From 3062d3c5cbf94764bc0b0fe0139fd1cc6d92fdb2 Mon Sep 17 00:00:00 2001
From: schnetter <schnetter@906471b6-c639-44d1-9ea0-3e3d6879f074>
Date: Wed, 4 May 2005 15:04:44 +0000
Subject: Define the known tensor type aliases

git-svn-id: http://svn.cactuscode.org/arrangements/CactusBase/SymBase/trunk@20 906471b6-c639-44d1-9ea0-3e3d6879f074
---
 doc/documentation.tex | 77 +++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 77 insertions(+)

diff --git a/doc/documentation.tex b/doc/documentation.tex
index b0fcbf8..7111d5b 100644
--- a/doc/documentation.tex
+++ b/doc/documentation.tex
@@ -471,6 +471,83 @@ obtained from \texttt{SymmetryRegister}.
 
 
 
+\section{Tensor Types}
+
+Cactus supports declaring the \emph{tensor type} of grid function
+groups.  These tensor types define how the grid functions, which are
+supposed to be tensor components, transform under various
+transformations, such as reflections and rotations.
+
+The tensor types are not declared directly; instead, a \emph{tensor
+type alias} is declared.  The following tensor type aliases are
+currently known and supported:
+\begin{description}
+\item[\texttt{scalar}:] a scalar $\rho$
+\item[\texttt{u}:] a vector $\beta^i$
+\item[\texttt{d}:] a covector $s_i$
+\item[\texttt{dd\_sym}:] a symmetric rank two tensor $\gamma_{ij}$
+\end{description}
+(More tensor type aliases are likely to be defined in the future.)
+
+In addition to the tensor type, one can also declare the \emph{tensor
+weight} and a \emph{tensor metric}.  The tensor weight specifies the
+behaviour under transformations that change the volume element.  The
+tensor metric specifies what metric has to be used to raise or lower
+indices for that quantity.
+
+Last but not least, a \emph{tensor special} can be defined for
+quantities that do not transform as tensor.  The currently supported
+tensor specials are
+\begin{description}
+\item[\texttt{Gamma}:]
+   for the transformation behaviour of the $\Gamma^i$ variables of the
+   BSSN formalism; it is $\Gamma^i := - \gamma^{jk} \Gamma^i_{jk}$
+   with $\Gamma^i_{jk} := \frac{1}{2} \gamma^{il} \left( \partial_k
+   \gamma_{lj} + \partial_j \gamma_{lk} - \partial_l \gamma_{jk}
+   \right)$
+\item[\texttt{log}:]
+   for the transformation behaviour of the variable $\phi$ of the BSSN
+   formalism; it is $\phi := \log \psi$ with $\psi^{12} := \det
+   \gamma_{ij}$.
+\end{description}
+
+\subsection{Example Tensor Type Declarations}
+
+From CactusWave/WaveToy:
+\begin{verbatim}
+CCTK_REAL scalarevolve TYPE=gf TAGS='tensortypealias="scalar"'
+\end{verbatim}
+
+From CactusEinstein/ADMBase:
+\begin{verbatim}
+CCTK_REAL metric TYPE=gf TAGS='tensortypealias="dd_sym" tensormetric="ADMBase::metric"'
+CCTK_REAL curv TYPE=gf TAGS='tensortypealias="dd_sym" tensormetric="ADMBase::metric"'
+CCTK_REAL lapse TYPE=gf TAGS='tensortypealias="scalar" tensormetric="ADMBase::metric"'
+CCTK_REAL shift TYPE=gf TAGS='tensortypealias="U" tensormetric="ADMBase::metric"'
+\end{verbatim}
+
+From AEIThorns/BSSN\_MoL:
+\begin{verbatim}
+CCTK_REAL ADM_BSSN_B TYPE=gf \
+    TAGS='tensortypealias="u" tensormetric="ADMBase::metric"'
+CCTK_REAL ADM_BSSN_dtlapse TYPE=gf \
+    TAGS='tensortypealias="scalar" tensormetric="ADMBase::metric"'
+CCTK_REAL ADM_BSSN_phi TYPE=gf \
+    TAGS='tensortypealias="scalar" tensormetric="BSSN_MoL::ADM_BSSN_metric" \
+          tensorweight=0.16666666666666667 tensorspecial="log"'
+CCTK_REAL ADM_BSSN_metric TYPE=gf \
+    TAGS='tensortypealias="dd_sym" tensormetric="BSSN_MoL::ADM_BSSN_metric" \
+          tensorweight=-0.66666666666666667'
+CCTK_REAL ADM_BSSN_K TYPE=gf \
+    TAGS='tensortypealias="scalar" tensormetric="BSSN_MoL::ADM_BSSN_metric"'
+CCTK_REAL ADM_BSSN_curv TYPE=gf \
+    TAGS='tensortypealias="dd_sym" tensormetric="BSSN_MoL::ADM_BSSN_metric" \
+          tensorweight=-0.66666666666666667'
+CCTK_REAL ADM_BSSN_gamma TYPE=gf \
+    TAGS='tensortypealias="u" tensormetric="BSSN_MoL::ADM_BSSN_metric" \
+          tensorweight=0.66666666666666667 tensorspecial="Gamma"'
+\end{verbatim}
+
 % Do not delete next line
 % END CACTUS THORNGUIDE
 
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