\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{StaticConformal}
\author{Tom Goodale et al}
\date{$ $Date$ $}

\maketitle

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\begin{abstract}
Base thorn to provide the variables for the static conformal factor
\end{abstract}

\section{Purpose}

This thorn provides the variables defining a static conformal factor
which is used to transform the physical metric.  If this thorn is
active and the {\tt ADMBase::metric\_type} parameter is set to
``{\tt static conformal}'' then the {\tt ADMBase::g...} variables are the
conformal values as opposed to the physical values.

The transformation is

$$ g_{ij}^{\mbox{physical}} = \psi^4 g_{ij}^{\mbox{conformal}} $$

The extrinsic curvature is not transformed.

Memory is provided for the conformal factor {\tt psi}, its first
derivatives {\tt psix}, {\tt psiy}, {\tt psiz}, and its second
derivatives {\tt psixx}, {\tt psixy}, {\tt psixz}, {\tt psiyy}, {\tt
psiyz}, and {\tt psizz} depending on the setting of the {\tt
conformal\_storage} parameter.

Note that the first and second ``derivative'' grid functions have an
additional factor of $1 / \psi$ normalisation since this is the most
common use of the derivative.  I.e. the grid functions are

\begin{eqnarray*}
 {\tt psi} &=& \psi, \\
 {\tt psix} &=& \psi_x/\psi, \qquad \mbox{etc}\\
 {\tt psixx} &=& \psi_{ij}/\psi \qquad \mbox{etc}
\end{eqnarray*}

Thorns need to check the value of the grid scalar 
{\tt conformal\_state} to determine how many levels of these variables have
actually been calculated before using the conformal factor:

\begin{description}
\item[{\tt conformal\_state=0}] 
No conformal factor has been calculated --- thorns may
assume the conformal factor is 1 at all points.  (I.e. the metric is physical.)
\item[{\tt conformal\_state=1}] 
The conformal factor has been calulated, but no derivatives.
\item[{\tt conformal\_state=2}]
The conformal factor and its first derivatives have been calculated.
\item[{\tt conformal\_state=3}]
The conformal factor and its first and second derivatives have been calculated.
\end{description}

Note that this means that if you only want to know whether {\tt psi} contains 
the values for the conformal factor you can check for {\tt conformal\_state > 0}.

\section{Utilities}

{\tt StaticConformal} provides functions to convert between physical and conformal 3-metric values. It is very important to understand that these functions 
apply the conversion {\it in place}. That is, if {\tt gxx} contains the conformal metric value, when the routine is exited it will now contain the physical metric value. These functions {\it do not} change the value of {\tt conformal\_state} and should be used with due care. (These functions are for example used
by some analysis thorns who work only with the physical metric, they apply the
transformation on entry to the analysis routine and switch it back on exit).

\begin{description}

\item[Convert from conformal to physical:]

{\tt
\begin{verbatim}


StaticConf_ConfToPhysInPlace (cctk_lsh(1),
                              cctk_lsh(2),
                              cctk_lsh(3),
                              psi,
                              gxx,
                              gxy,
                              gxz,
                              gyy,
                              gyz,
                              gzz)
\end{verbatim}
}

\item[Convert from physical to conformal:]

{\tt
\begin{verbatim}


StaticConf_PhysToConfInPlace (cctk_lsh(1),
                              cctk_lsh(2),
                              cctk_lsh(3),
                              psi,
                              gxx,
                              gxy,
                              gxz,
                              gyy,
                              gyz,
                              gzz)
\end{verbatim}
}

\end{description}

\section{Comments}

The {\tt StaticConformal} thorn itself does not calculate any conformal
factor, but does initialise the {\tt conformal\_state} variable to 0. 

Please note, no thorn should use the {\tt conformal\_state} variable
unless the parameter {\tt metric\_type} is {\tt ``static conformal''}.
The {\tt conformal\_state} variable is not assigned storage or
initialised by the StaticConformal thorn in any other case.

However thorns are free to themselves assign storage for {\tt
conformal\_state} and initialise it to zero if {\tt metric\_type} is
``physical''.  In this one case it is safe for them to use the
\texttt{conformal\_state} variable if \texttt{metric\_type} is not
``static conformal''.  This method allows them to use just one set of
ifs rather than first checking the metric\_type and then the
conformal\_state variable if the metric\_type is ``static conformal''.

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\end{document}