[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 50)

[eloisa.bentivegna at aei.mpg.de]

User: bentivegna
Date: 2011/04/11 03:32 AM

Modified:
 /
  ET.tex

Log:
 Added text and two figures for the Kasner example.
 Also included reference to local_bibtex/references.bib.

File Changes:

Directory: /
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File [modified]: ET.tex
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--- ET.tex	2011-04-11 05:05:01 UTC (rev 49)
+++ ET.tex	2011-04-11 08:32:32 UTC (rev 50)
@@ -1431,9 +1431,55 @@
 \subsection{Collapse\pages{2 Christian}}
 Show TOV collapse and BH formation
 
-\subsection{Cosmology\pages{2 Eloisa}}
-Convergence
+\subsection{Cosmology}
+The Einstein Toolkit is not only designed to evolve compact-object
+spacetimes, but it is also capable of solving the initial-value
+problem for spacetimes with radically different topology and global
+properties. In the following we illustrate the evolution of an
+initial-data set representing a constant-$t$ section of a
+spacetime from the Gowdy $T^3$ class~\cite{Gowdy71,New98}; models in
+this class have the line element:
+\begin{equation}
+\label{eq:gowdyT3}
+ds^2=\tau^{-1/2}e^{\lambda/2}(-d\tau^2+dz^2)+\tau[e^P(dx+Qdy)^2+e^{-P}dy^2]
+\end{equation}
+defined on a 3-torus $-x_0 \leq x \leq x_0$, $-y_0 \leq y \leq y_0$,
+$-z_0 \leq z \leq z_0$, with the functions $P$, $Q$ and $\lambda$ to be 
+determined by the Einstein equations. For $P=Q=\lambda=0$, a coordinate
+transformation $t=4/3 \tau^{3/4}$ (plus a rescaling of the spatial
+coordinates) casts the line element into the form:
+\begin{equation}
+\label{eq:kasner}
+ds^2=-dt^2+t^{4/3}(dx^2+dy^2)+t^{-2/3}dz^2
+\end{equation}
+which represents the familiar Kasner spacetime for a homogeneous but 
+anisotropically expanding universe. In the 3+1 decomposition described
+above, this reads:
+\begin{widetext}
+\begin{eqnarray}
+\alpha(t) &=& 1 \\
+\beta^i(t) &=& 0 \\
+\gamma_{ij}(t) &=& {\rm diag}(t^{4/3},t^{4/3},t^{-2/3}) \\
+K_{ij}(t) &=& - {\rm diag}(\frac{2}{3} t^{4/3},\frac{2}{3}t^{4/3},\frac{1}{3}t^{-2/3})
+\end{eqnarray}
+\end{widetext}
 
+In Figure\ref{fig:kasner}, we show the full evolution of the $t=1$ slice 
+of spacetime~\ref{eq:kasner}, along with the associated error for a sequence of 
+time resolutions.
+
+\begin{figure}
+\includegraphics[width=0.45\textwidth]{kasner.png}
+\includegraphics[width=0.45\textwidth]{err.png}
+\caption{Left: the evolution of a vacuum spacetime of the type~\ref{eq:gowdyT3},
+with $P=Q=\lambda=0$; the initial data are chosen as
+$\gamma_{ij}=\delta_{ij}$ and $K_{ij}={\rm diag}(-2/3,-2/3,1/3)$.
+Right: the numerical error for a sequence of four time resolutions $dt=[0.0125,0.025,0.05,0.1]$;
+the errors are scaled according to the expectation for fourth-order convergence.
+\label{fig:kasner}}
+\end{figure}
+
+
 \section{Future Work\pages{1 Frank}}
 This paper illustrated the current state of the ``Einstein Toolkit'',
 a collection of freely available and easy to use computational codes
@@ -1473,6 +1519,6 @@
 
 
 \bibliographystyle{amsplain-url}
-\bibliography{manifest/einsteintoolkit,local_bibtex/ott_references}
+\bibliography{manifest/einsteintoolkit,local_bibtex/ott_references,local_bibtex/references}
 
 \end{document}