
\documentstyle[11pt,epsf,titlepage]{article}

\begin{document}

\title { A Description of the Initial Value Formulation of Vacuum 
General Relativity for the Non-Specialist
\footnote{Work supported by NSF ASC 93 18152/PHY 93 18152 (ARPA supplemented)}}
\author{Mark Miller}
\date{October 10, 1994}
\maketitle


\section*{Introduction}
\pagestyle{myheadings}
In any specialized scientific endeavor, the workers in that specific
area develop their own language, or jargon, that can be confusing for
the non-specialist (and, at times, for the specialist, too!).  This
makes it difficult for scientists to ``latch--on'' to fields that
are outside their area of expertise.  General Relativity is certainly
no exception.  In light of the Computational Grand Challenge Program 
``Black Hole Binaries: Coalescence and Gravitational Radiation,''
there is a need for a ``jargonless'' description of the formulation of
General Relativity that is used in computer simulations (i.e. its 
initial value formulation).  This paper is an attempt at meeting this 
need.  

It should be stressed, however, that the main goal of this paper is to give
a precise mathematical statement (sans jargon) of the initial value
formulation of general relativity. The first section of the paper introduces
the reader to a few major conceptual issues in General Relativity.
In no way does this paper attempt
to give a full physical interpretation of General Relativity. 
 Interested readers
are referred elsewhere \cite{MTW,Wald}.

\vspace{10mm}

\noindent Note: This paper itself is a work in progress, and can be
added to/revised at any time . (Date of last revision: Oct 10, 1994).

\newpage

\section*{Conventions and Notation}

In this paper, standard index notation will be used.  The convention will be
that greek indices run from 0 to 3 (i.e. $\mu = 0,1,2,3$), and latin
indices run from 1 to 3 (i.e. $ i = 1,2,3$).  For example, $v^i$ denotes
the three numbers $(v^1,v^2,v^3)$ ({\bf not} v to the first, second, and third
power).  Also, we will differentiate between superscripted indices and
subscripted indices ( $v^i$ does {\bf not} equal $v_i$ in general).

The ``Einstein summation convention'' will also be employed.  This convention
dictates that indices that are repeated, once as a superscript and once
as a subscript, are summed over.  For example, $v_i w^i = v_1 w^1 +
v_2 w^2 + v_3 w^3$.  Another example is the standard matrix equation 
$M \vec{x} = \vec{b}$, where M is a (4 x 4) matrix and $\vec{x}$ and 
$\vec{b}$ are vectors.  This equation, using index notation, reads
$M^{\mu}_{\nu} x^{\nu} = b^{\mu}$.

The speed of light, c, is set equal to 1. Therefore, the time coordinate,
$x^0 = t$, is measured in meters. To convert to seconds, use the conversion
defined by $c = 1$: 1 second = $3 \times 10^8$ meters).

\newpage

\section{The Spirit of General Relativity}

General Relativity is a theory of gravitational interaction.  In a real
sense, it is a more accuarate theory than Newtonian gravity 
($F_g = G \frac {m_1 m_2}{{r_{12}}^2}$), because it has better agreement
with observations.  This is not to say that the Newtonian theory of
gravity is a bad theory of gravity.  It does very well in predicting how
fast a rock should fall or predicting how long it should take the moon
to circle the earth.  In simple cases like these, the Newtonian theory
of gravity and General Relativity agree to within obervational uncertainty.
In fact, Newtonian gravity is the ``weak-field limit'' of General 
Relativity.  However, very accurate measurements of other gravitational
phenomena reveal a deviation from the Newtonian theory of gravity.  To date
all of these deviations are explained by General Relativity \cite{Will}.
In addition to explaining weak-field deviations from Newtonian gravity,
General Relativity has recently (in the past decades) passed a
``strong-field'' test, that of the prediction of the rate of energy loss
due to gravitational waves emitted from a binary neutron star system.

Although Newtonian gravity and General Relativity are both theories of
gravitational interaction, they are conceptually very different.  Newton's
Universal Law of Gravitation gives the magnitude of the mutual force
between two point particles of mass $m_1$ and $m_2$ seperated by
a distance of $r_{12}$ as
\begin{displaymath}
F_{12} = G \frac {m_1 m_2}{{r_{12}}^2}
\end{displaymath}
where $G$ is a constant.  If we think of 3 dimensional space as the surface
of a flat 2 dimensional table, the picture is as follows:

\begin{figure}
\epsfxsize=5in \epsfbox{graphics.dir/newtongrav.eps}
\end{figure}

\vspace{1cm}

In the diagram, note the clock on the wall.  Its job is to keep track
of ``absolute time''.  In Newtonian gravity, time is seperate from space,
and the wrist watches on all of the observers in space agree with
the clock on the wall (absolute time).

In contrast to Newtonian gravity, with General Relativity, there 
{\it \bf is no} force due to gravity!!  Also, the ``table top'' that
the particles sit in is not flat, nor is it simply 3 dimensional space.
Pictorally, we can think of 4 dimensional spacetime as a 
stretched 2 dimensional rubber
sheet.  If we place a bowling ball on the sheet, the
bowling ball deforms the rubber sheet:

\epsfxsize=5in \epsfbox{graphics.dir/rubberbowl.eps}
\vspace{1cm}

\noindent Just as the bowling ball deforms the rubber sheet, General 
Relativity says that mass (or energy) will deform, or curve, the 
spacetime around it.  Also, there is no ``absolute time'' in General
Relativity.  Different observers will disagree as to how much
time passes between events.

Now, in addition to a bowling ball sitting on the rubber sheet, imagine a
marble rolling along the rubber sheet.  General Relativity says that there is
{\it no} gravitational force between the marble 
and the bowling ball!  The marble
will therefore travel in a straight line, since there are no forces acting
on it.  But what does it mean to travel in a straight line when the 
spacetime you are sitting in is curved?  In flat space, a straight line
between two points can be thought of as the unique path connecting the
two points that has the shortest length.  We take this concept of
``extremal distance'' over to the case of curved spacetime.  We now
say that the marble follows the path with extremal length (the path with
shortest distance).  These curves are called geodesics, and can be
intuitively thought of as the ``straightest possible path'' in curved
spacetimes.

\epsfxsize=5in \epsfbox{graphics.dir/rubbergeodesic.eps}
\vspace{1cm}

So, General Relativity is not a law which gives the gravitational force
between a bowling ball and a marble.  Instead, it is a theory that predicts
how the bowling ball curves the spacetime around it.  The marble in
``free fall'' is simply following a geodesic (path of shortest distance)
in this curved spacetime.  Therefore, the dynamical variables in
General Relativity are variables that determine the distance between any
two (infinitesimally) nearby points.  These variables are referred to as
``the metric.'' 

\section{Vacuum General Relativity}

Mathematically, the theory of vacuum General Relativity (General Relativity
without matter) is a set of 10 coupled, nonlinear, second--order,
partial differential equations involving ten functions of 4 independent 
variables or coordinates. The coordinates, $x^\mu$, can be thought of as
numbers labeling the ``where'' and ``when'' of events:

\begin{displaymath}
x^{\mu} = (\underbrace{t}_{when},\underbrace{x,y,z}_{where})
\end{displaymath}


\noindent The set of all events is the arena of General Relativity, 
and is referred
to as ``spacetime.''

The dynamical variables of the theory can be thought of as functions 
arranged as entries in a symmetric 4x4 matrix:
\begin{displaymath}
\left[
\begin{array}{cccc}
g_{00}(t,x,y,z) & g_{01}(t,x,y,z) & g_{02}(t,x,y,z) & g_{03}(t,x,y,z) \\
g_{01}(t,x,y,z) & g_{11}(t,x,y,z) & g_{12}(t,x,y,z) & g_{13}(t,x,y,z) \\
g_{02}(t,x,y,z) & g_{12}(t,x,y,z) & g_{22}(t,x,y,z) & g_{23}(t,x,y,z) \\
g_{03}(t,x,y,z) & g_{13}(t,x,y,z) & g_{23}(t,x,y,z) & g_{33}(t,x,y,z)
\end{array}
\right]
\end{displaymath}
where each entry is a function of the coordinates, $x^\mu$.

  In index notation, this
matrix is referred to as $g_{\mu \nu}$ (its symmetric property can be stated
as $g_{\mu \nu} = g_{\nu \mu}$).  In the literature, this matrix is known
as ``the metric,'' since it is used to calculate the distance between
nearby points in the physical spacetime.  Note that, although a 4x4 matrix
has 16 entries, the matrix is symmetric so that there are only 10 independent
functions.

Remember, $g_{\mu \nu}$ is a function of the coordinates, $x^\mu =
(t,x,y,z)$.  The Einstein field equations are 10 coupled, nonlinear, 
second order, partial differential equations involving the metric,
$g_{\mu \nu}$.  These equations would take tens of pages to explicitly list
as derivatives of the metric.  This fact
partially explains the need for jargon in General Relativity (relativists
refer to these equations as $R_{\mu \nu} = 0$.  Perhaps this is
``jargon-overkill'').

It should be stated that in General Relativity, one is free to use whatever 
coordinates (cartesian, spherical, hyperbolic, etc.) one wishes.  However,
the physical predictions of the theory are independent of the choice
of coordinates.  It is important to use this coordinate freedom to ones 
advantage when attacking a problem.  In fact, this coordinate freedom
introduces some ambiguity in the initial value formulation which must be
dealt with.

\section{General Relativity Admits an Initial Value Formulation}

The Einstein equations as described above are too complicated
to solve directly, except in cases of high symmetry.  In a realistic
problem (i.e. the coalescence of spiraling black holes), what one requires
is an initial value formulation of the theory.  The analogy in
classical mechanics is well known: once the position and velocity of each
particle is known, Newton's second law (along with rules for the interaction
between the particles) completely determines the trajectory of each
particle.  In that case, the initial data are the position and velocity of each
particle at an instant of time, $t_0$.

It turns out that General Relativity admits an initial value formulation that
is similar in spirit to that in classical mechanics.  What follows is a
description of that formulation.  The details of the calculations can 
be found in \cite{MTW,Wald}.

In the initial value formulation, spacetime ($x^\mu = (t,x,y,z)$) is 
separated into space ($x^i = (x,y,z)$) and time ($x^0 = t$).  This
``foliation'' can be viewed pictorally as follows: 

\epsfxsize=5in \epsfbox{graphics.dir/foliation.eps}

\noindent Space-time is sliced up into ``spatial surfaces,'' and
each different surface
is labeled by the (continuous) parameter, t.  These ``spatial surfaces''
can have any topology ($R^3$ ,$T^3$ ,$S^2 \times R$ , etc.).  
For our purposes, we will assume that each
surface is coordinatized by $x^i = (x,y,z)$.

The 10 Einstein equations are then ``projected'' onto this foliation.
What happens is that natural dynamical variables appear: the
so--called 3--metric, $q_{ij}$.  Like the dynamical variables for the
full Einstein equations $g_{\mu \nu}$, $q_{ij}$ can be thought of as a
symmetric matrix of functions:
\begin{displaymath}
\left[
\begin{array}{ccc}
q_{11}(t,x,y,z) & q_{12}(t,x,y,z)  & q_{13}(t,x,y,z) \\
q_{12}(t,x,y,z) & q_{22}(t,x,y,z) & q_{23}(t,x,y,z) \\
q_{13}(t,x,y,z) & q_{23}(t,x,y,z) & q_{33}(t,x,y,z)
\end{array}
\right]
\end{displaymath}
Unlike $g_{\mu \nu}$, $q_{ij}$ is only a 3x3 symmetric matrix, so there are
only 6 independent functions.

At first glance, it might appear that the system is over-determined.  Afterall,
there are 10 independent equations for 6 functions.  A closer look reveals
that 4 of the equations involve only first--order time derivatives of 
$q_{ij}$.  In the analogy with classical mechanics, these equations
involve only positions and velocities.  These equations are therefore
constraints on the initial data.  This leaves $ 10 - 4 = 6 $ equations
left over, which are called the evolution equations.  They completely 
determine the evolution of the 6 dynamical variables, $q_{ij}$.

\epsfxsize=5in \epsfbox{graphics.dir/eq1.eps}

When working with second order differential equations, it is usually
convenient (in analytical work, as well as in numerical work) to write
each second order (in time) differential equation as two first order
(in time) differential equations, treating the time derivative of the
dynamical variable as a separate dynamical variable.  In our case, the
new dynamical variable is $K_{ij},$ a symmetric 3x3 matrix of functions:
\begin{displaymath}
\left[
\begin{array}{ccc}
K_{11}(t,x,y,z) & K_{12}(t,x,y,z)  & K_{13}(t,x,y,z) \\
K_{12}(t,x,y,z) & K_{22}(t,x,y,z) & K_{23}(t,x,y,z) \\
K_{13}(t,x,y,z) & K_{23}(t,x,y,z) & K_{33}(t,x,y,z)
\end{array}
\right]
\end{displaymath}
and is called the ``extrinsic curvature.''  It derives its name from its
geometrical interpretation: it contains information on how each ``spatial
surface'' is ``embedded'' in the spacetime.  For our purposes, we will
think of $K_{ij}$ as ``the time derivative of $q_{ij}$'' (although,
$K_{ij}$ does {\bf not} strictly equal $\frac{\partial}{\partial t}
(q_{ij}(t,x,y,z))$, as we shall see explicitly).  The picture so far
looks like this:

\epsfxsize=6in \epsfbox{graphics.dir/eq2.eps}


\section{The Final Step}
Before writing down the constraint and evolution equations, let's
take a look back.  The way we ``slice up,'' or foliate, spacetime
is quite arbitrary:

\epsfxsize=5in \epsfbox{graphics.dir/foliation2.eps}

\vspace{10mm}
This is actually a restatement of the freedom to choose coordinates
in General Relativity.  This freedom results in ambiguity in the evolution
equations for $q_{ij}$ and $K_{ij}$.  Specifically, there are 4
undetermined functions in the evolution equations.  These functions are
given the names ``lapse'' (1 function) and ``shift'' (3 functions):

\begin{tabular}{lll}
Lapse: & $\alpha(t,x,y,z)$ & (1 function) \\
Shift: & $N^i(t,x,y,z)$ & (3 functions. Remember: $i=1,2,3$)
\end{tabular}

The geometrical interpretation of the lapse and shift is given in the 
picture below:

\epsfxsize=5in \epsfbox{graphics.dir/lapseshift.eps}

\vspace{10mm}

As shown in the diagram, the shift functions ($N^i$) are the components of the
vector that describe the ``tangential shift'' of coordinates as one moves from
the $t_0$ spatial surface to the $t_0 + \delta t$ spatial surface.  The
lapse describes how much physical time elapses between the two spatial
surfaces.  It should be emphasized again that the physical predictions
of the theory are completely independent of the choice of the lapse
and shift.  In numerical work, this freedom (if used wisely) can be used
to accomplish one or more of the following:
\begin{itemize}
   \item avoid singularities (regions of spacetime that generate infinite
          answers to physical questions)
   \item keep numerical algorithm stable
   \item insure the consistency of the constraints 
\end{itemize}

\section{Summary}
The initial value formulation of general relativity consists of 12
first order (in time), coupled, nonlinear differential equations
for the 12 dynamical variables, $q_{ij}(t,x,y,z)$ and $K_{ij}(t,x,y,z)$.
The initial data for the theory is specified by $q_{ij}(t_0,x,y,z)$ and
$K_{ij}(t_0,x,y,z)$ (that is, by assigning values to $q_{ij}$ and $K_{ij}$
at an instant of time, $t_0$) that satisfy 4 constraint equations. 
The constraint equations are consistent in the sense that if they
are solved on the initial spatial surface and that spatial surface is
evolved via the evolution equations, then the constraints are solved
automatically on any future spatial surface.
In addition, the evolution equations contain 4 arbitrary functions,
$\alpha $ (lapse) and $N^i$ (shift).  The physical predictions of
General Relativity are independent of the choice of these functions.

In the way of notation, I will define $q^{ij}$ (note the superscripted
indices) as the inverse of the 3--metric, $q_{ij}$.  That is,
\begin{displaymath}
{\left[
\begin{array}{ccc}
q^{11} & q^{12}  & q^{13} \\
q^{12} & q^{22} & q^{23} \\
q^{13} & q^{23} & q^{33}
\end{array}
\right]} = 
{\left[
\begin{array}{ccc}
q_{11} & q_{12}  & q_{13} \\
q_{12} & q_{22} & q_{23} \\
q_{13} & q_{23} & q_{33}
\end{array}
\right]}^{-1}
\end{displaymath}
I will also define
\begin{displaymath}
{\Gamma}^k_{ij} = \frac{1}{2} q^{km} ( \frac {\partial q_{im}}{\partial x^j}
                                    +  \frac {\partial q_{jm}}{\partial x^i}
                                    -  \frac {\partial q_{ij}}{\partial x^m})
\end{displaymath}

\subsection{Constraints}
The first constraint, $\cal C$, is called the scalar constraint 
(or sometimes the
Hamiltonian constraint).  It reads
\begin{displaymath}
0 = {\cal C} =  q^{ij}(\frac {\partial {\Gamma}^k_{ij}}{\partial x^k}
                      - \frac {\partial {\Gamma}^k_{jk}}{\partial x^i}
                      + {\Gamma}^m_{ij} {\Gamma}^k_{mk}
                      - {\Gamma}^m_{jk} {\Gamma}^k_{im} ) 
  + {(K_{ij} q^{ij})}^2 - K_{ik} K_{jm} q^{ij} q^{km}
\end{displaymath}
The other 3 constraints, ${\cal C}_i$, are called the vector constraints (or
sometimes the diffeomorphism constraints).  They read
\begin{displaymath}
0 = {\cal C}_i = q^{jk}(\frac {\partial K_{ik}}{\partial x^j} -
                        \frac {\partial K_{jk}}{\partial x^i} +
                        K_{jm} {\Gamma}^m_{ik} -
                        K_{im} {\Gamma}^m_{jk} )
\end{displaymath}

\subsection{Evolution Equations}
The evolution equations for the $q_{ij}$ are
\begin{displaymath}
\frac {\partial q_{ij}}{\partial t} = 2 \alpha K_{ij} +
   q_{jk}(\frac {\partial N^k}{\partial x^i} + N^m{\Gamma}^k_{im}) +
   q_{ik}(\frac {\partial N^k}{\partial x^j} + N^m{\Gamma}^k_{jm})
\end{displaymath}
The evolution equations for the $K_{ij}$ are
\begin{eqnarray*}
\frac {\partial K_{ij}}{\partial t} & = & 
   - \alpha (\frac {\partial {\Gamma}^k_{ij}}{\partial x^k} -
             \frac {\partial {\Gamma}^k_{jk}}{\partial x^i} +
             {\Gamma}^m_{ij} {\Gamma}^k_{km} -
             {\Gamma}^m_{jk} {\Gamma}^k_{im})
   + \alpha q^{km}(2 K_{ik} K_{jm} - K_{km} K_{ij}) + \\
  &  & + \frac { {\partial}^2 \alpha}{{\partial x^i} {\partial x^j}} -
       \frac {\partial \alpha}{\partial x^k} {\Gamma}^k_{ij} +
       N^k \frac {\partial K_{ij}}{\partial x^k} +
       K_{jk} \frac {\partial N^k}{\partial x^i} +
       K_{ik} \frac {\partial N^k}{\partial x^j}
\end{eqnarray*}

\section{Glossary of Terms}

\begin{description}
\item[Christoffel symbol] $({{\Gamma}^\mu}_{\alpha \beta})$  Mathematically,
these are simply combinations of first derivatives of the metric:
\begin{displaymath}
{\Gamma}^\mu_{\alpha \beta} = \frac{1}{2} g^{\mu \nu} ( 
              \frac {\partial g_{\alpha \nu}}{\partial x^\beta}
           +  \frac {\partial g_{\beta \nu}}{\partial x^\alpha}
           -  \frac {\partial g_{\alpha \beta}}{\partial x^\nu})
\end{displaymath}

\item[Einstein equations]  The Einstein equations are 10 coupled, nonlinear,
second order, partial differential equations in the 10 metric functions:
\begin{displaymath}
G_{\alpha \beta} = 8 \pi T_{\alpha \beta}
\end{displaymath}
where $G_{\alpha \beta}$ is the Einstein tensor and $T_{\alpha \beta}$ is
the stress-energy tensor.

\item[Einstein tensor] $(G_{\alpha \beta})$  The Einstein tensor is defined
as:
\begin{displaymath}
G_{\alpha \beta} = R_{\alpha \beta} - \frac {1}{2} R g_{\alpha \beta}
\end{displaymath}
where $R_{\alpha \beta}$ is the Ricci tensor, $R$ is the scalar curvature,
and $g_{\alpha \beta}$ is the metric.

\item[Geodesic] Geodesics are paths in spacetimes that have ``extremal
distance''.  They can be thought of intuitively as the ``straightest
possible paths'' in curved spacetime.  Physically, geodesics are trajectories
of particles in free-fall (no forces acting on the particles).

\item[Metric] $(g_{\mu \nu})$  The metric is a set of 10 functions that
can be thought of as entries in a symmetric matrix:
\begin{displaymath}
\left[
\begin{array}{cccc}
g_{00} & g_{01} & g_{02} & g_{03} \\
g_{01} & g_{11} & g_{12} & g_{13} \\
g_{02} & g_{12} & g_{22} & g_{23} \\
g_{03} & g_{13} & g_{23} & g_{33}
\end{array}
\right]
\end{displaymath}
Physically, the metric determines the distance between (infinitesimally)
nearby points.  All information concerning the geometry of the spacetime is
contained in the metric, and it is therefore the metric that is the
dynamical variable in General Relativity.

\item[Ricci tensor] $(R_{\alpha \beta})$  The Ricci tensor is a set
of 10 independent functions of the coordinates.  It is constructed by
contracting the second and fourth indices of the Riemann tensor:
$R_{\alpha \beta} = {R_{\alpha \mu \beta}}^\mu$.

\item[Riemann tensor] $({R_{\alpha \beta \mu}}^\nu)$  Mathematically, 
The Riemann tensor is
a set of 20 (not 256!!) independent functions of the
coordinates that completely describe the curvature of the spacetime.  
(if ${R_{\alpha \beta \mu}}^\nu = 0$, then the spacetime is said to
be completely flat.)
It
is constructed out of various first and second derivatives of the metric.

\item[Scalar Curvature] $(R)$ The scalar curvature is a function of the 
coordinates obtained by contracting the Ricci tensor with the metric:
$R = R_{\alpha \beta} g^{\alpha \beta}$.

\end{description}

\begin{thebibliography}{9}

\bibitem{MTW}C. Misner, K. Thorne, and J. Wheeler, {\it Gravitation}
(1973).

\bibitem{Wald} R. Wald, {\it General Relativity} (1984).

\bibitem{Will} C. M. Will, {\it Theory and Experiment in Gravitational
Physics} (1993).

\end{thebibliography}

\end{document}
