User: bentivegna Date: 2012/03/12 12:38 PM
Modified: / ET.tex /examples/cosmology/figs/ err.pdf, kasner.pdf, kasner.py
Log: Addressed first referee's comments to cosmology section. All the suggestions were incorporated. A few typos in the extrinsic curvature of Kasner have been fixed in the process.
File Changes:
Directory: / ============
File [modified]: ET.tex Delta lines: +14 -19 =================================================================== --- ET.tex 2012-03-12 17:12:48 UTC (rev 290) +++ ET.tex 2012-03-12 17:38:18 UTC (rev 291) @@ -3026,44 +3026,39 @@ problem for spacetimes with radically different topologies and global properties. In this section, we illustrate the evolution of an initial-data set representing a constant-$t$ section of a -spacetime from the Gowdy $T^3$ class~\cite{Gowdy:1971jh,New:1997me}. Models in -this class have the line element: +spacetime from the Gowdy $T^3$ class~\cite{Gowdy:1971jh,New:1997me}, namely +the Kasner model. This spacetime has 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 -in Section~\ref{sec:ADMBase}, this reads: +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 periodic boundary conditions. In the 3+1 decomposition +described in Section~\ref{sec:ADMBase}, 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}) +K_{ij}(t) &=& {\rm diag}(-\frac{2}{3} , t^{1/3},-\frac{2}{3} , t^{1/3},\frac{1}{3} , t^{-5/3}) \end{eqnarray} \end{widetext}
+This solution represents a vacuum, expanding universe with an homogeneous +but anisotropic metric tensor. In Figure~\ref{fig:kasner}, we show the full evolution of the $t=1$ slice of spacetime~\eref{eq:kasner}, along with the associated error for a sequence of -time resolutions. +time resolutions. We choose $x_0=y_0=z_0=5$ and spatial resolution equal to +$1$, and we run a set of four time resolutions equal to $[0.0125,0.025,0.05,0.1]$.
\begin{figure} \includegraphics[width=0.9\textwidth]{kasner.pdf} \includegraphics[width=0.9\textwidth]{err.pdf} - \caption{Top: the evolution of a vacuum spacetime of the type~\eref{eq:gowdyT3}, - with $P=Q=\lambda=0$; the initial data are chosen as + \caption{Top: the evolution of a vacuum spacetime of the type~\eref{eq:kasner}; + the initial data are chosen as $\gamma_{ij}=\delta_{ij}$ and $K_{ij}={\rm diag}(-2/3,-2/3,1/3)$. - Bottom: the numerical error for a sequence of four time resolutions $dt=[0.0125,0.025,0.05,0.1]$; + Bottom: the numerical error for the sequence of four time resolutions $[0.0125,0.025,0.05,0.1]$; + the superscripts $n$ and $e$ indicate the numerical and the exact solutions respectively, and the errors are scaled according to the expectation for fourth-order convergence. \label{fig:kasner}} \end{figure}
Directory: /examples/cosmology/figs/ ====================================
File [modified]: err.pdf Delta lines: +0 -0 =================================================================== (Binary files differ)
File [modified]: kasner.pdf Delta lines: +0 -0 =================================================================== (Binary files differ)
File [modified]: kasner.py Delta lines: +2 -0 =================================================================== --- examples/cosmology/figs/kasner.py 2012-03-12 17:12:48 UTC (rev 290) +++ examples/cosmology/figs/kasner.py 2012-03-12 17:38:18 UTC (rev 291) @@ -23,6 +23,7 @@ ax.plot(t[::5], ord2[::5], linestyle='none', marker='.', markersize=10, color='blue', label='$\gamma_{zz}$ (numerical)')
ax.set_xlabel(r'$t$') +ax.set_ylabel(r'$\gamma_{xx}$')
handles, labels = ax.get_legend_handles_labels() ax.set_yscale('log') @@ -44,6 +45,7 @@ axb.plot(t, (ord1-exa1)/16**3, linestyle='-', color='black', label='$(\gamma_{xx}^n-\gamma_{xx}^e)/16^3$')
axb.set_xlabel(r'$t$') +axb.set_ylabel(r'Error in $\gamma_{xx}$')
handles, labels = axb.get_legend_handles_labels() axb.legend(handles, labels, loc='best')