[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 291)

[eloisa.bentivegna at aei.mpg.de]

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/
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File [modified]: err.pdf
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File [modified]: kasner.pdf
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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')