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\begin{document}


\section{Initial Value Structure of Einstein's Equations}

\subsection{Standard System}

The traditional form of Einstein's field equations consists of the structure
equations 
\begin{equation}
Q^{\alpha \beta }{}_{\gamma }=0,\qquad T^{\rho }{}_{\alpha \beta }=0
\label{structure.eqns}
\end{equation}%
that are usually solved for the connection coefficients in terms of the
metric tensor components and the curvature equation 
\begin{equation}
G^{\mu \nu }=8\pi GT^{\mu \nu }  \label{curv.eqn}
\end{equation}%
where the trace-reversed Ricci tensor is given the special symbol%
\[
G^{\mu \nu }=\bar{R}^{\mu \nu }=R^{\mu \nu }-\frac{1}{2}Rg^{\mu \nu } 
\]%
and is called the \emph{Einstein Tensor}. In this form, the curvature
equation becomes a system of ten partial differential equations for the ten
metric tensor components $g_{\mu \nu }$.

Early research into the structure of these equations was greatly hampered by
the fact that the standard form of them is really quite complex. Once the
solution of the structure equations is substituted into the curvature
equation, the result is lengthy and involves both the metric components $%
g_{\mu \nu }$ and their inverse matrix $g^{\mu \nu }$. The presence of the
inverse matrix components makes the equations especially unpleasant because
they are not merely nonlinear but nonpolynomial as well.

One puzzle, which arises at once, is that there seems to be the wrong number
of equations. With ten independent equations for ten metric tensor
components, one would expect the metric tensor components to be fixed once
the appropriate boundary conditions have been given. However, the curvature
equation consists of components of a geometrical, coordinate-independent
condition. Thus, the coordinates must remain as four arbitrary functions per
spacetime point. That seems to contradict having the metric tensor
components determined throughout spacetime once boundary conditions have
been given.

\subsection{General Initial Value Problem (Fails)}

Suppose we wish to generate values for the functions $g_{\mu \nu }\left(
x\right) $ for each set $x=\left\{ x^{\text{a}},x^{\text{b}},x^{\text{c}%
},x^{\ast }\right\} $ of values of a particular set of spacetime
coordinates. Here, the coordinate $x^{\ast }$ is the one that has been
picked out for evolving a solution of the Einstein Field equations. Start by
specifying the appropriate values and derivatives of the metric tensor on
the surface $x^{\ast }=0$ and then use the Einstein equations to evolve the
metric tensor to other values of $x^{\ast }$. The problem of setting up the
appropriate values on the initial surface $x^{\ast }=0$ is called the \emph{%
initial value problem}. It can take several forms, depending on the nature
of the initial surface. If that initial surface is spacelike, then $x^{\ast
}=x^{0}$ is a time coordinate, $x^{\text{a}},x^{\text{b}},x^{\text{c}}$ are
space coordinates $x^{1},x^{2},x^{3}$ and the \emph{spacelike initial value
problem} is the result. If that initial surface is null or lightlike, then $%
x^{\ast }=x^{-}$ is a null coordinate that might, for example, be retarded
time, $x^{\text{c}}=x^{+}$ is another null coordinate that might be advanced
time, and one usually combines the remaining (spacelike) coordinates into a
complex coordinate $x^{\text{a}}+ix^{\text{b}}=z$. The result is called the 
\emph{characteristic initial value problem}, a name that comes from the
general theory of second order partial differential equations.%
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Start with the (incorrect) assumption that the curvature equations provide a
straightforward set of second order partial differential equations for the
metric tensor components. \emph{Solve} these equations for the ten second
derivative functions $g_{\mu \nu ,\ast \ast }$ with a result that we
represent as 
\[
g_{\mu \nu ,\ast \ast }=F_{\mu \nu }\left( g_{\alpha \beta },g_{\alpha \beta
,\gamma },g_{\alpha \beta ,q\ast },g_{\alpha \beta ,qs}\right) 
\]%
where the indexes $q,s$ take values a,b,c. To evolve a solution in the
variable $x^{\ast }$ specify $g_{\alpha \beta }$ and $g_{\alpha \beta ,\ast
} $ everywhere on the initial surface $x^{\ast }=0$. All of the other
derivatives that appear in the function $F_{\mu \nu }$ can then be found by
taking derivatives within the surface. At this point the system has been
reduced to first order form. Define $p_{\mu \nu }=g_{\mu \nu ,\ast }$ and
write the first order system%
\[
\begin{array}{c}
\dot{g}_{\mu \nu }=p_{\mu \nu }\smallskip \\ 
\dot{p}_{\mu \nu }=F_{\mu \nu }\left( g_{\alpha \beta },p_{\alpha \beta
},g_{\alpha \beta ,q},p_{\alpha \beta ,q},g_{\alpha \beta ,qs}\right)%
\end{array}%
\]%
where dots are being used to represent derivatives with respect to the
evolution variable $x^{\ast }$. Evolve this system on a computer by
converting the derivatives to differences so that the above system becomes%
\[
\begin{array}{c}
g_{\mu \nu }\left( \Delta x^{\ast }\right) =g_{\mu \nu }\left( 0\right)
+p_{\mu \nu }\left( 0\right) \Delta x^{\ast }\smallskip \\ 
p_{\mu \nu }\left( \Delta x^{\ast }\right) =p_{\mu \nu }\left( 0\right)
+F_{\mu \nu }\left( 0\right) \Delta x^{\ast }%
\end{array}%
\]

The critical assumption here is that all ten of the equations contain
independent second derivatives with respect to $x^{\ast }$ so that the \emph{%
solution} $F_{\mu \nu }$ can be found. That assumption is wrong and we can
find that out the hard way by writing out the equations in detail or we can
take a short cut by exploiting the geometrical nature of the curvature
equation.

\subsection{Implications of the Bianchi Identities}

In the process of finding the curvature equations we noted that the Einstein
tensor obeys the identities%
\[
G^{\mu \nu }{}_{;\nu }=0 
\]%
that can be obtained by contracting the curvature Bianchi identities twice.
The important thing to remember is that these equations are \emph{identities 
}--- if one expands the curvature tensor in terms of connection coefficients
and metric components and calculates the above expression, all of the terms
will cancel. Now write out the sum in the above expression%
\[
G^{\mu q}{}_{;q}+G^{\mu \ast }{}_{;\ast }=0 
\]%
and be more explicit about the derivatives%
\[
\frac{\partial G^{\mu q}}{\partial x^{q}}+\Gamma ^{\mu }{}_{\rho q}G^{\rho
q}+\Gamma ^{q}{}_{\rho q}G^{\mu \rho }+\frac{\partial G^{\mu \ast }}{%
\partial x^{\ast }}+\Gamma ^{\mu }{}_{\rho \ast }G^{\rho \ast }+\Gamma
^{\ast }{}_{\rho \ast }G^{\mu \rho }=0 
\]%
Pull one term to the left of the equation so that we can focus on it.%
\[
-\frac{\partial G^{\mu \ast }}{\partial x^{\ast }}=\frac{\partial G^{\mu q}}{%
\partial x^{q}}+\Gamma ^{\mu }{}_{\rho q}G^{\rho q}+\Gamma ^{q}{}_{\rho
q}G^{\mu \rho }+\Gamma ^{\mu }{}_{\rho \ast }G^{\rho \ast }+\Gamma ^{\ast
}{}_{\rho \ast }G^{\mu \rho } 
\]%
The Einstein tensor contains up to second derivatives of the metric tensor
components. The first term on the right in this last expression has third
derivatives in it, but the extra derivative is not with respect to the
evolution variable. Thus, the right-hand side of the equation has no third
derivatives with respect to $x^{\ast }$. Since this expression is an
identity, the left-hand side of the equation must also contain no third
derivatives with respect to $x^{\ast }$. One derivative with respect to $%
x^{\ast }$ can be plainly seen, so the Einstein tensor components $G^{\mu
\ast }$ cannot contain any second derivatives of the metric with respect to $%
x^{\ast }.$

The four curvature equations%
\[
G^{\mu \ast }=8\pi T^{\mu \ast } 
\]%
have a special character. They cannot be solved for $g_{\mu \nu ,\ast \ast }$
and can only depend on quantities $g_{\mu \nu },g_{\mu \nu ,\ast },g_{\mu
\nu ,q\ast },g_{\mu \nu ,qs}$ that can all be obtained from $g_{\mu \nu
},g_{\mu \nu ,\ast }$ on the initial surface. Thus, they contribute nothing
to evolving the metric tensor off the surface and simply constrain the
initial values that can be given for the functions $g_{\mu \nu },g_{\mu \nu
,\ast }$. For this reason they are called the \emph{initial value constraint
equations}.

The remaining six Einstein curvature equations can be solved for second
derivatives and contribute to the evolution of solutions in the usual way.
Thus, we have \emph{four constraints} on the initial data that must be
solved before the evolution begins and \emph{six dynamical equations} that
evolve the ten metric functions, leaving four metric functions arbitrary as
demanded by the coordinate-independent nature of the formalism.

One more point remains to be noted. From the twice contracted Bianchi
identity as we last wrote it above, the derivative $\frac{\partial G^{\mu
\ast }}{\partial x^{\ast }}$ is expressed in terms of the components of the
Einstein tensor and its derivatives with respect to coordinates other than $%
x^{\ast }$. Because the stress-energy tensor obeys an equation with the same
structure as the twice contracted Bianchi identity, the difference tensor 
\[
E^{\mu \nu }=G^{\mu \nu }-8\pi T^{\mu \nu } 
\]%
will also obey the same sort of equation:%
\[
-\frac{\partial E^{\mu \ast }}{\partial x^{\ast }}=\frac{\partial E^{\mu q}}{%
\partial x^{q}}+\Gamma ^{\mu }{}_{\rho q}E^{\rho q}+\Gamma ^{q}{}_{\rho
q}E^{\mu \rho }+\Gamma ^{\mu }{}_{\rho \ast }E^{\rho \ast }+\Gamma ^{\ast
}{}_{\rho \ast }E^{\mu \rho } 
\]%
Thus, if we start with initial data that satisfies the initial value
constraints $E^{\mu \ast }=0$ and use the dynamical equations $E^{pq}=0$ to
determine the evolution of the metric tensor, then all of the terms on the
right-hand side of the above equation will be zero. Consequently, for any
solution constructed in this way, 
\[
\frac{\partial E^{\mu \ast }}{\partial x^{\ast }}=0 
\]%
so that the initial value constraints do not need to be reimposed --- they
are preserved by the dynamical equations.

\subsection{The Spacelike Initial Value Problem}

To generate a solution of Einstein's equations from initial data on a
spacelike hypersurface (which I will take as $t=0$), follow these steps:

\begin{enumerate}
\item Solve the initial value equations $E^{\mu 0}=0$ on the initial
spacelike hypersurface.

\item Choose four conditions on the spacetime metric functions in order to
determine how the spacetime coordinates will evolve. For example, the three
conditions $g_{m0}=0$ cause the constant-position lines $\left( x^{m}=\text{%
const.}\right) $ to be orthogonal to the $x^{0}=$const. hypersurfaces while
the condition $g_{00}=-1$ ensures that these lines will be geodesics with $%
x^{0}=t$ the proper time along them. Another example of a set of \emph{%
coordinate conditions }would be $\partial _{\nu }g^{\mu \nu }=0$ which turns
out to be equivalent to using spacetime coordinates which obey the wave
equation $\Delta \left( x^{\mu }\right) =0$. (Show this.)

\item Combine the six dynamical Einstein equations $E^{ij}=0$ with the four
coordinate conditions to find a set of equations that can be integrated
forward in time.
\end{enumerate}

A useful trick, due to Arnowitt, Deser, and Misner, is to cast the combined
coordinate conditions and dynamical equations into the form of Hamilton's
equations%
\[
\begin{array}{c}
\dot{q}=\frac{\partial H}{\partial p} \\ 
\dot{p}=-\frac{\partial H}{\partial q}%
\end{array}%
\]%
so that, knowing the values of the variables $q,p$ everywhere at a given
time $t$ lets one estimate their values at a later time $t+\Delta t$ from%
\[
\begin{array}{c}
q\left( t+\Delta t\right) =q\left( t\right) +\Delta \frac{\partial H}{%
\partial p}\left( q\left( t\right) ,p\left( t\right) \right) \\ 
p\left( t+\Delta t\right) =p\left( t\right) -\Delta \frac{\partial H}{%
\partial q}\left( q\left( t\right) ,p\left( t\right) \right)%
\end{array}%
\]%
The appropriate $q$ variables to use in this formalism are the
``three-metric'' components $^{3}g_{ij}\left( x^{k},t\right) $ on the $%
x^{0}=t=$constant hypersurfaces. The momentum variables that are conjugate
to the metric components are denoted $\pi ^{ij}\left( x^{k},t\right) $ and
are combinations of the time derivative of the three-metric with space
derivatives of the mixed components $g_{0\mu }$ of the spacetime metric. We
will return to this scheme later in this chapter and again, when we discuss
how to project geometrical structures onto surfaces.

This method of generating solutions of Einstein's equations has been
implemented on large computers. It turns out to have several unpleasant
aspects.

\begin{enumerate}
\item One must start with a solution of the initial value equations and they
are not trivial to solve.

\item It is not always obvious that a given solution to the initial value
equations represents the physical situation that one wants it to represent.

\item Numerical methods are only approximate, so the initial value equations
are only approximately preserved as one evolves a solution in time. If one
is not careful, the numerical errors in the initial value equations will
grow exponentially in time, invalidating the solution.

\item A particular choice of coordinate conditions will often give rise to
coordinate singularities even though nothing singular is happening to the
spacetime. The numerical evolution stops at that point, often well before
anything of real interest has happened to the spacetime. For example, the
simplest coordinates, geodesic hypersurface-orthogonal coordinates, ($%
g_{00}=-1,\quad g_{0i}=0$) break down because their constant-position lines
are actually in free-fall and will intersect each other in finite amounts of
time. Near the Earth, such coordinates are good for about one hour.
\end{enumerate}

\subsection{Degrees of Freedom of the Gravitational Field}

How many functions per spacetime point characterize the gravitational field?
It is necessary to count carefully. We start by eliminating four spacetime
metric tensor components through four coordinate conditions that fix the
evolution of the spacetime coordinates once they are given on an initial
surface. The remaining six components and their time derivatives amount to
twelve functions for an apparent six dynamical degrees of freedom. These
twelve functions are subject to four initial value constraints, leaving
eight freely specifiable variables per spacetime point. We must not forget
that four of these variables correspond to the freedom to specify the
spacetime coordinates on the initial surface. Thus, only four freely
specifiable, physically significant functions can be given on an initial
hypersurface. Since these functions include both the variables and their
conjugate momenta, the final count is \emph{two degrees of freedom per
spacetime point}.

\subsection{Maxwell's Equations Initial Value Problem}

Maxwell's equations turn out to have a structure very similar to the one
that we have been exploring for Einstein's equations. It is somewhat
dangerous to use the properties of Maxwell's equations as a guide to the
properties of Einstein's equations because Maxwell's equations have many
nice properties (linearity for one) that Einstein's equations do not have.
However, now that we have discussed the structure of Einstein's equations,
it is helpful to see that Maxwell's equations have the same structure.

Maxwell's equations have the covariant form 
\begin{equation}
F_{[\mu \nu ;\gamma ]}=0  \label{maxwell.df}
\end{equation}%
corresponding to the familiar equations%
\[
\nabla \cdot B=0,\qquad \dot{B}-\nabla \times E=0 
\]%
and 
\begin{equation}
g^{\rho \sigma }F_{\mu \sigma ;\rho }=4\pi j_{\mu }  \label{maxwell.delta.f}
\end{equation}%
which corresponds to%
\[
\nabla \cdot E=4\pi \rho ,\qquad \dot{E}+\nabla \times B=4\pi j 
\]%
Equation (\ref{maxwell.df}) can be solved by introducing the vector
potential $A_{\mu }$%
\[
F_{\mu \nu }=A_{\mu ;\nu }-A_{\nu ;\mu } 
\]%
and the remaining Maxwell equations become%
\[
g^{\mu \nu }g^{\rho \sigma }\left( A_{\mu ;\sigma }-A_{\sigma ;\mu }\right)
_{;\rho }=4\pi j^{\nu } 
\]

Just as with Einstein's equations, there is an identity%
\[
\left[ g^{\mu \nu }g^{\rho \sigma }\left( A_{\mu ;\sigma }-A_{\sigma ;\mu
}\right) _{;\rho }\right] _{;\nu }=0 
\]%
which causes Maxwell's equations to guarantee the conservation of the source
current --- $g^{\mu \nu }\nabla _{\nu }j_{\mu }=0.$ This identity can be
used to argue that one of Maxwell's equations is an initial value
constraint, just as we used the twice contracted Bianchi identities to
analyze the nature of Einstein's equations. Here, however, the equations are
simple enough so that we can see their structure explicitly. Consider the $%
\mu =0$ equation%
\[
g^{\rho \sigma }\left( A_{0;\sigma }-A_{\sigma ;0}\right) _{;\rho }=4\pi
j_{0} 
\]%
and write out its components in a Lorentz frame in Minkowski spacetime%
\[
\begin{array}{c}
-\partial _{0}\left( \partial _{0}A_{0}-\partial _{0}A_{0}\right) \\ 
+\partial _{1}\left( \partial _{1}A_{0}-\partial _{0}A_{1}\right) \\ 
+\partial _{2}\left( \partial _{2}A_{0}-\partial _{0}A_{2}\right) \\ 
+\partial _{3}\left( \partial _{3}A_{0}-\partial _{0}A_{3}\right) =4\pi j_{0}%
\end{array}%
\]%
The second time derivatives in the first term clearly cancel so that this
equation involves only the vector potential components $A_{\mu }$ and their
first time derivatives. It is a constraint on the initial value data.

Where Einstein's theory has arbitrary coordinates, electromagnetic theory
has an arbitrary gauge. The vector potential can be modified according to%
\[
A_{\mu }^{\prime }=A_{\mu }+\Lambda _{;\mu } 
\]%
without changing the physically significant field tensor $F_{\mu \nu }$ at
all. Thus, we need to impose a condition that fixes the evolution of the
gauge as well as a condition on the initial data to fix the initial gauge. A
typical gauge evolution constraint is $A_{0}=0$ (the radiation gauge) which
is exactly analogous to the hypersurface orthogonal time condition $%
g_{m0}=0,g_{00}=-1$ in Einstein's theory.

Now count the degrees of freedom of an electromagnetic field. From now on, I
use a local Lorentz frame in Minkowski spacetime for all equations. Of the
four vector potential components, one is eliminated by a gauge evolution
constraint such as $A_{0}=0$. Six functions per spacetime point --- three
components $A_{j}$ and their three time derivatives $\partial _{0}A_{j}$ are
evolved by the three dynamical Maxwell equations. These six functions must
satisfy the constraint%
\[
\sum_{j=1}^{3}\left( \partial _{j}A_{0}-\partial _{0}A_{j}\right) _{;j}=4\pi
j_{0} 
\]%
which corresponds to $\nabla \cdot E=4\pi \rho $. Furthermore, some
condition is needed to eliminate the gauge freedom on the initial surface.
One good condition for specifying the initial gauge is%
\[
\sum_{j=1}^{3}\partial _{j}A_{j}=0 
\]%
which means that the six functions $A_{j},A_{j;0}$ are subject to two
conditions, leaving four freely specifiable variables per spacetime point or
two degrees of freedom, exactly as in the gravitational case.

\section{The Imperfect Analogy}

\subsection{Maxwell's Theory}

The structure of Einstein's Theory of Gravity bears strong similarities to
the structure of Maxwell's theory of electromagnetism but there are problems
in deciding just how to compare these theories. In Maxwell's theory, the
actual forces on charged particles are governed by the field tensor $F_{\mu
\nu }$. This tensor is the physically significant electromagnetic field. The
vector potential that, in a local Lorentz frame, is related to the field
tensor by 
\begin{equation}
F_{\mu \nu }=A_{\mu ;\nu }-A_{\nu ;\mu }  \label{f.vect.ptnl}
\end{equation}%
includes some physically irrelevant gauge information. Now consider
Einstein's theory and ask \textquotedblleft What fields govern the actual
forces on particles?\textquotedblright

\subsection{Gravity: The Force Analogy}

The apparent force in a particular coordinate system is given by the
connection coefficients $\Gamma ^{\rho }{}_{\sigma \delta }$. If we decide
that these are analogous to the electromagnetic field tensor components $%
F_{\mu \nu }$, then the equation%
\[
\Gamma ^{\alpha }{}_{\beta \gamma }=\frac{1}{2}g^{\alpha \sigma }\left(
g_{\sigma \beta ,\gamma }+g_{\sigma \gamma ,\beta }-g_{\beta \gamma ,\sigma
}\right) 
\]%
suggests that the metric tensor components are analogous to the vector
potentials. Notice, however, that the structure of this last equation is not
much like that of equation (\ref{f.vect.ptnl})

Now write out the curvature tensor components 
\[
\left[ R\right] _{\mu \nu }=\nabla _{e_{\mu }}\left[ \Gamma \right] _{\nu
}-\nabla _{e_{\nu }}\left[ \Gamma \right] _{\mu }+\left[ \Gamma \right]
_{\mu }\left[ \Gamma \right] _{\nu }-\left[ \Gamma \right] _{\nu }\left[
\Gamma \right] _{\mu }-c^{\sigma }{}_{\mu \nu }\left[ \Gamma \right]
_{\sigma } 
\]%
and assume a local Lorentz frame (where the connection coefficients vanish
but their derivatives do not) so that we get just%
\[
\left[ R\right] _{\mu \nu }=\left[ \Gamma \right] _{\nu ,\mu }-\left[ \Gamma %
\right] _{\mu ,\nu } 
\]%
and see if the structure of Einstein's equations looks like Maxwell's
equations in terms of this analogy. or 
\begin{equation}
R^{\alpha }{}_{\beta \mu \nu }=-2\Gamma ^{\alpha }{}_{\beta \left[ \mu ,\nu %
\right] }  \label{r.from.gamma}
\end{equation}%
where I have used the $\Gamma =0$ property of the local Lorentz frame. The
Einstein tensor is then%
\begin{eqnarray*}
G_{\mu \nu } &=&R^{\rho }{}_{\mu \rho \nu }-\frac{1}{2}g_{\mu \nu }R^{\rho
}{}_{\alpha \rho \beta }g^{\alpha \beta } \\
&=&-2\Gamma ^{\rho }{}_{\mu \left[ \rho ,\nu \right] }+g_{\mu \nu }\Gamma
^{\rho }{}_{\alpha \left[ \rho ,\beta \right] }g^{\alpha \beta }
\end{eqnarray*}%
and Einstein's Field Equations become%
\[
2\Gamma ^{\lambda }{}_{\mu \left[ \nu ,\lambda \right] }-g^{\rho \sigma
}\Gamma ^{\lambda }{}_{\rho \left[ \sigma ,\lambda \right] }g_{\mu \nu
}=8\pi GT_{\mu \nu }. 
\]%
Does this look like Maxwell's equations in terms of $F_{\mu \nu }$? If we
use harmonic coordinates, that obey the conditions%
\[
\Gamma ^{\lambda }{}_{\mu \lambda }=0 
\]%
and are compatible with our local Lorentz frame conditions, the Einstein
equations take the form%
\[
\Gamma ^{\lambda }{}_{\mu \nu ,\lambda }-\frac{1}{2}g^{\rho \sigma }\Gamma
^{\lambda }{}_{\rho \sigma ,\lambda }g_{\mu \nu }=8\pi GT_{\mu \nu } 
\]%
or%
\[
\Gamma ^{\lambda }{}_{\mu \nu ,\lambda }=8\pi G\bar{T}_{\mu \nu } 
\]%
which is not too different from Maxwell's equation.

One problem with interpreting the connection coefficients as analogous to
the electromagnetic field tensor is that the connection coefficients can be
made to vanish at any point by choosing a freely falling inertial coordinate
system. The freely falling observer has no local way to find out what the
connection coefficients are. The only thing that such an observer can
measure is the tidal force tensor that corresponds to components of the
curvature.

\subsection{Gravity: The Tidal Force Analogy}

Tidal forces are described by the Riemann tensor $R_{\alpha }{}^{\beta
}{}_{\mu \nu }$ which cannot be made to vanish at a point by any coordinate
transformation. Equation (\ref{r.from.gamma}) then corresponds to the
electromagnetic vector potential equation (\ref{f.vect.ptnl}) which can be
rewritten as%
\[
F_{\mu \nu }=2A_{\left[ \mu ,\nu \right] }. 
\]%
These two equations really do look a lot alike. The gravitational equation
just has a few extra indexes. Thus, the connection coefficients are
analogous to the vector potential components. Recall that vector potential
was introduced to solve the Maxwell equations%
\[
F_{[\mu \nu ;\gamma ]}=0. 
\]%
The curvature tensor constructed from the connection coefficients for a
zero-torsion connection obeys the curvature Bianchi identity that can be
written in the form%
\[
R^{\alpha }{}_{[\beta \mu \nu ;\gamma ]}=0. 
\]%
However, this analogy leaves the Einstein equations looking nothing at all
like the remaining Maxwell equation.

\subsection{An Extra Layer of Potentials}

Einstein's theory has three layers of variables, each a set of potentials
for the next: $g_{\mu \nu }\rightarrow \Gamma ^{\alpha }{}_{\beta \delta
}\rightarrow R_{\beta }{}^{\alpha }{}_{\mu \nu }$. Maxwell's theory has only
two such layers of variables, $A_{\mu }\rightarrow F_{\mu \nu }$. In each
case, only the last layer corresponds to observables that are well-defined
as soon as one picks a local set of basis vectors. Also, in each case, only
the first layer corresponds to freely specifiable variables. Each way of
comparing the Maxwell Theory's two layers to a pair of the Einstein Theory's
three layers has some good points. These days most physicists favor the
analogy%
\[
\begin{array}{c}
A_{\mu }\rightarrow F_{\mu \nu } \\ 
\Gamma ^{\alpha }{}_{\beta \delta }\rightarrow R_{\beta }{}^{\alpha }{}_{\mu
\nu }%
\end{array}%
\]%
so that the connection coefficients are analogous to the vector potential.
The main reason for favoring this analogy is that the vector potential can
be interpreted as a connection on a fibre bundle over spacetime and the
field tensor is the curvature of this fibre bundle. Nevertheless, the usual
formulation of Einstein's theory in terms of metric, connection, and
curvature, is not a very comfortable match to Maxwell's theory or to any
other field theory in physics.

\section{The 3+1 Split}

\subsection{Undoing Spacetime Covariance}

A long struggle was needed to cast gravitation theory into a form that
unifies space and time and does not single out any particular reference
frame. By the late 1950s this struggle was over and the formulation of
general relativity in terms of spacetime geometry was well understood. Thus
it came as a bit of a shock to veteran relativists when the path to
understanding the initial value problem involved splitting space and time in
a preferred reference frame.

The spacelike initial value problem asks one to specify data on an initial
spacelike hypersurface and then use Einstein's equations to evolve the rest
of the spacetime. The difficulty of the problem comes from the fact that
some of Einstein's equations constrain the initial data that can be given.
Arnowitt, Deser, and Misner realized that this problem involves a preferred
reference frame right from the start, the frame in which the initial
hypersurface is at constant time. Their program was as follows: Take full
advantage of that preferred frame. Express the full spacetime metric tensor
in terms of the metric tensor on the initial hypersurface. Express spacetime
covariant derivatives in terms of covariant derivatives within the
hypersurface. Express the spacetime curvature tensor in terms of the
three-dimensional curvature tensor of the hypersurface.

\subsection{The ADM Variables}

The best way to reexpress spacetime objects in terms of the geometry of a
hypersurface is to use the tensor that projects spacetime vectors onto the
hypersurface. We will discuss this projection tensor approach later. Here, I
will simply introduce the new hypersurface-covariant variables much as
Arnowitt, Deser, and Misner did. The metric tensor $h$ on the surface is
easiest to find from the contravariant components which express dot products
of basis one-forms. The unit one-form $d\tau $ normal to the surface,
defined by%
\[
d\tau \cdot d\tau =-1 
\]%
expresses changes in the proper time $\tau $ measured along curves
perpendicular to the surface and is related to changes in coordinate time by%
\[
d\tau =Ndx^{0} 
\]%
where $N$ is called the \emph{lapse function}. In terms of the spacetime
metric components,%
\[
N^{2}dx^{0}\cdot dx^{0}=-1 
\]%
or%
\[
g^{00}=-N^{-2}. 
\]

For the remaining components, find the rates of change of space coordinates
with respect to proper time $\tau $ along a curve perpendicular to the
surface. The spacetime metric supplies the tangent vector to this curve:%
\[
\frac{\partial }{\partial \tau }=g^{-1}\left( d\tau \right) 
\]%
so 
\[
\frac{\partial x^{i}}{\partial \tau }=g^{-1}\left( d\tau \right) \cdot
dx^{i}=g^{-1}\left( d\tau ,dx^{i}\right) =Ng^{-1}\left( dx^{0},dx^{i}\right)
=Ng^{0i} 
\]%
and the rate of change of the space coordinates with respect to coordinate
time along such a curve is%
\[
N^{i}=\frac{\partial x^{i}}{\partial t}=N\frac{\partial x^{i}}{\partial \tau 
}=N^{2}g^{0i}. 
\]%
These three components are referred to as the \textit{shift vector}. They
describe how the space coordinates shift relative to hypersurface-orthogonal
time lines as one moves from one value of the time coordinate $x^{0}$ to a
neighboring value.\ The spacetime metric tensor $g$ and its inverse $g^{-1}$
can now be expressed in terms of the lapse and shift function and the
three-dimensional metric $h$ that is induced on the constant-time surfaces.
The result is:%
\begin{eqnarray*}
\left[ g\right] &=&\left( 
\begin{array}{cc}
\left| \vec{N}\right| ^{2}-N^{2} & h\left( \vec{N}\right) ^{T} \\ 
h\left( \vec{N}\right) & \left[ h\right]%
\end{array}%
\right) \medskip \\
\left[ g^{-1}\right] &=&\left( 
\begin{array}{cc}
-1/N^{2} & \vec{N}^{T}/N^{2} \\ 
\vec{N}/N^{2} & \left[ h\right] ^{-1}-\vec{N}\vec{N}^{T}/N^{2}%
\end{array}%
\right)
\end{eqnarray*}%
where $\vec{N}$ is the set of shift-vector components $N^{i}$ arranged as a
column matrix. Verify by direct matrix multiplication that $\left[ g^{-1}%
\right] =\left[ g\right] ^{-1}$. Notice that the invariant four dimensional
volume element is connected to the invariant three-volume by%
\[
\sqrt{\left| g\right| }d^{4}x=N\sqrt{\left| h\right| }dx^{0}d^{3}x 
\]

The lapse and shift functions are clearly coordinate-dependent and say
nothing directly about the actual spacetime geometry. Thus, we do not expect
these functions to be dynamical variables. The three-dimensional metric
tensor $h$ specifies part of the initial data that we need by giving the
geometry of the initial hypersurface. The rate of change of this geometry
must also be given in order to have a full set of initial data. The time
derivatives $\partial _{x^{0}}h_{ij}$ would specify this rate of change but
in a way that depends on the evolution of the coordinate system.
Fortunately, the dependence can be separated out in the expression%
\[
\frac{\partial }{\partial x^{0}}h_{ij}=2NK_{ij}+N_{i|j}+N_{j|i} 
\]%
where the subscripts after `$|$' denote covariant derivatives with respect
to a connection compatible with the hypersurface metric $h$. The invariant
part of this quantity, $K_{ij}$ is called the extrinsic curvature or second
fundamental form of the hypersurface and is explained in the appendix to
this section. It turns out to be convenient to use a related quantity%
\[
\pi ^{ij}=-\sqrt{h}\left( K^{ij}-h^{ij}K\right) 
\]%
where $\sqrt{h}=\sqrt{\left\vert h\right\vert }$ is the square root of the
determinant of $h_{ij},$ latin indexes are raised using the inverse
hypersurface metric $h^{ij}$ and $K=K^{r}{}_{r}$ is the trace. In terms of
this quantity, Einstein's vacuum field equations take the form of four
constraints on the initial data%
\begin{equation}
\sqrt{h}\left[ {}^{3}R+h^{-1}\left( 
%TCIMACRO{\U{bd}}%
%BeginExpansion
{\frac12}%
%EndExpansion
\pi ^{2}-\pi ^{ij}\pi _{ij}\right) \right] =0  \label{hamiltonian}
\end{equation}%
\begin{equation}
\pi ^{ij}{}_{|j}=0  \label{momentum}
\end{equation}%
and a set of equations for evolving the data which take the form%
\[
\frac{\partial }{\partial x^{0}}\pi ^{ij}=\mathcal{F}^{ij}\left( h,\pi ,N,%
\vec{N}\right) . 
\]

For future reference, not that $\pi ^{ij}$ represents a tensor-density
rather than a simple tensor so that the covariant derivative in eq. \ref%
{momentum} has an extra term and can be written in terms of tensor
components as.%
\[
\left( K^{ij}-h^{ij}K\right) _{|j}=0. 
\]

This system of equations and variations of it can be used by a computer to
produce a numerically evolved solution to Einstein's equations. To start the
evolution, one needs a solution to the four constraints and coordinate
conditions that determine the lapse and shift functions.

\subsection{The Thin Sandwich Idea (Fails)}

Since there are four constraint equations to be solved and four
coordinate-dependent variables to be determined, it is tempting to simply
put those two things together. Specify both the space metric $h_{ij}$ and
its time derivative $\frac{\partial }{\partial x^{0}}h_{ij}$ $=\dot{h}_{ij}$
so that 
\begin{eqnarray*}
\pi ^{ij} &=&-\sqrt{h}\left( K^{ij}-h^{ij}K\right) \\
&=&-\sqrt{h}\frac{1}{2N}\left[ \left( \dot{h}^{ij}-N^{i|j}-N^{j|i}\right)
-h^{ij}h^{rs}\left( \dot{h}_{rs}-N_{r|s}-N_{s|r}\right) \right] \\
&=&-\sqrt{h}\frac{1}{2N}\left[ \left( \dot{h}^{ij}-h^{ij}h^{rs}\dot{h}%
_{rs}\right) +\left( N^{j|i}+N^{i|j}-2h^{ij}h^{rs}N_{r|s}\right) \right]
\end{eqnarray*}%
and attempt to solve the four constraints for the lapse and shift functions $%
N$ and $N^{i}$. Equation \ref{momentum} then takes the form of a system of
linear second order differential equations for the shift components%
\[
-\left[ \frac{1}{2N}\left( \dot{h}^{ij}-h^{ij}h^{rs}\dot{h}_{rs}\right) %
\right] _{|j}-\left[ \frac{1}{2N}\left(
D^{i}N^{j}+D^{j}N^{i}-2h^{ij}h^{rs}D_{s}N_{r}\right) \right] _{|j}=0 
\]%
The `thin sandwich' idea is to solve this linear system for the shift
functions $N^{i}$ and then solve the Hamiltonian constraint for the lapse
function $N$. The `thin sandwich' name comes from thinking of the space
metric $h_{ij}$ and its time derivative $\frac{\partial }{\partial x^{0}}%
h_{ij}$ $=\dot{h}_{ij}$ as describing a pair of neighboring constant-time
hypersurfaces and the lapse and shift functions as connecting them and
`filling' the sandwich.

At first, this system looks promising. It takes the form%
\[
\mathcal{\vec{D}}\left( \vec{N}\right) =\vec{j} 
\]%
where%
\[
j^{i}=-\left[ \frac{1}{2N}\left( \dot{h}^{ij}-h^{ij}h^{rs}\dot{h}%
_{rs}\right) \right] _{|j} 
\]%
and%
\[
\mathcal{D}^{i}\left( \vec{N}\right) =\left[ \frac{1}{2N}\left(
h^{ir}N_{|r}^{j}+h^{jr}N_{|r}^{i}-2h^{ij}N_{|r}^{r}\right) \right] _{|j} 
\]%
All one has to do is invert the second order operator $\mathcal{\vec{D}}$
and obtain the solution%
\[
\vec{N}=\mathcal{\vec{D}}^{-1}\vec{j}. 
\]%
That program does not seem too different from what one does to invert the
Laplacian operator to solve Poisson's equation%
\[
D^{2}\Phi =h^{rs}\Phi _{|sr}=q\qquad \Phi =D^{-2}q. 
\]%
The key to success is to establish that, with the right boundary conditions,
the solution of the equation is unique.

First, review how the uniqueness proof works for Poisson's equation. Let $%
\phi $ be the difference between two distinct solutions for $\Phi .$ That
difference must obey the homogeneous equation:%
\[
h^{rs}\phi _{|sr}=0. 
\]%
Now consider the integral over the constant-time hypersurface $\Sigma $%
\[
\int_{\Sigma }d^{3}x\sqrt{h}\phi h^{rs}\phi _{|sr}=0. 
\]%
Pull out a total divergence from the integrand and write this integral in
the form%
\[
\int_{\Sigma }d^{3}x\sqrt{h}\left[ \left( \phi h^{rs}\phi _{|s}\right)
_{|r}-\left( \phi _{|r}\right) h^{rs}\left( \phi _{|s}\right) \right] =0 
\]%
From a familiar theorem of differential geometry, the total divergence term
gives rise to a surface integral over the boundary of the hypersurface $%
\Sigma .$%
\[
\int_{\partial \Sigma }da\,n^{s}\phi \phi _{|s}-\int_{\Sigma }d^{3}x\sqrt{h}%
\phi _{|r}h^{rs}\phi _{|s}=0 
\]%
Here $n^{s}$ represents the normal to the boundary and $da$ is an area
element on the boundary. If the value of either $\Phi $ or the normal
derivative $n^{s}\Phi _{|s}$ is fixed on the boundary, then the
corresponding values for the difference $\phi $ will be zero and one obtains%
\[
\int_{\Sigma }d^{3}x\,\sqrt{h}\sqrt{h}\phi _{|r}h^{rs}\phi _{|s}=0 
\]%
Since the integrand is positive somewhere for any non-zero function $\phi $,
the integral will be positive unless $\phi $ is zero everywhere on the
hypersurface. As a result, the solution of the original Poisson equation is
unique and it is OK to represent it symbolically as $\Phi =D^{-2}q$.

For the shift-vector equation, the difference $\vec{\phi}$ between two
solutions will obey the integral equation%
\[
\int_{\Sigma }d^{3}x\sqrt{h}\vec{\phi}\cdot \mathcal{\vec{D}}\left( \vec{\phi%
}\right) =0 
\]%
or, in detail%
\[
\int_{\Sigma }d^{3}x\sqrt{h}\,\phi _{i}\left[ \frac{1}{2N}\left( h^{ir}\phi
^{j}{}_{|r}+h^{jr}\phi ^{i}{}_{|r}-2h^{ij}\phi ^{r}{}_{|r}\right) \right]
_{|j}=0 
\]%
which, after extracting a divergence, yields the relation%
\[
\begin{array}{c}
\int_{\partial \Sigma }da\,n_{j}\phi _{i}\left[ \frac{1}{2N}\left(
h^{ir}\phi ^{j}{}_{|r}+h^{jr}\phi ^{i}{}_{|r}-2h^{ij}\phi ^{r}{}_{|r}\right) %
\right] \medskip \\ 
-\int_{\Sigma }d^{3}x\sqrt{h}\,\phi _{i|j}\left[ \frac{1}{2N}\left(
h^{ir}\phi ^{j}{}_{|r}+h^{jr}\phi ^{i}{}_{|r}-2h^{ij}\phi ^{r}{}_{|r}\right) %
\right] =0%
\end{array}%
\]%
So long as the shift vector is fixed on the boundary of the region, the
difference vector $\vec{\phi}$ will vanish there and the surface integral
will be zero. \ However, when we inspect the resulting integral over the
constant-time hypersurface,%
\[
\int_{\Sigma }d^{3}x\,\frac{\sqrt{h}}{2N}\left[ h^{ir}\phi ^{j}{}_{|r}\phi
_{i|j}+h^{jr}\phi ^{i}{}_{|r}\phi _{i|j}-2h^{ij}\phi ^{r}{}_{|r}\phi _{i|j}%
\right] =0 
\]%
or%
\[
\int_{\Sigma }d^{3}x\,\frac{\sqrt{h}}{2N}\left[ \phi ^{j}{}_{|i}\phi
^{i}{}_{|j}+h^{jr}h_{is}\phi ^{i}{}_{|r}\phi ^{s}{}_{|j}-2\left( \phi
^{r}{}_{|r}\right) ^{2}\right] =0 
\]%
the integrand is simply not positive definite. \ Thus, one cannot conclude
that $\vec{\phi}$ is zero and the uniqueness of solutions for the shift
vector is not established. It appears that the thin sandwich approach to
solving the constraint equations is not well posed.

\section{Conformal Transformations}

\subsection{The Time-Symmetric Case}

Suppose that a spacetime has a constant-time hypersurface that represents an
instant of time symmetry. \ In that case, it should be possible to arrange
for all rates of change with respect to time to be zero at that instant.
Since the shift vector is essentially a rate of change of the space
coordinates with respect to hypersurface-orthogonal time, it too can be
chosen to be zero. The rate of change of the space metric can also be zero
which means that the second fundamental form $K_{ij}$ and therefore the
quantities $\pi _{ij}$ must be zero in this case. The vacuum constraints
then reduce to just one simple condition on the geometry of the
constant-time hypersurface:%
\[
^{3}R=0 
\]%
To solve this constraint, notice that there is a simple way to deform a
Riemannian geometry: Multiply the three-dimensional metric tensor by a
function.%
\[
h_{ij}=\phi ^{4}\,\tilde{h}_{ij} 
\]%
The scalar curvatures associated with the related metrics $h_{ij}$ and $%
\tilde{h}_{ij}$ are connected by a remarkable relation:%
\[
^{3}R=-8\phi ^{-5}\tilde{D}^{2}\phi +\phi ^{-4}\tilde{R} 
\]%
where 
\[
\tilde{D}^{2}\phi =\tilde{h}^{rs}\tilde{D}_{e_{r}}\tilde{D}_{e_{s}}\phi =%
\tilde{h}^{rs}\left( e_{s}\left( e_{r}\left( \phi \right) \right)
-e_{a}\left( \phi \right) \tilde{\Gamma}^{a}{}_{rs}\right) 
\]%
is the laplacian operator associated with the reference metric $\tilde{h}%
_{ij}$. The one remaining vacuum constraint then takes the form of a linear
wave equation:%
\[
\tilde{D}^{2}\phi +\frac{\tilde{R}}{8}\phi =0. 
\]

\subsection{Wormholes (Lots of them)}

A particularly interesting special case is a flat reference geometry. \ In
that case, the constraint becomes just%
\[
\tilde{\nabla}^{2}\phi =0
\]%
where $\tilde{\nabla}^{2}$ is the familiar flat-space laplacian operator.
Look for solutions with $\phi >0$ everywhere and $\phi \rightarrow 1$ at
infinity. These are asymptotically flat, regular space geometries. If the
constraint equation must hold everywhere, then the only solution is $\phi =1$
which corresponds to flat space. \ If we remove one point from the space and
permit $\phi $ to diverge near that point, then we get the solutions%
\[
\phi =1+\frac{2m}{\tilde{r}}
\]%
that correspond to placing a `charge' $+2m$ at the missing point. The area
of a sphere at constant radial coordinate $\tilde{r}$ is given by the usual
formula $4\pi \tilde{r}^{2}$ multiplied by the conformal factor $\phi ^{4}$
so that%
\[
A\left( \tilde{r}\right) =4\pi \left( 1+\frac{2m}{\tilde{r}}\right) ^{4}%
\tilde{r}^{2}
\]%
This area goes through a minimum at $\tilde{r}=2m$. As we will discuss
shortly, this is the initial data for a \emph{wormhole }which is the
maximally extended spacetime that matches the exterior geometry of a black
hole. 
\[
\FRAME{itbpF}{1.644in}{1.6959in}{0in}{}{}{1worm.jpg}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 1.644in;height 1.6959in;depth
0in;original-width 3.2335in;original-height 3.3347in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename
'graphics/1worm.jpg';file-properties "XNPEU";}}
\]%
The minimum area radius is called the wormhole throat and coincides with the
event horizon of the black hole.

One can continue this discussion and consider solutions of the form%
\[
\phi =1+\frac{2m_{1}}{\tilde{r}_{1}}+\frac{2m_{2}}{\tilde{r}_{2}}+\frac{%
2m_{3}}{\tilde{r}_{3}}+... 
\]%
where $\tilde{r}_{1},\tilde{r}_{2},\tilde{r}_{3},...$ are distances in the
flat metric from a set of points. The result is the initial data for a
collection of wormholes interacting with one another at an instant of time
symmetry.

\subsection{Conformal Weight Games}

This sort of game, due originally to James W. York, begins with a
three-dimensional ``reference metric'' $\tilde{h}_{ij}$ that generates the
actual three-dimensional metric on a hypersurface by the conformal
transformation $h_{ij}=\phi ^{4}\,\tilde{h}_{ij}$. Just as in the
time-symmetric case, the goal is to solve the scalar constraint (eq. \ref%
{hamiltonian}) for the conformal factor $\phi $. However, the general case
does not let us dispose of the second fundamental form $K^{ij}$ by setting
it equal to zero. Instead, split out the `scalar part' of this tensor by
defining a trace-part%
\[
K=h_{ij}K^{ij} 
\]%
and a trace-free part%
\[
A^{ij}=K^{ij}-\frac{1}{3}Kh^{ij}\text{.} 
\]%
Besides the second fundamental form, the general case also includes a
mass-energy density $\rho $ and a momentum current vector with components $%
j^{i}$ that also enter into the initial value constraints.

The object of the game is to start with a reference metric $\tilde{h}_{ij}$
and associated reference objects $\tilde{A}^{ij}$, $\tilde{K}$, $\tilde{\rho}
$ and $\tilde{j}^{i}$ and produce a metric $h_{ij}=\phi ^{4}\,\tilde{h}_{ij}$
and associated objects $A^{ij},K,\rho ,$ $j^{i}$ that makes it solve the
initial value constraint equations. Each of the associated objects is
assumed to be related to the corresponding reference objects by a conformal
transformation of the form $f=$ $\phi ^{s}f$. The exponent $s$ is called the 
\emph{conformal weight} of the object and is chosen to make the equations
come out as simply as possible. In particular, choose 
\begin{eqnarray*}
K &=&\tilde{K} \\
A^{ij} &=&\phi ^{-10}\tilde{A}^{ij} \\
\rho &=&\phi ^{-8}\tilde{\rho} \\
j^{i} &=&\phi ^{-10}\tilde{j}^{i}
\end{eqnarray*}%
and work out the resulting form of the constraint equations (\ref%
{hamiltonian}, \ref{momentum}). Start with the initial value constraints in
terms of these variables:%
\begin{eqnarray*}
^{3}R+K^{2}-K^{ij}K_{ij} &=&16\pi G\rho \\
D_{e_{j}}\left( K^{ij}-h^{ij}K\right) &=&8\pi G\,j^{i}
\end{eqnarray*}%
and some intermediate results:\vspace{0.05in}

$h=\det \left( \phi ^{4}\left[ \tilde{h}\right] \right) =\phi ^{16}\det
\left( \left[ \tilde{h}\right] \right) =\phi ^{16}\tilde{h},\qquad
h^{ij}=\phi ^{-4}\tilde{h}^{ij}$\vspace{0.05in}

$K^{ij}=A^{ij}+\frac{1}{3}Kh^{ij}=\phi ^{-10}\tilde{A}^{ij}+\frac{1}{3}K\phi
^{-4}\tilde{h}^{ij}$\vspace{0.05in}

$K_{ij}=\phi ^{-2}\tilde{A}_{ij}+\frac{1}{3}K\phi ^{4}\tilde{h}_{ij}$\vspace{%
0.05in}%
\begin{eqnarray*}
K^{ij}K_{ij} &=&\left( \phi ^{-10}\tilde{A}^{ij}+\frac{1}{3}K\phi ^{-4}%
\tilde{h}^{ij}\right) \left( \phi ^{-2}\tilde{A}_{ij}+\frac{1}{3}K\phi ^{4}%
\tilde{h}_{ij}\right) \\
&=&\phi ^{-12}\tilde{A}^{ij}\tilde{A}_{ij}+\frac{1}{3}K^{2}
\end{eqnarray*}

$\Gamma ^{i}{}_{jk}-\tilde{\Gamma}^{i}{}_{jk}=2\phi ^{-1}\left( \delta
_{j}^{i}\tilde{D}_{e_{k}}\phi +\delta _{k}^{i}\tilde{D}_{e_{j}}\phi -\tilde{h%
}_{jk}\tilde{h}^{is}\tilde{D}_{e_{s}}\phi \right) $\vspace{0.05in}

$D_{e_{j}}A^{ij}-\tilde{D}_{e_{j}}A^{ij}=\left( \Gamma ^{i}{}_{rj}-\tilde{%
\Gamma}^{i}{}_{rj}\right) A^{rj}+\left( \Gamma ^{j}{}_{rj}-\tilde{\Gamma}%
^{j}{}_{rj}\right) A^{ir}$

$=\left( \Gamma ^{i}{}_{rj}-\tilde{\Gamma}^{i}{}_{rj}\right) A^{rj}+\left(
\Gamma ^{j}{}_{rj}-\tilde{\Gamma}^{j}{}_{rj}\right) A^{ir}$\vspace{0.05in}

$=2\phi ^{-1}\left[ \left( \delta _{j}^{i}\tilde{D}_{e_{r}}\phi +\delta
_{r}^{i}\tilde{D}_{e_{j}}\phi -\tilde{h}_{jr}\tilde{h}^{is}\tilde{D}%
_{e_{s}}\phi \right) A^{rj}+\left( \delta _{r}^{j}\tilde{D}_{e_{j}}\phi
+\delta _{j}^{j}\tilde{D}_{e_{r}}\phi -\tilde{h}_{rj}\tilde{h}^{js}\tilde{D}%
_{e_{s}}\phi \right) A^{ir}\right] $\vspace{0.05in}

$=2\phi ^{-1}\left[ \left( \delta _{j}^{i}A^{rj}\tilde{D}_{e_{r}}\phi
+A^{rj}\delta _{r}^{i}\tilde{D}_{e_{j}}\phi -\tilde{h}_{jr}\tilde{h}%
^{is}A^{rj}\tilde{D}_{e_{s}}\phi \right) +\left( \tilde{D}_{e_{r}}\phi +3%
\tilde{D}_{e_{r}}\phi -\tilde{D}_{e_{r}}\phi \right) A^{ir}\right] $\vspace{%
0.05in}

$=2\phi ^{-1}\left( A^{ri}\tilde{D}_{e_{r}}\phi +A^{ij}\tilde{D}_{e_{j}}\phi
+3A^{ir}\tilde{D}_{e_{r}}\phi \right) =10\phi ^{-1}A^{ij}\tilde{D}%
_{e_{j}}\phi $\vspace{0.05in}

The reason for choosing weight $10$ for the trace-free part $A^{ij}$ can be
seen in the following calculation:%
\begin{eqnarray*}
D_{e_{j}}A^{ij} &=&\tilde{D}_{e_{j}}A^{ij}+10\phi ^{-1}A^{ij}\tilde{D}%
_{e_{j}}\phi \\
&=&\tilde{D}_{e_{j}}\left( \phi ^{-10}\tilde{A}^{ij}\right) +10\phi ^{-11}%
\tilde{A}^{ij}\tilde{D}_{e_{j}}\phi \\
&=&\phi ^{-10}\tilde{D}_{e_{j}}\tilde{A}^{ij}-10\phi ^{-11}\tilde{A}^{ij}%
\tilde{D}_{e_{j}}\phi +10\phi ^{-11}\tilde{A}^{ij}\tilde{D}_{e_{j}}\phi \\
&=&\phi ^{-10}\tilde{D}_{e_{j}}\tilde{A}^{ij}
\end{eqnarray*}%
so that%
\begin{eqnarray*}
D_{e_{j}}K^{ij} &=&D_{e_{j}}A^{ij}+\frac{1}{3}D_{e_{j}}\left( Kh^{ij}\right)
\\
&=&\phi ^{-10}\tilde{D}_{e_{j}}\tilde{A}^{ij}+\frac{1}{3}h^{ij}\tilde{D}%
_{e_{j}}K
\end{eqnarray*}%
With this choice of weight for $A^{ij}$, the divergence $D_{e_{j}}A^{ij}$
transforms conformally as well. The initial value constraints then become%
\begin{eqnarray*}
-8\phi ^{-5}\tilde{D}^{2}\phi +\phi ^{-4}\tilde{R}+\frac{2}{3}K^{2}-\phi
^{-12}\tilde{A}^{ij}\tilde{A}_{ij} &=&16\pi G\phi ^{-8}\tilde{\rho} \\
\phi ^{-10}\tilde{D}_{e_{j}}\tilde{A}^{ij}+\frac{1}{3}h^{ij}\tilde{D}%
_{e_{j}}K-h^{ij}\tilde{D}_{e_{j}}K &=&8\pi G\,\phi ^{-10}\tilde{j}^{i}
\end{eqnarray*}%
which take the form of a nonlinear, but elliptic equation for the conformal
factor $\phi $ 
\[
8\tilde{D}^{2}\phi -\tilde{R}\phi +\phi ^{-7}\tilde{A}^{ij}\tilde{A}_{ij}-%
\frac{2}{3}K^{2}\phi ^{5}+16\pi G\phi ^{-3}\tilde{\rho}=0 
\]%
and a simple equation that $\tilde{A}^{ij}$ needs to satisfy%
\begin{eqnarray*}
\tilde{D}_{e_{j}}\tilde{A}^{ij} &=&8\pi G\,\tilde{j}^{i}+\frac{2}{3}\phi
^{10}h^{ij}\tilde{D}_{e_{j}}K \\
&=&8\pi G\,\tilde{j}^{i}+\frac{2}{3}\phi ^{6}\tilde{h}^{ij}\tilde{D}%
_{e_{j}}K.
\end{eqnarray*}

To solve this last equation, decompose the tensor $\tilde{A}^{ij}$ into the
sum%
\[
\tilde{A}^{ij}=\tilde{A}_{\text{ref}}^{ij}+\tilde{h}^{ir}\tilde{D}%
_{e_{r}}W^{j}+\tilde{h}^{jr}\tilde{D}_{e_{r}}W^{i}-\frac{2}{3}\tilde{h}^{ij}%
\tilde{D}_{r}W^{r} 
\]%
of a \textquotedblleft first guess\textquotedblright\ $\tilde{A}_{\text{ref}%
}^{ij}$ and derivatives of a vector field $W$ so that the initial value
equation becomes a system of second order linear differential equations for
the vector field $W$. 
\[
\tilde{D}_{e_{j}}\left( \tilde{h}^{ir}\tilde{D}_{e_{r}}W^{j}+\tilde{h}^{jr}%
\tilde{D}_{e_{r}}W^{i}-\frac{2}{3}\tilde{h}^{ij}\tilde{D}_{e_{r}}W^{r}%
\right) =8\pi G\,\tilde{j}^{i}+\frac{2}{3}\phi ^{6}\tilde{h}^{ir}\tilde{D}%
_{e_{r}}K-\tilde{D}_{e_{j}}\tilde{A}_{\text{ref}}^{ij} 
\]%
This system has unique solutions for the vector $W$ because the second order
differential operator in this case can be derived from the variation of a
positive definite integral. A similar argument shows that one can always
relate an arbitrary given trace-free tensor $\tilde{A}_{\text{ref}}^{ij}$ to
a unique trace-free tensor $\tilde{A}_{\text{TT}}^{ij}$ that obeys%
\[
\tilde{D}_{e_{j}}\tilde{A}_{\text{TT}}^{ij}=0. 
\]%
Such a tensor is called \textquotedblleft transverse\textquotedblright . \
The idea is to find a vector field $U$ such that%
\[
\tilde{A}_{\text{ref}}^{ij}=\tilde{A}_{\text{TT}}^{ij}+\tilde{h}^{ir}\tilde{D%
}_{e_{r}}U^{j}+\tilde{h}^{jr}\tilde{D}_{e_{r}}U^{i}-\frac{2}{3}\tilde{h}^{ij}%
\tilde{D}_{r}U^{r} 
\]%
because the field $U$ then obeys the equation%
\[
\left( \tilde{\Delta}U\right) ^{i}=-\tilde{D}_{e_{j}}\tilde{A}_{\text{ref}%
}^{ij} 
\]%
where 
\[
\left( \tilde{\Delta}U\right) ^{i}=\tilde{D}_{e_{j}}\left( \tilde{h}^{ir}%
\tilde{D}_{e_{r}}U^{j}+\tilde{h}^{jr}\tilde{D}_{e_{r}}U^{i}-\frac{2}{3}%
\tilde{h}^{ij}\tilde{D}_{r}U^{r}\right) 
\]%
and there is a unique solution for each $\tilde{A}_{\text{ref}}^{ij}$.

\subsection{Slicing Conditions}

The results of the conformal analysis, suggest that the freely specifiable
initial data consists of the the conformal equivalence class of the space
geometry, which can be specified by giving a particular metric $\tilde{h}%
_{ij}$ that belongs to the class and the transverse-traceless part $\tilde{A}%
_{\text{TT}}^{ij}$ of the second fundamental form. A difficulty remains,
however, and can be seen as soon as the reduced forms of the initial value
constraints are placed together:%
\begin{eqnarray*}
8\tilde{D}^{2}\phi -\tilde{R}\phi +\phi ^{-7}\tilde{A}^{ij}\tilde{A}_{ij}-%
\frac{2}{3}K^{2}\phi ^{5}+16\pi G\phi ^{-3}\tilde{\rho} &=&0 \\
\left( \tilde{\Delta}W\right) ^{i}-8\pi G\,\tilde{j}^{i}-\frac{2}{3}\phi ^{6}%
\tilde{h}^{jr}\tilde{D}_{e_{r}}K &=&0
\end{eqnarray*}%
where%
\[
\tilde{A}^{ij}=\tilde{A}_{\text{TT}}^{ij}+\tilde{h}^{jr}\tilde{D}%
_{e_{r}}W^{j}+\tilde{h}^{ir}\tilde{D}_{e_{r}}W^{i}-\frac{2}{3}\tilde{h}^{ij}%
\tilde{D}_{e_{r}}W^{r}. 
\]%
The difficulty is that the equations are, in general, coupled so that the
equation for the conformal factor $\phi $ depends on the vector field $W$
(through the $\tilde{A}^{ij}\tilde{A}_{ij}$ term) and the equation for the
vector field $W$ depends on the conformal factor.

The coupling can be broken by placing a condition of the trace $K$ of the
second fundamental form of the spacelike hypersurfaces. Such a condition is
needed in any event because the choice of how to slice spactime into
constant-time hypersurfaces is completely open. In order to obtain a
definite time evolution for spacetime, some sort of condition is needed to
determine this choice. The condition 
\[
K=0 
\]%
corresponds to using surfaces of extremal volume and works quite well at
producing generalizations of the $t=$ constant surfaces of Minkowski
spacetime in spacetimes that are asymptotically flat. In spacetimes with
closed space sections, there is often only one extremal volume surface, but
the condition 
\[
\tilde{h}^{ir}\tilde{D}_{e_{r}}K=0 
\]%
which can also be written as $K=f\left( t\right) $ produces the
constant-time surfaces that are usually used and can be used to define a
natural time coordinate as well. When these conditions are used, it is
possible to solve a linear elliptic system of equations for the vector field 
$W$ and then solve a scalar elliptic equation for the conformal factor $\phi 
$. With the initial value equations solved in this way, one can then turn to
the dynamical Einstein equations to evolve the initial data to the next time
slice.

\section{Numerical Spacetimes}

\subsection{The Physical Problem}

With the initial value problem at least somewhat under control, it has
become possible to set up initial data on a spatial grid and evolve it by
using computers. The result is a numerical approximation to a solution of
Einstein's field equations. The first few attempts at numerical evolution
codes for Einstein's equations were written pretty much for their own sake.
However, there is now a pressing need for these codes to solve a real
physical problem and a massive investment of time and effort by a large
number of different research groups has already been made.

Black holes will be discussed more thoroughly in the next section of this
course. However, we have already seen some clues to just how easily they
form. Physically they can be regarded as manifestations of the fact that
pressure is a source of gravitation so that, when a star collapses, there
comes a point where increasing central pressure accelerates the collapse
instead of halting it. Mathematically we have already seen the seeds of
black hole solutions in the form of $1/r$ solutions of the elliptic equation
for the space geometry conformal factor. Astronomers have become convinced
that many massive stars end their lives as objects which have collapsed to
the point that light cannot escape from them. These collapsed objects are
well represented as black holes. The spacetime around an isolated black hole
is fairly easy to find using analytic methods and was, in fact, one of the
very first solutions of Einstein's field equations. The problem that has
spurred the development of numerical approaches to solving Einstein's
equations arises when both members of a binary star system collapse and form
black holes in orbit around one another. Such orbiting black holes are
expected to lose energy by generating gravitational waves, an effect which
has already been observed for a slightly less extreme system, the binary
pulsar, in which a pair of neutron stars orbit each other. As the orbiting
black holes spiral in toward one another, the gravitational radiation rate
increases and one expects a rising frequency ``chirp'' of radiation just
before the holes coalesce and form a single black hole.%
\[
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"1";cropright "1";cropbottom "0";filename
'graphics/spiral.jpg';file-properties "XNPEU";}} 
\]

Understanding this ``binary black hole'' system has become a matter of
urgency because several large-scale laser interferometers have just been
completed and will soon begin to search for the resulting gravitational
radiation. Precise predicted waveforms for the signal will be needed to
extract it from noise and extensive modeling of what happens will be needed
to extract information about the events that these systems detect.

\subsection{The Computer Problems}

The binary black hole system has no symmetry whatsoever. Thus, it requires a
full three-dimensional grid of evaluation points and evolution for a long
period of time in order to figure out what is happening. In addition to the
usual problems of any numerical simulation such as instabilities and
accumulating errors, there are problems peculiar to this system. Because the
coordinates evolve along with the spacetime, one cannot be sure that any
given mesh of evaluation points will continue to be a good one. Thus,
adaptive meshes are required. Because Einstein's equations generate
singularities, one must choose slicing conditions that generate spacelike
hypersurfaces that avoid these singularities so that the time evolution will
not founder on them. Finally, the goal of the calculation is to predict
gravitational radiation and that can be recognized only at a distance of a
large number of wavelengths from the coalescing holes.%
\[
\begin{array}{cc}
\FRAME{itbpF}{2.3263in}{1.625in}{0in}{}{}{waves1.jpg}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
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"1";cropright "1";cropbottom "0";filename
'graphics/waves1.jpg';file-properties "XNPEU";}} & \FRAME{itbpF}{2.3575in}{%
1.5541in}{0in}{}{}{waves2.jpg}{\special{language "Scientific Word";type
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2.3575in;height 1.5541in;depth 0in;original-width 3.1055in;original-height
2.0358in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'graphics/waves2.jpg';file-properties "XNPEU";}}%
\end{array}%
\]
Thus, one must use a large number of three-dimensional mesh points to
describe a far-field region in which very little is actually happening in
order to see the waves appear. Similarly, one should not start with initial
conditions that correspond to massive amounts of radiation pouring into the
system, and that also requires initial conditions over a large far-field
region.

The best way to evolve the outgoing gravitational waves is to use a system
in which the initial data is given on a lightlike or null hypersurface and
is evolved in a lightlike direction. In such a system it is very easy to be
sure that there is no radiation coming in to the system and extracting the
outgoing wave signal is also very simple. However, the best way to evolve
the spacetime near a pair of orbiting black holes is to use initial data on
spacelike hypersurfaces because lightlike surfaces tend to fold and develop
singularities in this region. Attempts to combine these two approaches
usually divide the spacetime into a near-field region where data is given on
spacelike surfaces, a far-field region where it is given on lightlike
surfaces, and a middle region where the two sets of data are matched.%
\[
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"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
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"1";cropright "1";cropbottom "0";filename
'graphics/waves3.jpg';file-properties "XNPEU";}} 
\]

A hybrid approach that would seem to make sense is to use hypersurfaces that
are actually spacelike everywhere, but approach outgoing null surfaces
asymptotically. This approach could use the well-developed machinery of the
spacelike initial data formalism but would gain the advantage of treating
the far-field region efficiently.%
\[
\FRAME{itbpF}{4.4105in}{2.0418in}{0in}{}{}{waves4.jpg}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
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"1";cropright "1";cropbottom "0";filename
'graphics/waves4.jpg';file-properties "XNPEU";}} 
\]

The condition $K=$ constant actually comes close because, in Minkowski
spactime, it produces spacelike hyperbolas that are indeed asymptotic to
null cones. The surfaces also turn out to be well-behaved in other
spacetimes but have not yet been used for serious numerical
calculations.\pagebreak

\section{Appendix}

\subsection{Fundamental Forms of a Hypersurface}

{}Spacetime coordinates 
\begin{eqnarray*}
x^{i};\qquad i &=&1,2,3 \\
x^{0} &=&t
\end{eqnarray*}%
Spacelike hypersurface $\Sigma \left( t\right) $%
\[
t=\text{constant} 
\]%
Basis vectors tangent to $\Sigma \left( t\right) $%
\[
\partial _{i}=\frac{\partial }{\partial x^{i}} 
\]

\subsection{First fundamental form for $\Sigma \left( t\right) $}

\begin{eqnarray*}
h\left( u,v\right) &=&g\left( u,v\right) \\
&&u,v\text{ both tangent to }\Sigma \left( t\right)
\end{eqnarray*}%
\[
h_{ij}=g\left( \partial _{i},\partial _{j}\right) =g_{ij} 
\]

\subsection{Unit vector normal to $\Sigma \left( t\right) $}

Start with the fact that the form $dt$ is already normal to the surface since%
\[
dt\cdot u=0 
\]%
for any vector $u$ tangent to $\Sigma \left( t\right) $. The form just needs
a normalization factor to make it a unit normal%
\[
d\tau =Ndt 
\]%
where we need%
\[
d\tau \cdot d\tau =-1 
\]%
But%
\[
Ndt\cdot Ndt=N^{2}dx^{0}\cdot dx^{0}=N^{2}g^{00} 
\]%
so%
\begin{eqnarray*}
N^{2}g^{00} &=&-1 \\
N &=&\left( -g^{00}\right) ^{-1/2}
\end{eqnarray*}%
or 
\[
g^{00}=-N^{-2} 
\]%
To get the unit vector normal to the surface, use the metric%
\[
n=\pm g^{-1}\left( d\tau \right) =\pm g^{-1}\left( Ndx^{0}\right) 
\]%
Either sign will work. The minus sign leads to the future-pointing normal
that is usually used.

\[
n=-g^{-1}\left( d\tau \right) =n^{0}\frac{\partial }{\partial t}%
+n^{i}\partial _{i} 
\]%
where%
\begin{eqnarray*}
n^{0} &=&-dx^{0}\cdot n=-dx^{0}\cdot g^{-1}\left( Ndx^{0}\right)
=-Ng^{-1}\left( dx^{0},dx^{0}\right) =-Ng^{00} \\
N^{2}g^{00} &=&-1 \\
Ng^{00} &=&N^{-1}
\end{eqnarray*}%
\[
n^{0}=N^{-1} 
\]

\[
n^{i}=-dx^{i}\cdot n=-dx^{i}\cdot g^{-1}\left( Ndx^{0}\right) =-Ng^{0i} 
\]%
In ADM notation, 
\[
N^{i}=N^{2}g^{0i} 
\]%
so%
\[
n^{i}=-N^{-1}N^{i} 
\]%
and%
\[
n=N^{-1}\left( \frac{\partial }{\partial t}-N^{i}\partial _{i}\right) 
\]%
\bigskip

\subsection{Second fundamental form for $\Sigma \left( t\right) $}

\begin{eqnarray*}
k\left( u,v\right) &=&-u\cdot D_{v}n \\
&&u,v\text{ both tangent to }\Sigma \left( t\right)
\end{eqnarray*}%
Side note: This definition is equivalent to%
\[
k\left( u,v\right) =u\cdot D_{v}g^{-1}\left( d\tau \right) =u\cdot
D_{v}\left( d\tau \right) 
\]%
and this is the form that should be used when the metric signature is in
doubt. For example, it gives the correct result for surfaces of constant $%
x^{2}+y^{2}+z^{2}$ in Cartesian space. \ The minus sign associated with the
normal vector $n$ is peculiar to spacelike hypersurfaces in spacetime.

Start by assuming hypersurface orthogonal coordinates with $x^{0}=\tau =$
proper time. These can always be chosen near a particular hypersurface.%
\[
n=\partial _{0} 
\]%
\[
\left[ g\right] =\left[ 
\begin{array}{cccc}
-1 & 0 & 0 & 0 \\ 
0 & g_{11} & g_{12} & g_{13} \\ 
0 & g_{21} & g_{22} & g_{23} \\ 
0 & g_{31} & g_{32} & g_{33}%
\end{array}%
\right] 
\]

\[
D_{\partial _{i}}n=D_{\partial _{i}}\partial _{0}=\Gamma ^{\rho
}{}_{0i}\partial _{\rho }=\Gamma ^{0}{}_{0i}\partial _{0}+\Gamma
^{r}{}_{0i}\partial _{r} 
\]%
\[
\Gamma ^{\alpha }{}_{\beta \gamma }=\frac{1}{2}g^{\alpha \sigma }\left(
e_{\gamma }g_{\sigma \beta }+e_{\beta }g_{\gamma \sigma }-e_{\sigma
}g_{\beta \gamma }\right) 
\]%
\begin{eqnarray*}
\Gamma ^{0}{}_{0i} &=&\frac{1}{2}g^{0\sigma }\left( \partial _{i}g_{\sigma
0}+\partial _{0}g_{i\sigma }-\partial _{\sigma }g_{0i}\right) \\
&=&\frac{1}{2}g^{00}\left( \partial _{i}g_{00}+\partial _{0}g_{i0}-\partial
_{0}g_{0i}\right) \\
&=&0
\end{eqnarray*}%
\begin{eqnarray*}
\Gamma ^{r}{}_{0i} &=&\frac{1}{2}g^{r\sigma }\left( \partial _{i}g_{\sigma
0}+\partial _{0}g_{i\sigma }-\partial _{\sigma }g_{0i}\right) \\
&=&\frac{1}{2}g^{rs}\partial _{0}g_{is}
\end{eqnarray*}%
\[
D_{\partial _{i}}n=\frac{1}{2}g^{rs}\partial _{0}g_{is}\partial _{r} 
\]%
Define the time derivative taken in this hypersurface orthogonal system as 
\[
\partial _{0}g_{is}=\frac{\partial g_{is}}{\partial \tau } 
\]%
\[
D_{v}n=\frac{1}{2}v^{i}g^{rs}\frac{\partial g_{is}}{\partial \tau }\partial
_{r} 
\]%
Now find the components of the second fundamental form%
\begin{eqnarray*}
k_{ab} &=&k\left( \partial _{a},\partial _{b}\right) =-\partial _{a}\cdot
D_{\partial _{b}}n \\
&=&-\partial _{a}\cdot \frac{1}{2}g^{rs}\frac{\partial g_{bs}}{\partial \tau 
}\partial _{r} \\
&=&-\frac{1}{2}g^{rs}\frac{\partial g_{bs}}{\partial \tau }g_{ar}=-\frac{1}{2%
}g^{rs}g_{ar}\frac{\partial g_{bs}}{\partial \tau } \\
k_{ab} &=&-\frac{1}{2}\frac{\partial g_{ba}}{\partial \tau }=-\frac{1}{2}%
\frac{\partial g_{ab}}{\partial \tau }
\end{eqnarray*}

Now state this expression in coordinate-independent language. The partial
derivative says to drag the space components of the metric tensor along the
vector field%
\[
n=\frac{\partial }{\partial \tau } 
\]%
and calculate their rate of change. That operation is the Lie derivative%
\[
\frac{\partial g_{ab}}{\partial \tau }=\left( \pounds _{n}g\right) \left(
\partial _{a},\partial _{b}\right) 
\]%
\[
g_{ab}=g\left( \partial _{a},\partial _{b}\right) 
\]%
Now use%
\[
n=N^{-1}\left( \frac{\partial }{\partial t}-N^{i}\partial _{i}\right) 
\]%
to obtain%
\[
\frac{\partial g_{ab}}{\partial \tau }=\left( \pounds _{N^{-1}\left( \frac{%
\partial }{\partial t}-N^{i}\partial _{i}\right) }g\right) \left( \partial
_{a},\partial _{b}\right) 
\]%
The Lie derivative works just like other derivatives that act on tensors, so
we subtract counter-terms for each argument.%
\[
\left( \pounds _{u}g\right) \left( \partial _{a},\partial _{b}\right)
=u\left( g_{ab}\right) -g\left( \pounds _{u}\partial _{a},\partial
_{b}\right) -g\left( \partial _{a},\pounds _{u}\partial _{b}\right) 
\]

\[
\left( \pounds _{\frac{\partial }{\partial t}}g\right) \left( \partial
_{a},\partial _{b}\right) =\frac{\partial g_{ab}}{\partial t}-g\left( 
\pounds _{\frac{\partial }{\partial t}}\partial _{a},\partial _{b}\right)
-g\left( \partial _{a},\pounds _{\frac{\partial }{\partial t}}\partial
_{b}\right) 
\]%
\begin{eqnarray*}
\frac{\partial g_{ab}}{\partial \tau } &=&\left( \pounds _{N^{-1}\left( 
\frac{\partial }{\partial t}-N^{i}\partial _{i}\right) }g\right) \left(
\partial _{a},\partial _{b}\right) \\
&=&N^{-1}\left( \frac{\partial }{\partial t}-N^{i}\partial _{i}\right)
g\left( \partial _{a},\partial _{b}\right) -g\left( \pounds _{N^{-1}\left( 
\frac{\partial }{\partial t}-N^{i}\partial _{i}\right) }\partial
_{a},\partial _{b}\right) -g\left( \partial _{a},\pounds _{N^{-1}\left( 
\frac{\partial }{\partial t}-N^{i}\partial _{i}\right) }\partial _{b}\right)
\\
&=&N^{-1}\left( \frac{\partial g_{ab}}{\partial t}-N^{i}\partial
_{i}g_{ab}\right) -g\left( \pounds _{N^{-1}\left( \frac{\partial }{\partial t%
}-N^{i}\partial _{i}\right) }\partial _{a},\partial _{b}\right) -g\left(
\partial _{a},\pounds _{N^{-1}\left( \frac{\partial }{\partial t}%
-N^{i}\partial _{i}\right) }\partial _{b}\right)
\end{eqnarray*}%
Recall that the Lie derivative of a vector field by another one is just the
commutator so that%
\begin{eqnarray*}
\pounds _{N^{-1}\left( \frac{\partial }{\partial t}-N^{i}\partial
_{i}\right) }\partial _{a} &=&\left[ N^{-1}\left( \frac{\partial }{\partial t%
}-N^{i}\partial _{i}\right) ,\partial _{a}\right] =N^{-1}\left( \frac{%
\partial }{\partial t}-N^{i}\partial _{i}\right) \partial _{a}-\partial
_{a}\left( N^{-1}\left( \frac{\partial }{\partial t}-N^{i}\partial
_{i}\right) \right) \\
&=&N^{-1}\left( \frac{\partial }{\partial t}-N^{i}\partial _{i}\right)
\partial _{a}+N^{-2}\left( \partial _{a}N\right) \left( \frac{\partial }{%
\partial t}-N^{i}\partial _{i}\right) -\left( N^{-1}\partial _{a}\left( 
\frac{\partial }{\partial t}-N^{i}\partial _{i}\right) \right) \\
&=&N^{-1}\left( \frac{\partial }{\partial t}-N^{i}\partial _{i}\right)
\partial _{a}+N^{-2}\left( \partial _{a}N\right) \left( \frac{\partial }{%
\partial t}-N^{i}\partial _{i}\right) -N^{-1}\left( \frac{\partial }{%
\partial t}-N^{i}\partial _{i}\right) \partial _{a}+N^{-1}\left( \partial
_{a}N^{i}\right) \partial _{i} \\
&=&+N^{-2}\left( \partial _{a}N\right) \left( \frac{\partial }{\partial t}%
-N^{i}\partial _{i}\right) +N^{-1}\left( \partial _{a}N^{i}\right) \partial
_{i}
\end{eqnarray*}%
\[
\pounds _{N^{-1}\left( \frac{\partial }{\partial t}-N^{i}\partial
_{i}\right) }\partial _{b}=+N^{-2}\left( \partial _{b}N\right) \left( \frac{%
\partial }{\partial t}-N^{i}\partial _{i}\right) +N^{-1}\left( \partial
_{b}N^{i}\right) \partial _{i} 
\]%
The derivative of $N$ looks like bad news, but persevere%
\begin{eqnarray*}
g\left( \pounds _{N^{-1}\left( \frac{\partial }{\partial t}-N^{i}\partial
_{i}\right) }\partial _{a},\partial _{b}\right) &=&g\left( N^{-2}\left(
\partial _{a}N\right) \left( \frac{\partial }{\partial t}-N^{i}\partial
_{i}\right) +N^{-1}\left( \partial _{a}N^{i}\right) \partial _{i},\partial
_{b}\right) \\
&=&g\left( N^{-2}\left( \partial _{a}N\right) \frac{\partial }{\partial t}%
-N^{-2}\left( \partial _{a}N\right) N^{i}\partial _{i}+N^{-1}\left( \partial
_{a}N^{i}\right) \partial _{i},\partial _{b}\right) \\
&=&N^{-2}\left( \partial _{a}N\right) g\left( \frac{\partial }{\partial t}%
,\partial _{b}\right) -N^{-2}\left( \partial _{a}N\right) N^{i}g\left(
\partial _{i},\partial _{b}\right) +N^{-1}\left( \partial _{a}N^{i}\right)
g_{ib} \\
&=&N^{-2}\left( \partial _{a}N\right) g_{0b}-N^{-2}\left( \partial
_{a}N\right) N^{i}g_{ib}+N^{-1}\left( \partial _{a}N^{i}\right) g_{ib}
\end{eqnarray*}%
Note, from the ADM decomposition of the spacetime metric, that%
\[
g_{0b}=N_{b}=N^{i}g_{ib} 
\]%
so%
\begin{eqnarray*}
g\left( \pounds _{N^{-1}\left( \frac{\partial }{\partial t}-N^{i}\partial
_{i}\right) }\partial _{a},\partial _{b}\right) &=&N^{-2}\left( \partial
_{a}N\right) N^{i}g_{ib}-N^{-2}\left( \partial _{a}N\right)
N^{i}g_{ib}+N^{-1}\left( \partial _{a}N^{i}\right) g_{ib} \\
&=&N^{-1}\left( \partial _{a}N^{i}\right) g_{ib}
\end{eqnarray*}%
The bad news has cancelled out!

The other term we have to calculate is just the same, with the indexes $a,b$
reversed:%
\[
g\left( \partial _{a},\pounds _{N^{-1}\left( \frac{\partial }{\partial t}%
-N^{i}\partial _{i}\right) }\partial _{b}\right) =N^{-1}\left( \partial
_{b}N^{i}\right) g_{ia} 
\]

Putting the various terms together,%
\begin{eqnarray*}
\frac{\partial g_{ab}}{\partial \tau } &=&N^{-1}\left( \frac{\partial g_{ab}%
}{\partial t}-N^{i}\partial _{i}g_{ab}\right) -g\left( \pounds %
_{N^{-1}\left( \frac{\partial }{\partial t}-N^{i}\partial _{i}\right)
}\partial _{a},\partial _{b}\right) -g\left( \partial _{a},\pounds %
_{N^{-1}\left( \frac{\partial }{\partial t}-N^{i}\partial _{i}\right)
}\partial _{b}\right) \\
&=&N^{-1}\left( \frac{\partial g_{ab}}{\partial t}-N^{i}\partial
_{i}g_{ab}\right) -N^{-1}\left( \partial _{a}N^{i}\right)
g_{ib}-N^{-1}\left( \partial _{b}N^{i}\right) g_{ia}
\end{eqnarray*}

We can work this out in detail, but the easiest thing to do at this point is
to note that we can always choose Riemann normal coordinates within the
hypersurface at a particular point. In those coordinates, the expression
becomes%
\[
\frac{\partial g_{ab}}{\partial \tau }=N^{-1}\left( \frac{\partial g_{ab}}{%
\partial t}-g_{ib}\partial _{a}N^{i}-g_{ia}\partial _{b}N^{i}\right) 
\]%
and, since the space metric tensor has vanishing derivatives and the
connection coefficients all vanish at our chosen point, the ordinary
derivatives can become covariant derivatives and the metric can be used to
lower indexes so that the expression can also be written as%
\[
\frac{\partial g_{ab}}{\partial \tau }=N^{-1}\left( \frac{\partial g_{ab}}{%
\partial t}-N_{a|b}-N_{b|a}\right) 
\]%
Since that form of the expression is coordinate invariant, it must hold in
any other coordinates. The final expression for the second fundamental form
is then%
\begin{eqnarray*}
k_{ab} &=&-\frac{1}{2}\frac{\partial g_{ab}}{\partial \tau } \\
&=&-\frac{1}{2N}\left( \frac{\partial g_{ab}}{\partial t}-N_{a|b}-N_{b|a}%
\right) .
\end{eqnarray*}

\end{document}

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    \noindent
    \hb@xt@1.8em{\hss\@makefnmark}#1}
\newcommand\contentsname{Contents}
\newcommand\listfigurename{List of Figures}
\newcommand\listtablename{List of Tables}
\newcommand\refname{References}
\newcommand\indexname{Index}
\newcommand\figurename{Figure}
\newcommand\tablename{Table}
\newcommand\partname{Part}
\newcommand\appendixname{Appendix}
\newcommand\abstractname{Abstract}
\def\today{\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \space\number\day, \number\year}
\setlength\columnsep{10\p@}
\setlength\columnseprule{0\p@}
\pagestyle{plain}
\pagenumbering{arabic}
\if@twoside
\else
  \raggedbottom
\fi
\if@twocolumn
  \twocolumn
  \sloppy
  \flushbottom
\else
  \onecolumn
\fi
\endinput
%%
%% End of file `article.cls'.

%%%%%%%%%%%%%%%%%% End /macros/LaTeX/base/article.cls %%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%% Start /macros/LaTeX/SWmacros/tcilatex.tex %%%%%%%%%%%%%%

% Macros for Scientific Word 4.0 documents saved with the LaTeX filter.
% Copyright (C) 2002 Mackichan Software, Inc.

\typeout{TCILATEX Macros for Scientific Word 5.0 <13 Feb 2003>.}
\typeout{NOTICE:  This macro file is NOT proprietary and may be 
freely copied and distributed.}
%
\makeatletter

%%%%%%%%%%%%%%%%%%%%%
% pdfTeX related.
\ifx\pdfoutput\relax\let\pdfoutput=\undefined\fi
\newcount\msipdfoutput
\ifx\pdfoutput\undefined
\else
 \ifcase\pdfoutput
 \else 
    \msipdfoutput=1
    \ifx\paperwidth\undefined
    \else
      \ifdim\paperheight=0pt\relax
      \else
        \pdfpageheight\paperheight
      \fi
      \ifdim\paperwidth=0pt\relax
      \else
        \pdfpagewidth\paperwidth
      \fi
    \fi
  \fi  
\fi

%%%%%%%%%%%%%%%%%%%%%
% FMTeXButton
% This is used for putting TeXButtons in the 
% frontmatter of a document. Add a line like
% \QTagDef{FMTeXButton}{101}{} to the filter 
% section of the cst being used. Also add a
% new section containing:
%     [f_101]
%     ALIAS=FMTexButton
%     TAG_TYPE=FIELD
%     TAG_LEADIN=TeX Button:
%
% It also works to put \defs in the preamble after 
% the \input tcilatex
\def\FMTeXButton#1{#1}
%
%%%%%%%%%%%%%%%%%%%%%%
% macros for time
\newcount\@hour\newcount\@minute\chardef\@x10\chardef\@xv60
\def\tcitime{
\def\@time{%
  \@minute\time\@hour\@minute\divide\@hour\@xv
  \ifnum\@hour<\@x 0\fi\the\@hour:%
  \multiply\@hour\@xv\advance\@minute-\@hour
  \ifnum\@minute<\@x 0\fi\the\@minute
  }}%

%%%%%%%%%%%%%%%%%%%%%%
% macro for hyperref and msihyperref
%\@ifundefined{hyperref}{\def\hyperref#1#2#3#4{#2\ref{#4}#3}}{}

\def\x@hyperref#1#2#3{%
   % Turn off various catcodes before reading parameter 4
   \catcode`\~ = 12
   \catcode`\$ = 12
   \catcode`\_ = 12
   \catcode`\# = 12
   \catcode`\& = 12
   \y@hyperref{#1}{#2}{#3}%
}

\def\y@hyperref#1#2#3#4{%
   #2\ref{#4}#3
   \catcode`\~ = 13
   \catcode`\$ = 3
   \catcode`\_ = 8
   \catcode`\# = 6
   \catcode`\& = 4
}

\@ifundefined{hyperref}{\let\hyperref\x@hyperref}{}
\@ifundefined{msihyperref}{\let\msihyperref\x@hyperref}{}




% macro for external program call
\@ifundefined{qExtProgCall}{\def\qExtProgCall#1#2#3#4#5#6{\relax}}{}
%%%%%%%%%%%%%%%%%%%%%%
%
% macros for graphics
%
\def\FILENAME#1{#1}%
%
\def\QCTOpt[#1]#2{%
  \def\QCTOptB{#1}
  \def\QCTOptA{#2}
}
\def\QCTNOpt#1{%
  \def\QCTOptA{#1}
  \let\QCTOptB\empty
}
\def\Qct{%
  \@ifnextchar[{%
    \QCTOpt}{\QCTNOpt}
}
\def\QCBOpt[#1]#2{%
  \def\QCBOptB{#1}%
  \def\QCBOptA{#2}%
}
\def\QCBNOpt#1{%
  \def\QCBOptA{#1}%
  \let\QCBOptB\empty
}
\def\Qcb{%
  \@ifnextchar[{%
    \QCBOpt}{\QCBNOpt}%
}
\def\PrepCapArgs{%
  \ifx\QCBOptA\empty
    \ifx\QCTOptA\empty
      {}%
    \else
      \ifx\QCTOptB\empty
        {\QCTOptA}%
      \else
        [\QCTOptB]{\QCTOptA}%
      \fi
    \fi
  \else
    \ifx\QCBOptA\empty
      {}%
    \else
      \ifx\QCBOptB\empty
        {\QCBOptA}%
      \else
        [\QCBOptB]{\QCBOptA}%
      \fi
    \fi
  \fi
}
\newcount\GRAPHICSTYPE
%\GRAPHICSTYPE 0 is for TurboTeX
%\GRAPHICSTYPE 1 is for DVIWindo (PostScript)
%%%(removed)%\GRAPHICSTYPE 2 is for psfig (PostScript)
\GRAPHICSTYPE=\z@
\def\GRAPHICSPS#1{%
 \ifcase\GRAPHICSTYPE%\GRAPHICSTYPE=0
   \special{ps: #1}%
 \or%\GRAPHICSTYPE=1
   \special{language "PS", include "#1"}%
%%%\or%\GRAPHICSTYPE=2
%%%  #1%
 \fi
}%
%
\def\GRAPHICSHP#1{\special{include #1}}%
%
% \graffile{ body }                                  %#1
%          { contentswidth (scalar)  }               %#2
%          { contentsheight (scalar) }               %#3
%          { vertical shift when in-line (scalar) }  %#4

\def\graffile#1#2#3#4{%
%%% \ifnum\GRAPHICSTYPE=\tw@
%%%  %Following if using psfig
%%%  \@ifundefined{psfig}{\input psfig.tex}{}%
%%%  \psfig{file=#1, height=#3, width=#2}%
%%% \else
  %Following for all others
  % JCS - added BOXTHEFRAME, see below
    \bgroup
	   \@inlabelfalse
       \leavevmode
       \@ifundefined{bbl@deactivate}{\def~{\string~}}{\activesoff}%
        \raise -#4 \BOXTHEFRAME{%
           \hbox to #2{\raise #3\hbox to #2{\null #1\hfil}}}%
    \egroup
}%
%
% A box for drafts
\def\draftbox#1#2#3#4{%
 \leavevmode\raise -#4 \hbox{%
  \frame{\rlap{\protect\tiny #1}\hbox to #2%
   {\vrule height#3 width\z@ depth\z@\hfil}%
  }%
 }%
}%
%
\newcount\@msidraft
\@msidraft=\z@
\let\nographics=\@msidraft
\newif\ifwasdraft
\wasdraftfalse

%  \GRAPHIC{ body }                                  %#1
%          { draft name }                            %#2
%          { contentswidth (scalar)  }               %#3
%          { contentsheight (scalar) }               %#4
%          { vertical shift when in-line (scalar) }  %#5
\def\GRAPHIC#1#2#3#4#5{%
   \ifnum\@msidraft=\@ne\draftbox{#2}{#3}{#4}{#5}%
   \else\graffile{#1}{#3}{#4}{#5}%
   \fi
}
%
\def\addtoLaTeXparams#1{%
    \edef\LaTeXparams{\LaTeXparams #1}}%
%
% JCS -  added a switch BoxFrame that can 
% be set by including X in the frame params.
% If set a box is drawn around the frame.

\newif\ifBoxFrame \BoxFramefalse
\newif\ifOverFrame \OverFramefalse
\newif\ifUnderFrame \UnderFramefalse

\def\BOXTHEFRAME#1{%
   \hbox{%
      \ifBoxFrame
         \frame{#1}%
      \else
         {#1}%
      \fi
   }%
}


\def\doFRAMEparams#1{\BoxFramefalse\OverFramefalse\UnderFramefalse\readFRAMEparams#1\end}%
\def\readFRAMEparams#1{%
 \ifx#1\end%
  \let\next=\relax
  \else
  \ifx#1i\dispkind=\z@\fi
  \ifx#1d\dispkind=\@ne\fi
  \ifx#1f\dispkind=\tw@\fi
  \ifx#1t\addtoLaTeXparams{t}\fi
  \ifx#1b\addtoLaTeXparams{b}\fi
  \ifx#1p\addtoLaTeXparams{p}\fi
  \ifx#1h\addtoLaTeXparams{h}\fi
  \ifx#1X\BoxFrametrue\fi
  \ifx#1O\OverFrametrue\fi
  \ifx#1U\UnderFrametrue\fi
  \ifx#1w
    \ifnum\@msidraft=1\wasdrafttrue\else\wasdraftfalse\fi
    \@msidraft=\@ne
  \fi
  \let\next=\readFRAMEparams
  \fi
 \next
 }%
%
%Macro for In-line graphics object
%   \IFRAME{ contentswidth (scalar)  }               %#1
%          { contentsheight (scalar) }               %#2
%          { vertical shift when in-line (scalar) }  %#3
%          { draft name }                            %#4
%          { body }                                  %#5
%          { caption}                                %#6


\def\IFRAME#1#2#3#4#5#6{%
      \bgroup
      \let\QCTOptA\empty
      \let\QCTOptB\empty
      \let\QCBOptA\empty
      \let\QCBOptB\empty
      #6%
      \parindent=0pt
      \leftskip=0pt
      \rightskip=0pt
      \setbox0=\hbox{\QCBOptA}%
      \@tempdima=#1\relax
      \ifOverFrame
          % Do this later
          \typeout{This is not implemented yet}%
          \show\HELP
      \else
         \ifdim\wd0>\@tempdima
            \advance\@tempdima by \@tempdima
            \ifdim\wd0 >\@tempdima
               \setbox1 =\vbox{%
                  \unskip\hbox to \@tempdima{\hfill\GRAPHIC{#5}{#4}{#1}{#2}{#3}\hfill}%
                  \unskip\hbox to \@tempdima{\parbox[b]{\@tempdima}{\QCBOptA}}%
               }%
               \wd1=\@tempdima
            \else
               \textwidth=\wd0
               \setbox1 =\vbox{%
                 \noindent\hbox to \wd0{\hfill\GRAPHIC{#5}{#4}{#1}{#2}{#3}\hfill}\\%
                 \noindent\hbox{\QCBOptA}%
               }%
               \wd1=\wd0
            \fi
         \else
            \ifdim\wd0>0pt
              \hsize=\@tempdima
              \setbox1=\vbox{%
                \unskip\GRAPHIC{#5}{#4}{#1}{#2}{0pt}%
                \break
                \unskip\hbox to \@tempdima{\hfill \QCBOptA\hfill}%
              }%
              \wd1=\@tempdima
           \else
              \hsize=\@tempdima
              \setbox1=\vbox{%
                \unskip\GRAPHIC{#5}{#4}{#1}{#2}{0pt}%
              }%
              \wd1=\@tempdima
           \fi
         \fi
         \@tempdimb=\ht1
         %\advance\@tempdimb by \dp1
         \advance\@tempdimb by -#2
         \advance\@tempdimb by #3
         \leavevmode
         \raise -\@tempdimb \hbox{\box1}%
      \fi
      \egroup%
}%
%
%Macro for Display graphics object
%   \DFRAME{ contentswidth (scalar)  }               %#1
%          { contentsheight (scalar) }               %#2
%          { draft label }                           %#3
%          { name }                                  %#4
%          { caption}                                %#5
\def\DFRAME#1#2#3#4#5{%
  \vspace\topsep
  \hfil\break
  \bgroup
     \leftskip\@flushglue
	 \rightskip\@flushglue
	 \parindent\z@
	 \parfillskip\z@skip
     \let\QCTOptA\empty
     \let\QCTOptB\empty
     \let\QCBOptA\empty
     \let\QCBOptB\empty
	 \vbox\bgroup
        \ifOverFrame 
           #5\QCTOptA\par
        \fi
        \GRAPHIC{#4}{#3}{#1}{#2}{\z@}%
        \ifUnderFrame 
           \break#5\QCBOptA
        \fi
	 \egroup
  \egroup
  \vspace\topsep
  \break
}%
%
%Macro for Floating graphic object
%   \FFRAME{ framedata f|i tbph x F|T }              %#1
%          { contentswidth (scalar)  }               %#2
%          { contentsheight (scalar) }               %#3
%          { caption }                               %#4
%          { label }                                 %#5
%          { draft name }                            %#6
%          { body }                                  %#7
\def\FFRAME#1#2#3#4#5#6#7{%
 %If float.sty loaded and float option is 'h', change to 'H'  (gp) 1998/09/05
  \@ifundefined{floatstyle}
    {%floatstyle undefined (and float.sty not present), no change
     \begin{figure}[#1]%
    }
    {%floatstyle DEFINED
	 \ifx#1h%Only the h parameter, change to H
      \begin{figure}[H]%
	 \else
      \begin{figure}[#1]%
	 \fi
	}
  \let\QCTOptA\empty
  \let\QCTOptB\empty
  \let\QCBOptA\empty
  \let\QCBOptB\empty
  \ifOverFrame
    #4
    \ifx\QCTOptA\empty
    \else
      \ifx\QCTOptB\empty
        \caption{\QCTOptA}%
      \else
        \caption[\QCTOptB]{\QCTOptA}%
      \fi
    \fi
    \ifUnderFrame\else
      \label{#5}%
    \fi
  \else
    \UnderFrametrue%
  \fi
  \begin{center}\GRAPHIC{#7}{#6}{#2}{#3}{\z@}\end{center}%
  \ifUnderFrame
    #4
    \ifx\QCBOptA\empty
      \caption{}%
    \else
      \ifx\QCBOptB\empty
        \caption{\QCBOptA}%
      \else
        \caption[\QCBOptB]{\QCBOptA}%
      \fi
    \fi
    \label{#5}%
  \fi
  \end{figure}%
 }%
%
%
%    \FRAME{ framedata f|i tbph x F|T }              %#1
%          { contentswidth (scalar)  }               %#2
%          { contentsheight (scalar) }               %#3
%          { vertical shift when in-line (scalar) }  %#4
%          { caption }                               %#5
%          { label }                                 %#6
%          { name }                                  %#7
%          { body }                                  %#8
%
%    framedata is a string which can contain the following
%    characters: idftbphxFT
%    Their meaning is as follows:
%             i, d or f : in-line, display, or floating
%             t,b,p,h   : LaTeX floating placement options
%             x         : fit contents box to contents
%             F or T    : Figure or Table. 
%                         Later this can expand
%                         to a more general float class.
%
%
\newcount\dispkind%

\def\makeactives{
  \catcode`\"=\active
  \catcode`\;=\active
  \catcode`\:=\active
  \catcode`\'=\active
  \catcode`\~=\active
}
\bgroup
   \makeactives
   \gdef\activesoff{%
      \def"{\string"}%
      \def;{\string;}%
      \def:{\string:}%
      \def'{\string'}%
      \def~{\string~}%
      %\bbl@deactivate{"}%
      %\bbl@deactivate{;}%
      %\bbl@deactivate{:}%
      %\bbl@deactivate{'}%
    }
\egroup

\def\FRAME#1#2#3#4#5#6#7#8{%
 \bgroup
 \ifnum\@msidraft=\@ne
   \wasdrafttrue
 \else
   \wasdraftfalse%
 \fi
 \def\LaTeXparams{}%
 \dispkind=\z@
 \def\LaTeXparams{}%
 \doFRAMEparams{#1}%
 \ifnum\dispkind=\z@\IFRAME{#2}{#3}{#4}{#7}{#8}{#5}\else
  \ifnum\dispkind=\@ne\DFRAME{#2}{#3}{#7}{#8}{#5}\else
   \ifnum\dispkind=\tw@
    \edef\@tempa{\noexpand\FFRAME{\LaTeXparams}}%
    \@tempa{#2}{#3}{#5}{#6}{#7}{#8}%
    \fi
   \fi
  \fi
  \ifwasdraft\@msidraft=1\else\@msidraft=0\fi{}%
  \egroup
 }%
%
% This macro added to let SW gobble a parameter that
% should not be passed on and expanded. 

\def\TEXUX#1{"texux"}

%
% Macros for text attributes:
%
\def\BF#1{{\bf {#1}}}%
\def\NEG#1{\leavevmode\hbox{\rlap{\thinspace/}{$#1$}}}%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% macros for user - defined functions
\def\limfunc#1{\mathop{\rm #1}}%
\def\func#1{\mathop{\rm #1}\nolimits}%
% macro for unit names
\def\unit#1{\mathord{\thinspace\rm #1}}%

%
% miscellaneous 
\long\def\QQQ#1#2{%
     \long\expandafter\def\csname#1\endcsname{#2}}%
\@ifundefined{QTP}{\def\QTP#1{}}{}
\@ifundefined{QEXCLUDE}{\def\QEXCLUDE#1{}}{}
\@ifundefined{Qlb}{\def\Qlb#1{#1}}{}
\@ifundefined{Qlt}{\def\Qlt#1{#1}}{}
\def\QWE{}%
\long\def\QQA#1#2{}%
\def\QTR#1#2{{\csname#1\endcsname {#2}}}%
\long\def\TeXButton#1#2{#2}%
\long\def\QSubDoc#1#2{#2}%
\def\EXPAND#1[#2]#3{}%
\def\NOEXPAND#1[#2]#3{}%
\def\PROTECTED{}%
\def\LaTeXparent#1{}%
\def\ChildStyles#1{}%
\def\ChildDefaults#1{}%
\def\QTagDef#1#2#3{}%

% Constructs added with Scientific Notebook
\@ifundefined{correctchoice}{\def\correctchoice{\relax}}{}
\@ifundefined{HTML}{\def\HTML#1{\relax}}{}
\@ifundefined{TCIIcon}{\def\TCIIcon#1#2#3#4{\relax}}{}
\if@compatibility
  \typeout{Not defining UNICODE  U or CustomNote commands for LaTeX 2.09.}
\else
  \providecommand{\UNICODE}[2][]{\protect\rule{.1in}{.1in}}
  \providecommand{\U}[1]{\protect\rule{.1in}{.1in}}
  \providecommand{\CustomNote}[3][]{\marginpar{#3}}
\fi

\@ifundefined{lambdabar}{
      \def\lambdabar{\errmessage{You have used the lambdabar symbol. 
                      This is available for typesetting only in RevTeX styles.}}
   }{}

%
% Macros for style editor docs
\@ifundefined{StyleEditBeginDoc}{\def\StyleEditBeginDoc{\relax}}{}
%
% Macros for footnotes
\def\QQfnmark#1{\footnotemark}
\def\QQfntext#1#2{\addtocounter{footnote}{#1}\footnotetext{#2}}
%
% Macros for indexing.
%
\@ifundefined{TCIMAKEINDEX}{}{\makeindex}%
%
% Attempts to avoid problems with other styles
\@ifundefined{abstract}{%
 \def\abstract{%
  \if@twocolumn
   \section*{Abstract (Not appropriate in this style!)}%
   \else \small 
   \begin{center}{\bf Abstract\vspace{-.5em}\vspace{\z@}}\end{center}%
   \quotation 
   \fi
  }%
 }{%
 }%
\@ifundefined{endabstract}{\def\endabstract
  {\if@twocolumn\else\endquotation\fi}}{}%
\@ifundefined{maketitle}{\def\maketitle#1{}}{}%
\@ifundefined{affiliation}{\def\affiliation#1{}}{}%
\@ifundefined{proof}{\def\proof{\noindent{\bfseries Proof. }}}{}%
\@ifundefined{endproof}{\def\endproof{\mbox{\ \rule{.1in}{.1in}}}}{}%
\@ifundefined{newfield}{\def\newfield#1#2{}}{}%
\@ifundefined{chapter}{\def\chapter#1{\par(Chapter head:)#1\par }%
 \newcount\c@chapter}{}%
\@ifundefined{part}{\def\part#1{\par(Part head:)#1\par }}{}%
\@ifundefined{section}{\def\section#1{\par(Section head:)#1\par }}{}%
\@ifundefined{subsection}{\def\subsection#1%
 {\par(Subsection head:)#1\par }}{}%
\@ifundefined{subsubsection}{\def\subsubsection#1%
 {\par(Subsubsection head:)#1\par }}{}%
\@ifundefined{paragraph}{\def\paragraph#1%
 {\par(Subsubsubsection head:)#1\par }}{}%
\@ifundefined{subparagraph}{\def\subparagraph#1%
 {\par(Subsubsubsubsection head:)#1\par }}{}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% These symbols are not recognized by LaTeX
\@ifundefined{therefore}{\def\therefore{}}{}%
\@ifundefined{backepsilon}{\def\backepsilon{}}{}%
\@ifundefined{yen}{\def\yen{\hbox{\rm\rlap=Y}}}{}%
\@ifundefined{registered}{%
   \def\registered{\relax\ifmmode{}\r@gistered
                    \else$\m@th\r@gistered$\fi}%
 \def\r@gistered{^{\ooalign
  {\hfil\raise.07ex\hbox{$\scriptstyle\rm\text{R}$}\hfil\crcr
  \mathhexbox20D}}}}{}%
\@ifundefined{Eth}{\def\Eth{}}{}%
\@ifundefined{eth}{\def\eth{}}{}%
\@ifundefined{Thorn}{\def\Thorn{}}{}%
\@ifundefined{thorn}{\def\thorn{}}{}%
% A macro to allow any symbol that requires math to appear in text
\def\TEXTsymbol#1{\mbox{$#1$}}%
\@ifundefined{degree}{\def\degree{{}^{\circ}}}{}%
%
% macros for T3TeX files
\newdimen\theight
\@ifundefined{Column}{\def\Column{%
 \vadjust{\setbox\z@=\hbox{\scriptsize\quad\quad tcol}%
  \theight=\ht\z@\advance\theight by \dp\z@\advance\theight by \lineskip
  \kern -\theight \vbox to \theight{%
   \rightline{\rlap{\box\z@}}%
   \vss
   }%
  }%
 }}{}%
%
\@ifundefined{qed}{\def\qed{%
 \ifhmode\unskip\nobreak\fi\ifmmode\ifinner\else\hskip5\p@\fi\fi
 \hbox{\hskip5\p@\vrule width4\p@ height6\p@ depth1.5\p@\hskip\p@}%
 }}{}%
%
\@ifundefined{cents}{\def\cents{\hbox{\rm\rlap c/}}}{}%
\@ifundefined{tciLaplace}{\def\tciLaplace{\ensuremath{\mathcal{L}}}}{}%
\@ifundefined{tciFourier}{\def\tciFourier{\ensuremath{\mathcal{F}}}}{}%
\@ifundefined{textcurrency}{\def\textcurrency{\hbox{\rm\rlap xo}}}{}%
\@ifundefined{texteuro}{\def\texteuro{\hbox{\rm\rlap C=}}}{}%
\@ifundefined{euro}{\def\euro{\hbox{\rm\rlap C=}}}{}%
\@ifundefined{textfranc}{\def\textfranc{\hbox{\rm\rlap-F}}}{}%
\@ifundefined{textlira}{\def\textlira{\hbox{\rm\rlap L=}}}{}%
\@ifundefined{textpeseta}{\def\textpeseta{\hbox{\rm P\negthinspace s}}}{}%
%
\@ifundefined{miss}{\def\miss{\hbox{\vrule height2\p@ width 2\p@ depth\z@}}}{}%
%
\@ifundefined{vvert}{\def\vvert{\Vert}}{}%  %always translated to \left| or \right|
%
\@ifundefined{tcol}{\def\tcol#1{{\baselineskip=6\p@ \vcenter{#1}} \Column}}{}%
%
\@ifundefined{dB}{\def\dB{\hbox{{}}}}{}%        %dummy entry in column 
\@ifundefined{mB}{\def\mB#1{\hbox{$#1$}}}{}%   %column entry
\@ifundefined{nB}{\def\nB#1{\hbox{#1}}}{}%     %column entry (not math)
%
\@ifundefined{note}{\def\note{$^{\dag}}}{}%
%
\def\newfmtname{LaTeX2e}
% No longer load latexsym.  This is now handled by SWP, which uses amsfonts if necessary
%
\ifx\fmtname\newfmtname
  \DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
  \DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
  \DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
  \DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
  \DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
  \DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
  \DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
\fi

%
% Greek bold macros
% Redefine all of the math symbols 
% which might be bolded	 - there are 
% probably others to add to this list

\def\alpha{{\Greekmath 010B}}%
\def\beta{{\Greekmath 010C}}%
\def\gamma{{\Greekmath 010D}}%
\def\delta{{\Greekmath 010E}}%
\def\epsilon{{\Greekmath 010F}}%
\def\zeta{{\Greekmath 0110}}%
\def\eta{{\Greekmath 0111}}%
\def\theta{{\Greekmath 0112}}%
\def\iota{{\Greekmath 0113}}%
\def\kappa{{\Greekmath 0114}}%
\def\lambda{{\Greekmath 0115}}%
\def\mu{{\Greekmath 0116}}%
\def\nu{{\Greekmath 0117}}%
\def\xi{{\Greekmath 0118}}%
\def\pi{{\Greekmath 0119}}%
\def\rho{{\Greekmath 011A}}%
\def\sigma{{\Greekmath 011B}}%
\def\tau{{\Greekmath 011C}}%
\def\upsilon{{\Greekmath 011D}}%
\def\phi{{\Greekmath 011E}}%
\def\chi{{\Greekmath 011F}}%
\def\psi{{\Greekmath 0120}}%
\def\omega{{\Greekmath 0121}}%
\def\varepsilon{{\Greekmath 0122}}%
\def\vartheta{{\Greekmath 0123}}%
\def\varpi{{\Greekmath 0124}}%
\def\varrho{{\Greekmath 0125}}%
\def\varsigma{{\Greekmath 0126}}%
\def\varphi{{\Greekmath 0127}}%

\def\nabla{{\Greekmath 0272}}
\def\FindBoldGroup{%
   {\setbox0=\hbox{$\mathbf{x\global\edef\theboldgroup{\the\mathgroup}}$}}%
}

\def\Greekmath#1#2#3#4{%
    \if@compatibility
        \ifnum\mathgroup=\symbold
           \mathchoice{\mbox{\boldmath$\displaystyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\textstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptscriptstyle\mathchar"#1#2#3#4$}}%
        \else
           \mathchar"#1#2#3#4% 
        \fi 
    \else 
        \FindBoldGroup
        \ifnum\mathgroup=\theboldgroup % For 2e
           \mathchoice{\mbox{\boldmath$\displaystyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\textstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptscriptstyle\mathchar"#1#2#3#4$}}%
        \else
           \mathchar"#1#2#3#4% 
        \fi     	    
	  \fi}

\newif\ifGreekBold  \GreekBoldfalse
\let\SAVEPBF=\pbf
\def\pbf{\GreekBoldtrue\SAVEPBF}%
%

\@ifundefined{theorem}{\newtheorem{theorem}{Theorem}}{}
\@ifundefined{lemma}{\newtheorem{lemma}[theorem]{Lemma}}{}
\@ifundefined{corollary}{\newtheorem{corollary}[theorem]{Corollary}}{}
\@ifundefined{conjecture}{\newtheorem{conjecture}[theorem]{Conjecture}}{}
\@ifundefined{proposition}{\newtheorem{proposition}[theorem]{Proposition}}{}
\@ifundefined{axiom}{\newtheorem{axiom}{Axiom}}{}
\@ifundefined{remark}{\newtheorem{remark}{Remark}}{}
\@ifundefined{example}{\newtheorem{example}{Example}}{}
\@ifundefined{exercise}{\newtheorem{exercise}{Exercise}}{}
\@ifundefined{definition}{\newtheorem{definition}{Definition}}{}


\@ifundefined{mathletters}{%
  %\def\theequation{\arabic{equation}}
  \newcounter{equationnumber}  
  \def\mathletters{%
     \addtocounter{equation}{1}
     \edef\@currentlabel{\theequation}%
     \setcounter{equationnumber}{\c@equation}
     \setcounter{equation}{0}%
     \edef\theequation{\@currentlabel\noexpand\alph{equation}}%
  }
  \def\endmathletters{%
     \setcounter{equation}{\value{equationnumber}}%
  }
}{}

%Logos
\@ifundefined{BibTeX}{%
    \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
                 T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}}{}%
\@ifundefined{AmS}%
    {\def\AmS{{\protect\usefont{OMS}{cmsy}{m}{n}%
                A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}}{}%
\@ifundefined{AmSTeX}{\def\AmSTeX{\protect\AmS-\protect\TeX\@}}{}%
%

% This macro is a fix to eqnarray
\def\@@eqncr{\let\@tempa\relax
    \ifcase\@eqcnt \def\@tempa{& & &}\or \def\@tempa{& &}%
      \else \def\@tempa{&}\fi
     \@tempa
     \if@eqnsw
        \iftag@
           \@taggnum
        \else
           \@eqnnum\stepcounter{equation}%
        \fi
     \fi
     \global\tag@false
     \global\@eqnswtrue
     \global\@eqcnt\z@\cr}


\def\TCItag{\@ifnextchar*{\@TCItagstar}{\@TCItag}}
\def\@TCItag#1{%
    \global\tag@true
    \global\def\@taggnum{(#1)}%
    \global\def\@currentlabel{#1}}
\def\@TCItagstar*#1{%
    \global\tag@true
    \global\def\@taggnum{#1}%
    \global\def\@currentlabel{#1}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\def\QATOP#1#2{{#1 \atop #2}}%
\def\QTATOP#1#2{{\textstyle {#1 \atop #2}}}%
\def\QDATOP#1#2{{\displaystyle {#1 \atop #2}}}%
\def\QABOVE#1#2#3{{#2 \above#1 #3}}%
\def\QTABOVE#1#2#3{{\textstyle {#2 \above#1 #3}}}%
\def\QDABOVE#1#2#3{{\displaystyle {#2 \above#1 #3}}}%
\def\QOVERD#1#2#3#4{{#3 \overwithdelims#1#2 #4}}%
\def\QTOVERD#1#2#3#4{{\textstyle {#3 \overwithdelims#1#2 #4}}}%
\def\QDOVERD#1#2#3#4{{\displaystyle {#3 \overwithdelims#1#2 #4}}}%
\def\QATOPD#1#2#3#4{{#3 \atopwithdelims#1#2 #4}}%
\def\QTATOPD#1#2#3#4{{\textstyle {#3 \atopwithdelims#1#2 #4}}}%
\def\QDATOPD#1#2#3#4{{\displaystyle {#3 \atopwithdelims#1#2 #4}}}%
\def\QABOVED#1#2#3#4#5{{#4 \abovewithdelims#1#2#3 #5}}%
\def\QTABOVED#1#2#3#4#5{{\textstyle 
   {#4 \abovewithdelims#1#2#3 #5}}}%
\def\QDABOVED#1#2#3#4#5{{\displaystyle 
   {#4 \abovewithdelims#1#2#3 #5}}}%
%
% Macros for text size operators:
%
\def\tint{\mathop{\textstyle \int}}%
\def\tiint{\mathop{\textstyle \iint }}%
\def\tiiint{\mathop{\textstyle \iiint }}%
\def\tiiiint{\mathop{\textstyle \iiiint }}%
\def\tidotsint{\mathop{\textstyle \idotsint }}%
\def\toint{\mathop{\textstyle \oint}}%
\def\tsum{\mathop{\textstyle \sum }}%
\def\tprod{\mathop{\textstyle \prod }}%
\def\tbigcap{\mathop{\textstyle \bigcap }}%
\def\tbigwedge{\mathop{\textstyle \bigwedge }}%
\def\tbigoplus{\mathop{\textstyle \bigoplus }}%
\def\tbigodot{\mathop{\textstyle \bigodot }}%
\def\tbigsqcup{\mathop{\textstyle \bigsqcup }}%
\def\tcoprod{\mathop{\textstyle \coprod }}%
\def\tbigcup{\mathop{\textstyle \bigcup }}%
\def\tbigvee{\mathop{\textstyle \bigvee }}%
\def\tbigotimes{\mathop{\textstyle \bigotimes }}%
\def\tbiguplus{\mathop{\textstyle \biguplus }}%
%
%
%Macros for display size operators:
%
\def\dint{\mathop{\displaystyle \int}}%
\def\diint{\mathop{\displaystyle \iint}}%
\def\diiint{\mathop{\displaystyle \iiint}}%
\def\diiiint{\mathop{\displaystyle \iiiint }}%
\def\didotsint{\mathop{\displaystyle \idotsint }}%
\def\doint{\mathop{\displaystyle \oint}}%
\def\dsum{\mathop{\displaystyle \sum }}%
\def\dprod{\mathop{\displaystyle \prod }}%
\def\dbigcap{\mathop{\displaystyle \bigcap }}%
\def\dbigwedge{\mathop{\displaystyle \bigwedge }}%
\def\dbigoplus{\mathop{\displaystyle \bigoplus }}%
\def\dbigodot{\mathop{\displaystyle \bigodot }}%
\def\dbigsqcup{\mathop{\displaystyle \bigsqcup }}%
\def\dcoprod{\mathop{\displaystyle \coprod }}%
\def\dbigcup{\mathop{\displaystyle \bigcup }}%
\def\dbigvee{\mathop{\displaystyle \bigvee }}%
\def\dbigotimes{\mathop{\displaystyle \bigotimes }}%
\def\dbiguplus{\mathop{\displaystyle \biguplus }}%

\if@compatibility\else
  % Always load amsmath in LaTeX2e mode
  \RequirePackage{amsmath}
\fi

\def\ExitTCILatex{\makeatother\endinput}

\bgroup
\ifx\ds@amstex\relax
   \message{amstex already loaded}\aftergroup\ExitTCILatex
\else
   \@ifpackageloaded{amsmath}%
      {\if@compatibility\message{amsmath already loaded}\fi\aftergroup\ExitTCILatex}
      {}
   \@ifpackageloaded{amstex}%
      {\if@compatibility\message{amstex already loaded}\fi\aftergroup\ExitTCILatex}
      {}
   \@ifpackageloaded{amsgen}%
      {\if@compatibility\message{amsgen already loaded}\fi\aftergroup\ExitTCILatex}
      {}
\fi
\egroup

%Exit if any of the AMS macros are already loaded.
%This is always the case for LaTeX2e mode.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% NOTE: The rest of this file is read only if in LaTeX 2.09 compatibility
% mode. This section is used to define AMS-like constructs in the
% event they have not been defined.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\typeout{TCILATEX defining AMS-like constructs in LaTeX 2.09 COMPATIBILITY MODE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Macros to define some AMS LaTeX constructs when 
%  AMS LaTeX has not been loaded
% 
% These macros are copied from the AMS-TeX package for doing
% multiple integrals.
%
\let\DOTSI\relax
\def\RIfM@{\relax\ifmmode}%
\def\FN@{\futurelet\next}%
\newcount\intno@
\def\iint{\DOTSI\intno@\tw@\FN@\ints@}%
\def\iiint{\DOTSI\intno@\thr@@\FN@\ints@}%
\def\iiiint{\DOTSI\intno@4 \FN@\ints@}%
\def\idotsint{\DOTSI\intno@\z@\FN@\ints@}%
\def\ints@{\findlimits@\ints@@}%
\newif\iflimtoken@
\newif\iflimits@
\def\findlimits@{\limtoken@true\ifx\next\limits\limits@true
 \else\ifx\next\nolimits\limits@false\else
 \limtoken@false\ifx\ilimits@\nolimits\limits@false\else
 \ifinner\limits@false\else\limits@true\fi\fi\fi\fi}%
\def\multint@{\int\ifnum\intno@=\z@\intdots@                          %1
 \else\intkern@\fi                                                    %2
 \ifnum\intno@>\tw@\int\intkern@\fi                                   %3
 \ifnum\intno@>\thr@@\int\intkern@\fi                                 %4
 \int}%                                                               %5
\def\multintlimits@{\intop\ifnum\intno@=\z@\intdots@\else\intkern@\fi
 \ifnum\intno@>\tw@\intop\intkern@\fi
 \ifnum\intno@>\thr@@\intop\intkern@\fi\intop}%
\def\intic@{%
    \mathchoice{\hskip.5em}{\hskip.4em}{\hskip.4em}{\hskip.4em}}%
\def\negintic@{\mathchoice
 {\hskip-.5em}{\hskip-.4em}{\hskip-.4em}{\hskip-.4em}}%
\def\ints@@{\iflimtoken@                                              %1
 \def\ints@@@{\iflimits@\negintic@
   \mathop{\intic@\multintlimits@}\limits                             %2
  \else\multint@\nolimits\fi                                          %3
  \eat@}%                                                             %4
 \else                                                                %5
 \def\ints@@@{\iflimits@\negintic@
  \mathop{\intic@\multintlimits@}\limits\else
  \multint@\nolimits\fi}\fi\ints@@@}%
\def\intkern@{\mathchoice{\!\!\!}{\!\!}{\!\!}{\!\!}}%
\def\plaincdots@{\mathinner{\cdotp\cdotp\cdotp}}%
\def\intdots@{\mathchoice{\plaincdots@}%
 {{\cdotp}\mkern1.5mu{\cdotp}\mkern1.5mu{\cdotp}}%
 {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}%
 {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}}%
%
%
%  These macros are for doing the AMS \text{} construct
%
\def\RIfM@{\relax\protect\ifmmode}
\def\text{\RIfM@\expandafter\text@\else\expandafter\mbox\fi}
\let\nfss@text\text
\def\text@#1{\mathchoice
   {\textdef@\displaystyle\f@size{#1}}%
   {\textdef@\textstyle\tf@size{\firstchoice@false #1}}%
   {\textdef@\textstyle\sf@size{\firstchoice@false #1}}%
   {\textdef@\textstyle \ssf@size{\firstchoice@false #1}}%
   \glb@settings}

\def\textdef@#1#2#3{\hbox{{%
                    \everymath{#1}%
                    \let\f@size#2\selectfont
                    #3}}}
\newif\iffirstchoice@
\firstchoice@true
%
%These are the AMS constructs for multiline limits.
%
\def\Let@{\relax\iffalse{\fi\let\\=\cr\iffalse}\fi}%
\def\vspace@{\def\vspace##1{\crcr\noalign{\vskip##1\relax}}}%
\def\multilimits@{\bgroup\vspace@\Let@
 \baselineskip\fontdimen10 \scriptfont\tw@
 \advance\baselineskip\fontdimen12 \scriptfont\tw@
 \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
 \lineskiplimit\lineskip
 \vbox\bgroup\ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil\crcr}%
\def\Sb{_\multilimits@}%
\def\endSb{\crcr\egroup\egroup\egroup}%
\def\Sp{^\multilimits@}%
\let\endSp\endSb
%
%
%These are AMS constructs for horizontal arrows
%
\newdimen\ex@
\ex@.2326ex
\def\rightarrowfill@#1{$#1\m@th\mathord-\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill
 \mkern-6mu\mathord\rightarrow$}%
\def\leftarrowfill@#1{$#1\m@th\mathord\leftarrow\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill\mkern-6mu\mathord-$}%
\def\leftrightarrowfill@#1{$#1\m@th\mathord\leftarrow
\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill
 \mkern-6mu\mathord\rightarrow$}%
\def\overrightarrow{\mathpalette\overrightarrow@}%
\def\overrightarrow@#1#2{\vbox{\ialign{##\crcr\rightarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\let\overarrow\overrightarrow
\def\overleftarrow{\mathpalette\overleftarrow@}%
\def\overleftarrow@#1#2{\vbox{\ialign{##\crcr\leftarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\def\overleftrightarrow{\mathpalette\overleftrightarrow@}%
\def\overleftrightarrow@#1#2{\vbox{\ialign{##\crcr
   \leftrightarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\def\underrightarrow{\mathpalette\underrightarrow@}%
\def\underrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil
  $\crcr\noalign{\nointerlineskip}\rightarrowfill@#1\crcr}}}%
\let\underarrow\underrightarrow
\def\underleftarrow{\mathpalette\underleftarrow@}%
\def\underleftarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil
  $\crcr\noalign{\nointerlineskip}\leftarrowfill@#1\crcr}}}%
\def\underleftrightarrow{\mathpalette\underleftrightarrow@}%
\def\underleftrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th
  \hfil#1#2\hfil$\crcr
 \noalign{\nointerlineskip}\leftrightarrowfill@#1\crcr}}}%
%%%%%%%%%%%%%%%%%%%%%

\def\qopnamewl@#1{\mathop{\operator@font#1}\nlimits@}
\let\nlimits@\displaylimits
\def\setboxz@h{\setbox\z@\hbox}


\def\varlim@#1#2{\mathop{\vtop{\ialign{##\crcr
 \hfil$#1\m@th\operator@font lim$\hfil\crcr
 \noalign{\nointerlineskip}#2#1\crcr
 \noalign{\nointerlineskip\kern-\ex@}\crcr}}}}

 \def\rightarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
  $#1\copy\z@\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
  \mkern-6mu\mathord\rightarrow$}
\def\leftarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
  $#1\mathord\leftarrow\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\copy\z@\mkern-2mu$}\hfill
  \mkern-6mu\box\z@$}


\def\projlim{\qopnamewl@{proj\,lim}}
\def\injlim{\qopnamewl@{inj\,lim}}
\def\varinjlim{\mathpalette\varlim@\rightarrowfill@}
\def\varprojlim{\mathpalette\varlim@\leftarrowfill@}
\def\varliminf{\mathpalette\varliminf@{}}
\def\varliminf@#1{\mathop{\underline{\vrule\@depth.2\ex@\@width\z@
   \hbox{$#1\m@th\operator@font lim$}}}}
\def\varlimsup{\mathpalette\varlimsup@{}}
\def\varlimsup@#1{\mathop{\overline
  {\hbox{$#1\m@th\operator@font lim$}}}}

%
%Companion to stackrel
\def\stackunder#1#2{\mathrel{\mathop{#2}\limits_{#1}}}%
%
%
% These are AMS environments that will be defined to
% be verbatims if amstex has not actually been 
% loaded
%
%
\begingroup \catcode `|=0 \catcode `[= 1
\catcode`]=2 \catcode `\{=12 \catcode `\}=12
\catcode`\\=12 
|gdef|@alignverbatim#1\end{align}[#1|end[align]]
|gdef|@salignverbatim#1\end{align*}[#1|end[align*]]

|gdef|@alignatverbatim#1\end{alignat}[#1|end[alignat]]
|gdef|@salignatverbatim#1\end{alignat*}[#1|end[alignat*]]

|gdef|@xalignatverbatim#1\end{xalignat}[#1|end[xalignat]]
|gdef|@sxalignatverbatim#1\end{xalignat*}[#1|end[xalignat*]]

|gdef|@gatherverbatim#1\end{gather}[#1|end[gather]]
|gdef|@sgatherverbatim#1\end{gather*}[#1|end[gather*]]

|gdef|@gatherverbatim#1\end{gather}[#1|end[gather]]
|gdef|@sgatherverbatim#1\end{gather*}[#1|end[gather*]]


|gdef|@multilineverbatim#1\end{multiline}[#1|end[multiline]]
|gdef|@smultilineverbatim#1\end{multiline*}[#1|end[multiline*]]

|gdef|@arraxverbatim#1\end{arrax}[#1|end[arrax]]
|gdef|@sarraxverbatim#1\end{arrax*}[#1|end[arrax*]]

|gdef|@tabulaxverbatim#1\end{tabulax}[#1|end[tabulax]]
|gdef|@stabulaxverbatim#1\end{tabulax*}[#1|end[tabulax*]]


|endgroup
  

  
\def\align{\@verbatim \frenchspacing\@vobeyspaces \@alignverbatim
You are using the "align" environment in a style in which it is not defined.}
\let\endalign=\endtrivlist
 
\@namedef{align*}{\@verbatim\@salignverbatim
You are using the "align*" environment in a style in which it is not defined.}
\expandafter\let\csname endalign*\endcsname =\endtrivlist




\def\alignat{\@verbatim \frenchspacing\@vobeyspaces \@alignatverbatim
You are using the "alignat" environment in a style in which it is not defined.}
\let\endalignat=\endtrivlist
 
\@namedef{alignat*}{\@verbatim\@salignatverbatim
You are using the "alignat*" environment in a style in which it is not defined.}
\expandafter\let\csname endalignat*\endcsname =\endtrivlist




\def\xalignat{\@verbatim \frenchspacing\@vobeyspaces \@xalignatverbatim
You are using the "xalignat" environment in a style in which it is not defined.}
\let\endxalignat=\endtrivlist
 
\@namedef{xalignat*}{\@verbatim\@sxalignatverbatim
You are using the "xalignat*" environment in a style in which it is not defined.}
\expandafter\let\csname endxalignat*\endcsname =\endtrivlist




\def\gather{\@verbatim \frenchspacing\@vobeyspaces \@gatherverbatim
You are using the "gather" environment in a style in which it is not defined.}
\let\endgather=\endtrivlist
 
\@namedef{gather*}{\@verbatim\@sgatherverbatim
You are using the "gather*" environment in a style in which it is not defined.}
\expandafter\let\csname endgather*\endcsname =\endtrivlist


\def\multiline{\@verbatim \frenchspacing\@vobeyspaces \@multilineverbatim
You are using the "multiline" environment in a style in which it is not defined.}
\let\endmultiline=\endtrivlist
 
\@namedef{multiline*}{\@verbatim\@smultilineverbatim
You are using the "multiline*" environment in a style in which it is not defined.}
\expandafter\let\csname endmultiline*\endcsname =\endtrivlist


\def\arrax{\@verbatim \frenchspacing\@vobeyspaces \@arraxverbatim
You are using a type of "array" construct that is only allowed in AmS-LaTeX.}
\let\endarrax=\endtrivlist

\def\tabulax{\@verbatim \frenchspacing\@vobeyspaces \@tabulaxverbatim
You are using a type of "tabular" construct that is only allowed in AmS-LaTeX.}
\let\endtabulax=\endtrivlist

 
\@namedef{arrax*}{\@verbatim\@sarraxverbatim
You are using a type of "array*" construct that is only allowed in AmS-LaTeX.}
\expandafter\let\csname endarrax*\endcsname =\endtrivlist

\@namedef{tabulax*}{\@verbatim\@stabulaxverbatim
You are using a type of "tabular*" construct that is only allowed in AmS-LaTeX.}
\expandafter\let\csname endtabulax*\endcsname =\endtrivlist

% macro to simulate ams tag construct


% This macro is a fix to the equation environment
 \def\endequation{%
     \ifmmode\ifinner % FLEQN hack
      \iftag@
        \addtocounter{equation}{-1} % undo the increment made in the begin part
        $\hfil
           \displaywidth\linewidth\@taggnum\egroup \endtrivlist
        \global\tag@false
        \global\@ignoretrue   
      \else
        $\hfil
           \displaywidth\linewidth\@eqnnum\egroup \endtrivlist
        \global\tag@false
        \global\@ignoretrue 
      \fi
     \else   
      \iftag@
        \addtocounter{equation}{-1} % undo the increment made in the begin part
        \eqno \hbox{\@taggnum}
        \global\tag@false%
        $$\global\@ignoretrue
      \else
        \eqno \hbox{\@eqnnum}% $$ BRACE MATCHING HACK
        $$\global\@ignoretrue
      \fi
     \fi\fi
 } 

 \newif\iftag@ \tag@false
 
 \def\TCItag{\@ifnextchar*{\@TCItagstar}{\@TCItag}}
 \def\@TCItag#1{%
     \global\tag@true
     \global\def\@taggnum{(#1)}%
     \global\def\@currentlabel{#1}}
 \def\@TCItagstar*#1{%
     \global\tag@true
     \global\def\@taggnum{#1}%
     \global\def\@currentlabel{#1}}

  \@ifundefined{tag}{
     \def\tag{\@ifnextchar*{\@tagstar}{\@tag}}
     \def\@tag#1{%
         \global\tag@true
         \global\def\@taggnum{(#1)}}
     \def\@tagstar*#1{%
         \global\tag@true
         \global\def\@taggnum{#1}}
  }{}

\def\tfrac#1#2{{\textstyle {#1 \over #2}}}%
\def\dfrac#1#2{{\displaystyle {#1 \over #2}}}%
\def\binom#1#2{{#1 \choose #2}}%
\def\tbinom#1#2{{\textstyle {#1 \choose #2}}}%
\def\dbinom#1#2{{\displaystyle {#1 \choose #2}}}%

% Do not add anything to the end of this file.  
% The last section of the file is loaded only if 
% amstex has not been.
\makeatother
\endinput

%%%%%%%%%%%%%%% End /macros/LaTeX/SWmacros/tcilatex.tex %%%%%%%%%%%%%%%

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%% This is a generated file.
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%% 
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%% --------------------------------------------------------------
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%% 
%% This is a generated file.
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%% Copyright 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004
%% The LaTeX3 Project and any individual authors listed elsewhere
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%% 
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%% --------------------------------------------------------------
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