[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 125)

[diener at cct.lsu.edu]

User: diener
Date: 2011/08/24 11:35 AM

Modified:
 /
  ET.tex

Log:
 Add text about the QLM measurements of mass and spin.

File Changes:

Directory: /
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File [modified]: ET.tex
Delta lines: +28 -1
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--- ET.tex	2011-08-24 16:34:33 UTC (rev 124)
+++ ET.tex	2011-08-24 16:35:07 UTC (rev 125)
@@ -2323,7 +2323,7 @@
 
 Since there is non-trivial gravitational wave content in the initial data
 the mass of the black hole changes when evolved. In Figure~\ref{fig:ah_mass}
-we show in the left plot the irreducible mass as calculates by
+we show in the left plot the irreducible mass as calculated by
 \codename{AHFinderDirect} as function of time at the low (red solid curve),
 medium (green long dashed curve) and high (blue short dashed curve) resolutions.
 \begin{figure}
@@ -2355,6 +2355,33 @@
 be due to non-convergent errors from the puncture being able to propagate
 to the horizon location at the lowest resolution.
 
+Finally, in Figure~\ref{fig:ah_mass_spin} we show the total
+mass (left plot) and the change in the spin, $\Delta S = S(t) - S(t=0)$, as
+calculated by \codename{QuasiLocalMeasures}.
+\begin{figure}
+ \includegraphics[width=0.33\textwidth,angle=-90]{examples/kerr/figs/qlm_mass}
+ \includegraphics[width=0.33\textwidth,angle=-90]{examples/kerr/figs/qlm_spin}
+ \caption{The left plot shows the total mass and the right plot shows the changein spin (i.e.\ $\Delta S=S(t)-S(t=0)$ of the black hole as function of time.
+In both plots the red (solid) curve is for low, green (long dashed) for medium
+and blue (short dashed) for high resolution. In the right plot the pink (dotted)
+curve shows the high resolution result scaled for second order convergence wit
+the medium resolution.}
+\label{fig:ah_mass_spin}
+\end{figure}
+In both cases the red (solid) curve is low, green (long dashed) is medium and
+blue (short dashed) is high resolution results. Since the spacetime is 
+axi-symmetric the gravitational waves can not radiate angular momentum. Thus
+any change in the spin must be due to numerical error and $\Delta S$ should
+converge to zero with increasing resolution. This is clearly seen in the
+right plot of Figure~\ref{fig:ah_mass_spin} by the fact that the pink 
+(dotted) curve (the high resolution result scaled by a factor of $1.78$ for
+second order convergence to the resolution of the medium resolution) and the
+green (long dashed) curve are on top of each other. The reason that we only
+see second order convergence in the spin is due to the fact that thorn 
+\codename{QuasiLocalMeasures} still have some parts of the algorithm that are
+only second order accurate.  The increase of about 0.22\% in the mass of the
+black hole is caused solely by the increase in the irreducible mass.
+
 \subsection{BHB\pages{2 Bruno}}
 \label{sec:bbh-example}
 %BCM: DONE:ID parameters, TP number of collocation points, grid structure,