User: diener Date: 2012/03/05 03:01 PM

Modified: / ET.tex

Log: Make the relationship between M_{bh}, M_{AH} and M_{IH} more clear.

File Changes:

Directory: / ============

File [modified]: ET.tex Delta lines: +10 -3 =================================================================== --- ET.tex 2012-03-05 20:31:26 UTC (rev 257) +++ ET.tex 2012-03-05 21:01:13 UTC (rev 258) @@ -2333,7 +2333,7 @@ \includegraphics[width=0.9\textwidth]{ah_mass_conv} \caption{The top plot shows the irreducible mass of the apparent horizon as a function of time at low (black solid curve), medium (blue dashed curve) -and high (red dotted curve) resolutions. The inset is a zoom in on the +and high (red dotted curve) resolutions. The inset is a zoom in on the $y$-axis to more clearly show the differences between the resolutions. The bottom plot shows the convergence of the irreducible mass. The black (solid) curve shows the difference between the medium and low resolution results, @@ -2342,6 +2342,9 @@ difference between the high and medium resolutions scaled according to fourth and third-order convergence respectively.} \label{fig:ah_mass} \end{figure} +Note that the irreducible mass +$M_{\mathrm{AH}}$ is smaller than the initial mass $M_{\mathrm{bh}}$ due to +the spin of the black hole. The inset shows in more detail the differences between the different resolutions. The irreducible mass increases by about 0.3% during the first $40\mathrm{M}$ of evolution and then remains constant (within numerical error) for the @@ -2358,8 +2361,12 @@ to the horizon location at the lowest resolution.

Finally, in Figure~\ref{fig:ah_mass_spin} we show the total -mass (top plot) and the change in the spin, $\Delta S = S(t) - S(t=0)$, as -calculated by \codename{QuasiLocalMeasures}. +mass $M_{\mathrm{IH}}$ (top plot) and the change in the spin, $\Delta S = S(t) - S(t=0)$, as +calculated by \codename{QuasiLocalMeasures}. The total mass $M_{\mathrm{IH}}$ +is calculated using the Christodoulou formula (see +equation~(\ref{eq:christodoulou})) and initially agrees with +$M_{\mathrm{bh}}$ but then starts to increase due to the flux of gravitational +wave energy through the horizon. \begin{figure} \includegraphics[width=0.9\textwidth]{qlm_mass} \includegraphics[width=0.9\textwidth]{qlm_spin}