[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 255)

[diener at cct.lsu.edu]

User: diener
Date: 2012/03/05 02:01 PM

Modified:
 /
  ET.tex

Log:
 Added the Christodoulou formula (with reference) and slightly changed
 the notation.

File Changes:

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

File [modified]: ET.tex
Delta lines: +13 -2
===================================================================
--- ET.tex	2012-03-05 18:40:12 UTC (rev 254)
+++ ET.tex	2012-03-05 20:01:40 UTC (rev 255)
@@ -1866,8 +1866,10 @@
 There are also the quasi-local measures of mass and angular momentum, from 
 any \ahz{s} found during the spacetime.  Both 
 \codename{AHFinderDirect} and \codename{AHFinder} output the 
-corresponding irreducible mass derived from the area of the horizon $m_H = 
-\sqrt{A/(16\pi)}$.
+corresponding irreducible mass derived from the area of the horizon 
+$M_{\mathrm{ir}} = 
+\sqrt{A/(16\pi)}$. Note that the irreducible mass is smaller than the total
+mass if the black hole is spinning.
 
 The module \codename{QuasiLocalMeasures} implements
 the calculation of mass and spin multipoles from the isolated 
@@ -1885,6 +1887,15 @@
 can calculate useful quantities such as the Weyl or Ricci scalars
 or the three-volume element of the horizon world tube
 in addition to physical observables such as mass and momenta. 
+For the calculation of the total mass \codename{QuasiLocalMeasures} uses the
+Christodoulou mass~\cite{Christodoulou:1970wf}
+\begin{equation}
+  M^2 = M_{\mathrm{ir}}^2+\frac{S^2}{4 M_{\mathrm{ir}}^2},
+\label{eq:christodoulou}
+\end{equation}
+that takes into account the
+contributions of the irreducuble mass $M_{\mathrm{ir}}$ and angular
+momemtum $S$ to the total black hole mass $M$.
 
 Finally, the module \codename{HydroAnalysis} additionally locates the 
 (coordinate) center of mass as well as the point of maximum rest mass density of a