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Diffstat (limited to 'doc')
| -rw-r--r-- | doc/documentation.tex | 143 |
1 files changed, 129 insertions, 14 deletions
diff --git a/doc/documentation.tex b/doc/documentation.tex index e3730fc..4ac63f5 100644 --- a/doc/documentation.tex +++ b/doc/documentation.tex @@ -1,4 +1,26 @@ \documentclass{article} + +\newif\ifpdf +\ifx\pdfoutput\undefined +\pdffalse % we are not running PDFLaTeX +\else +\pdfoutput=1 % we are running PDFLaTeX +\pdftrue +\fi + +\ifpdf +\usepackage[pdftex]{graphicx} +\else +\usepackage{graphicx} +\fi + +\parskip = 0 pt +\parindent = 0pt +\oddsidemargin = 0 cm +\textwidth = 16 cm +\topmargin = -1 cm +\textheight = 24 cm + \begin{document} \title{ADMAnalysis} @@ -10,33 +32,126 @@ \section{Purpose} -This thorn calculates +This thorn provides analysis routines to calculate the following quantities: \begin{itemize} \item -The trace of the extrinsic curvature ({\bf trK}). +The trace of the extrinsic curvature ($trK$). \item -The determinant of the metric ({\bf detg}). +The determinant of the 3-metric ($detg$). \item -The components of the metric in spherical coordinates -({\bf grr,grq,grp,gqq,gqp,gpp}). +The components of the 3-metric in spherical coordinates \\ +($g_{rr},g_{r\theta},g_{r\phi},g_{\theta\theta},g_{\phi\theta},g_{\phi\phi}$). \item -The components of the extrinsic curvature in spherical coordinates -({\bf Krr,Krq,Krp,Kqq,Kqp,Kpp}). +The components of the extrinsic curvature in spherical coordinates \\ +($K_{rr},K_{r\theta},K_{r\phi},K_{\theta\theta},K_{\theta\phi},K_{\phi\phi}$). \end{itemize} -if output is requested for them. +\section{Trace of Extrinsic Curvature} + +The trace of the extrinsic curvature at each point on the grid is placed in +the grid function {\tt trK}. The algorithm for calculating the trace +uses the physical metric, that is it includes any conformal factor. + +\begin{equation} +{\tt trK} \equiv tr K = \frac{1}{\psi^4} g^{ij} K_{ij} +\end{equation} -\section{Comments} +\section{Determinant of 3-Metric} -If the parameter {\bf rsquared\_in\_sphm} is set, it squares the -radial coordinate before applying the tranformation. +The determinant of the 3-metric at each point on the grid is placed in +the grid function {\tt detg}. This is always the determinant of the +conformal metric, that is it does not include any conformal factor. -In the spherical transormation, the $\theta$ coordinate is referred to +\begin{equation} +{\tt detg} \equiv det g = +-g_{13}^2*g_{22}+2*g_{12}*g_{13}*g_{23}-g_{11}*g_{23}^2- +g_{12}^2*g_{33}+g_{11}*g_{22}*g_{33} +\end{equation} + + +\section{Transformation to Spherical Cooordinates} + +The values of the metric and/or extrinsic curvature in a spherical +polar coordinate system $(r,\theta,\phi)$ evaluated at each point on +the computational grid are placed in the grid functions ({\tt grr}, +{\tt grt}, {\tt grp}, {\tt gtt}, {\tt gtp}, {\tt gpp}) and ({\tt krr}, +{\tt krt}, {\tt krp}, {\tt ktt}, {\tt ktp}, {\tt kpp}). +In the spherical transformation, the $\theta$ coordinate is referred to as {\bf q} and the $\phi$ as {\bf p}. -This thorn knows how to handle `physical' and `static conformal' -metric types. +The general transformation from Cartesian to Spherical for such tensors +is + +\begin{eqnarray*} +A_{rr}&=& +\sin^2\theta\cos^2\phi A_{xx} ++\sin^2\theta\sin^2\phi A_{yy} ++\cos^2\theta A_{zz} ++2\sin^2\theta\cos\phi\sin\phi A_{xy} +\\ +&& ++2\sin\theta\cos\theta\cos\phi A_{xz} ++2\sin\theta\cos\theta\sin\phi A_{yz} +\\ +A_{r\theta}&=& +r(\sin\theta\cos\theta\cos^2\phi A_{xx} ++2*\sin\theta\cos\theta\sin\phi\cos\phi A_{xy} ++(\cos^2\theta-\sin^2\theta)\cos\phi A_{xz} +\\ +&& ++\sin\theta\cos\theta\sin^2\phi A_{yy} ++(\cos^2\theta-\sin^2\theta)\sin\phi A_{yz} +-\sin\theta\cos\theta A_{zz}) +\\ +A_{r\phi}&=& +r\sin\theta(-\sin\theta\sin\phi\cos\phi A_{xx} +-\sin\theta(\sin^2\phi-\cos^2\phi)A_{xy} +-\cos\theta\sin\phi A_{xz} +\\ +&& ++\sin\theta\sin\phi\cos\phi A_{yy} ++\cos\theta\cos\phi A_{yz}) +\\ +A_{\theta\theta}&=& +r^2(\cos^2\theta\cos^2\phi A_{xx} ++2\cos^2\theta\sin\phi\cos\phi A_{xy} +-2\sin\theta\cos\theta\cos\phi A_{xz} ++\cos^2\theta\sin^2\phi A_{yy} +\\ +&& +-2\sin\theta\cos\theta\sin\phi A_{yz} ++\sin^2\theta A_{zz}) +\\ +A_{\theta\phi}&=& +r^2\sin\theta(-\cos\theta\sin\phi\cos\phi A_{xx} +-\cos\theta(\sin^2\phi-\cos^2\phi)A_{xy} ++\sin\theta \sin\phi A_{xz} +\\ +&& ++\cos\theta\sin\phi\cos\phi A_{yy} +-\sin\theta\cos\phi A_{yz}) +\\ +A_{\phi\phi}&=& +r^2\sin^2\theta(\sin^2\phi A_{xx} +-2\sin\phi\cos\phi A_{xy} ++\cos^2\phi A_{yy}) +\end{eqnarray*} + +If the parameter {\tt normalize\_dtheta\_dphi} is set to {\tt yes}, +the angular components are projected onto the vectors $(r d\theta, r \sin\theta d \phi)$ instead of the default vector $(d \theta, d\phi)$. That is, + +\begin{eqnarray*} +A_{\theta\theta} & \rightarrow & A_{\theta\theta}/r^2 +\\ +A_{\phi\phi}& \rightarrow & A_{\phi\phi}/(r^2\sin^2\theta) +\\ +A_{r\theta} & \rightarrow & A_{r\theta}/r +\\ +A_{r\phi} & \rightarrow & A_{r\phi}/(r\sin\theta) +\\ +A_{\theta\phi} & \rightarrow & A_{\theta\phi}/r^2\sin\theta) +\end{eqnarray*} % Automatically created from the ccl files by using gmake thorndoc \include{interface} |