\documentclass{article}
\begin{document}

\title{StaticConformal}
\author{Tom Goodale et al}
\date{April 2002}
\maketitle

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

\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 {\bf ADMBase::metric\_type} parameter is set to
`static conformal' then the {\bf 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 {\bf psi}, its first
derivatives {\bf psix}, {\bf psiy}, {\bf psiz}, and its second
derivatives {\bf psixx}, {\bf psixy}, {\bf psixz}, {\bf psiyy}, {\bf
psiyz}, and {\bf psizz} depending on the setting of the {\bf
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

$$ \psi $$
$$ \psi_i/\psi$$
and

$$ \psi_{ij}/\psi $$

Thorns need to check the value of the grid scalar 
{\bf conformal\_state} to determine how many levels of these variables have
actually been calculated.

\begin{itemize}
\item[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[1] 
The conformal factor has been calulated, but no derivatives.
\item[2]
The conformal factor and its first derivatives have been calculated.
\item[3]
The conformal factor and its first and second derivatives have been calculated.
\end{itemize}

\section{Comments}

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

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

However thorns are free to themselves assigned storage for
conformal\_state and initialised it to zero if metric\_type is
"physical".  In this one case is it safe for them to use
the conformal\_state variable if 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".

% Automatically created from the ccl files by using gmake thorndoc
\include{interface}
\include{param}
\include{schedule}

\end{document}