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\begin{document}

\title{ADMBase}
\author{Tom Goodale}
\date{$ $Date$ $}

\maketitle

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\begin{abstract}
Provides the basic ADM variables used in the $3+1$ formalism
\end{abstract}

\section{Purpose}

Thorn {\tt ADMBase} provides core infrastructure for thorns
implementing general relativity on a 3D grid in the $3+1$
formalism. It provides the basic variables (3-metric, extrinsic
curvature, lapse and shift vector) for the $3+1$ formalism, in
addition to a set of parameters to regulate the methods used for their
evolution. These variables are used to communicate between thorns
providing initial data, evolution methods and analysis routines for
the $3+1$ formalism. In addition, the variables can be used as a
mechanism to interact with alternative formalisms, as long as routines
can be written to transform alternative variables into these $3+1$
variables.


\section{Using ADMBase}

\subsection{3+1 Variables}

The variables provided by {\tt ADMBase} are:

\begin{itemize}
\item
The 3-metric tensor, $g_{ij}$

 {\tt gxx}, {\tt gxy}, {\tt gxz},{\tt gyy}, {\tt gyz},{\tt gzz}

\item The extrinsic curvature tensor, $K_{ij}$ 

{\tt kxx}, {\tt kxy}, {\tt kxz},{\tt kyy},{\tt kyz},{\tt kzz}

\item The lapse function, $\alpha$

 {\tt alp}

\item The (optional) shift vector $\beta^i$ 

{\tt betax}, {\tt betay},{\tt betaz}

\end{itemize}

By default the metric and extrinsic curvature tensors are assumed to
be {\it physical}, however these semantics can be changed by use of
the {\tt metric\_type} parameter. {\tt  ADMBase} provides the default value
of {\tt physical}, however another thorn can extend this parameter, 
for example to specify that the variables {\tt gxx} etc actually refer
to the {\it conformal} 3-metric.


\subsection{Initial Data}

Initial data for the $3+1$ variables is specified by the {\tt
initial\_data} (3-metric and extrinsic curvature), {\tt
initial\_lapse} (lapse), and {\tt initial\_shift} (shift) parameters.
By default, {\tt ADMBase} initialises the 3-metric and extrinsic
curvature to Minkowski and the lapse to one. Initial data thorns
override these defaults by extending the parameters. To see which
initial data sets are available in your executable run for example

{\tt
\begin{verbatim}
./cactus_<config> -o admbase::initial_data | grep Range
\end{verbatim}
}

The CactusEinstein arrangement includes thorns providing initial data
for various black hole combinations, perturbed black holes and linear
gravitational waves.

\subsection{Evolution Methods}

Analogous to specifying initial data, evolution methods are chosen by
the {\tt evolution\_method} (3-metric and extrinsic curvature), {\tt
lapse\_evolution\_method} (lapse), and {\tt shift\_evolution\_method}
(shift) parameters.  By default, {\tt ADMBase} does not evolve the
3-metric or extrinsic curvature, and holds the lapse and shift static.



\section{Programming With ADMBase}

\subsection{3+1 Variables}

It is highly recommended that all thorns which inherit from ADMBase
check the value of the {\tt metric\_type} parameter in a routine
scheduled at {\tt CCTK\_PARAMCHECK} and signal an error if the metric
type is not recognised. (See the source file {\tt ParamCheck.c} in any
of the thorns in the {\tt CactusEinstein} arrangement for examples of
this, and note that the {\tt PARAMCHECK} time bin is a good place to 
check for illegal/bad combinations of parameters, and also to inform
the user of any relevent details of the parameters she has chosen).

ADMBase allocates one timelevel of memory for all variables, except
the shift, which is only allocated if the {\tt initial\_shift}
parameter is set to a value other than `none'.  (`none' is the
default.)  The state of the shift storage is indicated by the {\tt
shift\_state} grid scalar.  This is 1 if there is storage for the
shift, and 0 otherwise.

The thorn provides, on request, initial data to set the metric and
extrinsic curvature to flat space in cartesian coordinates, to set the
initial lapse to one and the initial shift to zero.


\subsection{Initial Data} 

To include your initial data sets for the 3-metric, extrinsic
curvature, lapse and shift in the {\tt ADMBase} infrastructure, extend
the keyword parameters {\tt initial\_data}, {\tt initial\_lapse} and
{\tt initial\_shift}. For example, in the {\tt param.ccl} file of {\tt CactusEinstein/IDAnalyticBH},

{\tt
\begin{verbatim}
shares: ADMBase

EXTENDS KEYWORD initial_data 
{
  "schwarzschild"      :: "One Schwarzshild black hole"
  "bl_bh"              :: "Brill Lindquist black holes"
  "misner_bh"          :: "Misner black holes"
  "multiple_misner_bh" :: "Multiple Misner black holes"
  "kerr"	       :: "One Kerr black hole"	
} 
\end{verbatim}
}

{\tt ADMBase} also schedules two groups {\tt ADMBase\_InitialData} and
{\tt ADMBase\_InitialGauge} in this order at {\tt CCTK\_INITIAL}.
Initial data and initial gauge thorns should schedule their routines
to run in this group, for example

{\tt
\begin{verbatim}
if (CCTK_Equals(initial_data,"schwarzschild")) 
{ 	
   schedule Schwarzschild in ADMBase_InitialData
   {
     LANG: C
   } "Construct initial data for a single Schwarzschild black hole"
}
\end{verbatim}
}

{\tt ADMBase} also schedules a group {\tt ADMBase\_PostInitial} at
{\tt CCTK\_INITIAL} after both {\tt ADMBase\_InitialData} and {\tt
  ADMBase\_InitialGauge}.  This group is meant for thorns that modify
the initial data, such as e.g.\ adding noise to an exact solution.



\section{Shift Vector}

It is only relatively recently that numerical relativists have started
to use a shift vector in 3D calculations, and previously, to save
space, storage for the shift vector was not allocated. If the parameter
{\tt initial\_shift} is set to {\tt none}, {\tt ADMBase} does not allocate
storage for {\tt betax}, {\tt betay}, {\tt betaz} and sets the grid scalar
{\tt shift\_state} to 0. In all other cases the {\tt shift\_state} parameter
is set to 1. 

Thorns using the shift should always check that storage for the shift is
allocated before using it.

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\end{document}
