves.
\section{Plane Waves}
A full description of plane waves can be found in the PhD Thesis of
Malcolm Tobias, {\it The Numerical Evolution of Gravitational Waves},
which can be found at {\tt 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 {\it plus} solution
$$
h_{xx}=-h_{yy}=f(t\pm z), h_{xy}=h_{xz}=h_{yz}=h_{zz} = 0
$$
and the {\it cross} solution
$$
h_{xy}=h_{yx}=f(t\pm z), h_{yz}=h_{xx}=h_{yy}=h_{zz}=0
$$
This thorn implements the {\tt plus} solution, with the waveform
$f(t\pm z)$ having the form of a Gaussian modulated sine function.
Now working with a general direction of propagation $k$ we have the
plane wave solution:
$$
f(t,x,y,z) = A_{in} e^{-(k_i^p x^i + \omega_p(t-r_a) )^2} \cos(k_ix^i+\omega t)
+ A_{out} e^{-(k_i^p x^i -\omega_p(t-r_a))^2} \cos(k_i x^i - \omega t)
$$
and
\begin{eqnarray*}
g_{xx}&=& 1 + f[\cos^2\phi - \cos^\theta\sin^2\phi]
\\
g_{xy}&=& - f \sin^2 \theta \sin \phi \cos \phi
\\
g_{xz} &=& f \sin\theta \cos\theta \sin\phi
\\
g_{yy} &=& 1+f [\sin^2\phi - cos^2\theta \cos^2\phi]
\\
g_{yz} &=& f \sin\theta \cos\theta \cos\phi
\\
g_{zz} &=& 1-f\sin^2\theta
\end{eqnarray*}
The extrinsic curvature is then calculated from
\begin{equation}
K_{ij} = - \frac{1}{2\alpha} \dot{g}_{ij}
\end{equation}
\section{Teukolsky waves}
Teukolsky waves are quadrupole wave solutions to the linearized
Einstein equations. For a full description, see: PRD 26:745 (1982).
\section{Comments}
The extrinsic curvature is initialized assuming the initial lapse is one.
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% END CACTUS THORNGUIDE
\end{document}
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