\documentstyle[aps,prd,12pt]{revtex}
\begin{document}
\def\R4{{}^{(4)}R}
\def\g4{{}^{(4)}g}
\def\half{\frac{1}{2}}
\def\lb{\left(}
\def\rb{\right)}
\def\lie{\mbox{\pounds}}
\newcommand{\lp}{\left[}
\newcommand{\rp}{\right]}
\newcommand{\HALF}{ \frac{1}{2} }
\newcommand{\SIXTH}{ \frac{1}{6} }
\newcommand{\lc}{\bigl(}
\newcommand{\rc}{\bigl)}
\newcommand{\Lc}{\Bigl(}
\newcommand{\Rc}{\Bigl)}
\newcommand{\lC}{\biggl(}
\newcommand{\rC}{\biggl)}
\newcommand{\LC}{\biggl(}
\newcommand{\RC}{\biggl)}

\section{Kerr-Schild metric}
The Kerr-Schild metric is given by,
\begin{equation}
g_{\mu \nu} = \eta_{\mu \nu} + 2 H l_{\mu} l_{\nu},
\label{eqn:kerrschild.metric}
\end{equation}
where $\eta_{\mu \nu} = {\rm diag}(-1,1,1,1)$ is 
the Minkowski spacetime metric, $H$ is a scalar function of the coordinates 
and $l^{\mu}$ is a null vector with respect to both $\eta_{\mu \nu}$ and
$g_{\mu \nu}$. This metric describes both the 
Schwarzschild and Kerr spacetimes in a slicing that is not pathological at 
the horizon. For the former spacetime, (\ref{eqn:kerrschild.metric}) leads to
the Schwarzschild metric in Eddington-Finkelstein coordinates and for the 
latter it leads to the Kerr metric. In both cases the coordinates that I use
are $(t,x,y,z)$.
For a Kerr spacetime the choice of $H$ and $l_{\mu}$ are
\begin{equation}
H = \frac{M r^3}{r^4 + a^2 z^2}
\label{eqn:kerr.hdefn}
\end{equation}
and 
\begin{eqnarray}
l_{\mu} = \left( 1, \frac{rx + ay}{r^2 + a^2}, 
\frac{ry - ax}{r^2 + a^2}, \frac{z}{r}\right).
\label{eqn:kerr.ldefn}
\end{eqnarray}
Where $M$ is the mass of the Kerr black hole and $a = J/M$ is the angular
momentum of the black hole. $r(x,y,z)$ is obtained from the relation,
\begin{equation}
\frac{x^2 + y^2}{r^2 + a^2} + \frac{z^2}{r^2} = 1
\end{equation}
as
\begin{equation}
r^2 = \frac{1}{2}\left(\rho^2 - a^2 \right) + \sqrt{\frac{1}{4}
\left(\rho^2 - a^2 \right) + a^2 z^2},
\end{equation}
with $\rho = \sqrt{x^2 + y^2 + z^2}$.

Taking the $a\rightarrow 0$ limit, the Schwarzschild spacetime can be obtained.
This leads to 
\begin{equation}
H = \frac{M}{r}
\label{eqn:schw.hdefn}
\end{equation}
and
\begin{equation}
l_{\mu} = \left(1, \frac{x}{r}, \frac{y}{r}, \frac{z}{r} \right)
\label{eqn:schw.ldefn}
\end{equation}
where $r = \sqrt{x^2 + y^2 + z^2}$ and $M$ is the mass as before. The resulting
metric in Schwarschild coordinates is the ingoing Eddington-Finkelstein form 
of the Schwarzschild metric,
\begin{equation}
ds^2 = - \lb 1-\frac{2M}{r} \rb dt^2 + \frac{4M}{r} dr dt + 
\lb 1 + \frac{2M}{r} \rb dr^2 + r^2 \lb d\theta^2 + \sin^2{\theta} d\phi^2 \rb.
\label{eqn:eddfink.line}
\end{equation}

Another useful property of (\ref{eqn:kerrschild.metric}) is its form-invariance
under a Lorentz transformation. Under a Lorentz transformation, by definition
the Minkowski spacetime metric remains invariant, 
$\eta_{\mu \nu} \rightarrow \eta_{\mu \nu}$. $H \rightarrow \hat{H}$ goes to a
new scalar function and $l_{\mu} \rightarrow \hat{l}_{\mu}$ remains a null 
vector.  Hence, the form of the metric is left invariant with new definitions 
for $H$ and $l_{\mu}$. This property is computationally advantageous in that
one may code the general form of the metric and then specify 
$H$, $l_{\mu}$ and their spatial derivatives for the problem of interest.
The algebraic form of (\ref{eqn:kerrschild.metric}) can be imposed during the
course of a numerical evolution of boosted black hole spacetimes.
There is a relationship between the lapse, shift and the 3-metric in
the 3+1 form of (\ref{eqn:kerrschild.metric}) that essentially amounts to a
gauge choice that leads to a form of the spacetime metric 
(\ref{eqn:kerrschild.metric}). But the Kerr-Schild metric is a restricted
subset of solutions to Einstein's equations, so not all metrics can be 
written in the form(\ref{eqn:kerrschild.metric}).

The null cone structure of this metric\cite{boyer.1,MTW} is very well known 
for the 
Schwarzschild and Kerr spacetimes. In particular we know the outgoing null
vector $l^{\mu}$ everywhere in the spacetime. This makes this metric a good
candidate for testing apparent horizon location algorithms as well as for
studying differencing near the horizon in the black hole evolution problem.
As stated earlier, this metric has no pathological behaviour at the horizon.
In the 3+1 splitting the lapse, shift, 3-metric and extrinsic curvature are
all regular across the horizon and the slices of constant coordinate time
cover the region $0 < r < \infty$. Because of the null cone structure of this
metric and the non-pathological behaviour at the horizon it is expected this
coordinate system is a good one to use in numerical evolutions of black hole 
spacetimes.

\subsection{3+1 Splitting of the Kerr-Schild metric}

The decomposition of the Kerr-Schild metric into a 3+1 form in Cartesian
coordinates is straight-forward and was carried out using a tensor package 
for Maple\cite{choptuik.3}. 
Comparing (\ref{eqn:kerrschild.metric}) to the standard 3+1 form of the 
spacetime metric,
\begin{equation}
ds^2 = -\alpha^2 dt^2 + 
g_{ij} \lb dx^i + \beta^i dt \rb \lb dx^j + \beta^j dt \rb
\label{eqn:3+1.form}
\end{equation}
and using the property that 
$\eta^{\mu\nu}l_{\mu}l_{\nu} = 0 \rightarrow l_t^2 = l_x^2 + l_y^2 + 
l_z^2$ it is possible to show that the lapse is
\begin{eqnarray}
\alpha & = & \frac{1}{\sqrt{1 + 2H(l_x^2 + l_y^2 + l_z^2})} \\
 & = & \frac{1}{\sqrt{1 + 2 H l_t^2}},
\label{eqn:kerrschild.lapse}
\end{eqnarray}
the shift is 
\begin{equation}
\beta_i = 2 H l_t l_i
\label{eqn:kerrschild.shift}
\end{equation}
and the 3-metric is 
\begin{equation}
g_{ij} = \eta_{ij} + 2Hl_i l_j.
\label{eqn:kerrschild.3metric}
\end{equation}

Using the definition of the extrinsic curvature, $K_{ij}$, we have,
\begin{eqnarray}
K_{\mu\nu} = -\perp \nabla_{\mu} n_{\nu},
\nonumber
\end{eqnarray}
where $n_{\nu} = \left(-\alpha,0,0,0\right)$. The results for $K_{ij}$
for general $H$ and $l_{\mu}$ are lengthy and have not been included in this
discussion.

\begin{eqnarray*}
\lefteqn{{\it K_{xx}} :=  -  \left( {\vrule 
height0.79em width0em depth0.79em} \right. \! \!  - 4\,{H}^{2}\,
{\it lz}^{2}\,{\it lx}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{t}}}\,{\it lx}\, \!  \right)  -  \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{H}\, \!  \right) \,{\it lx}^{2} + 4
\,{H}^{2}\,{\it lz}^{2}\,{\it lx}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{x}}}\,{\it lt}\, \!  \right) } \\
 & & \mbox{} - 2\,{H}\,{\it lz}^{2}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{H}\, \!  \right) \,{\it lx}^{2} + 2
\,{H}\,{\it lz}\,{\it lt}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{z}}}\,{H}\, \!  \right) \,{\it lx}^{2} + 4\,{H}^{2}\,
{\it lz}\,{\it lt}\,{\it lx}\, \left( \! \,{\frac {{ \partial}}{{
 \partial}{z}}}\,{\it lx}\, \!  \right)  \\
 & & \mbox{} - 4\,{H}^{2}\,{\it lz}\,{\it lt}\,{\it lx}\, \left( 
\! \,{\frac {{ \partial}}{{ \partial}{x}}}\,{\it lz}\, \! 
 \right)  + 4\,{H}^{2}\,{\it ly}\,{\it lt}\,{\it lx}\, \left( \! 
\,{\frac {{ \partial}}{{ \partial}{y}}}\,{\it lx}\, \!  \right) 
 - 4\,{H}^{2}\,{\it ly}\,{\it lt}\,{\it lx}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{x}}}\,{\it ly}\, \!  \right)  \\
 & & \mbox{} + 2\,{H}\,{\it ly}\,{\it lt}\, \left( \! \,{\frac {{
 \partial}}{{ \partial}{y}}}\,{H}\, \!  \right) \,{\it lx}^{2} + 
4\,{H}^{2}\,{\it ly}^{2}\,{\it lx}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{x}}}\,{\it lt}\, \!  \right)  + 2\,{H}\,
{\it lx}^{3}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{x}}}
\,{H}\, \!  \right) \,{\it lt} \\
 & & \mbox{} - 2\,{H}\,{\it ly}^{2}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{H}\, \!  \right) \,{\it lx}^{2} - 4
\,{H}^{2}\,{\it ly}^{2}\,{\it lx}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{\it lx}\, \!  \right)  - 2\,{H}\,
{\it lx}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{t}}}\,
{\it lx}\, \!  \right)  \\
 & & \mbox{} + 2\,{H}\,{\it lx}\, \left( \! \,{\frac {{ \partial}
}{{ \partial}{x}}}\,{\it lt}\, \!  \right)  + 2\,{H}\, \left( \! 
\,{\frac {{ \partial}}{{ \partial}{x}}}\,{\it lx}\, \!  \right) 
\,{\it lt} + 2\, \left( \! \,{\frac {{ \partial}}{{ \partial}{x}
}}\,{H}\, \!  \right) \,{\it lx}\,{\it lt} - 4\,{H}^{2}\,{\it lx}
^{3}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{t}}}\,{\it lx
}\, \!  \right)  \\
 & & \mbox{} - 2\,{H}\,{\it lx}^{4}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{H}\, \!  \right)  + 4\,{H}^{2}\,
{\it lx}^{3}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{x}}}
\,{\it lt}\, \!  \right)  \! \! \left. {\vrule 
height0.79em width0em depth0.79em} \right)  \left/ {\vrule 
height0.41em width0em depth0.41em} \right. \! \!  \left( {\vrule 
height0.41em width0em depth0.41em} \right. \! \!  \\
 & & (\, - 2\,{H}\,{\it lz}^{2} - 1 - 2\,{H}\,{\it ly}^{2} - 2\,{
H}\,{\it lx}^{2} + 2\,{H}\,{\it lt}^{2}\,)\,\sqrt {1 + 2\,{H}\,
{\it lt}^{2}}\, \! \! \left. {\vrule 
height0.41em width0em depth0.41em} \right) 
\end{eqnarray*}
\begin{eqnarray*}
\lefteqn{{\it K_{xy}} :=  \left( {\vrule 
height0.79em width0em depth0.79em} \right. \! \!  - {H}\,{\it lx}
\, \left( \! \,{\frac {{ \partial}}{{ \partial}{t}}}\,{\it ly}\,
 \!  \right)  - {H}\, \left( \! \,{\frac {{ \partial}}{{ \partial
}{t}}}\,{\it lx}\, \!  \right) \,{\it ly} -  \left( \! \,{\frac {
{ \partial}}{{ \partial}{t}}}\,{H}\, \!  \right) \,{\it lx}\,
{\it ly} + {H}\,{\it ly}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{x}}}\,{\it lt}\, \!  \right) } \\
 & & \mbox{} + {H}\, \left( \! \,{\frac {{ \partial}}{{ \partial}
{x}}}\,{\it ly}\, \!  \right) \,{\it lt} +  \left( \! \,{\frac {{
 \partial}}{{ \partial}{x}}}\,{H}\, \!  \right) \,{\it ly}\,{\it 
lt} + {H}\,{\it lx}\, \left( \! \,{\frac {{ \partial}}{{ \partial
}{y}}}\,{\it lt}\, \!  \right)  + {H}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{y}}}\,{\it lx}\, \!  \right) \,{\it lt} \\
 & & \mbox{} +  \left( \! \,{\frac {{ \partial}}{{ \partial}{y}}}
\,{H}\, \!  \right) \,{\it lx}\,{\it lt} - 2\,{H}\,{\it lz}^{2}\,
 \left( \! \,{\frac {{ \partial}}{{ \partial}{t}}}\,{H}\, \! 
 \right) \,{\it lx}\,{\it ly} - 2\,{H}^{2}\,{\it lz}^{2}\,
 \left( \! \,{\frac {{ \partial}}{{ \partial}{t}}}\,{\it lx}\,
 \!  \right) \,{\it ly} \\
 & & \mbox{} + 2\,{H}^{2}\,{\it lz}^{2}\,{\it lx}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{y}}}\,{\it lt}\, \!  \right)  + 
2\,{H}^{2}\,{\it ly}^{2}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{y}}}\,{\it lx}\, \!  \right) \,{\it lt} + 2\,{H}^{2}\,
{\it ly}^{2}\,{\it lx}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{y}}}\,{\it lt}\, \!  \right)  \\
 & & \mbox{} - 2\,{H}^{2}\,{\it lz}^{2}\,{\it lx}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{t}}}\,{\it ly}\, \!  \right)  + 
2\,{H}^{2}\,{\it lz}^{2}\,{\it ly}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{x}}}\,{\it lt}\, \!  \right)  - 2\,{H}^{2}
\,{\it lx}^{3}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{t}
}}\,{\it ly}\, \!  \right)  \\
 & & \mbox{} + 2\,{H}^{2}\,{\it lz}\,{\it lt}\,{\it lx}\, \left( 
\! \,{\frac {{ \partial}}{{ \partial}{z}}}\,{\it ly}\, \! 
 \right)  + 2\,{H}^{2}\,{\it lz}\,{\it lt}\, \left( \! \,{\frac {
{ \partial}}{{ \partial}{z}}}\,{\it lx}\, \!  \right) \,{\it ly}
 + 2\,{H}\,{\it lx}^{2}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{x}}}\,{H}\, \!  \right) \,{\it ly}\,{\it lt} \\
 & & \mbox{} + 2\,{H}^{2}\,{\it lx}^{2}\,{\it ly}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{x}}}\,{\it lt}\, \!  \right)  - 
2\,{H}\,{\it lx}^{3}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{t}}}\,{H}\, \!  \right) \,{\it ly} + 2\,{H}^{2}\,{\it 
ly}^{3}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{x}}}\,
{\it lt}\, \!  \right)  \\
 & & \mbox{} - 2\,{H}^{2}\,{\it ly}^{3}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{\it lx}\, \!  \right)  + 2\,{H}^{2}
\,{\it lx}^{3}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{y}
}}\,{\it lt}\, \!  \right)  + 2\,{H}^{2}\,{\it lx}^{2}\, \left( 
\! \,{\frac {{ \partial}}{{ \partial}{x}}}\,{\it ly}\, \! 
 \right) \,{\it lt} \\
 & & \mbox{} - 2\,{H}^{2}\,{\it lx}^{2}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{\it lx}\, \!  \right) \,{\it ly} - 
2\,{H}\,{\it ly}^{3}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{t}}}\,{H}\, \!  \right) \,{\it lx} - 2\,{H}^{2}\,{\it 
ly}^{2}\,{\it lx}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{
t}}}\,{\it ly}\, \!  \right)  \\
 & & \mbox{} - 2\,{H}^{2}\,{\it ly}^{2}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{x}}}\,{\it ly}\, \!  \right) \,{\it lt} - 
2\,{H}^{2}\,{\it lx}^{2}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{y}}}\,{\it lx}\, \!  \right) \,{\it lt} - 2\,{H}^{2}\,
{\it lz}\,{\it lt}\,{\it lx}\, \left( \! \,{\frac {{ \partial}}{{
 \partial}{y}}}\,{\it lz}\, \!  \right)  \\
 & & \mbox{} - 2\,{H}^{2}\,{\it lz}\,{\it lt}\,{\it ly}\, \left( 
\! \,{\frac {{ \partial}}{{ \partial}{x}}}\,{\it lz}\, \! 
 \right)  + 2\,{H}\,{\it lz}\,{\it lt}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{z}}}\,{H}\, \!  \right) \,{\it lx}\,{\it 
ly} + 2\,{H}\,{\it ly}^{2}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{y}}}\,{H}\, \!  \right) \,{\it lx}\,{\it lt} \\
 & &  \! \! \left. {\vrule height0.79em width0em depth0.79em}
 \right)  \left/ {\vrule height0.41em width0em depth0.41em}
 \right. \! \!  \left( \! \,(\,2\,{H}\,{\it lz}^{2} + 1 + 2\,{H}
\,{\it ly}^{2} + 2\,{H}\,{\it lx}^{2} - 2\,{H}\,{\it lt}^{2}\,)\,
\sqrt {1 + 2\,{H}\,{\it lt}^{2}}\, \!  \right) 
\end{eqnarray*}
\begin{eqnarray*}
\lefteqn{{\it K_{xz}} :=  -  \left( {\vrule 
height0.79em width0em depth0.79em} \right. \! \!  - 2\,{H}\,{\it 
lx}^{2}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{x}}}\,{H}
\, \!  \right) \,{\it lz}\,{\it lt} - 2\,{H}^{2}\,{\it lz}^{3}\,
 \left( \! \,{\frac {{ \partial}}{{ \partial}{x}}}\,{\it lt}\,
 \!  \right)  - {H}\, \left( \! \,{\frac {{ \partial}}{{ \partial
}{z}}}\,{\it lx}\, \!  \right) \,{\it lt}} \\
 & & \mbox{} -  \left( \! \,{\frac {{ \partial}}{{ \partial}{z}}}
\,{H}\, \!  \right) \,{\it lx}\,{\it lt} + {H}\,{\it lx}\,
 \left( \! \,{\frac {{ \partial}}{{ \partial}{t}}}\,{\it lz}\,
 \!  \right)  + {H}\, \left( \! \,{\frac {{ \partial}}{{ \partial
}{t}}}\,{\it lx}\, \!  \right) \,{\it lz} +  \left( \! \,{\frac {
{ \partial}}{{ \partial}{t}}}\,{H}\, \!  \right) \,{\it lx}\,
{\it lz} \\
 & & \mbox{} - {H}\,{\it lz}\, \left( \! \,{\frac {{ \partial}}{{
 \partial}{x}}}\,{\it lt}\, \!  \right)  - {H}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{x}}}\,{\it lz}\, \!  \right) \,
{\it lt} -  \left( \! \,{\frac {{ \partial}}{{ \partial}{x}}}\,{H
}\, \!  \right) \,{\it lz}\,{\it lt} - {H}\,{\it lx}\, \left( \! 
\,{\frac {{ \partial}}{{ \partial}{z}}}\,{\it lt}\, \!  \right) 
 \\
 & & \mbox{} - 2\,{H}^{2}\,{\it ly}\,{\it lt}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{y}}}\,{\it lx}\, \!  \right) \,
{\it lz} - 2\,{H}^{2}\,{\it ly}\,{\it lt}\,{\it lx}\, \left( \! 
\,{\frac {{ \partial}}{{ \partial}{y}}}\,{\it lz}\, \!  \right) 
 - 2\,{H}\,{\it lz}^{2}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{z}}}\,{H}\, \!  \right) \,{\it lx}\,{\it lt} \\
 & & \mbox{} + 2\,{H}^{2}\,{\it ly}\,{\it lt}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{x}}}\,{\it ly}\, \!  \right) \,
{\it lz} - 2\,{H}\,{\it ly}\,{\it lt}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{y}}}\,{H}\, \!  \right) \,{\it lx}\,{\it 
lz} + 2\,{H}^{2}\,{\it lz}^{3}\, \left( \! \,{\frac {{ \partial}
}{{ \partial}{t}}}\,{\it lx}\, \!  \right)  \\
 & & \mbox{} - 2\,{H}^{2}\,{\it lx}^{3}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{z}}}\,{\it lt}\, \!  \right)  + 2\,{H}^{2}
\,{\it lx}^{3}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{t}
}}\,{\it lz}\, \!  \right)  - 2\,{H}^{2}\,{\it lx}^{2}\, \left( 
\! \,{\frac {{ \partial}}{{ \partial}{x}}}\,{\it lz}\, \! 
 \right) \,{\it lt} \\
 & & \mbox{} - 2\,{H}^{2}\,{\it lx}^{2}\,{\it lz}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{x}}}\,{\it lt}\, \!  \right)  + 
2\,{H}\,{\it lx}^{3}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{t}}}\,{H}\, \!  \right) \,{\it lz} + 2\,{H}^{2}\,{\it 
lx}^{2}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{t}}}\,
{\it lx}\, \!  \right) \,{\it lz} \\
 & & \mbox{} + 2\,{H}^{2}\,{\it ly}\,{\it lt}\,{\it lx}\, \left( 
\! \,{\frac {{ \partial}}{{ \partial}{z}}}\,{\it ly}\, \! 
 \right)  - 2\,{H}^{2}\,{\it lz}^{2}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{z}}}\,{\it lx}\, \!  \right) \,{\it lt} - 
2\,{H}^{2}\,{\it lz}^{2}\,{\it lx}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{z}}}\,{\it lt}\, \!  \right)  \\
 & & \mbox{} + 2\,{H}^{2}\,{\it lz}^{2}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{x}}}\,{\it lz}\, \!  \right) \,{\it lt} + 
2\,{H}\,{\it lz}^{3}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{t}}}\,{H}\, \!  \right) \,{\it lx} + 2\,{H}^{2}\,{\it 
lz}^{2}\,{\it lx}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{
t}}}\,{\it lz}\, \!  \right)  \\
 & & \mbox{} - 2\,{H}^{2}\,{\it ly}^{2}\,{\it lx}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{z}}}\,{\it lt}\, \!  \right)  - 
2\,{H}^{2}\,{\it ly}^{2}\,{\it lz}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{x}}}\,{\it lt}\, \!  \right)  + 2\,{H}\,
{\it ly}^{2}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{t}}}
\,{H}\, \!  \right) \,{\it lx}\,{\it lz} \\
 & & \mbox{} + 2\,{H}^{2}\,{\it ly}^{2}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{\it lx}\, \!  \right) \,{\it lz} + 
2\,{H}^{2}\,{\it ly}^{2}\,{\it lx}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{\it lz}\, \!  \right)  + 2\,{H}^{2}
\,{\it lx}^{2}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{z}
}}\,{\it lx}\, \!  \right) \,{\it lt} \! \! \left. {\vrule 
height0.79em width0em depth0.79em} \right)  \left/ {\vrule 
height0.41em width0em depth0.41em} \right. \! \!  \left( {\vrule 
height0.41em width0em depth0.41em} \right. \! \!  \\
 & & (\,2\,{H}\,{\it lz}^{2} + 1 + 2\,{H}\,{\it ly}^{2} + 2\,{H}
\,{\it lx}^{2} - 2\,{H}\,{\it lt}^{2}\,)\,\sqrt {1 + 2\,{H}\,
{\it lt}^{2}}\, \! \! \left. {\vrule 
height0.41em width0em depth0.41em} \right) 
\end{eqnarray*}
\begin{eqnarray*}
\lefteqn{{\it K_{yy}} :=  \left( {\vrule 
height0.79em width0em depth0.79em} \right. \! \!  -  \left( \! \,
{\frac {{ \partial}}{{ \partial}{t}}}\,{H}\, \!  \right) \,{\it 
ly}^{2} - 2\,{H}\,{\it ly}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{t}}}\,{\it ly}\, \!  \right)  + 2\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{y}}}\,{H}\, \!  \right) \,{\it 
ly}\,{\it lt} + 2\,{H}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{y}}}\,{\it ly}\, \!  \right) \,{\it lt}} \\
 & & \mbox{} + 2\,{H}\,{\it ly}\, \left( \! \,{\frac {{ \partial}
}{{ \partial}{y}}}\,{\it lt}\, \!  \right)  + 4\,{H}^{2}\,{\it lz
}\,{\it lt}\,{\it ly}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{z}}}\,{\it ly}\, \!  \right)  - 4\,{H}^{2}\,{\it lz}\,
{\it lt}\,{\it ly}\, \left( \! \,{\frac {{ \partial}}{{ \partial}
{y}}}\,{\it lz}\, \!  \right)  \\
 & & \mbox{} - 4\,{H}^{2}\,{\it ly}\,{\it lt}\,{\it lx}\, \left( 
\! \,{\frac {{ \partial}}{{ \partial}{y}}}\,{\it lx}\, \! 
 \right)  + 2\,{H}\,{\it lx}\,{\it lt}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{x}}}\,{H}\, \!  \right) \,{\it ly}^{2} + 4
\,{H}^{2}\,{\it ly}\,{\it lt}\,{\it lx}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{x}}}\,{\it ly}\, \!  \right)  \\
 & & \mbox{} + 2\,{H}\,{\it lz}\,{\it lt}\, \left( \! \,{\frac {{
 \partial}}{{ \partial}{z}}}\,{H}\, \!  \right) \,{\it ly}^{2} - 
4\,{H}^{2}\,{\it ly}^{3}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{t}}}\,{\it ly}\, \!  \right)  + 4\,{H}^{2}\,{\it lz}^{2
}\,{\it ly}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{y}}}\,
{\it lt}\, \!  \right)  \\
 & & \mbox{} - 2\,{H}\,{\it lz}^{2}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{H}\, \!  \right) \,{\it ly}^{2} - 4
\,{H}^{2}\,{\it lz}^{2}\,{\it ly}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{\it ly}\, \!  \right)  + 2\,{H}\,
{\it ly}^{3}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{y}}}
\,{H}\, \!  \right) \,{\it lt} \\
 & & \mbox{} + 4\,{H}^{2}\,{\it lx}^{2}\,{\it ly}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{y}}}\,{\it lt}\, \!  \right)  - 
2\,{H}\,{\it ly}^{2}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{t}}}\,{H}\, \!  \right) \,{\it lx}^{2} - 4\,{H}^{2}\,
{\it lx}^{2}\,{\it ly}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{t}}}\,{\it ly}\, \!  \right)  \\
 & & \mbox{} + 4\,{H}^{2}\,{\it ly}^{3}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{y}}}\,{\it lt}\, \!  \right)  - 2\,{H}\,
{\it ly}^{4}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{t}}}
\,{H}\, \!  \right)  \! \! \left. {\vrule 
height0.79em width0em depth0.79em} \right)  \left/ {\vrule 
height0.41em width0em depth0.41em} \right. \! \!  \left( {\vrule 
height0.41em width0em depth0.41em} \right. \! \!  \\
 & & (\,2\,{H}\,{\it lz}^{2} + 1 + 2\,{H}\,{\it ly}^{2} + 2\,{H}
\,{\it lx}^{2} - 2\,{H}\,{\it lt}^{2}\,)\,\sqrt {1 + 2\,{H}\,
{\it lt}^{2}}\, \! \! \left. {\vrule 
height0.41em width0em depth0.41em} \right) 
\end{eqnarray*}
\begin{eqnarray*}
\lefteqn{{\it K_{yz}} :=  -  \left( {\vrule 
height0.79em width0em depth0.79em} \right. \! \!  - {H}\,{\it lz}
\, \left( \! \,{\frac {{ \partial}}{{ \partial}{y}}}\,{\it lt}\,
 \!  \right)  - {H}\, \left( \! \,{\frac {{ \partial}}{{ \partial
}{y}}}\,{\it lz}\, \!  \right) \,{\it lt} -  \left( \! \,{\frac {
{ \partial}}{{ \partial}{y}}}\,{H}\, \!  \right) \,{\it lz}\,
{\it lt} - {H}\,{\it ly}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{z}}}\,{\it lt}\, \!  \right) } \\
 & & \mbox{} - {H}\, \left( \! \,{\frac {{ \partial}}{{ \partial}
{z}}}\,{\it ly}\, \!  \right) \,{\it lt} -  \left( \! \,{\frac {{
 \partial}}{{ \partial}{z}}}\,{H}\, \!  \right) \,{\it ly}\,{\it 
lt} + {H}\,{\it ly}\, \left( \! \,{\frac {{ \partial}}{{ \partial
}{t}}}\,{\it lz}\, \!  \right)  + {H}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{\it ly}\, \!  \right) \,{\it lz} \\
 & & \mbox{} +  \left( \! \,{\frac {{ \partial}}{{ \partial}{t}}}
\,{H}\, \!  \right) \,{\it ly}\,{\it lz} - 2\,{H}\,{\it lz}^{2}\,
 \left( \! \,{\frac {{ \partial}}{{ \partial}{z}}}\,{H}\, \! 
 \right) \,{\it ly}\,{\it lt} - 2\,{H}^{2}\,{\it lz}^{2}\,
 \left( \! \,{\frac {{ \partial}}{{ \partial}{z}}}\,{\it ly}\,
 \!  \right) \,{\it lt} \\
 & & \mbox{} - 2\,{H}^{2}\,{\it lz}^{2}\,{\it ly}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{z}}}\,{\it lt}\, \!  \right)  + 
2\,{H}^{2}\,{\it lz}^{2}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{y}}}\,{\it lz}\, \!  \right) \,{\it lt} + 2\,{H}\,{\it 
lz}^{3}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{t}}}\,{H}
\, \!  \right) \,{\it ly} \\
 & & \mbox{} + 2\,{H}^{2}\,{\it lz}^{2}\,{\it ly}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{t}}}\,{\it lz}\, \!  \right)  + 
2\,{H}^{2}\,{\it ly}^{2}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{z}}}\,{\it ly}\, \!  \right) \,{\it lt} - 2\,{H}\,{\it 
ly}^{2}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{y}}}\,{H}
\, \!  \right) \,{\it lz}\,{\it lt} \\
 & & \mbox{} - 2\,{H}^{2}\,{\it ly}^{2}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{y}}}\,{\it lz}\, \!  \right) \,{\it lt} - 
2\,{H}^{2}\,{\it ly}^{2}\,{\it lz}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{y}}}\,{\it lt}\, \!  \right)  + 2\,{H}\,
{\it ly}^{3}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{t}}}
\,{H}\, \!  \right) \,{\it lz} \\
 & & \mbox{} + 2\,{H}^{2}\,{\it ly}^{2}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{\it ly}\, \!  \right) \,{\it lz} - 
2\,{H}^{2}\,{\it lx}^{2}\,{\it ly}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{z}}}\,{\it lt}\, \!  \right)  - 2\,{H}^{2}
\,{\it lx}^{2}\,{\it lz}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{y}}}\,{\it lt}\, \!  \right)  \\
 & & \mbox{} + 2\,{H}\,{\it lx}^{2}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{H}\, \!  \right) \,{\it ly}\,{\it 
lz} + 2\,{H}^{2}\,{\it lx}^{2}\, \left( \! \,{\frac {{ \partial}
}{{ \partial}{t}}}\,{\it ly}\, \!  \right) \,{\it lz} + 2\,{H}^{2
}\,{\it lx}^{2}\,{\it ly}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{t}}}\,{\it lz}\, \!  \right)  \\
 & & \mbox{} + 2\,{H}^{2}\,{\it lx}\,{\it lt}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{z}}}\,{\it lx}\, \!  \right) \,
{\it ly} + 2\,{H}^{2}\,{\it lx}\,{\it lt}\, \left( \! \,{\frac {{
 \partial}}{{ \partial}{y}}}\,{\it lx}\, \!  \right) \,{\it lz}
 - 2\,{H}\,{\it lx}\,{\it lt}\, \left( \! \,{\frac {{ \partial}}{
{ \partial}{x}}}\,{H}\, \!  \right) \,{\it ly}\,{\it lz} \\
 & & \mbox{} - 2\,{H}^{2}\,{\it lx}\,{\it lt}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{x}}}\,{\it ly}\, \!  \right) \,
{\it lz} - 2\,{H}^{2}\,{\it lx}\,{\it lt}\,{\it ly}\, \left( \! 
\,{\frac {{ \partial}}{{ \partial}{x}}}\,{\it lz}\, \!  \right) 
 - 2\,{H}^{2}\,{\it lz}^{3}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{y}}}\,{\it lt}\, \!  \right)  \\
 & & \mbox{} + 2\,{H}^{2}\,{\it lz}^{3}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{\it ly}\, \!  \right)  - 2\,{H}^{2}
\,{\it ly}^{3}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{z}
}}\,{\it lt}\, \!  \right)  + 2\,{H}^{2}\,{\it ly}^{3}\, \left( 
\! \,{\frac {{ \partial}}{{ \partial}{t}}}\,{\it lz}\, \! 
 \right)  \! \! \left. {\vrule height0.79em width0em depth0.79em}
 \right)  \left/ {\vrule height0.41em width0em depth0.41em}
 \right. \! \!  \left( {\vrule height0.41em width0em depth0.41em}
 \right. \! \!  \\
 & & (\,2\,{H}\,{\it lz}^{2} + 1 + 2\,{H}\,{\it ly}^{2} + 2\,{H}
\,{\it lx}^{2} - 2\,{H}\,{\it lt}^{2}\,)\,\sqrt {1 + 2\,{H}\,
{\it lt}^{2}}\, \! \! \left. {\vrule 
height0.41em width0em depth0.41em} \right) 
\end{eqnarray*}
\begin{eqnarray*}
\lefteqn{{\it K_{zz}} :=  -  \left( {\vrule 
height0.79em width0em depth0.79em} \right. \! \!  \left( \! \,
{\frac {{ \partial}}{{ \partial}{t}}}\,{H}\, \!  \right) \,{\it 
lz}^{2} + 2\,{H}\,{\it lz}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{t}}}\,{\it lz}\, \!  \right)  + 4\,{H}^{2}\,{\it lz}\,
{\it lt}\,{\it ly}\, \left( \! \,{\frac {{ \partial}}{{ \partial}
{z}}}\,{\it ly}\, \!  \right) } \\
 & & \mbox{} - 2\,{H}\,{\it ly}\,{\it lt}\, \left( \! \,{\frac {{
 \partial}}{{ \partial}{y}}}\,{H}\, \!  \right) \,{\it lz}^{2} - 
4\,{H}^{2}\,{\it lz}\,{\it lt}\,{\it ly}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{y}}}\,{\it lz}\, \!  \right)  + 4\,{H}^{2}
\,{\it ly}^{2}\,{\it lz}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{t}}}\,{\it lz}\, \!  \right)  \\
 & & \mbox{} - 4\,{H}^{2}\,{\it lx}^{2}\,{\it lz}\, \left( \! \,
{\frac {{ \partial}}{{ \partial}{z}}}\,{\it lt}\, \!  \right)  + 
2\,{H}\,{\it lz}^{2}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{t}}}\,{H}\, \!  \right) \,{\it lx}^{2} + 4\,{H}^{2}\,
{\it lx}^{2}\,{\it lz}\, \left( \! \,{\frac {{ \partial}}{{ 
\partial}{t}}}\,{\it lz}\, \!  \right)  \\
 & & \mbox{} + 4\,{H}^{2}\,{\it lz}\,{\it lt}\,{\it lx}\, \left( 
\! \,{\frac {{ \partial}}{{ \partial}{z}}}\,{\it lx}\, \! 
 \right)  - 2\,{H}\,{\it lx}\,{\it lt}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{x}}}\,{H}\, \!  \right) \,{\it lz}^{2} - 4
\,{H}^{2}\,{\it lz}\,{\it lt}\,{\it lx}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{x}}}\,{\it lz}\, \!  \right)  \\
 & & \mbox{} - 2\,{H}\,{\it lz}^{3}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{z}}}\,{H}\, \!  \right) \,{\it lt} - 4\,{H
}^{2}\,{\it ly}^{2}\,{\it lz}\, \left( \! \,{\frac {{ \partial}}{
{ \partial}{z}}}\,{\it lt}\, \!  \right)  - 4\,{H}^{2}\,{\it lz}
^{3}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{z}}}\,{\it lt
}\, \!  \right)  \\
 & & \mbox{} + 2\,{H}\,{\it lz}^{4}\, \left( \! \,{\frac {{ 
\partial}}{{ \partial}{t}}}\,{H}\, \!  \right)  + 4\,{H}^{2}\,
{\it lz}^{3}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{t}}}
\,{\it lz}\, \!  \right)  - 2\, \left( \! \,{\frac {{ \partial}}{
{ \partial}{z}}}\,{H}\, \!  \right) \,{\it lz}\,{\it lt} - 2\,{H}
\, \left( \! \,{\frac {{ \partial}}{{ \partial}{z}}}\,{\it lz}\,
 \!  \right) \,{\it lt} \\
 & & \mbox{} - 2\,{H}\,{\it lz}\, \left( \! \,{\frac {{ \partial}
}{{ \partial}{z}}}\,{\it lt}\, \!  \right)  + 2\,{H}\,{\it lz}^{2
}\, \left( \! \,{\frac {{ \partial}}{{ \partial}{t}}}\,{H}\, \! 
 \right) \,{\it ly}^{2} \! \! \left. {\vrule 
height0.79em width0em depth0.79em} \right)  \left/ {\vrule 
height0.41em width0em depth0.41em} \right. \! \!  \left( {\vrule 
height0.41em width0em depth0.41em} \right. \! \!  \\
 & & (\,2\,{H}\,{\it lz}^{2} + 1 + 2\,{H}\,{\it ly}^{2} + 2\,{H}
\,{\it lx}^{2} - 2\,{H}\,{\it lt}^{2}\,)\,\sqrt {1 + 2\,{H}\,
{\it lt}^{2}}\, \! \! \left. {\vrule 
height0.41em width0em depth0.41em} \right) 
\end{eqnarray*}

\subsection{Eddington-Finkelstein metric}

Substituting (\ref{eqn:schw.hdefn}) and (\ref{eqn:schw.ldefn}) in the 
expressions for the lapse, shift, 3-metric and extrinsic curvature we get
the following results.
The lapse is 
\begin{equation}
\alpha = \frac{1}{\sqrt{1 + 2M/r}}.
\label{eqn:eddfink.lapse}
\end{equation}
The shift is 
\begin{equation}
\beta_i  = \frac{2M}{r^2}x_i.
\end{equation}
The 3-metric is 
\begin{equation}
g_{ij} = \eta_{ij} + \frac{2M}{r^3} x_i x_j,
\label{eqn:3metric.eddfink}
\end{equation}
where $x_i = (x,y,z)$.
The extrinsic curvature is 
\begin{equation}
K_{ij} = \frac{2M}{r^4} \frac{1}{\sqrt{1+2M/r}}
\left[ r^2 \eta_{ij} - (2 + \frac{M}{r}) x_i x_j \right].
\label{eqn:extcurv.eddfink}
\end{equation}

The event horizon in this spacetime is at $r=2M$. Since this is a static
spacetime the apparent horizon also will be located at $r=2M$. We shall use
this in later chapters.

\subsection{Kerr Metric}

Using (\ref{eqn:kerr.hdefn}) and (\ref{eqn:kerr.ldefn}) the 3+1 form of the
Kerr metric can be obtained.
The lapse is 
\begin{equation}
\alpha = 
\sqrt {{\frac {\left ({r}^{2}+{a}^{2}\right )\left ({r}^{4}+{a}^{2}{z}^{2}\right )}{{r}^{6}+{a}^{2}{r}^{4}+2
\,{r}^{3}M{x}^{2}+2\,{r}^{3}M{z}^{2}+2\,M{r}^{3}{y}^{2}+{a}^{2}{z}^{2}{r}^{2}+2\,M{z}^{2}{a}^{2}r+{a}^{4}{z}
^{2}}}}
\end{equation}
The components of the shift are
\begin{eqnarray}
\beta_x & = & 2\,\sqrt {{\frac {{M}^{2}{r}^{6}\left (rx+ay\right )^{2}}{\left ({r}^{4}+{a}^{2}{z}^{2}\right )^{2}\left ({r}^{2}+{a}^{2}\right )^{2}}}}
\\
\beta_y & = & 2\,\sqrt {{\frac {{M}^{2}{r}^{6}\left (-ry+ax\right )^{2}}{\left ({r}^{4}+{a}^{2}{z}^{2}\right )^{2}\left ({r}^{2}+{a}^{2}\right )^{2}}}}
\\
\beta_z & = & 2\,\sqrt {{\frac {{M}^{2}{r}^{4}{z}^{2}}{\left ({r}^{4}+ {a}^{2}{z}^{2}\right )^{2}}}},
\end{eqnarray}
and the components of the 3-metric are 
\begin{eqnarray}
^{(3)}g_{xx}  & = &
\left({r}^{8}+2\,{r}^{6}{a}^{2}+{r}^{4}{a}^{4}+{a}^{2}{z}^{2}{r}^{4}+
2\,{a}^{4}{z}^{2}{r}^{2}+{a}^{6}{z}^{2}+2\,{r}^{5}M{x}^{2}+ \right.
\nonumber \\
& & \left.
 4\,{r}^{4}Mxay+2\,{r}^{3}M{a}^{2}{y}^{2}\right)/\left(\left ({r}^{4}+{a}^{2}{z}^{2}\right )\left (
{r}^{2}+{a}^{2}\right )^{2}\right),
\end{eqnarray}
\begin{eqnarray}
^{(3)}g_{xy} = -{\frac {2\,M{r}^{3}\left (rx+ay\right )\left (-ry+ax\right )}{\left ({r}^{4}+{a}^{2}{z}^{2}\right )\left ({
r}^{2}+{a}^{2}\right )^{2}}},
\end{eqnarray}
\begin{eqnarray}
^{(3)}g_{xz} =    {\frac {2\,M{r}^{2}\left (rx+ay\right )z}{\left ({r}^{4}+{a}^{2}{z}^{2}\right )\left ({r}^{2}+{a}^{2}\right 
)}},
\end{eqnarray}
\begin{eqnarray}
^{(3)}g_{yy} &=& \left({r}^{8}+2\,{r}^{6}{a}^{2}+{r}^{4}{a}^{4}+{a}^{2}{z}^{2}{r}^{4}+2\,{a}^{4}{z}^{2}{r}^{2}+{a}^{6}{z}^{
2}+2\,{r}^{5}M{y}^{2} \right. \nonumber \\
& & \left. 
-4\,{r}^{4}Mxay+2\,{r}^{3}M{a}^{2}{x}^{2}\right)/\left[\left({r}^{4}+{a}^{2}{z}^{2}\right )\left (
{r}^{2}+{a}^{2}\right )^{2}\right],
\end{eqnarray}
\begin{equation}
^{(3)}g_{yz} = -{\frac {2\,M{r}^{2}\left (-ry+ax\right )z}{\left ({r}^{4}+{a}^{2}{z}^{2}\right )\left ({r}^{2}+{a}^{2}
\right )}},
\end{equation}
\begin{equation}
^{(3)}g_{zz} = {\frac {2\,Mr{z}^{2}+{r}^{4}+{a}^{2}{z}^{2}}{{r}^{4}+{a}^{2}{z}^{2}}}.
\end{equation}

The Kerr spacetime has a number of interesting properties\cite{boyer.1,MTW}. 
The first is the presence of a region of the spacetime known as the ergosphere. In this region any timelike observer is dragged in the direction of the 
rotation of the black hole. This effect is known as frame-dragging. The next
property is that the event horizon is an oblate spheroid located at
\begin{equation}
r_+ = M + \sqrt{M^2 - a^2}.
\label{eqn:kerr.horizon.location}
\end{equation}
Once again, since this metric is stationary, we would expect the apparent 
horizon to be located at the same location as the event horizon. In later
chapters we show that this is the case for this form of the Kerr spacetime
metric.

\section{Boosted Kerr-Schild metric}
We noted that the Kerr-Schild metric,
\begin{equation}
g_{\mu \nu} = \eta_{\mu \nu}  + 2Hl_{\mu} l_{\nu},
\end{equation}
was form-invariant under a Lorentz transformation. In this section we take 
advantage of this in our analysis of boosted Kerr black hole apparent horizons.
The 3+1 splitting of the Kerr-Schild metric results in a 3-metric and 
extrinsic curvature that depend on $l_{\mu}$, $\partial_{\mu} l_{\nu}$,
$H$ and $\partial_{\mu} H$, each of which can be a function of $(t,x,y,z)$.
Let $\bar{\cal O}$ be the
rest frame of the black hole. Then $(\bar{t},\bar{x^i})$ are the coordinates
in the rest frame of the black hole. Here $\bar{l}_{\mu}$, 
$\partial_{\mu} \bar{l}_{\nu}$, $\bar{H}$ and $\partial_{\mu} \bar{H}$ are
functions of $\bar{x^i}$. Note there is no time dependence in the unboosted
case. In our approach to carrying out the 3+1 decomposition of the 
Kerr-Schild metric the form of the 3-metric and extrinsic curvature in terms of
$l_{\mu}$, $H$ and their space-time derivatives remains fixed. 
That is, we can take the $\bar{\cal O}$ frame
and carry out a boost to the ${\cal O}$ frame in the direction
$\hat{v}_i = (\hat{v}_x,\hat{v}_y, \hat{v}_z)$ using the transformation matrix,
\begin{eqnarray}
\Lambda^{\bar{t}}_{t} & = & \gamma \\
\Lambda^{\bar{t}}_{i} & = & -v \gamma \hat{v}_i \\
\Lambda^{\bar{i}}_{j} & = &  (\gamma - 1) \hat{v}_i \hat{v}_j + \delta_{ij}.
\end{eqnarray}
Here $\delta_{ij}\hat{v}^i \hat{v}^j = 1$, $v$ is the boost velocity and
$\gamma = 1/\sqrt{1-v^2}$. This transformation leads to a new
null vector, $l_{\mu}$ and a new function $H=\bar{H}(x^{\mu})$. 
\begin{eqnarray}
l_{\mu} & = & \Lambda^{\nu}_{\bar{\mu}} \bar{l}_{\nu} 
\label{eqn:l.trans} \\
H & = & \bar{H}(\Lambda^{\bar{\nu}}_{\mu} \bar{x}_{\nu}).
\label{eqn:H.trans}
\end{eqnarray}
These preserve the form of the spacetime metric and its 3+1 decomposition in 
terms of $l_{\mu}$, $H$ and their spacetime derivatives.  We take advantage
of this form invariance in constructing our model problem. The code that
evaluates the 3-metric and extrinsic curvature remains fixed due to this
invariance. All that we need to specify are the spacetime derivatives of 
$l_{\mu}$ and $H$.
\subsection{Boosted Eddington-Finkelstein case}
For the Schwarzschild spacetime we use $\bar{H}$ and $\bar{l}_{\mu}$ given
by equations (\ref{eqn:schw.hdefn}) and (\ref{eqn:schw.ldefn}). To boost
this spacetime we carry out the coordinate transformation prescribed earlier
on $\bar{H}$ and $\bar{l}_{\mu}$ given by 
equations (\ref{eqn:l.trans}) and (\ref{eqn:l.trans}).

If we consider boosts only in the z-direction then the coordinates 
transform as
\begin{eqnarray}
\bar{t} & = & \gamma (t - vz) \\
\bar{x} & = & x \\
\bar{y} & = & y \\
\bar{z} & = & \gamma (z - vt),
\label{eqn:bst.trans}
\end{eqnarray}
and the radial coordinate becomes, $r=\sqrt{x^2 + y^2 + 
\gamma^2 ( z - vt)^2}$. Then $\bar{H} \rightarrow H=M/r$ with $r$ as 
just defined. The components of the null vector in terms of the old are now
\begin{eqnarray}
l_t & = & \gamma (\bar{l}_t - v\bar{l}_z) \\
l_x & = & \bar{l}_x \\
l_y & = & \bar{l}_y \\
l_z & = & \gamma (\bar{l}_z - v\bar{l}_t).
\label{eqn:bst.nullvect}
\end{eqnarray}
Where
\begin{eqnarray}
\bar{l}_t & = & 1 \\
\bar{l}_x & = & x / r \\
\bar{l}_y & = & y / r \\
\bar{l}_z & = & \gamma (z - vt)/r
\label{eqn:bst.nullvect.base}
\end{eqnarray}
with $r$ as defined above.

>From (\ref{eqn:bst.nullvect}), $H$ and their spacetime derivatives, the metric, 
extrinsic curvature, lapse and shift may be constructed.

\end{document}
------------------------------------------------------------------------
Mijan Huq                          mijan@hoffmann.ph.utexas.edu
Center for Relativity,             Phone 512 471 4700
Department of Physics,
University of Texas at Austin,
Austin, TX 78712-1081

URL: http://wwwrel.ph.utexas.edu/Members/mijan/welcome.html
------------------------------------------------------------------------


