document Hisa-aki Shinkai's Kastor-Traschen maximally charged multiple BHs

solution


git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinInitialData/Exact/trunk@66 e296648e-0e4f-0410-bd07-d597d9acff87

1 files changed, 50 insertions, 6 deletions

diff --git a/doc/documentation.tex b/doc/documentation.tex
index d1591a3..791378b 100644
--- a/doc/documentation.tex
+++ b/doc/documentation.tex
@@ -80,7 +80,7 @@ parameter:
 	\item[{\tt "fakebinary"}]
 		Thorne's ``fake binary'' approximate spacetime
 	\item[{\tt "multiBH"}]
-		Maximally charged multi BH solutions
+		Kastor-Traschen Maximally charged multi BH solutions
 	\end{description}
 \item[Cosmological spacetimes]\mbox{}\\[-\baselineskip]
 	\begin{description}
@@ -395,9 +395,51 @@ black hole's center; the default setting should be ok for most purposes.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\subsection{Maximally Charged Multi-BH Solutions}
+\subsection{Kastor-Traschen Maximally Charged Multi-BH Solutions}
 
-FIXME: get more info from Hisa-aki
+\verb|Exact::exactmodel = "multiBH"| specifies the Kastor-Traschen (KT)
+solution.  The file \verb|KTsol.tex| in the documentation directory
+of this thorn gives more details/references about this solution and
+the realated Majumdar-Papapetrou (MP) solution.
+
+Basically, the Kastor-Traschen solution is a multi-black-hole solution
+to Einstein's equation with cosmological constant, containing arbitrarily
+many maximally charged ($Q=M$) quasi-Schwarzschild black holes that
+participate in an overall de Sitter expansion or contraction.  (It
+vaguely resembles a cosmological generalization of the Brill-Lindquist
+initial data.)  In the $\Lambda \rightarrow 0$ limit, the KT solutions
+reduce to the MP solution.
+
+To write the KT metric, we first choose $(x_i, y_i, z_i) \in \Re^3$,
+$i=1,2,\cdots,N$ for locations of black holes.  Then the line element is
+\begin{equation}
+ds^2=-\frac{1}{\Omega^2} dt^2+a(t)^2 \Omega^2(dx^2+dy^2+dz^2)
+\end{equation}
+where
+\begin{align}
+\Omega	&= 1+\sum_{i=1}^N {\frac{M_i}{a r_i}}				\\
+a	&= e^{Ht}							\\
+H	&= \pm \sqrt{\frac{\Lambda}{3}}					\\
+r_i	&= \sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}			%%%\\
+\end{align}
+We interpret $M_i$ as the mass of the $i{\rm th}$ black hole,
+although we have neither an asymptotically flat region nor event
+horizons available to convert this naive interpretation into a rigorous
+one.
+
+This thorn supports up to 4~black holes; the physics parameters are
+\begin{align}
+N	& = \text{\tt KT\_nBH}						\\
+H	& = \text{\tt KT\_Hubble}					%%%\\
+\end{align}
+and
+\begin{align}
+M_i	& = \text{\tt m\_bh$i$}						\\
+x_i	& = \text{\tt co\_bh$i$x}					\\
+y_i	& = \text{\tt co\_bh$i$y}					\\
+z_i	& = \text{\tt co\_bh$i$z}					%%%\\
+\end{align}
+for each $i = 1$, $2$, $3$, $4$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
@@ -636,9 +678,11 @@ These parameters are useful to hide the spherical symmetry of the metric.
 \section{Acknowledgments}
 
 Many different people have contributed code to this thorn.
-Jonathan Thornburg wrote this documentation in May 2002 based on
-the comments in the code, some reverse-engineering, and querying
-various people about how the code works.
+Jonathan Thornburg wrote most of this documentation in May 2002 based
+on the comments in the code, some reverse-engineering, and querying
+various people about how the code works.  The description of the
+Kastor-Traschen maximally charged multi-BH solutions is adapted from
+the file \verb|KTsol.tex| in this same directory, by Hisa-aki Shinkai.
 
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