[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 66)

[cott at tapir.caltech.edu]

User: cott
Date: 2011/04/27 12:58 PM

Modified:
 /
  ET.tex

Log:
 * more intro text to curvature evolution

File Changes:

Directory: /
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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