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

Modified: / ET.tex

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

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File [modified]: ET.tex Delta lines: +28 -1 =================================================================== --- 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,