To proj_IVP and proj_ELLIPTIC subscribers:

While during working in thorn_IVP, I began to think that we should 
treat boundary conditions for momentum constraint equations more
carefully.  Attached LaTeX file is a note of Bowen's expression 
[GRG14(1982)1183], which was also applied to Baumgarte et al, and 
Wilson et al.  

I am now coding up his multipole expansion in general case as a part
of thorn_outerBC.  This will work as preparing outer boundary values
by summing up momentums all over the grid, and pass data to elliptic
solver as a Dirichlet problem.  

Such Dirichlet treatment may only work with (m*2^n+1)^3 grid in Bernd's
BAM_Elliptic.  So I am not sure this implimented initial data is 
immediately applicable for evolutions using Bernd's Robin BC, which 
requires (m*2^n+3)^3 grid structure. 

Anyway, if I finished coding, I will post again. 


Hisaaki SHINKAI
-----------------------------------------------------------
E-mail: shinkai@null.wustl.edu
URL:    http://wugrav.wustl.edu/People/HISAAKI/shinkai.html
Office: Dept. of Physics, Washington University
        Campus Box 1105, One Brookings Dr.
        St.Louis, MO 63130-4899, USA
        phone:  +1-314-935-4617  fax:  +1-314-935-6219
-----------------------------------------------------------


PS. I also put postscript version at
http://wugrav.wustl.edu/People/HISAAKI/notes/momBC.ps


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\begin{document}

\begin{center}
{\Large\bf Boundary Conditions for momentum constraints }\\
{\large Conformally flat case: Bowen's multipole expansion}

\end{center}
\begin{flushright}
version 0.1 ~~~
19990205 Hisaaki Shinkai \\
{\tt shinkai@null.wustl.edu}
\end{flushright}

While during working in {\tt thorn\_IVP}, I began to think that we should 
treat boundary conditions for momentum constraint equations more
carefully.  This is a note of Bowen's expression [GRG14(1982)1183],
which was also applied to Baumgarte {\it et al}, Wilson {\it et al} 
directly.

%=======================================================================
\section{What Bowen wrote}
Under the constant-mean-curvature ($\nabla^i K=0$) condition
and conformally flatness, longitudinal part of the 
momentum constraints become
\be
\Delta W^i + {1 \over 3} \nabla^i \nabla_j W^j = 8 \pi S^i. \label{mom}
\en
By introducing a decomposition of $W^i$ into vector and gradient terms
\be
W^i=V^i-{1 \over 4}\nabla^i \theta,
\en
we can express (\ref{mom}) as
\bera
\Delta V^i &=& 8 \pi S^i, \\
\Delta \theta &=& \nabla_i V^i, 
\enra
and the longitudinal part of $K_{ij}$ as
\be
K^{ij}_L=\nabla^i V^j + \nabla^j V^i -{1 \over 2} \delta^{ij}\nabla_k V^k
-{1 \over 2} \nabla^i\nabla^j \theta.
\en

If the source is of finite extent, then the solutions in the 
outside vacuum region are expressed as
\bear
V^i &=& -2 \sum_{l=0}^\infty Q^{ij_1\cdots j_l} n_{j_1} \cdots n_{j_l}
{1 \over r^{l+1}},\\
\theta &=& - \sum_{l=1}^\infty Q^{ \{ ij_1\cdots j_{l-1} \} } 
n_in_{j_1}\cdots n_{j_{l-1}} {1 \over r^{l-1}} \nonumber \\
&&+\sum_{l=0}^\infty { 2 (l+1) \over (2l+1)(2l+3)}
Q_k^{kj_1 \cdots j_l} n_{j_1}\cdots n_{j_l} {1 \over r^{l+1}}
\nonumber \\
&&+ \sum_{l=1}^\infty {2l-1 \over 2l+1} M^{ \{ ij_1 \cdots j_{l-1} \} }
n_in_{j_1}\cdots n_{j_{l-1}}{1 \over r^{l+1}}
\enar
where $n^i = x^i r^{-1}$ in the Cartesian cordinate, the 
multipoles $Q$ and $M$ are defined as 
\bera
Q^{ij_1 \cdots j_l} &\equiv& {(2l-1)!! \over l! } \int S^i ({\bf r})
x^{ \{ j_1 }x^{j_2} \cdots x^{j_l \} } dV, \\
M^{ij_1 \cdots j_l} &\equiv& {(2l-1)!! \over l! } \int r^2 S^i ({\bf r})
x^{ \{ j_1 }x^{j_2} \cdots x^{j_l \} } dV, 
\enra
and where brackets denote the completely symmetric trace-free part
\be
Z^{ \{ ij_1 \cdots j_l \} } = Z^{(ij_1 \cdots j_l)} - {l \over 2l+1}
Z_k^{k(j_1 \cdots j_{l-1}}\delta^{j_l i)}
\en

%-----------------------------------------------------------------------
\section{Which term is the lowest order? (my calculations)}
More direct expressions are:
\bera
V^i &=& -{2 \over r} \int S^i dV - {2 \over r^3} x^j \int S^i x^j dV
-{3 \over r^5} x^j x^k \int S^i x^{ \{j } x^{k \} } dV \nonumber \\&&
   -{5 \over r^7} x^j x^k x^l \int S^i x^{ \{j } x^k x^{l \} } dV 
- \cdots
\label{expand1}\\
\theta &=& - {x^i \over r} \int S^i dV 
- {2\over 3} {1 \over r} \int S^k x^k dV
- {x^i x^j \over r^3 } \int S^{ \{ i } x^{j \} } dV
- \cdots
\label{expand2}
\enra

\subsection{Inspiral Binary of equal mass stars}
For the case of inspiral binary (circular motion) of equal mass, 
which locate on $x$-axis and the orbital rotating vector 
aligns to $z$-axis, 
the lowest falling-off behavior is
\bera
V^x &=& -2 {y \over r^3}  \int S^x y dV \\
V^y &=& -2 {x \over r^3}  \int S^y x dV \\
V^z &=& -5 {xyz \over r^7}  \int S^z xyz dV \\
\theta &=&  - {1 \over 2} {xy \over r^3} [ \int S^x y dV + \int S^y x dV]
\enra
\subsection{Head-on situation of equal mass stars}
For the case of head-on two equal mass stars, which move
along $x$-axis and with the center of mass is at the origin, then 
the lowest falling-off behavior is
\bera
V^x &=& -2 {x \over r^3}  \int S^x x dV  \\
V^y &=& -5 {xyz \over r^7}  \int S^y xyz dV  \\
V^z &=& -5 {xyz \over r^7}  \int S^z xyz dV  \\
\theta &=& - {2\over 3} {1 \over r} \int S^k x^k dV
\enra

%-----------------------------------------------------------------------
\section{How we treat these?}
I am now coding (\ref{expand1}) and (\ref{expand2}) for general situation as a part of
{\tt thorn\_outerBC}.  If it is ready, I will announce again.

\end{document}

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