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.

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File [modified]: ET.tex Delta lines: +49 -3 =================================================================== --- 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}