User: cott Date: 2011/04/27 12:58 PM
Modified: / ET.tex
Log: * more intro text to curvature evolution
File Changes:
Directory: / ============
File [modified]: ET.tex Delta lines: +12 -6 =================================================================== --- ET.tex 2011-04-27 17:47:39 UTC (rev 65) +++ ET.tex 2011-04-27 17:58:18 UTC (rev 66) @@ -935,10 +935,12 @@ and Centrella et al.~\cite{centrella:10}. GR hydrodynamics has been reviewed by Font~\cite{font:08}. In the following, we assume the reader to be familiar with general relativity. +We assume $G = c = M_\odot = 1$ throughout.
The Einstein Toolkit provides code to evolve the Einstein equations \begin{equation} G^{\mu\nu} = 8 \pi T^{\mu\nu},, +\label{eq:einstein} \end{equation} in the $3+1$ split, foliating 4D spacetime into sequences of spacelike 3-hypersurfaces (slices) connected by timelike normal vectors. In the @@ -949,12 +951,16 @@ observer\footnote{A normal observer follows a wordline tanget to the unit normal on the 3-hypersurface.} and the shift vector $\beta^i$ that describes how spatial coordinates change from one slice to the -next. +next. $T^{\mu\nu}$ in equation~\ref{eq:einstein} is the stress-energy +tensor, which we choose to be that of an ideal relativistic fluid, +\begin{equation} +T^{\mu\nu} = \rho h u^\mu u^\nu - g^{\mu\nu} P,,, +\end{equation} +where $\rho$ is the rest-mass density, $u^\mu$ is the 4-velocity, +$g^{\mu\nu}$ is the 4-metric, and $h = 1 + \epsilon + P/\rho$ is the +relativistic specific enthalpy with $\epsilon$ and $P$ being the +specific internal energy and the pressure, respectively.
- -\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} @@ -967,7 +973,7 @@
The evolved variables are the conformal factor $\Phi$, the conformal -3-metric $\tilde{gamma}_{ij}$, the trace $K$ of the extrinsic curvature, +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