\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{ADMMacros}
\author{Tom Goodale}
\date{$ $Date$ $}

\maketitle

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\begin{abstract}
Provides macros for other ADM based thorns to use
\end{abstract}

\section{Purpose}

This thorn provides various macros which can be used to calculate
quantities, such as the Christoffel Symbol or Riemann Tensor
components, using the basic variables of thorn {\tt ADMBase}.
The macros can be used in both Fortran and C code. The macros work 
pointwise to calculate quantities at specific grid points, which are 
always labelled by {\tt i}, {\tt j} and {\tt k}. They are written in 
such a way that needed quantities which are already calculated are 
reused.

\section{Using ADM Macros}

Each macro described in Section~\ref{admmacros:macros} is implemented using three include 
files. 

\begin{itemize}

\item {\tt <MACRONAME>\_declare.h} sets up the declarations for the internal macro 
	variables. All the internal (hidden) variables have names beginning with 
        the macro name. This file should be included in the declarations section
	of your routine.

\item {\tt <MACRONAME>\_guts.h} is the actual included source code which will calculate
       the quantities.

\item {\tt <MACRONAME>\_undefine.h} resets the macros. This file {\bf must be included}
	at the end of every loop using macros. Without an undefine file, a second loop
  	using macros will assume that quantities have already been calculated.

\end{itemize}

To use the macros, first make sure that you really want to use the macro pointwise, and 
that you have already set the indices $i$, $j$, and $k$ to identify the correct grid point. 

Find the name of the macro from the table in Section~\ref{admmacros:macros} and put the include 
files in the correct place following the instructions above. Note that all ADMMacro include files
are in the directory {\tt CactusEinstein/ADMMacros/src/macros}, so this means adding lines such 
as

{\tt
\begin{verbatim}
#include "CactusEinstein/ADMacros/src/macro/<MACRONAME>_<TYPE>.h"
\end{verbatim}
}

Each variable that the macro calculates is listed in the table of
Section~\ref{admmacros:macros}.  Note that these variable names are
themselves macros and are case sensitive. {\bf Always use the macro
variables on the right hand sides of equations, never redefine them
yourself, since they may be used in later (hidden) calculations.}

\section{Example}

This example comes from thorn {\tt CactusEinstein/Maximal} and uses the $trK$ macro to 
calculate the trace of the extrinsic curvature. 

{\tt
\begin{verbatim}
c     Declarations for macros.
#include "CactusEinstein/ADMMacros/src/macro/TRK_declare.h"

<CUT> 

c     Add the shift term: N = B^i D_i(trK).
      if ((maxshift).and.(shift_state.eq.1)) then
         do k=1,nz
            do j=1,ny
               do i=1,nx
#include "CactusEinstein/ADMMacros/src/macro/TRK_guts.h"
                  K_temp(i,j,k) = TRK_TRK
                end do
             end do
         end do
#include "CactusEinstein/ADMMacros/src/macro/TRK_undefine.h"
\end{verbatim}
}


\section{Macros}
\label{admmacros:macros}
Macros exist for the following quantities

\begin{tabular}{p{5cm}p{5cm}p{5cm}}
{\bf Calculates} & {\bf Macro Name} & {\bf Sets variables} \\
All first spatial derivatives of lapse, $\alpha_{,i}$: & DA & DA\_DXDA, DA\_DYDA, DA\_DZDA\\
All second spatial derivatives of lapse, $\alpha_{,ij}$: & DDA & DDA\_DXXDA, DDA\_DXYDA, DDA\_DXZDA, DDA\_DYYDA, DDA\_DYZDA, DDA\_DZZDA\\
All second covariant spatial derivatives of lapse, $\alpha_{;ij}$: & CDCDA &\\
All first spatial derivatives of shift, $\beta^{i}_{\;\;j}$: & DB &\\
All first covariant derivatives of the extrinsic curvature, $K_{ij;kl}$ & CDK &\\
First covariant derivatives of the extrinsic curvature, $K_{ij;x}$, $K_{ij;y}$, $K_{ij;z}$ & CDXCDK, CDYCDK, CDZCDK &\\
Determinant of 3-metric: & DETG &\\
Upper 3-metric, $g{ij}$:& UPPERMET &\\
Trace of extrinsic curvature $trK$: & TRK &\\
Trace of stress energy tensor: & TRT &\\
Hamiltonian constraint: & HAMADM & \\
Partial derivatives of extrinsic curvature, $K_{ij,x}$, $K_{ij,y}$, $K_{ij,z}$: & DXDK, DYDK, DZDK &\\
First partial derivatives of 3-metric, $g_{ij,x}$, $g_{ij,y}$, $g_{ij,z}$: & DXDG, DYDG, DZDG & \\
All first partial derivatives of 3-metric, $g_{ij,k}$: & DG &\\
First covariant derivatives of 3-metric, $g_{ij;x}$, $g_{ij;y}$, $g_{ij;z}$: & DXDCG, DYDCG, DZDCG &\\
Second partial derivatives of 3-metric, $g_{ij,xx}$, $g_{ij,xy}$, $g_{ij,xz}$: & DXXDG, DXYDG, DXZDG, DYYDG, DYZDG, DZZDG& \\
All second partial derivative of 3-metric, $g_{ij,lm}$ & DDG &\\
Ricci tensor $R_{ij}$: & RICCI &\\
Trace of Ricci tensor $R$: & TRRICCI &\\
Christoffel symbols of first kind: $\Gamma_{cab}$ & CHR1&\\
Christoffel symbols of second kind $\Gamma^{c}_{\;\;ab}$: & CHR2& \\
Momentum constraints & MOMX, MOMY, MOMZ&\\
Source term in evolution equation for conformal metric, $\tilde{g}_{ij,t}$: & DCGDT &\\

\end{tabular}

\section{Definitions}

\begin{equation}
\Gamma_{cab} = \frac{1}{2}\left(g_{ac,b} + g_{bc,a} - g_{ab,c}\right)
\end{equation}

\begin{equation}
\Gamma^{c}_{\;\;ab} = g^{cd}\Gamma_{dab} = \frac{1}{2} g^{cd} \left(g_{ad,b} + g_{bd,a} - g_{ab,d}\right)
\end{equation}

%\section{Comments}

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