From 0b25b268183c088584dbeead3240260ebd492c08 Mon Sep 17 00:00:00 2001 From: jthorn Date: Mon, 13 Jun 2005 15:32:57 +0000 Subject: clarify wording when we define logarithmic radial coordinate $\eta$ git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinInitialData/IDAxiBrillBH/trunk@74 0a4070d5-58f5-498f-b6c0-2693e757fa0f --- doc/documentation.tex | 21 ++++++++------------- 1 file changed, 8 insertions(+), 13 deletions(-) diff --git a/doc/documentation.tex b/doc/documentation.tex index 1f6c948..781430b 100644 --- a/doc/documentation.tex +++ b/doc/documentation.tex @@ -78,22 +78,17 @@ q = Q_0 \sin^n \theta \left [ \exp\left(\frac{\eta - \eta_0^2}{\sigma^2}\right ) \right ]. \end{equation} Note regularity along the axis requires that the exponent $n$ must be -even. Choose a logarithmic radial coordinate $\eta$, which is -related to the -asymptoticlly flat coordinate $r$ by $\eta = \ln (2r/m)$, where $m$ is a -scale parameter. One can rewrite (\ref{IDAxiBrillBH/eqn:sph-coord}) as -\begin{equation} -ds^2 = \psi(\eta)^4 [ e^{2 q} (d \eta^2 + d\theta^2) + \sin^2 -\theta d\phi^2]. -\end{equation} - -In the previous Bernstein work, the above $r$ is transformed to a -logarithmic radial coordinate - +even. Choosing a logarithmic radial coordinate \begin{equation} \label{IDAxiBrillBH/eta-coord} \eta = \ln{\frac{2r}{m}}. \end{equation} +(where $m$ is a scale parameter), one can rewrite +(\ref{IDAxiBrillBH/eqn:sph-coord}) as +\begin{equation} +ds^2 = \psi(\eta)^4 [ e^{2 q} (d \eta^2 + d\theta^2) + \sin^2 +\theta d\phi^2]. +\end{equation} The scale parameter $m$ is equal to the mass of the Schwarzschild black hole, if $q=0$. In this coordinate, the 3-metric is @@ -153,7 +148,7 @@ and outer boundary condition, a Robin condition: This thorn solves equation~(\ref{IDAxiBrillBH/eqn:ham-linear}) on a 2-D $(\eta,\theta)$ grid. However, Cactus needs a 3-D grid, typically with -Cartesian coordinates. Therefore, this thorn interpolate $\psi$ and its +Cartesian coordinates. Therefore, this thorn interpolates $\psi$ and its $(\eta,\theta)$ derivatives to the Cartesian grid. The parameters \verb|neta| and \verb|nq| specify the resolution of -- cgit v1.2.3