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% \filename{gem1.tex} \version{9-jun-2004< 26-may-2003 < 5-jun-2001<-4-jun-1997 <-16-apr-1992}
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\Chapter{Introduction}
\Section{Motivation: Local Special Relativity plus Rotating Coordinates}
Most of us know special relativity pretty well and are quite happy switching
back and forth between the spacetime picture of 4-vector algebra and the
space plus time picture of events occurring in space as time elapses,
using 3-vector algebra. We have no difficulty using Lorentz
transformations to transform 3-dimensional quantities from those measured
by one inertial observer to another.
We also have no problem extending our splitting algebra to
spacetime derivative operators in inertial coordinates yielding the
space plus time equivalent of time derivatives and spatial derivative
operators.
After all, these are the derivatives we began with before
learning special relativity.
Figure \ref{fig:11} suggestively compares these two pictures.
%\fig{Spacetime and Space plus Time}%
%{Spacetime and Space plus Time. Although spacetime is the arena
%where calculations are simpler, we always interpret them through our
%space plus time worldview, which depends on the choice of inertial
%observer.}%
%{saspt}
\begin{figure}\typeout{figure gemfig11}
\label{fig:11}
\input fig/gemfig11
\caption{Spacetime and Space plus Time. Although spacetime is the arena
where calculations are simpler, we always interpret them through our
space plus time worldview, which depends on the choice of inertial
observer.}
\end{figure}
Since old habits die hard,
there is a strong incentive to push this habit
into general relativity. This helps us interpret spacetime information
in a curved spacetime using the same intuition that we have about
classical 3-dimensional physics.
The catch is that one no longer has a privileged
class of ``global inertial frames" as in special relativity which
effectively allows a single inertial observer in flat spacetime to set up
a preferred class of global inertial coordinate systems that may be
used to interpret observations at all other points in spacetime.
Essentially, one inertial observer in flat spacetime
uniquely determines a family
of inertial observers filling the spacetime with no relative velocities,
and the ``observations" of the original observer of an event not on his
own worldline are understood to be
those of the companion observer who is present.
The only option allowing us to continue to make a spacetime splitting
without relying on flatness
is to give up the global splitting associated with a single preferred observer
and settle for the local splitting of each member of a family
of observers filling the spacetime and in arbitrary motion in the absence of
any preference. Such a splitting takes place in the tangent space to each
event in spacetime, describing the locally Minkowskian neighborhood of
each observer in the family. If we agree to split each such tangent space
based on the 4-velocity of the observer at a given event in the same way
that we split flat spacetime globally based on the worldline of a single
inertial observer, then all of our familiarity with special relativity
can be transferred to general relativity in a rather straightforward way.
The only difference is that what we did globally before, with the splitting
at every spacetime point determined by the splitting at a single spacetime
point, namely any point on the worldline of a chosen inertial observer,
must be
abandoned in favor of doing the same thing
independently at each spacetime point,
modulo continuity/differentiability conditions.
There is one catch. Being creatures of habit, we like global splittings,
so even though in general
there is no preferred way of doing a global splitting, we can just do it
arbitrarily, though clearly matters may be simplified if such a splitting
can be adapted to any special structure that may exist in a particular
spacetime.
One then has to reconcile the local observer splittings with
such a global splitting. This too is not anything particularly deep, and
it involves using the linear algebra of nonorthogonal bases independently
at each spacetime point (since such a global splitting will in general be
nonorthogonal)
to represent the orthogonal splitting of the
local observers on spacetime.
The new complication in general relativity is that in general one
must deal with a family of so called ``test observers" in arbitrary
motion, and this introduces the well known effects that
accompany noninertial (i.e., accelerated)
observers even in nonrelativistic physics.
However, in the classical example of a ``rigidly rotating" family of
noninertial observers in nonrelativistic physics, one still has a global
correlation between the different members of the family of observers
since one has a global (though not physical)
Cartesian coordinate system which rotates with the
passage of time. In the extension to general relativity one must consider
such effects locally at each spacetime point.
%\fig{Noninertial forces}%
%{Noninertial forces experienced in a rotating frame are illustrated
%with a merry-go-round. For a body at rest on the merry-go-round,
%there is a radially outward force, and if that body begins to walk on
%the merry-go-round, a force perpendicular to the direction of motion
%on the merry-go-round will be felt as well.
%}%
%{nif}
The effects of rigid rotation are well known and familiar.
%as playfully illustrated in Figure \ref{fig:nif}. % drawing of cat on merry-go
% -round
There is the centrifugal %(or ``merry-go-round")
force that a body feels in the rotating frame
even if it is not moving with respect to that frame
due to the frame's acceleration
and the Coriolis %(``mobile merry-go-round")
force that a body feels if it is in motion with respect to that frame
due to the frame's rotation.
Quantitatively in a system rotating with constant angular velocity
$\vec\Omega$,
there is a force on an otherwise free body
\beq
\ddot{\vec x} = \vec F / m = \vec g + \vec v \times \vec H \ ,
\eeq
where the ``gravitoelectric" force (per unit mass)
\beq
\vec g = -\vec A = - \vec\Omega \times \vec V
= - \vec\Omega \times ( \vec\Omega \times \vec x)
\eeq
is the negative of the acceleration field $\vec A$
of the rotating observers,
whose velocity field is $V=\vec\Omega \times \vec x$.
Similarly the ``gravitomagnetic" force is the
cross product of the body's velocity $\vec v = \dot{\vec x}$
in the rotating system
with the ``gravitomagnetic" vector field
$\vec H = 2 \vec \Omega = \curl \vec V$,
which is twice the angular velocity vector of the rotating frame
and equals the local vorticity of the velocity field.
The ``gravitoelectromagnetic"
terminology due to Thorne
is in direct analogy with the Lorentz force of
electromagnetism in a nonrotating system
\beq
(m/q) \ddot{\vec x} = \vec F / q = \vec E + \vec v \times \vec B \ .
\eeq
Note that the ``gravitomagnetic" vector field $\vec H = \curl \vec V$
admits a vector potential $\vec V$
in the same way that the magnetic field $\vec B= \curl \vec A$
locally admits a vector potential.
Similarly the ``gravitoelectric"
vector field
\beq
\vec g
= -\grad [-\half \vec V \cdot \vec V]
\eeq
admits a scalar potential in this time-independent case just
like the conservative electric field $\vec E$ in electrostatics.
The velocity field of the rotating observers and minus half its length squared
(once multiplied by the mass $m$)
serve as
vector and scalar potentials for the noninertial forces we often
call ``fictitious" forces in classical mechanics, and they are
directly interpretable in terms of kinematical
properties of the velocity field of the family of noninertial observers
used to describe the motion of the body being studied. These forces
vanish as soon as we require the members of the family of observers
to be inertial.
In a curved spacetime such global frames are not immediately available,
so one must analyse the situation in the local rest space of each observer
in the family of test observers used to describe the physics in
3-dimensional form. The kinematical properties of the 4-velocity field
of these observers in spacetime, with some extra complication,
directly generalize the above problem, leading to the introduction
of ``gravitoelectromagnetic" forces which enter into the force
equation when expressed in terms of a family of noninertial observers.
Rather than doing a global comparison of their motion with respect
to a global inertial frame which does not exist in curved spacetime,
it must be a local comparison at each point of spacetime
with a suitable inertial
observer having the same 4-velocity and a set of nonrotating spatial axes.
Of course there are new features in curved spacetime which have no analogy
in electromagnetism and this has to do with the spatial metric which
describes the relative distances between nearby test observers
as well as the different proper times that different observers
use at the same event in spacetime, both of which depend on the gravitational
field.
These features, as well as the
potentials for the gravitoelectromagnetic vector forces, are contained
in the spacetime metric.
Taking into account the local proper time complicates slightly
the above analogy which relies on a global proper time function on flat
spacetime.
Thus we need relativistic definitions of acceleration,
gradient, curl, time derivative,
etc. This together with the details of the way in which an observer
in spacetime measures quantities at a given event
will enable us to push our present knowledge about special relativity
and noninertial motion to the case of general relativity.
It helps explain the so called ADM or three-plus-one approach
to general relativity
as well as the slightly different
Landau-Lifshitz approach
and shows their similarities and differences,
both of which involve the structure of a family of test observers together
with a global time function on spacetime.
It also shows how both relate to the
Ehlers-Hawking-Ellis
splitting approach, which is based only on a family of test observers
with no global time function assumed to be available.
\Section{Why bother?}
You might say, why invest a lot of time into understanding the details
of this way of looking at all the different splitting approaches used
in general relativity? After all, relativity physics liberated us from the
prison of 3-dimensional language into the arena of spacetime where
the true nature of kinematics and dynamics became much simpler to
understand. Why try to climb back into the cage of 3-dimensional physics?
Wasn't centuries of solitary confinement enough?
Well, many spacetimes and idealized problems
we use in understanding gravitational theory
practically beg us to do this. A rotating black hole spacetime,
for example, has two very different privileged families of test
observers, one of which is suited to the ADM picture and the other
to the Landau-Lifshitz picture. The one we use depends on the
question we want to study. Both turn out to be useful.
The Ehlers-Hawking-Ellis picture provides the means to relate these
two different pictures to each other, which is important if
they both turn out to be needed, as they in fact do. The failure to
investigate this ``relativity of spacetime splitting formalisms"
has kept most of us from having better intuition not only about
black holes, or even the relativistic picture of the nonrelativistic
problem of rotating coordinate systems (which continues to confuse
people even now),
but other interesting rotating spacetimes like the
G\"odel universe.
Many of us are somewhat familiar with (or at least aware of) the initial
value problem for the ADM splitting of spacetime
based on a family of spacelike hypersurfaces,
but the corresponding problem for the splitting of Landau and Lifshitz
based on a timelike congruence is almost unknown. In fact the exact solution
industry for stationary spacetimes studies precisely this problem
without the difficulties that breaking the stationarity symmetry introduces,
since only the initial value problem remains of the Einstein equations
in that case. The choice of metric variables made in studying this problem
is adapted to the Landau-Lifshitz
splitting, though few stop to think about its geometric significance
or realize it is a decomposition parallel to the ADM splitting.
Even if none of these special spacetimes interests you,
if you are interested in any post-Newtonian calculations of
more realistic, say isolated self-gravitating systems,
then understanding the splitting game
helps to make a little more sense out of what is universally done
in that field. In recent years many references to
``the gravitomagnetic field" have sprung up, but since no one
took the time to unambiguously define just what this field was,
controversy has blossomed between different schools of thought about
just what a ``real gravitomagnetic field" is. %\Cite{Ciufolini 1991}
% (See Figure \ref{fig:chesh}.)
%
People can agree on a naive definition of the field in this
stationary weak-field limit
but for strong fields many different definitions are possible depending
on the choices one makes for the way in which observers make measurements.
All of these different choices can be fit into a general framework
and related to each other.
Of course the big difficulty in general relativity has always been
the limited analytical computations that can be done exactly. An entire
industry has grown up around the approximation schemes for general relativity
which in practice involves rather complicated details. Certainly the naive
things one can do for relatively simple exact solutions in terms of
interpreting them in 3-dimensional form cannot be extended easily to the
more realistic calculations done in post-Newtonian approximations to
general relativity. However, one can interpret features of the approximate
analysis in terms of the exact spacetimes which it is meant to approximate.
Unfortunately many aspects of this exact geometry being approximated seem
to be lost in the details of the approximation scheme itself. The language
of the present discussion can help put these details into some perspective.
% drawing from korean paper:
%\fig{Cheshire Cat}%
%{{\it ``The"\/} gravitomagnetic field of an isolated rotating
%self-gravitating
%body illustrated as the Cheshire cat from {\it Alice and Wonderland\/},
%whose smile might represent the commonly accepted
%naive definition in the weak field limit,
%while the rest of the cat remains obscure otherwise.
%}%
%{chesh}
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% \filename{em1b.tex} \version{22-mar-1992}{5-mar-1993}
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%\Message{.......................bridge work necessary.}
%\Section{Starting vocabulary}
The splitting of the gravitational field is closely related to the
splitting of the electromagnetic field, though historically reversed in
direction, since the individual electric and magnetic fields of classical
Newtonian physics were unified
into a single spacetime field by special relativity,
while the spacetime metric of general relativity only later gave birth
to the electric- and magnetic-like gravitational fields accompanied by
the spatial metric.
%
The four-dimensional form of Maxwell's equations in a curved spacetime
is very elegant
and powerful, perhaps because it is independent of any particular
observers or local coordinates.
However, in many practical applications, spacetime is endowed
at least locally
with either a preferred congruence of integral curves of a timelike
vector field or a preferred slicing by a family of spacelike
hypersurfaces or both, and it is convenient to decompose the electromagnetic
field in some way using this additional structure.
Of course the electric
and magnetic fields measured by an observer in spacetime are well
defined and easily expressed in an orthonormal frame adapted to the
observer's local rest space, but often coordinate systems and
nonorthonormal frames prove more convenient for studying the field
equations. In this context one must reinterpret the
computational quantities
which are naturally introduced in terms of some family of test observers.
When the present work began, a clear discussion
of the various approaches to formulating Maxwell's equations in terms
of three-dimensional quantities and their relationship to each other
did not exist. %%%% appear to exist. %%%%%%
Moreover,
the analogous discussion for the gravitational field itself was even more
conspicuously absent in the literature.
It therefore seemed useful to
carefully develop the mathematics of the
splitting formalisms in general relativity which provide the foundation
for the subsequent splittings of the electromagnetic field as well as
other matter fields on spacetime.
%It therefore
%seems useful to provide such a discussion %%%%%%
%for the electromagnetic field %%%%%
%together with a clear development
%of the splitting formalisms which provide its foundation.
In so doing one obtains a precise description of spatial gravitational
force fields in the different points of view and of their
interrelationships, as well as a clear exposition of the similarities
between those nonlinear fields and the linear electric and magnetic
fields.
In view of the evolution of terminology which has taken place, it seems
natural to refer to this analogy by the name of
gravitoelectromagnetism.
The analogy between the linear theory of electromagnetism and the linearized theory of general relativity was noted by Einstein even before arriving at his final formulation of the Einstein equations \Cite{Verbin and Nielsen, 2004}, who compared the geodesic equations with the Lorentz force law. This analogy for the final theory was then soon spelled out explicitly by Thirring \Cite{1918}.
Although the analogy in practice is usually considered for the linearized gravitational field,
the implications and limitations of this analogy are best seen
in the fully nonlinear context of general relativity.
Of course as previously noted,
an obvious question to ask is ``Why bother to split spacetime at all?"
Certainly the idea of a four-dimensional spacetime and its local Lorentzian
geometry has been an important advance of this century. However,
our intuition and experience are decidedly three-dimensional in character,
and splittings of spacetime into space plus time
allow us to interface better with the
four-dimensional information, even when a splitting does not occur
naturally. When it does, it can considerably simplify the presentation
and interpretation of both the gravitational field and whatever matter
fields are present.
This is not to say that space-plus-time splittings are always useful.
Sometimes $2+2$-splittings are important, examples of which occur
in spherical symmetry or in the null initial value problem.
In many other instances splittings are actually
unproductive, obscuring the spacetime structure of a problem. However,
when splittings are useful, they are worth doing carefully.
Many different points of view may be taken in splitting spacetime
into space plus time
but their interrelationships are rarely considered.
This is a useful thing to do since our intuition is based on the standard
splitting of flat spacetime, but the three-dimensional quantities which serve
as a foundation for this intuition often find themselves associated with
distinct points of view in the context of a general splitting of an
arbitrary spacetime. The result is that no single point of view captures
all aspects of our three-dimensional intuition, and the particular application
really should determine the choice which is most appropriate.
However, no common mathematical framework and no common notation yet
exist to enable one to easily switch point of view or compare different
points of view. Most relativists are consequently
prisoners of the language of one particular choice.
The goal of the present work is to establish such a common mathematical
framework to help break down the barriers which exist between different
schools of relativists who have settled upon a single choice of point of view.
Unfortunately no text on general relativity can spare the space to
do justice to this idea of a relativity of splitting formalisms,
so we learn general relativity
from one school or another
but rarely appreciate more than one approach in our working lives
even if we are relativists.
%However, if something is worth doing, it is worth
%doing right.
The present text attempts to provide both a universal language and the
detailed formulas which describe the relatively straightforward
but systematic analysis of these various approaches.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Starting vocabulary}
An overriding necessity in this enterprise is a careful definition of the
vocabulary to be employed since the standard labels which occur in these
discussions do not have universally accepted interpretations.
All of the various splitting points of view can be nicely classified and
will be assigned labels according to a neutral scheme independent of
any particular surnames, thus sidestepping the issue
of who ``owns" which ideas. Each splitting point of view is based on two
fundamental concepts: {\it measurement\/} and {\it evolution\/}, the realization
of which differ for each of the possible choices.
Before sketching the
history of this topic, it is useful to establish some of the basic
terminology to be used in what follows.
Of course in order to encompass all of the approaches one finds in the
literature in a simple scheme,
one must be a bit loose about exactly what mathematical details
characterize each of the basic categories into which these approaches will be
divided.
The notions of time and space are complementary since a ``time line" represents
``time elapsing at a point fixed in space" while a ``time hypersurface"
represents ``space at a moment of time". These two different notions of time,
the first which focuses on measuring time at a single point of space and the
second which is associated with some kind of synchronization of times at
different points of space,
will be assigned the labels ``time" and ``space" respectively. These divide the
splitting points of view into two categories, those in which a local time
direction is fundamental, and those in which a nonlocal correlation of
local times, i.e., space is fundamental.
In the time category, the ``time lines" must be timelike in order to
represent a local time direction at each event in spacetime, while in
the space category, the ``time hypersurfaces" or ``spaces" must be
spacelike in order to be associated with a moment of time in the usual sense
of a Riemannian space (alternatively, in order that orthogonality define
a local time direction).
Given this
division, one may consider a partial splitting or a full splitting
depending on whether any additional structure is assumed.
Table 1.1\message{Table 1.1}
establishes this general classification of points of view and the
terminology that will be used to describe it.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\typeout{table 1.1}
\begin{table}
\typeout{4 pictex pictures for table 2}
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% vertically centered vbox:
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\long\def\vcV#1{\vbox to .7truein{\hsize= 1.7truein\parindent=0pt\vfil
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\long\def\vcs#1{\vbox to .7truein{\hsize= 1.5truein\parindent=0pt\vfil
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% modification of 12pt strutbox of plain.tex:
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\def\strutfifteen{\relax \ifmmode\copy\strutboxfifteen
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\let\strutf=\strutfifteen
\let\strutF=\strut
%
\newbox\strutboxhigh
\setbox\strutboxhigh=\hbox{\vrule height 20pt depth 8pt width0pt}
\def\struthigh{\relax \ifmmode\copy\strutboxhigh
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%
\setbox0=\ylap{\vbox{\halign{\strut\hfil # \hfil\cr
without\cr causality\cr condition\cr}}}
\ht0=0pt \dp0=0pt
$$
\vbox{\tabskip=0pt \offinterlineskip
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& \multispan2 & & \VVVtop{time: 1\strutf\\(``single-observer" time)\strutf }& &
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parametrized nonlinear\cr
reference frame: $\{t,x^a\}$\cr
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& \cr\trule
& & \vcs{TIME gauge} & &
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\vcV{\VVVtop{\strutF arbitrary synchronization\\ of observer times\strutF\\
$\leftrightarrow$ slicing\strutF}} & &
\vcV{\VVVtop{\strutF arbitrary identification\\ of
``points of space"\strutF\\
$\leftrightarrow$ threading\strutF}} & \cr\trule
}}$$
\caption{A characterization of the different points of view (p.o.v.)
that may be
adopted in splitting spacetime. Solid lines in diagrams imply the use
of the appropriate causality condition while dashed lines indicate
that no causality condition is assumed. The hypersurface p.o.v.\
is essentially equivalent to the vorticity-free congruence p.o.v.
The reference p.o.v.\ corresponds to a full splitting in which no causality assumptions are made.}
%
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Given no additional structure, one has only a partial splitting of spacetime,
splitting off either the time or the space alone. In the first case,
to be called the ``{\it congruence point of view\/}," one has only a timelike
congruence at one's disposal, with a unit timelike tangent vector field $u$.
Spacetime will be assumed to be time-oriented as well as oriented, so
one may assume that $u$ is future-pointing. It may then be interpreted
as the 4-velocity of a family of test observers whose worldlines are the
curves of the congruence, and it determines the {\it local time direction\/}
at each point of spacetime. The orthogonal complement of this local time
direction in the tangent space is the {\it local rest space\/} $LRS_u$ of the
test observer at that event.
It is exactly this structure that one needs for the {\it measurement process\/}
which will be the same for the full and partial splittings in a given
category in Table 1. The orthogonal decomposition of the tensor algebra
induced by this decomposition of the tangent space at each event will
define the measurement process, modulo a final step in which projection
along the local time direction is replaced by contraction with $u$, yielding
a collection
of ``{\it spatial tensor fields\/}" of different rank for each spacetime
tensor field that is split.
In general the splitting of the tangent spaces does not extend to the
spacetime manifold. Only in the special case that the rotation of $u$
vanishes does such an extension exist and one has a family of orthogonal
spacelike hypersurfaces which slice the spacetime, leading to a full
splitting in this category to be discussed below. The second case in the
category of partial splittings of spacetime,
that of a spacelike slicing of spacetime with no additional structure,
is essentially equivalent to the special case of a nonrotating congruence
since every spacelike slicing admits a family of timelike orthogonal
trajectories. These are the integral curves of the (rotation-free) unit normal
vector field $n$ to the slicing, which may be assumed to be future-pointing.
The accompanying point of view, for the sake of completeness,
might be called the ``{\it hypersurface point of view\/}". Its measurement
process is associated with the normal congruence, taking $u=n$ as the
4-velocity of the family of test observers who do the measuring.
The local rest spaces of this family are integrable and coincide with
the subspaces of the tangent space which are tangent to the slicing.
A full splitting of spacetime at the manifold level requires both
a {\it slicing\/} of the spacetime and a congruence, to be referred to as a
``{\it threading\/}" of the spacetime, together with a compatibility
condition that the two families be everywhere transversal.
Such a structure will be called a ``{\it nonlinear reference frame\/}"
in order to distinguish it first from the terms ``reference frame,"
``frame of reference," ``reference system" and ``system of reference"
that one finds in the literature, second from the connotation of
``frame" in the context of a linear frame of vector fields, and third
from related terminology which occurs in the discussion of globally
constant frames in flat spacetime.
A ``{\it parametrized nonlinear reference frame\/}" will consist of a
nonlinear reference frame together with a choice of parametrization of the
family of slices. Such a parametrization defines a specific ``time function" $t$
on the spacetime which in turn provides an obvious parametrization for each
curve in the threading congruence.
In the category of full splittings,
the distinguishing criterion is the causality condition imposed on
the nonlinear reference frame.
In the ``{\it slicing point of view\/}" the slicing is assumed to be
spacelike, but no assumption is made about the causality properties
of the threading, which serves only as a way of identifying the points
on different slices.
In the ``{\it threading point of view\/}," the threading is assumed to be
timelike, but no assumption is made about the causality properties of the
slicing, which serves only to synchronize in some arbitrary fashion points
on different curves in the congruence. If both causality conditions hold,
then both points of view hold and one can transform between them.
On the other hand it can also be useful in the case that at least one
of the two causality conditions holds to not take advantage of that condition
and exploit only the structure of the nonlinear reference frame that does not
depend on it. This leads to the ``{\it reference point of view\/}," whose
measurement process is associated with the nonorthogonal decomposition of
the tangent space into the direct sum of a 1-dimensional subspace tangent
to the threading and a three-dimensional subspace tangent to the slicing.
One can always relate either the slicing or threading points of view to this
acausal approach,
which is the way in which they are usually represented in a local
coordinate system adapted to the nonlinear reference frame.
The partial splittings may be related to the full splittings in different ways.
In the threading point of view one may define a (future-pointing) unit timelike
tangent vector field $m$ along the threading congruence, while in the slicing
point of view one has the (future-pointing) timelike unit normal vector field
$n$. By making the respective choices $o=m$ and $o=n$ (``o" for ``observer")
of the 4-velocity of
a privileged family of test observers in these two
points of view, the identification $u=o$
relates each of them to a corresponding congruence point of view described
above, defining for each a measurement process.
When both the slicing and threading
points of view hold, then a unique boost in each tangent space relates the
two timelike unit vectors $m$ and $n$ and this may be extended to a
transformation of the measurement process.
In the special case of an {\it orthogonal nonlinear reference frame\/}
(one for which both the causality conditions hold and the slicing and
threading are everywhere orthogonal),
then $m=n$ and the two points of view coincide.
{\it Evolution\/} is defined first by a choice of a 1-parameter group of
diffeomorphisms of the spacetime into itself which in some sense advances
into the future (either its orbits are timelike
or it pushes certain spacelike hypersurfaces
into their future), and second by a choice of transport along its orbits
for the spatial fields of the given point of view.
For a partial splitting only one congruence is available and it is timelike.
In the absense of additional structure one can take $u$ or $n$ respectively
in the congruence or hypersurface points of view as the generator of
such a group, and choose either spatially-projected (``{\it spatial\/}")
Lie transport
(``noncovariant" but integrable)
or spatially-projected parallel transport (``covariant" but in general
nonintegrable)
along this congruence. The latter transport of spatial fields coincides
with Fermi-Walker transport which defines locally nonrotating axes along
a worldline.
Each of these choices may be extended to the full splitting in its
category but it is the spatial Lie transport along the threading
congruence which defines the evolution relative to the nonlinear reference
frame, since fields which are ``rigidly" attached to this frame do
not evolve with this choice.
However, unlike Fermi-Walker transport, spatial Lie transport is in
general incompatible with orthonormal frames.
A compromise between the two kinds of transport
leads to {\it co-rotating Fermi-Walker transport\/}, which
is the closest one can get to attaching an
orthonormal frame to the nonlinear reference frame. \message{awkward??}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Historical background}
Armed with this initial vocabulary, the historical background may be sketched
in a way that puts the different formalisms into some perspective.
The slicing and threading points of view today are introduced to most of us
through two leading textbooks, respectively {\it Gravitation\/} by Misner,
Thorne, and Wheeler \Cite{1973}
and {\it The Classical Theory of Fields\/} by Landau
and Lifshitz \Cite{1975}, each of which carefully avoids mention of the
``competing" point of view. Both points of view can be traced back to the
early forties when the first edition of the Landau-Lifshitz text \Cite{1941}
introduced the threading point of view splitting of the spacetime metric
and, in the stationary case, of the spacetime connection to yield
spatial gravitational forces, as still described
in their last edition. Soon after,
Lichnerowicz \Cite{1944} introduced the beginnings of
the slicing point of view with an article discussing the initial value
problem in an orthogonal nonlinear reference frame. His later book on
general relativity and electromagnetism \Cite{Lichnerowicz 1955} curiously
enough makes use of the threading split, but in actual applications uses
an orthogonal nonlinear reference frame in which the two points of
view agree.
The threading point of view apparently
dominated during the fifties when much interest was focused on the equations
of motion for test particles.
M\o ller discussed a parametrization-dependent definition of spatial
gravitational forces for a general spacetime
in the first edition of his text {\it The Theory of Relativity\/}
\Cite{M\o ller 1952}
at the beginning of the decade. This was then
refined to a parametrization-independent splitting later in the decade
by Zel'manov \Cite{1956, 1959}
in the Soviet Union and then independently by Cattaneo \Cite{1958,1959a,b,c}
in Italy. Unfortunately most of the few references to these works that
do appear in the literature cite papers written in Russian, Italian or French,
so one must dig to find English versions.
The most accessible discussion of much of
this material is the second edition of
M\o ller's text \Cite{1972} which describes it in detail and
also contrasts it with his original splitting. However, for some reason
this text itself is
not very prominent among the relativity texts one usually encounters
for those of us who have entered the field since the sixties,
perhaps because of its old-fashioned viewpoint. Of these authors,
only Zel'manov \Cite{1956}
discussed the splitting of Einstein's equations in the general case.
Meanwhile the slicing point of view was further developed during the fifties by
Choquet-Bruhat \Cite{1956} %who split the spacetime metric
and Dirac \Cite{1959}.
Choquet-Bruhat extended Lichnerowicz's initial value discussion to the
general case, identifying but not naming the lapse and shift variables
of the slicing point of view using orthonormal frame techniques.
Dirac
recognized the significance of the slicing point of view metric decomposition
for the Hamiltonian dynamics
of general relativity and its relation to his theory of constrained
Hamiltonian systems.
This was then refined in a series of papers at the
turn of the decade by Arnowit, Deser and Misner who used the Hamiltonian
formulation permitted by the slicing point of view to study the true degrees
of freedom of the gravitational field, culminating in an often cited
review article \Cite{Arnowit, Deser and Misner 1962}.
This ushered in the new era of domination of the
splitting scene by the slicing point of view, pushed by the problem of quantum
gravity where Hamiltonian techniques have played a rather important role in
an endless quest that has not yet met success.
The notation of Arnowit, Deser and
Misner, soon labeled by Wheeler's lapse and shift terminology
\Cite{Wheeler 1964}
and later effectively propagated by the text of Misner, Thorne and Wheeler
\Cite{1973},
has found widespread acceptance.
The slicing point of view is also commonly referred to as
the ``$3+1$" or ADM formalism.
A number of useful reviews of various aspects of this formalism exist,
among them
being articles by York \Cite{1979}, Isenberg and Nester \Cite{1980},
Fischer and Marsden \Cite{1978, 1979} and Gotay et al \Cite{1991}.
The term ``$1+3$" formalism with its obvious change in emphasis has been
suggested %by Damour
as an alternative label for the threading point of view.
Although Cattaneo stopped his analysis of the threading point of view
at the connection level,
Cattaneo-Gasperini \Cite{1961, 1963} and Ferrarese \Cite{1963, 1965}
continued it to the curvature level, studying the splitting of the
spacetime curvature and of Einstein's equations, and the various
definitions of spatial curvature that are possible.
This previously unexplored area of differential geometry dealing with
a degenerate but nonintegrable connection, namely
the spatial covariant derivative or transverse covariant derivative
which occurs in the threading and congruence points of view, has
never been fully understood in the context of the initial value
problem for those points of view. This problem, which has been
discussed by Stachel \Cite{1980} and
Ferrarese \Cite{1987,1988, 1989},
is of a completely different character
than in the slicing point of view where it is rather well understood,
and its resolution is still an open problem.
The approach of Cattaneo and Ferrarese to the threading point of view was
reformulated by Massa \Cite{1974a,b,c} and used to discuss
gyroscope precession \Cite{Massa and Zordan, 1975}.
This latter problem, more than any other, has focused
attention on the effects of spatial gravitational fields.
More recently Perjes \Cite{1988}
and Abramowicz \Cite{1988, 1990} have considered variations of
M\o ller's parametrization-dependent threading approach.
In the slicing point of view
the natural extension of the splitting of the metric to the splitting of the
connection and the discussion of spatial gravitational forces has only
recently been considered,
perhaps forgotten in the emphasis on the initial value problem and
Hamiltonian dynamics.
The roots of this
discussion can be traced back to the original threading point of view work,
although this link is not made apparent in citations.
At the crossover point between the popularity of the threading and slicing
points of view, Forward \Cite{1961}
described the analogy between electromagnetism
and linearized general relativity using the reference point of view, in a
slight variation of M\o ller's formalism. In the late seventies
this article then inspired a reference
point of view discussion of the PPN formalism by Braginsky, Caves and
Thorne \Cite{1977} who introduced an ``electric-type" gravitational field and
a ``magnetic-type" gravitational field,
the latter of which became the
``gravitomagnetic" field of the eighties
in a discussion of linearized general relativity by Braginsky, Polnarev and
Thorne \Cite{1984}. In linearized gravity, in the usual weak field slow motion
discussions,
the corresponding spatial gravitational fields
defined in the threading, slicing and reference points of view
are very closely related and agree to the lowest order,
although not to full post-Newtonian order.
Curiously enough, it is always the threading fields which are used in
post-Newtonian discussions even when the slicing point of view is
advocated before linearization.
%the distinction only becoming apparent in the case of strong
%fields or fast motion.
The introduction of the terminology ``gravitoelectric" field
and the first discussion of
slicing spatial gravitational forces finally appeared in the text
{\it Black Holes: The Membrane Paradigm\/}
by Thorne et al \Cite{1986},
but only in the case of a stationary gravitational field in their
treatment of black hole spacetimes.
This may be extended in an
obvious way to general spacetimes in a notation which shows the close
relationship to the threading point of view as will be described below.
The congruence point of view is briefly introduced in an article by Hawking
\Cite{1966} and at great length in a pair of articles by
Ellis \Cite{1971, 1973} at the beginning of the seventies (later updated by Ellis and van Elst \Cite{1998}),
all based on
earlier unifying work of Ehlers \Cite{1961} and of
Kundt and Tr\"umper \Cite{1962} unavailable in English
until the Ehlers article alone finally appeared in tranlation
over three decades later \Cite{1994}.
Completing the congruence vector field $u$ to an orthonormal frame
leads to the explicit orthonormal frame approach of
%This point of view
%abstracts out the orthogonal decomposition from the later discussion of the
%full orthonormal frame approach by
Estabrook and Wahlquist \Cite{1964},
who in a note added in proof in their article
thank Pirani for calling their attention
to Cattaneo's work and cite articles in French; they were
not the first or the last to have been
as they say ``completely unaware of this work."
(Cattaneo himself appears to have been completely unaware of Zel'manov's work,
while the present authors were completely unaware of both when this project
was begun.)
The hypersurface point of view is described in the article by
Zel'manov \Cite{1973} in the early seventies in terms of a
nonlinear reference frame, but some work is required to decipher his notation.
Ehlers \Cite{1961} and Ellis \Cite{1971}
also treat the hypersurface point of view as a special case of the
congruence point of view.
These various splittings of spacetime are particularly interesting in the case
of electromagnetism, where all of our intuition is tied to individual
electric and magnetic fields, and astrophysical applications can be
aided by allowing this intuition to find expression in the context of
a splitting.
Early work by Ruffini and collaborators
\Cite{Hanni and Ruffini 1973, Hanni and Ruffini 1975,
Ruffini and Wilson 1975, Damour and Ruffini 1975, Hanni 1977,
Damour et al 1978, Ruffini 1978}
followed up in more detail by Damour \Cite{1978, 1982}
revealed the utility
of introducing the concept of electric and magnetic fields
in studying black hole systems. %More recently
This was
discussed later in great detail from the slicing point of view
by Thorne and Macdonald \Cite{1982},
who summarize the history of the different
splittings in general and as applied to Maxwell's equations, and by
Thorne et al \Cite{1986}
for application to black hole systems.
Maxwell's equations may be expressed in the congruence point of view
as done by Ellis \Cite{1973}, in the threading point of view as done
by Benvenuti \Cite{1960} in an application of Cattaneo's formalism
unfortunately appearing only in Italian,
and in the slicing point of view as described by Misner, Thorne and Wheeler
\Cite{1973}
(for the correct vector potential splitting, see Isenberg and Nester
\Cite{1980}, for example).
The reference point of view splitting is much older, dating back to the
beginnings of general relativity in an article by Tamm \Cite{1924},
as noted by Skrotsky \Cite{1957} and Plebanski \Cite{1960}.
This appears in a peculiar mix of the reference and threading points of view
in an exercise in the Landau and Lifshitz text \Cite{1975}.
The mystery of this
latter approach has caused its share of confusion about how one should
define electric and magnetic fields in applications in general relativity.
Hanni \Cite{1977}
has given the complementary version which mixes the reference and
slicing points of view.
It is rather interesting to compare each of these numerous splittings
of Maxwell's equations with the others.
Maxwell's equations from the slicing point of view,
first considered by Misner and Wheeler \Cite{1957}
using the language of differential forms,
are discussed in great detail by Thorne and Macdonald \Cite{1982}
and by Thorne et al \Cite{1986}.
An enormous language barrier exists at present between the slicing and
threading points of view, preventing those versed in the formalism
and notation of one from easily penetrating the other or understanding
how the two are related.
This is exactly the problem the present exposition hopes
to address, namely the lack of a common
mathematical framework to discuss both approaches on an equal footing.
Both of these splitting formalisms can be developed in a completely
parallel way as complementary aspects of a single geometrical structure imposed
on spacetime (the nonlinear reference frame),
aspects which in a close way are related by the same duality
that links contravariant and covariant fields on the spacetime manifold.
Furthermore, each of these approaches has important ties with the congruence
point of view which invariantly describes the geometry of the observer
congruence and with the reference point of view which links these
discussions to adapted coordinate systems in practice.
The style of the threading approach as usually presented
is somewhat more
cumbersome than the slicing one, so it will be recast in the slicing style,
generalizing
from adapted coordinate systems to adapted local frames. In the special
case of an orthogonal slicing and threading of a spacetime,
the two descriptions will then coincide.
This standardization of ideas and notation can also prove useful in
approximation techniques, including the recent axiomatization of the
idea of a Newtonian limit of general relativity based on a family of
spacetime splittings as discussed by Ehlers \Cite{1989},
Lottermoser \Cite{1989}, and Schmidt \Cite{Ehlers, Schmidt and Lottermoser
1990}.
%Much of the complication of this work arises simply because
%the geometry of the threading point of view is not well known.
The relatively unknown geometry of the spatial connection of the congruence
and threading points of view is also currently of interest in the
reformulation of gauge-invariant perturbation theory for
Friedmann-Robertson-Walker spacetimes by Ellis and coworkers \Cite{Ellis
and Bruni 1989, Ellis, Hwang, and Bruni 1989}.
Each of these limits concern special cases of the problem of perturbation
theory for a general spacetime from the somewhat unfamiliar
alternative points of view to be discussed in this monograph.
Chapter 2 describes the congruence point of view, which is later used
to define the measurement process for the slicing and threading points of
view.
Chapter 3 studies the splitting geometry associated with a nonlinear
reference frame and discusses both the slicing and
threading points of view and their relationship to the congruence point
of view and the reference point of view.
%This leads to a precise definition of spatial gravitational
%forces in each of the various points of view.
Chapter 4 discusses electromagnetism in detail from each of the points of
view while Chapter 5 considers the very important case of
stationary spacetimes and the Sagnac effect and synchronization
questions. The special case of flat spacetime in rotating
coordinates clarifies the relationship of the spatial gravitational forces
to the centrifugal and Coriolis forces.
Chapter 6 discusses the weak field, slow motion limit and almost
Friedmann-Robertson-Walker spacetimes.
Although this text contains many formulas, only a few simple ideas
applied in a methodical way are behind much of the detail. Orthogonal
decomposition, by itself or
represented in the context of a nonorthogonal decomposition,
is at the heart of the algebra of respectively partial and
full splittings of spacetime. Orthogonal projection of differential
operators in the same spirit then provides the differential tools necessary
to extend the splitting algebra to include derivatives.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\Section{Orthogonalization in the Lorentzian plane}
The heart of linking spacetime splittings to coordinate systems is a simple idea, but one which receives little attention in our academic preparation: the use of nonorthogonal coordinate systems. It is very useful to look at the Euclidean and Lorentzian 2-planes $E^2$ and $M^2$ to recall first a case for which our geometric intuition holds and then see how it differs in the spacetime arena.
\subsubsection{The Euclidean example: general linear coordinates}
Given any two nonzero linearly independent vectors $(\Vec X,\Vec Y)$ in $R^2$ with its usual Euclidean structure, then expressing the position vector $\Vec r = x\Vec X +y\Vec Y$ defines its coordinates $(x,y)$, which may be thought of as functions on $R^2$. Then the self-dot product of the position vector defines the quadratic distance formula
\beq
\vec r \cdot \vec r = A x^2 + B x y + C y^2\ ,
\eeq
where
\beq
A = \Vec X \cdot \Vec X\ ,
B = \Vec X \cdot \Vec Y\ ,
C = \Vec Y \cdot \Vec Y\ .
\eeq
Assume this is positive-definite: $B^2-4AC<0, A>0,C>0$. Figure \ref{fig:1E} illustrates the geometry.
\typeout{Figure 1E}
\def\Unit{1cm}
\begin{figure}[t]\label{fig:1E}
$$\vbox{
\beginpicture
\setcoordinatesystem units <\Unit,\Unit> point at 0 0
\plot -1 0 5 0 / % x axis
\plot -0.5 -1 1.75 3.5 / % y axis
\arrow <.3\Unit> [.15,.4] from 3.7 0 to 4.0 0 % X
\arrow <.3\Unit> [.15,.4] from 4.7 0 to 5.0 0 % x-axis
\arrow <.3\Unit> [.15,.4] from 0 0 to 1.25 2.5 % Y
\arrow <.3\Unit> [.15,.4] from 0 0 to 1.75 3.5 % y-axis
\arrow <.3\Unit> [.15,.4] from 0 0 to 2 1 % r
\setdashes
\plot 1.25 2.5 5.25 2.5 / % x axis
\plot 4 0 5.25 2.5 / % y axis
\setdots
\plot 0.5 1 2 1 / % x axis
\plot 1.5 0 2 1 / % y axis
\put {\mathput{x}} [t] at 4.75 -0.3
\put {\mathput{y}} [Br] at 1.35 3.2
\put {\mathput{X}} [t] at 2.2 -0.3
\put {\mathput{Y}} [Br] at .5 1.5
\put {\mathput{\vec r}} [Bl] at 2.15 1.15
\endpicture}
$$
%\vglue \Unit
$$\vbox{
\beginpicture
\setcoordinatesystem units <\Unit,\Unit> point at 0 0
\plot -1 0 5 0 / % x axis
\plot 0 2.5 1.25 2.5 / % proj
\plot 0 -1 0 3.5 / % y axis
\arrow <.3\Unit> [.15,.4] from 3.7 0 to 4.0 0 % X
\arrow <.3\Unit> [.15,.4] from 4.7 0 to 5.0 0 % x-axis
\arrow <.3\Unit> [.15,.4] from 1.25 2.5 to 0 2.5 % proj
\arrow <.3\Unit> [.15,.4] from 0 0 to 1.25 2.5 % Y
\arrow <.3\Unit> [.15,.4] from 0 0 to 0 3.5 % y-axis
\arrow <.3\Unit> [.15,.4] from 0 0 to 2 1 % r
\setdashes
\plot 1.25 2.5 4 2.5 / % x axis
\plot 4 0 4 2.5 / % y axis
\setplotsymbol ({\huge.}) \plotsymbolspacing=5pt
\setdots
\plot 0 1 2 1 / % x axis
\plot 2 0 2 1 / % y axis
\setplotsymbol ({\small.}) \plotsymbolspacing=1pt
\put {\mathput{\overline x}} [t] at 4.75 -0.3
\put {\mathput{\overline y}} [Br] at -0.3 3.3
\put {\mathput{\overline X = X}} [t] at 2.2 -0.3
\put {\mathput{Y}} [B] at 1.3 1.5
\put {\mathput{\overline Y}} [B] at -0.5 1.5
\put {\mathput{-\textstyle\frac{B}{2A} X}} [B] at .75 3.0
\put {\mathput{\vec r}} [Bl] at 2.15 1.15
\setsolid
\setcoordinatesystem units <\Unit,\Unit> point at -7 0
\plot -1 0.5 3.2 -1.6 / % x axis
\plot 4 0 3.2 -1.6 / % proj
\plot -0.5 -1 1.75 3.5 / % y axis
\arrow <.3\Unit> [.15,.4] from 0 0 to 4.0 0 % X
\arrow <.3\Unit> [.15,.4] from -1 0.5 to 3.2 -1.6 % =X
\arrow <.3\Unit> [.15,.4] from -1 0.5 to 4 -2 % x-axis
\arrow <.3\Unit> [.15,.4] from 4 0 to 3.2 -1.6 % proj
\arrow <.3\Unit> [.15,.4] from 0 0 to 1.25 2.5 % Y
\arrow <.3\Unit> [.15,.4] from -0.5 -1 to 1.75 3.5 % y-axis
\arrow <.3\Unit> [.15,.4] from 0 0 to 2 1 % r
\setdashes
\plot 1.25 2.5 4.45 0.9 / % x axis
\plot 4 0 4.45 0.9 / % y axis
\setplotsymbol ({\huge.}) \plotsymbolspacing=5pt
\setdots
\plot .8 1.6 2 1 / % x axis
\plot 1.2 -0.6 2 1 / % y axis
\setplotsymbol ({\small.}) \plotsymbolspacing=1pt
\put {\mathput{\overeq x}} [tr] at 3.5 -2
\put {\mathput{\overeq y}} [Br] at 1.27 3.2
\put {\mathput{X}} [t] at 2.2 -0.3
\put {\mathput{\overeq X}} [t] at 1.5 -1
\put {\mathput{\overeq Y = Y}} [rB] at 0.2 1.5
\put {\mathput{-\textstyle\frac{B}{2C} Y}} [l] at 4.0 -0.8
\put {\mathput{\vec r}} [Bl] at 2.15 1.15
\endpicture}
$$
%\vglue \Unit
\caption{General linear coordinates on $E^2$ and their two orthogonalizations. The dashed lines indicate the unit coordinate rectangle and the dotted lines the projections of a typical vector along the coordinate axes. }
\end{figure}
There are two choices for completing the square on the quadratic form:
\beq
\meqalign{
\Vec r \cdot \Vec r
&= A (x+\frac{B}{2A} y)^2 + (C-\frac{B^2}{4A}) y^2\qquad
&= (A -\frac{B^2}{2C})x^2 + C(y+\frac{B}{2C}x)^2\ ,\cr
&= A (\overline x)^2 + (C-\frac{B^2}{4A}) (\overline y)^2 \qquad
&= (A -\frac{B^2}{4C})(\overeq x)^2 + C(\overeq y)^2\ ,\cr
}
\eeq
where new adapted coordinates are defined by
\beq
\meqalign{
&\overline x = x+\frac{B}{2A} y \ ,\quad
& x = \overline x - \frac{B}{2A} y\ ,
\qquad
& \overeq x = x\ ,\
& x = \overeq x\ ,\cr
& \overline y = y \ ,\
& y = \overline y \ ,
& \overeq y = y + \frac{B}{2C} x\ ,\quad
& y = \overeq y - \frac{B}{2C} \overeq x\ , \cr
}
\eeq
The new coordinate systems correspond to the new bases
\beq
\Vec r =x \Vec X +y \Vec Y
=\overline x \Vec{\overline X} + \overline y \Vec{\overline Y}
=\overeq x \Vec{\overeq X} + \overeq y \Vec{\overeq Y}
\eeq
which transform in a complimentary way
\beq
\meqalign{
&\Vec{\overline X} = \Vec X \ ,\quad
& \Vec X = \Vec{\overline X} \ ,
\qquad
& \Vec{\overeq X} = \Vec X -\frac{B}{2C} \Vec Y\ ,\
& \Vec X = \Vec{\overeq X} +\frac{B}{2C} \Vec{\overeq Y}\ ,
\cr
& \Vec{\overline Y} = \Vec Y -\frac{B}{2A} \Vec X \ ,\quad
& \Vec Y = \Vec{\overline Y} +\frac{B}{2A} \Vec{\overline X} \ ,\qquad
& \Vec{\overeq Y} = \Vec Y \ ,\quad
& \Vec Y = \Vec{\overeq Y} \ , \cr
}
\eeq
In each case one basis vector is retained and the additional new basis vector is obtained by orthogonal projection of the other, a process complimentary to completing the square on the corresponding coordinates.
Figure \ref{fig:1E} illustrates these projections and the new unit coordinate rectangles.
\subsubsection{The Lorentzian case: general linear coordinates}
The Lorentzian case of two-dimensional Minkowski space $M^2$ is similar but our geometric intuition no longer holds.
Given a pair of vectors $(\Vec X,\Vec T)$ in $M^2$, one spacelike, the other timelike, then expressing the position vector $\Vec r = x\Vec X +t\Vec T$ defines its coordinates $(x,t)$, which may be thought of as functions on $M^2$. Then the self-dot product of the position vector defines the quadratic distance formula
\beq
\vec r \cdot \vec r = A x^2 + B x t + C t^2\ ,
\eeq
where
\beq
A = \Vec X \cdot \Vec X\ ,
B = \Vec X \cdot \Vec T\ ,
C = \Vec T \cdot \Vec T\ .
\eeq
Assume this is a Lorentzian inner product: $B^2-4AC>0, A>0,C<0$. Figure \ref{fig:1M} illustrates the geometry.
Again there are two choices for completing the square on the quadratic form:
\beq
\meqalign{
\Vec r \cdot \Vec r
&= A (x+\frac{B}{2A} t)^2 + (C-\frac{B^2}{4A}) t^2\qquad
&= (A -\frac{B^2}{2C})x^2 + C(t+\frac{B}{2C}x)^2\ ,\cr
&= A (\overline x)^2 + (C-\frac{B^2}{4A}) (\overline t)^2 \qquad
&= (A -\frac{B^2}{4C})(\overeq x)^2 + C(\overeq t)^2\ ,\cr
}
\eeq
where new adapted coordinates are defined by
\beq
\meqalign{
&\overline x = x+\frac{B}{2A} t \ ,\quad
& x = \overline x - \frac{B}{2A} t\ ,
\qquad
& \overeq x = x\ ,\
& x = \overeq x\ ,\cr
& \overline t = t \ ,\
& t = \overline t \ ,
& \overeq t = t + \frac{B}{2C} x\ ,\quad
& t = \overeq t - \frac{B}{2C} \overeq x\ , \cr
}
\eeq
The new coordinate systems correspond to the new bases
\beq
\Vec r =x \Vec X +t \Vec T
=\overline x \Vec{\overline X} + \overline t \Vec{\overline T}
=\overeq x \Vec{\overeq X} + \overeq t \Vec{\overeq T}
\eeq
which transform in a complimentary way
\beq
\meqalign{
&\Vec{\overline X} = \Vec X \ ,\quad
& \Vec X = \Vec{\overline X} \ ,
\qquad
& \Vec{\overeq X} = \Vec X -\frac{B}{2C} \Vec T\ ,\
& \Vec X = \Vec{\overeq X} +\frac{B}{2C} \Vec{\overeq T}\ ,
\cr
& \Vec{\overline T} = \Vec T -\frac{B}{2A} \Vec X \ ,\quad
& \Vec T = \Vec{\overline T} +\frac{B}{2A} \Vec{\overline X} \ ,\qquad
& \Vec{\overeq T} = \Vec T \ ,\quad
& \Vec T = \Vec{\overeq T} \ , \cr
}
\eeq
In each case one basis vector is retained and the additional new basis vector is obtained by orthogonal projection of the other, a process complimentary to completing the square on the corresponding coordinates.
Figure \ref{fig:1M} illustrates these projections and the new unit coordinate rectangles. The first orthogonalization remains the same, but the second orthogonalization changes due to the change in sign of $C$.
\typeout{Figure 1M}
\begin{figure}[t]\label{fig:1M}
%a)
$$\vbox{
\beginpicture\
\setcoordinatesystem units <\Unit,\Unit> point at 0 100
\plot -1 0 5 0 / % x axis
\plot -0.5 -1 1.75 3.5 / % y axis
\arrow <.3\Unit> [.15,.4] from 3.7 0 to 4.0 0 % X
\arrow <.3\Unit> [.15,.4] from 4.7 0 to 5.0 0 % x-axis
\arrow <.3\Unit> [.15,.4] from 0 0 to 1.25 2.5 % Y
\arrow <.3\Unit> [.15,.4] from 0 0 to 1.75 3.5 % y-axis
\arrow <.3\Unit> [.15,.4] from 0 0 to 2 2 % r
\setdashes
\plot 1.25 2.5 5.25 2.5 / % x axis
\plot 4 0 5.25 2.5 / % y axis
\setplotsymbol ({\huge.}) \plotsymbolspacing=5pt
\setdots
\plot 1 2 2 2 / % x axis
\plot 1 0 2 2 / % y axis
\put {\mathput{x}} [t] at 4.75 -0.3
\put {\mathput{t}} [Br] at 1.35 3.2
\put {\mathput{X}} [t] at 2.2 -0.3
\put {\mathput{T}} [Br] at .5 1.5
\put {\mathput{\vec r}} [l] at 2.15 2
\endpicture}
$$
%\vglue \Unit
%b)
$$\vbox{
\beginpicture
\setcoordinatesystem units <\Unit,\Unit> point at 0 100
\plot -1 0 5 0 / % x axis
\plot 0 2.5 1.25 2.5 / % proj
\plot 0 -1 0 3.5 / % y axis
\arrow <.3\Unit> [.15,.4] from 3.7 0 to 4.0 0 % X
\arrow <.3\Unit> [.15,.4] from 4.7 0 to 5.0 0 % x-axis
\arrow <.3\Unit> [.15,.4] from 1.25 2.5 to 0 2.5 % proj
\arrow <.3\Unit> [.15,.4] from 0 0 to 1.25 2.5 % Y
\arrow <.3\Unit> [.15,.4] from 0 0 to 0 3.5 % y-axis
\arrow <.3\Unit> [.15,.4] from 0 0 to 2 2 % r
\setdashes
\plot 1.25 2.5 4 2.5 / % x axis
\plot 4 0 4 2.5 / % y axis
\setplotsymbol ({\huge.}) \plotsymbolspacing=5pt
\setdots
\plot 0 2 2 2 / % x axis
\plot 2 0 2 2 / % y axis
\put {\mathput{\overline x}} [t] at 4.75 -0.3
\put {\mathput{\overline t}} [Br] at -0.3 3.3
\put {\mathput{\overline X = X}} [t] at 2.2 -0.3
\put {\mathput{T}} [B] at 1.3 1.5
\put {\mathput{\overline T}} [B] at -0.5 1.5
\put {\mathput{-\textstyle\frac{B}{2A} X}} [B] at .75 3.0
\put {\mathput{\vec r}} [l] at 2.15 2
%c)
\setplotsymbol ({\small.}) \plotsymbolspacing=1pt
\setsolid
\setcoordinatesystem units <\Unit,\Unit> point at -7 100
%\plot -1 0.5 3.2 -1.6 / % x axis
%\plot 4 0 3.2 -1.6 / % proj
\plot -0.5 -1 1.75 3.5 / % y axis
\arrow <.3\Unit> [.15,.4] from 0 0 to 5.33 2.67 % =X
\arrow <.3\Unit> [.15,.4] from 0 0 to 4 0 % X
\arrow <.3\Unit> [.15,.4] from -1 -0.5 to 6.5 3.25 % x-axis
\arrow <.3\Unit> [.15,.4] from 4 0 to 5.33 2.67 % proj
\arrow <.3\Unit> [.15,.4] from 0 0 to 1.25 2.5 % Y
\arrow <.3\Unit> [.15,.4] from -0.5 -1 to 1.75 3.5 % y-axis
\arrow <.3\Unit> [.15,.4] from 0 0 to 2 2 % r
\setdashes
\plot 1.25 2.5 6.58 5.17 / % x axis
\plot 4 0 6.58 5.17 / % y axis
\setplotsymbol ({\huge.}) \plotsymbolspacing=5pt
\setdots
\plot .67 1.33 2 2 / % x axis
\plot 1.33 0.67 2 2 / % y axis
\put {\mathput{\overeq x}} [t] at 6.25 2.75
\put {\mathput{\overeq t}} [Br] at 1.27 3.2
\put {\mathput{X}} [t] at 2.2 -0.3
\put {\mathput{\overeq X}} [t] at 3.5 1.4
\put {\mathput{\overeq T = T}} [rB] at 0.2 1.5
\put {\mathput{-\textstyle\frac{B}{2C} T}} [l] at 5.1 1.25
\put {\mathput{\vec r}} [l] at 2.15 2
\endpicture}
$$
%\vglue \Unit
\caption{General linear coordinates on $M^2$ and their two orthogonalizations. The dashed lines indicate the unit coordinate rectangle and the dotted lines the projections of a typical vector along the coordinate axes. }
\end{figure}
This whole discussion may be transferred to a tangent space where the starting basis vectors are two frame vectors and the coordinate functions correspond exactly to the dual 1-forms. If one pictures a 1-form $\sigma$ geometrically in the tangent space by associating with it the two subspaces $\sigma(\vec r)=0,\sigma(\vec r)=1$, then the parallel sides of the unit coordinate rectangle serve this purpose for the two dual 1-forms. Note that the dual 1-form parallel sides for one basis vector are parallel to the other basis vector so that the natural evaluation gives 0.
If we assume that the original basis of the tangent space is a coordinate frame $X=\partial_x, T=\partial_t$, then the dual 1-forms are $dx$ and $dt$. The figure shows that the first orthogonalization does not change $dt$, while the second one does not change $dx$.
The quadratic form is then the line element
\beq
ds^2 = g_{xx}\, dx^2 + 2 g_{tx}\, dt\, dx + g_{tt}\, dt^2\ ,
\eeq
and the space orthogonalization is
\begin{eqnarray}
ds^2 &=& g_{xx}\, (dx + \frac{g_{tx}}{g_{xx}}\, dt)^2
+ (g_{tt}-\frac{g_{tx}^2}{g_{xx}})\, dt^2
\nonumber\\
&=& g_{xx}\, (dx + N^x\, dt)^2
-N^2\, dt^2\ ,
\end{eqnarray}
while the time orthogonalization is
\begin{eqnarray}
ds^2 &=& (g_{xx} -\frac{g_{tx}^2}{g_{tt}})\, dx^2
+ g_{tt}\,(dt +\frac{g_{tx}^2}{g_{tt}}\,dx)^2
\nonumber\\
&=& \gamma_{xx}\, dx^2
-M^2\,(dt +M_x\,dx)^2\ .
\end{eqnarray}
These two distinct completions of the square lead immediately to a suggestive lapse function and shift vector notation for each case which describes the geometry of the associated projections in the context of the original coordinate frame
\beq
\meqalign{
& \overline X =\partial_x\ ,
& \overline X \cdot \overline X = g_{xx}\ ,\cr
& \overline T = \partial_t + N^x \partial_x\ ,\qquad
& \overline T \cdot \overline T = -N^2\ ,\cr
& & & & \cr
& \overeq X =\partial_x + M_x \partial_t\ ,
& \overeq X \cdot \overeq X = \gamma_{xx}\ ,\cr
& \overeq T = \partial_t\ ,\qquad
& \overeq T \cdot \overeq T = -M^2\ ,\cr
}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Notation and conventions}
The conventions of Misner, Thorne and
Wheeler \Cite{1973} will be followed unless stated otherwise or unless
an ambiguity arises. Lower case Greek indices will take the values 0,1,2,3
and lower case Latin indices the values 1,2,3. The signature of the
spacetime metric will be ({\tt -+++}).
A convenient mix of index and index-free notation
will be used in our discussion in order to bridge the gap between
those who feel comfortable without indices and those who do not.
This leads to a problem when in the index notation
the kernel symbol of an object is
used to denote its trace, determinant, norm or other scalar property
or if the same kernel symbol is used for two different objects.
In these cases obvious
distinguishing marks must be added to the index-free symbols which in
our applications will turn out not to be too cumbersome.
For example, ``index shifting" with the spacetime
metric associates a single kernel symbol with many
different objects.
The ``index lowering" and ``index raising" maps associated with the
spacetime metric will be denoted by $\flat$ and $\sharp$ respectively,
and the symbols $S^\flat$ and $S^\sharp$ will refer to the
fully covariant and fully contravariant forms respectively of a
given tensor field $S$. This gives a unique symbol for one-index objects
if we identify the symbol alone with the index position which
best characterizes the properties of the object
or for two-index objects
if we agree to use the kernel symbol alone for the mixed object.
The spacetime metric tensor itself $\four g_{\alpha\beta}$
will be denoted by
$\four \gm$ in an index-free notation (and the inverse or contravariant
metric tensor $\four g^{\alpha\beta}$ by $\four\gm^{-1} = \four \gm^\sharp$)
so that the symbol
$\four g \equiv |\det(\four g_{\alpha\beta})|$ can denote
(the absolute value of) its determinant as is customary in the index notation.
The prefix ``$\,\four\,\,$"
will distinguish certain four-dimensional objects from
related three-dimensional objects for which it is convenient to use the same kernel
symbol.
Similarly the lapse function $N$ and the shift vector field $N^\alpha$ of the
slicing formalism lead to the need for using $\vec N$ to indicate the shift
in an index-free notation.
The same problem will exist with the lapse function $M$ and
shift 1-form $M_\alpha$ to be introduced below for the threading point of view,
so the 1-form will be
distinguished by the index-free symbol $\Overeq M$. The two parallel lines
over the kernel symbol recall the geometrical interpretation of a covector
in terms of a family of parallel planes in its related vector space
in the same way that the arrow
oversymbol recalls the geometrical interpretation of a vector.
These symbols interact with those for index-shifting in an obvious way,
with
$\Overeq N\equiv \vec N{}^\flat$ and $\vec M\equiv \Overeq M{}^\sharp$.
One also needs an index-free notation for contraction of tensors.
A generalized vector contraction notation will prove useful. Let the
left contraction $S\leftcontract T$ denote the tensor product of the
two tensors $S$ and $T$ with a contraction between the rightmost contravariant
index of $S$ with the leftmost covariant index of $T$
(i.e., $ S^{\ldots\alpha}_{\ \ldots} T^{\ldots}_{\ \alpha\ldots} $),
and the right
contraction $S\rightcontract T$ the tensor product with a contraction
between the leftmost contravariant index of $T$ with the rightmost
covariant index of $S$
(i.e., $ S^{\ldots}_{\ \ldots\alpha} T^{\alpha\ldots}_{\ \ldots} $),
assuming in each case that such indices exist.
Each of these contractions may themselves be generalized to an ordered
contraction of sets of $p$ adjacent indices, indicated by
$\leftcontractp{p}$ and $\rightcontractp{p}$ respectively, reducing to the
previous contractions when $p=1$.
Finally the spacetime or portion of spacetime under
discussion will always be assumed to be orientable and time-orientable
and local coordinates and frames will be assumed to be compatible with
these orientations when appropriate.
The unit oriented volume 4-form $\four \eta$ is defined by the
component expression in an oriented frame
$\four\eta_{\alpha\beta\delta\gamma} =g^{1/2}
\epsilon_{\alpha\beta\delta\gamma} $ with $\epsilon_{0123}=1$,
so that in an (oriented) orthonormal frame one has
$\eta_{0123} =1= -\eta^{0123}$.
The dual of a $p$-form will then be represented in index form as
a left contraction with $\four\eta$ as in Misner, Thorne and Wheeler.
For a 2-form $F$ the order does not matter and the dual is
\beq
\dual F_{\alpha\beta} = \half F_{\gamma\delta}
\four\eta^{\gamma\delta}{}_{\alpha\beta} \ .
\eeq
One must be careful when comparing formulas involving $\four\eta$ with the
convention which uses the ordered indices 1,2,3,4 as in
Hawking and Ellis \Cite{1973}
rather than 0,1,2,3. This problem (among others) affects formulas for the
vorticity of a timelike congruence.
The one exception to the conventions of Misner, Thorne and Wheeler made in
the present text is a reversal of the order of the covariant indices on the
symbol for the components of the connection in a spacetime
frame $\{e_\alpha\}$ following Hawking and Ellis \Cite{1977}
\beq
\four\del\sub{e_\alpha} e_\beta
= \four\Gamma^\gamma{}_{\alpha\beta} e_\gamma
= e_{\beta;\alpha}\ .
\eeq
This convention might be called the ``del" convention in contrast with the
``semicolon" convention in which the order of the covariant indices
instead follows the semicolon notation.
Both the comma and subscripted partial notation will be used to indicate
the frame derivatives of functions
\beq
e_\alpha f = \partial_\alpha f = f_{,\alpha}\ .
\eeq
One notational conflict between the old fashioned component notation
and modern frame notation is the convention which doesn't distinguish
between the derivatives of components and the components of derivatives.
For example, $\Lie\sub{X} Y^\alpha = [X,Y]^\alpha $ conventionally
denotes the components of the Lie derivative of the vector field $Y$
and not the Lie derivative of its components, which are instead
represented by $ X Y^\alpha = Y^\alpha{}_{,\beta} X^\beta$.
On the other hand in the frame notation it is natural to write
for the product rule
\beq
\Lie\sub{X} (Y^\alpha e_\alpha) = (\Lie\sub{X} Y^\alpha) e_\alpha
+ Y^\alpha (\Lie\sub{X} e_\alpha) = [\Lie\sub{X} Y]^\alpha e_\alpha\ ,
\eeq
and here the symbol $ \Lie\sub{X} Y^\alpha = X Y^\alpha$ instead really
does stand for the Lie derivative of the components. The context
should always make clear which meaning is intended, and if it doesn't,
an explicit comment will.
The notation introduced here may be found to be somewhat cumbersome
in a particular application by someone already familiar with that particular
application and accustomed to a more streamlined choice of symbols.
However,
the point is not to develop a notation that is the simplest to use in
a specific application, but one which is capable of describing
unambiguously all possible applications
and their relationships.
One may always later adopt
an abbreviated notation for specific calculations by elimination
of some of the qualifying marks on the kernel symbols.
Given this introduction to the notational philosophy
to be employed in the text, one can find in Appendix A
a list of actual formulas from differential geometry that will
be required. All discussion will be local in character. The
region of four-dimensional spacetime where the discussion is valid
will be designated by $\four M$, and this region will be referred to
simply as spacetime.
A good example to keep in mind is
the exterior of the event horizon in a black hole spacetime, for example,
where the Boyer-Lindquist coordinates are valid. The threading point of
view is valid outside the ergosphere (the time coordinate lines are
are timelike),
while the slicing point of view is valid outside the event horizon
(the time coordinate hypersurfaces are spacelike).