From fd4a87f87c975a360d694af9c1913d39e196d7fd Mon Sep 17 00:00:00 2001 From: jthorn Date: Sun, 12 May 2002 14:42:59 +0000 Subject: adding Hisa-aki Shinkai's documentation for for the "multiBH" solution git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinInitialData/Exact/trunk@65 e296648e-0e4f-0410-bd07-d597d9acff87 --- doc/KTsol.tex | 200 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 200 insertions(+) create mode 100644 doc/KTsol.tex (limited to 'doc') diff --git a/doc/KTsol.tex b/doc/KTsol.tex new file mode 100644 index 0000000..e3b5e3a --- /dev/null +++ b/doc/KTsol.tex @@ -0,0 +1,200 @@ +% Hisaaki Shinkai shinkai@atlas.riken.go.jp <- PL use this address +%----------------------------------------------------------------------- +% KTsol.tex version 0.1 19980603 +% Hisaaki Shinkai shinkai@wurel.wustl.edu +%----------------------------------------------------------------------- +%00000000111111111122222222223333333333444444444455555555556666666666777 +%----------------------------------------------------------------------- +\documentstyle[11pt]{article} +% A4 tate ----------------------------- +\topmargin 0.3in +\oddsidemargin 0.0cm +\evensidemargin 0.0cm +\textwidth 16.0cm +\textheight 23.0cm +\headsep 0.0in +%--- US letter size +\topmargin -0.0in +\headsep 0.0in +\oddsidemargin 0.0in +\evensidemargin 0.0in +\textwidth 6.5in +\textheight 8.5in +%----------------------------------------------------------------------- +%------------------------- macro for begin-eqs +\def\non{\nonumber \\} +\def\nonn{\nonumber \\ &&} +\def\be{\begin{equation}} +\def\en{\end{equation}} +\def\bear{\begin{eqnarray}} +\def\enar{\end{eqnarray}} +\def\beas{\begin{eqnarray*}} +\def\enas{\end{eqnarray*}} +%--------------->>>>>>>>>>>> commands for number of eqs. (1.1a)(1.1b)... +% following definition is for book or report style +%\renewcommand{\theequation}{\thechapter.\theenumi\alph{equation}} +\def\bera{ \setcounter{enumi}{\value{equation}} + \addtocounter{enumi}{1} + \setcounter{equation}{0} + \renewcommand{\theequation}{\theenumi\alph{equation}} + \begin{eqnarray} } +\def\enra{ \end{eqnarray} + \setcounter{equation}{\value{enumi}} + \renewcommand{\theequation}{\arabic{equation}} } +%----------------------------------------------------------------------- +\def\non{\nonumber \\} +\def\nonn{\nonumber \\ &&} +\def\dsp{\displaystyle} +\def\mova{\left( M \over a\right)} +%------------------------- macro for lists +\def\been{\begin{enumerate}} +\def\enen{\end{enumerate}} +\def\beit{\begin{itemize}} +\def\enit{\end{itemize}} + +\def\pl{ \partial} +\def\half{{1 \over 2}} +%----------------------------------------------------------------------- +\begin{document} +\noindent +filename="KTsol.tex" HShinkai ({\tt shinkai@atlas.riken.go.jp}) + +\begin{center} + +{\Large\bf Multi Black Hole solutions} + +\end{center} +\begin{flushright} +version 0.1 ~~~ +19980603 Hisaaki Shinkai +\end{flushright} + +\section{Majumdar-Papapetrou solution} +Majumdar-Papapetrou (MP) +solutions\cite{MP} is a multi-black-hole solution to Einstein's +equation. +Each black holes are charged maximally, $Q=M$, and +the balance between gravitational attraction and +electrostatic repulsion among the black holes causes each to maintain +its position relative to the others eternally. +The MP solutions are given by +\be +ds^2=-{1 \over \Omega^2} dt^2+ \Omega^2(dx^2+dy^2+dz^2), +\label{MPmetric} +\en +$$\mbox{where}~~ +\Omega=1+\sum_{i=1}^N {M_i \over r_i},~~ \mbox{and}~~ + r_i=\sqrt{(x-x_i)^2+(y-y_i)^2+(z-z_i)^2}. +$$ +where $M_i$ and $(x_i, y_i, z_i) \in {\bf R}^3$ are masses and +locations of black holes. + + +\section{Kastor-Traschen solution} + + +The Kastor-Traschen (KT) solutions \cite{KT} is the cosmological +version of MP solution. This is a multi-black-hole solution + to Einstein's +equation with cosmological constant, contains arbitrary many +$Q=M$ black holes that participate in an overall de Sitter expansion +or contraction. In the $\Lambda \rightarrow 0$ +limit, the KT solutions reduce to the MP solution. + +To write the KT +metric, we first choose $(x_i, y_i, z_i) \in {\bf R}^3$, +$i=1,2,\cdots,N$ for locations of black holes, then +\be +ds^2=-{1 \over \Omega^2} dt^2+a(t)^2 \Omega^2(dx^2+dy^2+dz^2), +\label{KTmetric} +\en +$$\mbox{where}~~ +\Omega=1+\sum_{i=1}^N {M_i \over a r_i},~~ a=e^{Ht}, ~~~ +H=\pm \sqrt{\Lambda \over 3}.$$ +$$\mbox{and}~~ r_i=\sqrt{(x-x_i)^2+(y-y_i)^2+(z-z_i)^2}$$ +where 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. + +If $H<0$, then the solution represents ``incoming" charged BHs. +If $H>0$, then the solution represents ``outgoing" charged WHs. + + +The principal null directions of KT solutions are illustrated in +\cite{KTpnd}. +The horizon structure of KT solutions are discussed in \cite{KThorizon}. + + +%----------------------------------------------------------------------- +\newpage +\appendix +\section{Reissner-Nordstr{\o}m-de Sitter solution } +It will be interesting to see such a cosmological extention for +a single black hole case. +The global structure of this solution is discussed in +\cite{RNdShorizon}. + +\subsection{Reissner-Nordstr{\o}m-de Sitter solution (static coord) } +\be +ds^s=-V(R)dT^2 + {1 \over V(R)} dR^2 + R^2 d\Omega^2 +\en +$$\mbox{where}~~ +V(R)=1-{2M\over R}+{Q^2 \over R^2}-{\Lambda \over 3} R^2. $$ +\subsection{Reissner-Nordstr{\o}m-de Sitter solution (cosmological coord) } +By the transformation +$$ +a(t) r = R-M, ~~~ +t= T+h(R), ~~~ +{dh \over dR} = - {HR^2 \over (R-M)V(R)} +$$ +and setting $Q=M$, we will get the cosmological coordinate version of +RNdS as +\be +ds^2=-{1\over F^2}dt^2 + {a^2(t) F^2} ( dr^2 + r^2 d\Omega^2) +\en +$$\mbox{where}~~ +F=1+{M\over ar}, ~~~ a=e^{Ht}, ~~~ +H=\pm \sqrt{\Lambda \over 3}.$$. + +The horizons appeare at +$$ +r_\pm = {1\over 2 a(t) |H| } ( 1 - 2 M |H| \pm \sqrt{1 - 4 M |H|} ) +$$ +which are corresponds to de Sitter horizon ($r_+$) and +outer BH horizon ($r_-$), respectivly. + + + + +%%********************************************************************** +%23456789012345678901234567890123456789012345678901234567890123456789012 + +\baselineskip .15in +\begin{thebibliography}{99} + +\bibitem{MP} +S. D. Majumdar, Phys. Rev. {\bf 72}, 930 (1947); A. Papapetrou, +Proc. R. Ir. Acad. Sect. {\bf A51}, 191 (1947); +J. B. Hartle and S. W. Hawking, Commun. Math. Phys. {\bf 26}, 87 +(1972). + + +\bibitem{KT} +D. Kastor and J. Traschen, Phys. Rev. {\bf D47}, 5370 (1993). + +\bibitem{KTpnd} +L. Gunnarsen, H. Shinkai and K. Maeda, Class. Quantum Grav. {\bf 12}, +133 (1995). + +\bibitem{KThorizon} +K. Nakao, T. Shiromizu and S. A. Hayward, +Phys. Rev. {\bf D52}, 796 (1995). + +\bibitem{RNdShorizon} +D. R. Brill and S. A. Hayward +Class. Quant. Grav. {\bf 11}, 359 (1994). + +\end{thebibliography} + +\end{document} -- cgit v1.2.3