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\section{Model Problem(Kerr-Schild metric)}
\begin{eqnarray}
g_{\mu\nu} = \eta_{\mu \nu} + 2 H l_{\mu} l_{\nu} \nonumber
\end{eqnarray}
where $H$ is a scalar, and $l^{\mu}l_{\mu} = 0$ wrt $\eta_{\mu\nu}$ and
$g_{\mu\nu}$.
\begin{itemize}
\item Schwarzschild spacetime(ingoing Eddington-Finkelstein coordinates)
\item Kerr
\item Boosted Schwarzschild/Kerr
\end{itemize}

%%\section{3+1 Decomposition}
\begin{itemize}
\item 3+1 Decomposition of Kerr-Schild metric
\begin{eqnarray}
\alpha & = & 1 / \sqrt{1+ 2Hl_t^2} \nonumber \\
\beta_i &= & 2 H l_t l_i \nonumber \\
g_{ij} & = &  \delta_{ij} + 2 H l_i l_j \nonumber \\
K_{ij} & = & -\frac{1}{2\alpha} \left( \partial_t g_{ij} - D_i \beta_j -
D_j \beta_i \right). \nonumber
\end{eqnarray}
\item Algebraic condition on lapse derived from $l^{\mu}l_{\mu} = 0$ as
\begin{eqnarray}
\alpha = \sqrt{1 / \left( 1 + \sqrt{\beta_x^2 + \beta_y^2 + \beta_z^2}\right)}
\nonumber
\end{eqnarray}
\item For each case we specify $H$, $l_{\mu}$, $\partial_{\mu} H$ and
$\partial_{\mu} l_{\nu}$ and use the resulting 3-metric and extrinsic
curvature as initial data.
\item $\beta^i$ specified analytically for now
\item $\alpha$ is specified algebraically in terms of the shift and 
3-metric components.
\end{itemize}
\newpage
\section{Model Problems}
\begin{itemize}
\item Schwarzschild spacetime
\begin{eqnarray}
H & = & M /r \nonumber \\
l_t & = & 1 \nonumber \\
l_i & = & x_i /r \nonumber \\
r & = & \sqrt{x^2 + y^2 + z^2} \nonumber
\end{eqnarray}
\item Kerr
\begin{eqnarray}
H & = & M r^3 / \left(r^4 + a^2 z^2\right) \nonumber \\
l_t & = & 1 \nonumber \\
l_x & = & \left(rx + ay\right)/\left(r^2 + a^2\right)\nonumber \\
l_y & = & \left(ry - ax\right)/\left(r^2 + a^2\right)\nonumber \\
l_z & = & z / r \nonumber 
\end{eqnarray}
where $r$ is now determined from
\begin{eqnarray}
\frac{x^2 + y^2}{r^2 + a^2} + \frac{z^2}{r^2} = 1 \nonumber
\end{eqnarray}
\item Boosted Schwarzschild
 \begin{enumerate}
  \item The Kerr-Schild spacetime metric is form-invariant under a boost. \\
         $\bar{l}_{\mu} \rightarrow l_{\mu}$ and $\bar{H} \rightarrow H$. The 
         spacetime metric remains of the Kerr-Schild form.
    \begin{eqnarray}
    H & = & M /r \nonumber \\
    l_t & = & \gamma (\bar{l}_t - v\bar{l}_z) \nonumber \\
    l_x & = & \bar{l}_x \nonumber \\
    l_y & = & \bar{l}_y \nonumber \\
    l_z & = & \gamma (\bar{l}_z - v\bar{l}_t)\nonumber
    \end{eqnarray}
    where $r=\sqrt{x^2 + y^2 + \gamma^2 ( z - vt)^2}$ and
    \begin{eqnarray}
    \bar{l}_t & = & 1 \nonumber \\
    \bar{l}_x & = & x / r \nonumber \\
    \bar{l}_y & = & y / r \nonumber \\
    \bar{l}_z & = & \gamma (z - vt)/r \nonumber
    \end{eqnarray}

 \end{enumerate}
\end{itemize}
\end{document}
