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\documentclass{article}
\begin{document}
\title{IDAnalyticBH}
\author{Steve Brandt, Carsten Gundlach, Joan Masso, Ed Seidel, Paul Walker}
\date{1997-1998}
\maketitle
\abstract{Analytic initial data for vacuum black hole spacetimes}
\section{Purpose}
Initial data is provided for the 3-metric, extrinsic curvature, and
if appropriate the conformal factor and it's spatial derivatives as
well as the lapse and shift. The current initial data sets are:
\begin{enumerate}
\item single (Schwarzschild) black hole in isotropic
coordinates $$ g_{ij} = $$
\item up to four Brill-Lindquist black holes,
\item Any number of Misner-type black holes.
\item A single Kerr black hole
\end{enumerate}
\section{Solutions}
\subsection{Schwarzschild}
\noindent{\bf 3-Metric}
\begin{eqnarray*}
g_{xx} &=& 1 + \frac{1}{2Mr^4}\\
g_{yy} &=& 1 + \frac{1}{2Mr^4}\\
g_{zz} &=& 1 + \frac{1}{2Mr^4}\\
g_{xy} &=& 0\\
g_{xz} &=& 0\\
g_{yz} &=& 0
\end{eqnarray*}
\noindent{\bf Conformal 3-Metric}
\begin{eqnarray*}
g_{xx} &=& 1 \\
g_{yy} &=& 1 \\
g_{zz} &=& 1 \\
g_{xy} &=& 0\\
g_{xz} &=& 0\\
g_{yz} &=& 0
\end{eqnarray*}
\noindent{\bf Conformal Factor}
$$
\phi = 1+\frac{1}{2r}
$$
\noindent{\bf Conformal Factor}
\begin{eqnarray*}
K_{xx} &=& 0\\
K_{yy} &=& 0\\
K_{zz} &=& 0\\
K_{xy} &=& 0\\
K_{xz} &=& 0\\
K_{yz} &=& 0
\end{eqnarray*}
\noindent{\bf Lapse}
$$
\alpha = \frac{2r-M}{2r+M}
$$
\section{Comments}
This thorn uses no special numerical methods, however two points
are worth noting
\begin{itemize}
\item The solution for Misner is obtained by summing a sequence
\item The spatial derivatives of the conformal metric (when required)
are calculated accurately using finite differencing of the
exact solution by a very small spacing </item>
\end{itemize}
It is still to be decided how initial data should be supplied to
for systems of evolution equations which use more or different
variables to those in the einstein implementation
% Automatically created from the ccl files by using gmake thorndoc
\include{interface}
\include{param}
\include{schedule}
\end{document}
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