[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 38)

[cott at tapir.caltech.edu]

User: cott
Date: 2011/03/26 09:28 PM

Modified:
 /
  ET.tex

Log:
 * more evolution text

File Changes:

Directory: /
============

File [modified]: ET.tex
Delta lines: +19 -11
===================================================================
--- ET.tex	2011-03-26 15:14:54 UTC (rev 37)
+++ ET.tex	2011-03-27 02:28:19 UTC (rev 38)
@@ -553,6 +553,9 @@
 Alcubierre~\cite{alcubierre:08} and Baumgarte \& Shapiro~\cite{baumgarte:10}.
 GR hydrodynamics has been reviewed by Font~\cite{font:08}.
 
+\todo{state Einstein equations, mention conventions, give stress
+energy tensor}
+
 \subsubsection{Spacetime Curvature Evolution} The Einstein Toolkit
 curvature evolution code \codename{McLachlan}~\cite{brown:09} is
 auto-generated from tensor equations via \codename{Kranc}
@@ -561,26 +564,31 @@
 \cite{york:79}. For this, the Baumgarte-Shapiro-Shibata-Nakamura
 (BSSN) conformal-tracefree reformulation
 \cite{shibata:95,baumgarte:95,alcubierre:00} of the original
-Arnowitt-Deser-Misner formalism~\cite{adm:62} is employed.
+Arnowitt-Deser-Misner (ADM) formalism~\cite{adm:62} is employed.
 
-This leads to the following set of evolved variables:
 
-\todo{continue editing here}
+The evolved variables are the conformal factor $\Phi$, the conformal
+3-metric $\tilde{gamma}_{ij}$, the trace $K$ of the extrinsic curvature,
+the trace free extrinsic curvature $A_{ij}$ and the conformal connection
+functions $\tilde{\Gamma}^i$. These are defined in terms of the
+standard ADM 4-metric $g_{ij}$, 3-metric $\gamma{ij}$, and extrinsic
+curvature $K_{ij}$ by
 
+
 \begin{eqnarray}
-  \phi & := & \log \left[ \frac{1}{12} \det \gamma_{ij} \right]
+  \phi & := & \log \left[ \frac{1}{12} \det \gamma_{ij} \right]\,,
   \\
-  \tilde\gamma_{ij} & := & e^{-4\phi}\; \gamma_{ij}
+  \tilde\gamma_{ij} & := & e^{-4\phi}\; \gamma_{ij}\,,
   \\
-  K & := & g^{ij} K_{ij}
+  K & := & g^{ij} K_{ij}\,,
   \\
-  \tilde A_{ij} & := & e^{-4\phi} \left[ K_{ij} - \frac{1}{3} g_{ij} K
+  \tilde A_{ij} & := & e^{-4\phi} \left[ K_{ij} - \frac{1}{3} g_{ij} K\,,
     \right]
   \\
   \tilde\Gamma^i & := & \tilde\gamma^{jk} \tilde\Gamma^i_{jk} .
 \end{eqnarray}
-Our exact evolution equations are as described by Eqs.~(3) to (10) of
-\cite{ES-Brown2007b}, which we list here for completeness:
+
+The evolution equations are then:
 \begin{widetext}
 \begin{eqnarray}
   \partial_0 \alpha & = & -\alpha^2 f(\alpha, \phi, x^\mu) (K -
@@ -661,13 +669,13 @@
 variables are lapse $\alpha$, shift $\beta^i$, and a quantity $B^i$
 related to the time derivative of the shift. The gauge parameters $f$,
 $G$, $H$, and $\eta$ are determined by our choice of a $1+\log$
-slicing:
+\cite{alcubierre:03a} slicing:
 \begin{eqnarray}
   f(\alpha,\phi,x^\mu) & := & 2/\alpha
   \\
   K_0(x^\mu) & := & 0
 \end{eqnarray}
-and $\Gamma$-driver shift condition:
+and $\Gamma$-driver shift condition \cite{alcubierre:03a}:
 \begin{eqnarray}
   G(\alpha,\phi,x^\mu) & := & (3/4)\, \alpha^{-2}
   \\