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} \