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\Chapter{The slicing and threading points of view}
\Section{Introduction}
In order to solve tensor field equations on spacetime, one has to deal either
with local coordinates or certain equivalence classes of local coordinates
reflecting some kind of geometrical properties assumed to hold for the
spacetime, i.e., reflecting either spacetime symmetries or algebraic
properties of curvature or something of this nature.
Without any such assumptions one must tie the geometry to the differential
structure of the spacetime manifold to have actual partial differential
equations to solve.
The nonlinear reference frame provides a way of anchoring the congruence
or hypersurface point of view to the spacetime manifold and makes available
a complete set of ``potentials" whose various derivatives yield each of the
pieces of the spacetime connection. In a sense it establishes an
``integrable" structure on the spacetime manifold which is used to represent
the (in general) ``nonintegrable" structure of the congruence alone.
The hypersurface point of view is integrable, but the nonlinear reference
frame must be adapted to this integrable structure or a situation similar
to the nontrivial congruence point of view results.
Perhaps the most important function of the nonlinear reference frame is that
it converts the temporal Lie derivative into an ordinary partial
derivative with respect to the time coordinate
in the congruence point of view and the integrable spatial
derivatives in the hypersurface point of view into ordinary partial derivatives
with respect to the spatial coordinates in an adapted coordinate system.
The remaining derivatives in each point of view involve both
spatial and time derivatives, except in the special case of an
orthogonal nonlinear reference frame where no mixing occurs.
The action of the gravitational field on test bodies is described in the
context of a partial splitting of spacetime by the spatial gravitational force
fields and the spatial part of the spatial connection, all of which
together represent the spacetime connection.
In a certain sense the spatial metric $P(u)^\flat$ is a potential for the
spatial part of the spatial connection $\del(u) \circ P(u)$ and
for the expansion tensor, while
$u^\flat$ itself acts as a potential for the vector spatial gravitational
force fields through equation (\ref{eq:obsgpot}).
However, the latter relationship does not involve the
spatial or temporal derivatives of spatial quantities, like the scalar
and vector potentials that result from the splitting of the electromagnetic
4-potential.
One needs a full splitting in order
to introduce a 4-potential for the gravitoelectromagnetic vector fields
in a way analogous to the electromagnetic case.
\Section{Algebra}
\Subsection{The nonlinear reference frame}
Suppose spacetime or a portion of spacetime $\four M$ has a nonlinear
reference frame as described in the introduction. This amounts to an
equivalence class of adapted coordinate systems $\{t,x^a\}$, the time
coordinate $t$ parametrizing the slicing and the spatial coordinates
$\{x^a\}$ parametrizing the threading. One may reparametrize the time
function $t\mapsto f^a(t)$ and change the spatial coordinates in a time
independent way $ x^a\mapsto f^a(x)$ without changing the nonlinear reference
frame to which these adapted coordinates correspond. This is an implicit
representation of the nonlinear reference frame.
On the other hand one may explicitly represent the nonlinear reference frame
by a parametrization map $I: R\times \Sigma \to \four M$ which is a
diffeomorphism from the ``\emf{parameter spacetime}"
$R\times \Sigma$ onto $\four M$,
where $\Sigma$ is a 3-manifold diffeomorphic to the quotient
$\four M/{\Thd}$ of $\four M$ by the threading congruence.
This may be thought of as a 1-parameter family of imbeddings $I_t : \Sigma
\to \four M$ or as a 3-parameter family of imbeddings $I_x: R \to
\four M$ yielding respectively the slices $\Sigma_t = I_t(\Sigma)$
and the parametrized threading curves $I_x(R)$, where
$I_t(x) = I_x(t) = I(t,x) $ and $(t,x)\in R\times \Sigma$.
A particular parametrization map defines a parametrized nonlinear reference
frame, while the original nonlinear reference frame is the equivalence
class of such parametrized nonlinear reference frames under repararametrization
of the time parameter. A given parametrization map defines in a natural
way a time function $t$ on $\four M$, whose value on a given slice equals
the value of the time parameter which describes it, and selects naturally
a choice of parametrization for each curve in the threading congruence
(by restriction of the time function to the curve).
One may introduce two ``\emf{computational 3-spaces}": 1) the slicing
computational 3-space $\Sigma$ associated with an explicit parametrization map
and 2) the threading computational 3-space $\four M/\Thd$
which is the quotient of $\four M$ by the threading congruence.
Let $\pi: \four M \to \four M/\Thd$ be the projection map
associated with this quotient and let $\pi_t: \Sigma_t \to
\four M/\Thd$ be its restriction to the slice $\Sigma_t$.
The composed maps $\pi \circ I_t = \pi_t \circ I_t :\Sigma
\to \four M/\Thd$ provide a natural identification between
the two computational 3-spaces (which is of course independent of $t$),
each of which can be identified in turn with the hypersurface $\Sigma_t$ for
each value of $t$. Any one of these may be thought of as a representation
of an abstract computational 3-space. This space will be understood when the
qualifier slicing or threading is omitted.
Figure \ref{fig:comp}\message{Figure 3.1 in text.}
illustrates the relationship of the various spaces and maps.
\fig{Computational 3-spaces}{The relationship between the slicing and
threading computational 3-spaces and the nonlinear reference frame.}{comp}
The nonlinear reference frame induces a direct sum decomposition of the
tangent and cotangent bundles (and all higher tensor bundles) called
the \emf{reference decomposition}. Each tangent space is decomposed into
the 1-dimensional \emf{threading subspace} tangent to the threading congruence
and the 3-dimensional \emf{slicing subspace } tangent to the slicing.
The differential $dt=\omega^0$ of a time function for the slicing defines
the ``\emf{slicing 1-form}" $\omega^0$ whose kernel determines the distribution
of slicing subspaces. The parametrization of the threading curves induced by
this time function defines a tangent vector field $e_0$ called the
``\emf{threading vector field}", whose 1-parameter group of diffeomorphisms
has the threading congruence as its family of orbits. Given an
explicit parametrization map $I$, this group corresponds to translation
in the parameter $t$ in the parameter spacetime $R \times\Sigma$.
This group provides the realization of the concept of ``evolution"
with respect to the nonlinear reference frame.
The dual role of $t$ as a time function and as a parameter along the
integral curves of $e_0$ is represented by the compatibility condition
$\omega^0(e_0)=1$.
Figure \ref{fig:comptancot}\message{Figure 3.2}
illustrates the reference decomposition of the tangent and cotangent
spaces.
\fig{Reference decomposition of tangent and cotangent spaces}%
{Reference decomposition of tangent and cotangent spaces.}{comptancot}
\Subsection{Measurement and the lapse function}
The causality property of the nonlinear reference frame introduces the
equally important concept of measurement. In each point of view,
three-dimensional quantities must be interpreted in terms of the orthogonal
decomposition of the tangent space induced by the 4-velocity of a family
of \emf{test observers} and their associated local rest spaces,
which have yet to be identified.
When both
points of view hold, the case of a spacelike slicing and a timelike threading,
then a well-defined transformation between the two families of
observers will exist.
In the slicing point of view, the slicing 1-form is timelike and can
be normalized
\be
N^{-2} =-\four \gm^{-1}(\omega^0,\omega^0)\ ,\qquad
\omega^{\bot} = N \omega^0\ ,\qquad
\four \gm^{-1}(\omega^{\bot},\omega^{\bot}) =-1\ ,
\ee
reversed in sign, and its index raised to obtain the future-pointing
unit normal $n\equiv e_{\bot}= -(\omega^\bot)^\sharp$
to the slicing. This may be interpreted as
the 4-velocity field of a family of test observers whose worldlines coincide
with the normal congruence. The unit 1-form $\omega^\bot =-n^\flat$ has
as its kernel
the integrable distribution of local rest spaces $LRS_n$ of these observers.
This 1-form also measures the differential of proper time along their
individual worldlines. All of the notation from the hypersurface point of
view for the normal vector field $n$ may be taken over intact.
Recall that the ``perp" index ``$\bot$" indicates
perpendicularity to the slicing in this context.
In the threading point of view, the threading vector field is timelike and
can be normalized
\be
M^{2} =-\four \gm(e_0,e_0)\ ,\qquad
e_\top= M^{-1} e_0\ ,\qquad
\four \gm(e_\top,e_\top) =-1\ ,
\ee
to obtain a unit tangent vector field $e_\top\equiv m$
which may be interpreted as the 4-velocity of a family of test observers.
The sign-reversed index-lowered
1-form $\omega^{\top}=-m^\flat = -(e_\top)^\flat$ has as its kernel the
(in general) nonintegrable family of local rest spaces of this family of
observers, to be denoted by $LRS_m$.
This 1-form measures the differential of proper time along their worldlines.
The identification $u=m$ enables all of the notation from the congruence
point of view to be taken over intact. Recall that the
``tan" index symbol ``$\top$" indicates tangency to the
threading.
The normalization factors $N$ and $M$,
expressable as
\be
N = \omega^\bot (e_0) \ , \qquad M^{-1} = \omega^0(e_\top)\ ,
\ee
or equivalently as
\be
N^{-1} = \omega^0 (e_\bot) \ , \qquad M = \omega^\top(e_0)\ ,
\ee
will both be referred to as the
\emf{lapse function}, with the clarifier
``slicing" or ``threading" preceding this term when necessary to distinguish
the two points of view.
The slicing point of view notation
for the lapse and shift is that of Arnowit, Deser and
Misner \Cite{1962}, while the terminology is due to Wheeler \Cite{1964}.
The lapse function in each case describes the rate of change
with respect to the coordinate time
of the observer proper time measured along the observer congruence.
To allow a simultaneous discussion of both points of view, let $o$
stand respectively for $n$ and $m$.
Correspondingly let $L(n) = N$ and $L(m) = M$ so that $L(o)$ denotes
the lapse function in each point of view. Then the proper time along
each curve in the
the observer congruence is related to the parametrization by the
ccordinate time function by
\be
d\tau_o / dt = L(o) \ .
\ee
Given the family of test observers in each point of view, one can decompose
all tensor fields on the spacetime using the orthogonal decomposition of
the tangent space induced by their unit 4-velocity $o$.
This measurement process has already been discussed at length for
the congruence point of view. By making the identification $u=o$,
one can carry over that discussion to the slicing and threading points of view.
Of course
in the slicing point of view, it is technically the hypersurface point of
view that describes the measurement process, and the
notation will reflect this distinction.
Thus the observer decomposition (respectively slicing decomposition and
threading decomposition) of the tangent space
is accomplished by two projections
\be
\eqalign{
X=& T(o)X + P(o)X\ ,\cr
& [T(o)X]^\alpha = T(o)^\alpha{}_\beta X^\beta =
(-o^\alpha o_\beta) X^\beta\ ,\cr
& [P(o)X]^\alpha = P(o)^\alpha{}_\beta X^\beta =
(\delta^\alpha{}_\beta +o^\alpha o_\beta) X^\beta\ ,\cr
}\ee
where $X$ is an arbitrary vector field.
The temporal projection $T(o)$
projects along the local time direction of the observer
with 4-velocity $o$, while the spatial projection
$P(o)$ projects into the local rest space
$LRS_o$ associated with that observer.
A single vector field $X$ can either
be represented as a sum of the vector fields $T(o)X$ and $P(o)X$, or as
a pair consisting of the scalar $X^{(o)} =-o_\beta X^\beta$ and the
vector $P(o)X$. This scalar will be denoted respectively by
$X^\bot$ and $X^\top$ in the slicing and threading points of view,
suppressing the argument $o$ for a slight simplification of the notation.
Thus the family of spatial fields which represents the
vector field is $\{X^\bot, P(n)X\}$ in the slicing point of view and
$\{X^\top, P(m)X\}$ in the threading point of view.
The notation and discussion is identical for the measurement process
in the hypersurface and congruence points of view respectively.
It is most easily described in terms of an
observer-adapted frame, but
with a nonlinear reference frame it is more natural to consider first
frames which are adapted to the nonlinear reference frame. These are
partially-observer-adapted frames, which may then be
projected to obtain observer-adapted frames.
\Subsection{The Shift}
Splitting the metric tensor in each point of view still only yields the
spatial metric $P(o)^\flat$ as the only nontrivial spatial field in
the family of spatial fields into which it decomposes. The missing pieces
come from the decomposition of the ``\emf{spatial-gauge field}" of the
parametrized nonlinear reference frame: $e_0$ in the slicing point of view
and $\omega^0$ in the threading point of view. These fields contain
information about the
tilting of the threading subspace and slicing subspace
respectively away from the
local time direction and the local rest space of the family of test observers
respectively in the two points of view, which are determined instead by
the timelike ``\emf{causality fields}" $\omega^0$ and $e_0$ respectively.
Although it is common to see the spatial gauge field assumed to timelike
as well, this is not necessary.
In this way
one recovers the lapse function already discussed as the normalizing factor
for the slicing 1-form and threading vector field respectively
from the scalar part and one obtains the
slicing \emf{shift vector field} or threading \emf{shift 1-form} respectively from
the rank one part
\be
\meqalign{
& e_0= T(n)e_0 + P(n)e_0 = N n + \vec N = N e_\bot + \vec N
& \quad \leftrightarrow \quad \{N, \vec N\}\ ,\cr
& \omega^0= T(m)\omega^0 + P(m)\omega^0 = M^{-1} (-m^\flat) + \Overeq M
= M^{-1} \omega^\top + \Overeq M
&\quad \leftrightarrow \quad \{M^{-1}, \covec M\}\ .\cr
}\ee
Again the modifier ``slicing" or ``threading" will precede the term ``shift"
when necessary to distinguish the two points of view.
In slicing point of view the shift is most naturally considered as
a vector field $\vec N$,
and it determines the tilting of the threading curves away
from the normal direction $n$.
In the threading point of view the shift is most naturally considered as
a 1-form $\Overeq M$,
and it determines the tilting of the threading local rest spaces
$LRS_m$ away from the directions tangent to the slicing.
Let $\Overeq N= \vec N {}^\flat$ and $\vec M=\Overeq M{}^\sharp$
denote the slicing shift 1-form and the threading
shift vector field respectively.
The vector oversymbol and double overbar notation is necessary
when an index-free notation is used in order
to conform with the identical kernel symbols
for the lapse and shift established by the slicing point of view
conventions.
The choice of the word ``shift" in the slicing point of view refers to
a ``shifting of the points of space" between the normal congruence
and the threading congruence. In that point of view the threading
serves only as an identification map between the different slices
in the slicing, and differing threadings lead to different identifications
which are described by a diffeomorphism on each slice.
For example, if one has two different parametrization maps
$I$ and $\bar I$ which correspond to
the same parametrized slicing of $\four M$ but different threading curves,
then as illustrated in
Figure \ref{fig:compshift},\message{Figure 3.3}
the map $I_t \circ \bar I{}^{-1}_t :
\Sigma_t \to \Sigma_t$ shifts the points of the slice $\Sigma_t$
from the old points to the new points, while the map
$I_t^{-1} \circ \bar I_t : \Sigma \to \Sigma$ does the same on the
computational 3-space $\Sigma$. If $I$ describes the orthogonal threading
and $\bar I$ any other threading with shift vector field $\vec N$,
then the time-dependent diffeomorphism on $\Sigma$ representing the
relative shifting of the points of space is just the
1-parameter group of diffeomorphisms of the time-dependent
vector field $I^{-1}_t{}_\ast \vec N$ there.
\fig{Shifting of points of the computational 3-space.}{Two different
parametrization maps lead to a relative shifting of the points of
the computational 3-space.}{compshift}
In the threading point of view the ``shift" connotation is less appropriate,
although the alternative term ``tilt" applies equally well to both
points of view.
There is no natural slicing with respect to which the slices are shifted
for a given threading congruence (except in the nonrotating case), and
different slicings correspond to a relative shifting of the time parameters
along each threading curve (rather than a shifting of the ``spatial
parameters" along each slice as occurs in the slicing point of view).
If one interprets the threading and slicing points of view as the mathematical
realizations of the two complementary notions of time, then it is natural
to associate the remaining freedom in the nonlinear reference frame
for a given choice of ``time" (slicing in the slicing point of view
and threading in the threading point of view) with the ``\emf{spatial
gauge freedom}" in the nonlinear reference frame. It is in this
sense that the fields $e_0$ and $\omega^0$ respectively in the slicing
and threading points of view have been referred to as spatial gauge
fields. For a given parametrized slicing or threading respectively,
it is the shift vector field and 1-form respectively whose variation
corresponds to this spatial gauge freedom.
\Subsection{Computational frames and the reference decomposition}
Of course indices can be very useful and a particular choice of frame
used to introduce them is wise if one is interested in viewing the
spacetime from the point of view of the nonlinear reference frame,
as indeed we are. Given an explicit parametrization of the nonlinear
reference frame, it is natural to complete the threading vector field
$e_0$ to a frame adapted
to both the threading and slicing and which may be thought of as
a linear extension of the given parametrized nonlinear reference frame.
It suffices to choose a frame $\{e_a\}$ for the 3-dimensional subspace
of the tangent space which is tangent to the slicing
and which is Lie dragged along $e_0$
\be
[e_a,e_b]= C^c{}_{ab}e_c\ ,\qquad
[e_0,e_a]= 0\ .
\ee
An unambiguous term for such a frame already exists, named a
``\emf{computational frame}" by York \Cite{1979}, generalizing the
coordinate frame of adapted coordinates $\{t,x^a\}$ which
occurs as the special case $C^c{}_{ab}=0$.
Adapted coordinate frames often prove useful for local computations.
Note that only the ``spatial" structure functions of the computational
frame are nonzero: $ C^\gamma{}_{\alpha\beta}= \delta^\gamma{}_c
\delta^a{}_\alpha \delta^b{}_\beta C^c{}_{ab}$.
In terms of the congruence point of view terminology, the computational
frame is a partially-observer-adapted frame which is comoving with
the threading vector field $e_0$.
The dual frame $\{\omega^0,\omega^a\}$ consists of the slicing 1-form
(whose kernel is the slicing tangent subspace)
and three 1-forms which annihilate the threading vector field.
Expressing a tensor field in terms of components with respect to
a computational frame
leads to a ``\emf{reference decomposition}" of the field according to the
reference temporal index 0 and the reference spatial indices 1,2,3,
corresponding to the (in general)
nonorthogonal decomposition of the tangent space
into the direct sum of the slicing subspace and the threading subspace.
For example, the reference
decomposition of a vector field and a 1-form defines the
``\emf{reference temporal projection}" $\Tscr$ and the
``\emf{reference spatial projection}" $\Pscr$ for one-index tensors
\be
\eqalign{
&X=X^0 e_0 + X^a e_a = [\Tscr X]+ [\Pscr X]\ ,\cr
&X^\flat =X_0 \omega^0 + X_a \omega^a = [\Tscr X^\flat]+ [\Pscr X^\flat]\ .\cr
}\ee
One can even speak of a collection of ``\emf{reference spatial fields}"
into which a tensor decomposes as in the orthogonal decomposition associated
with the observers. For a given type of tensor field, there is in fact
an isomorphism between the orthogonal decomposition and
certain pieces of the reference decomposition of
all the tensors related by index raising and lowering.
To see this it is enough to project the computational frame as necessary
to obtain a frame adapted to the observer orthogonal decomposition, which
then establishes the appropriate isomorphism with the reference decomposition.
This observer-adapted frame will be called the \emf{projected computational
frame}.
In the slicing point of view, the reference spatial vector fields $e_a$
are already adapted to the observer local rest space $LRS_n$ and it
suffices to project $e_0$ along the normal direction, while the slicing
1-form $\omega^0$ is proportional to the covariant unit normal $n^\flat$
and one need only project the reference
spatial 1-forms $\omega^a$ orthogonally to the normal direction
\be
\eqalign{
\hbox{slicing definitions:\qquad}
& \epsilon_0 =T(n)e_0\ ,\qquad \theta^a=P(n)\omega^a\ , \cr
\hbox{projected computational frame:\qquad}
& \{\epsilon_0,e_a\}\ ,\cr
\hbox{dual frame:\qquad} & \{\omega^0,\theta^a\}\ .\cr
}\ee
In the threading point of view, the threading vector field
$e_0$ is already tangent to the threading subspace and it
suffices to project the reference spatial vectors $e_a$
into the local rest space $LRS_m$, while the reference spatial 1-forms
$\omega^a$ are already to orthogonal to the local time direction and
one need only project the slicing 1-form along it
\be
\eqalign{
\hbox{threading definitions:\qquad}
& \epsilon_a =P(m)e_a\ ,\qquad \theta^0=T(m)\omega^0\ , \cr
\hbox{projected computational frame:\qquad}
& \{e_0,\epsilon_a\}\ ,\cr
\hbox{dual frame:\qquad}
& \{\theta^0,\omega^a\}\ .\cr
}\ee
In each case the isomorphisms from the reference projection eigenspaces
to those of the observer orthogonal projection spaces are just the
corresponding orthogonal projections themselves.
The inverse isomorphisms are just
the reference projections $\Tscr$ and $\Pscr$, whose nontrivial
values on the projected computational frame and dual frame are
\be \eqalign{
\hbox{slicing:}\qquad & \Tscr \epsilon_0 = e_0\ ,\qquad
\Pscr \theta^a = \omega^a\ ,\cr
\hbox{threading:}\qquad & \Tscr \theta^0 = \omega^0\ ,\qquad
\Pscr \epsilon_a = e_a\ .\cr
}\ee
Thus
one can continue to work with the computational frame using these
isomorphisms rather than explicitly introducing new components with
respect to the projected computational frame, although for certain
purposes like differentiation the latter is essential.
Figure \ref{fig:compframe}
illustrates the relationships between the computational
frame vectors and 1-forms and those of the projected computational
frame.\message{Figure 3.4}
\fig{Projected computational frame}{The relationship between the
computational frame and the projected computational frame.}{compframe}
By duality the lapse function in each point of view also normalizes
the temporal projection of the spatial gauge field
\be \meqalign{
N^2 &= - \four \gm (\epsilon_0,\epsilon_0)\ ,\qquad &
e_\bot=n= N^{-1} \epsilon_0\ , \cr
M^{-2} &= - \four \gm^{-1} (\theta^0,\theta^0)\ ,\qquad &
\theta^\top = -m^\flat = M \theta^0\ .\cr
}\ee
\Subsection{Decomposing the metric}
The orthogonal projections of the computational frame vectors and 1-forms
define the shift components in each point of view; alternatively the
matrix of the linear transformation between the two frames determines the
shift components
\be
\eqalign{
\hbox{slicing:}\qquad
\epsilon_0=
& T(n)e_0 = e_0 - N^a e_a\ , \qquad
\theta^a = P(n)\omega^a =\omega^a +N^a \omega^0\ ,\cr
& P(n)e_0= N^a e_a =\vec N\ ,\cr
\hbox{threading:}\qquad
\epsilon_a=
& P(m)e_0 = e_a + M_a e_0\ , \qquad
\theta^0 = T(m)\omega^0 =\omega^0 -M_a \omega^a\ ,\cr
& P(m)\omega^0= M_a \omega^a =\covec M\ .\cr
}\ee
The two choices of shift correspond to the two possible ways
of completing the square in the metric to reduce it to block-diagonal form,
which is the form it takes in the projected computational frame.
The lapse then determines the metric on the 1-dimensional space along the
time direction and the spatial metric on the 3-dimensional spatial subspace.
Indices on components
taken with respect to the projected computational
frame can then be shifted with the appropriate
subblock of the metric.
Given these definitions,
the spacetime metric and inverse metric in the computational and
projected computational frame in the slicing point of view are
\be \eqalign{
\four \gm&= \four g_{\alpha\beta}\omega^\alpha\otimes\omega^\beta
= -N^2\omega^0\otimes\omega^0
+g_{ab}(\omega^a +N^a\omega^0)\otimes(\omega^b +N^b\omega^0) \cr
& \equiv -N^2 \omega^0\otimes\omega^0 +g_{ab}\theta^a\otimes\theta^b\ ,\cr
\four \gm{}^{-1}&= \four g^{\alpha\beta}e_\alpha\otimes e_\beta
= -N^{-2}(e_0-N^ae_a)\otimes (e_0-N^be_b)
+g^{ab}e_a \otimes e_b\cr
& \equiv -N^{-2} \epsilon_0\otimes\epsilon_0 +g^{ab} e_a\otimes e_b
\ ,\cr}
\ee
i.e., in components
\be \imeqalign{
\four g_{00} &=-(N^2-N_cN^c)\ , \qquad & \four g_{0a} &=N_a\ , \qquad &
\four g_{ab} &= g_{ab}\ ,\cr
\four g^{00} &=-N^{-2}\ ,\qquad & \four g^{0a}=N^{-2}N^a\ , \qquad &
\four g^{ab} &=g^{ab}-N^{-2}N^aN^b\ ,\cr}
\ee
where $(g^{ab})$ is the matrix inverse of the positive-definite
matrix $(g_{ab})$ and the indices
on the shift vector field $\vec N=N^ae_a$ are lowered and raised
using $g_{ab}$ and $g^{ab}$.
The single independent component of the volume 4-form, i.e., the
(absolute value of the) square root of the metric determinant, is
$\four g^{1/2} = N g^{1/2}$.
%Misner, Thorne and Wheeler\Cite{10} avoid this problem by referring instead
%to the line element $ds^2$.
In the threading point of view they are instead given by
\be \eqalign{
\four \gm&= -M^2(\omega^0 -M_a\omega^a)\otimes(\omega^0 -M_b\omega^b)
+\gamma_{ab}\omega^a \otimes \omega^b \cr
& \equiv -M^2 \theta^0\otimes \theta^0 + \gamma_{ab}\omega^a\otimes\omega^b
\ ,\cr
\four \gm{}^{-1}&= -M^{-2}e_0 \otimes e_0
+\gamma^{ab}(e_a+M_ae_0) \otimes (e_b+M_be_0)\cr
&\equiv -M^{-2}e_0\otimes e_0 +\gamma^{ab}\epsilon_a\otimes\epsilon_b
\ ,\cr}
\ee
i.e., in components
\be \eqalign{
& \four g_{00}=-M^2\ ,\quad \four g_{0a}=M^2 M_a\ , \quad
\four g_{ab}=\gamma_{ab}- M^2 M_aM_b\ ,\cr
& \four g^{00}=-(M^{-2}-M_cM^c)\ ,\quad \four g^{0a}=M^a\ , \quad
\four g^{ab}=\gamma^{ab}\ .\cr}
\ee
Here
the spatial metric matrix $(\gamma_{ab})$ is positive-definite, with
inverse $(\gamma^{ab})$. These are used to lower and raise indices on
the shift 1-form.
Letting $\gamma=\det(\gamma_{ab})>0$, one has
$\four g^{1/2} =M\gamma^{1/2}$.
For the slicing and threading
parametrizations of the spacetime metric, it is
precisely the two explicit terms in the final representation of $\four \gm$
and $\four \gm^{-1}$ above
which correspond to the covariant and contravariant form of the
orthogonal projections along the local time and space directions.
The spatial metric in each case is just the covariant form of the
projection. In the slicing point of view, its restriction to a slice
yields the induced metric on the slice submanifold making it into
a Riemannian manifold,
while in the threading
point of view it yields the projected metric on the slice submanifold,
making the slice into a different Riemannian manifold representing the
projected geometry rather than the induced geometry, which is not
necessarily Riemannian in the threading point of view without an
additional causality assumption.
In each point of view, however,
this spatial metric describes the relative distances of the
worldlines of nearby observers at a given time $t$.
Spatial tensor fields in each approach have only spatially indexed components
nonzero when expressed in the projected computational frame.
These spatial indices may be raised and lowered with the respective spatial
metric matrix to yield the same result as index shifting with the spacetime
metric on the computational frame indices.
This equivalence follows from the relations
\be \eqalign{
& e_a{}^\flat =g_{ab}\theta^b\ ,\qquad
\theta^{a\,\sharp} = g^{ab} e_b\ ,\cr
& \epsilon_a{}^\flat =\gamma_{ab}\omega^b\ ,\qquad
\omega^{a\,\sharp} = \gamma^{ab} \epsilon_b\ ,\cr}
\ee
which hold for the bases of spatial vector fields and 1-forms in each
point of view. In fact on the respective spatial subspace of the tangent
space, the bases $\{\theta^{a\,\sharp}\}$ and
$\{\omega^{a\,\sharp}\}$ are respectively \emf{reciprocal} to the bases
$\{e_a\}$ and $\{\epsilon_a\}$ in the classical
terminology \Cite{Sokolnikoff 1964}.
The spatial metric matrices are projected computational
frame components of the fully covariant and fully contravariant forms of the
spatial projection tensors, themselves spatial tensors in each point of view
\be \meqalign{\label{eq:sis}
& P(n)^\flat = g_{ab} \theta^a\otimes\theta^b\ ,\qquad
& P(n)^\sharp = g^{ab} e_a\otimes e_b\ ,\cr
& P(m)^\flat = \gamma_{ab} \omega^a\otimes\omega^b\ ,\qquad
& P(m)^\sharp = \gamma^{ab} \epsilon_a\otimes \epsilon_b\ .\cr}
\ee
Similar remarks apply to index raising and lowering of purely
temporal fields using the lapse function
\be \eqalign{
& \epsilon_0{}^\flat =- N^2 \omega^0\ ,\qquad
\omega^{0\,\sharp} = -N^{-2} \epsilon_0\ ,\cr
& e_0{}^\flat =- M^2 \theta^0\ ,\qquad
\theta^{0\,\sharp} = -M^{-2} e_0\ .\cr
}\ee
However, the measurement process reduces all fields to families of
spatial fields so this is not needed.
The projected computational frame in each point of view is an observer-adapted
frame for the corresponding congruence of test observers. As such one
can take over all of the results derived in such a frame. One need only
make the appropriate identifications between the present notation and
the congruence point of view notation. This correspondence is as
follows
\be \label{eq:sctcorr}
\vbox{\tabskip=10pt
\halign{#&&\hfil $#$ \hfil \cr
& n & \leftarrow & u=o & \rightarrow & m \cr
& \{\epsilon_0, e_a\} & \leftarrow & \{e_0,e_a\} & \rightarrow &
\{e_0, \epsilon_a\} \cr
& \{\omega^0, \theta^a\} & \leftarrow & \{\omega^0,\omega^a\}
& \rightarrow & \{\theta^0, \omega^a\} \cr
& g_{ab} & \leftarrow & h_{ab} & \rightarrow & \gamma_{ab} \cr
& N & \leftarrow & \L & \rightarrow & M \ .\cr}}
\ee
\label{sec:metricsplit}
\Subsection{Relationship between the reference and observer decompositions}
\label{sec:refobsrel}
The explicit isomorphisms between the reference decomposition subspaces and the
orthogonal decomposition subspaces can easily be seen from the following
explicit form of both
projections themselves and their effect on a vector field
$X$ and its corresponding 1-form
\be \meqalign{
Id &= \Tscr +\Pscr &= T(n)+P(n) &= T(m)+P(m) \ ,\cr
\delta^\alpha{}_\beta e_\alpha\otimes \omega^\beta
&= {e_0 \otimes \omega^0} + e_a\otimes \omega^a
&= \underbrace{\epsilon_0\otimes \omega^0}
\sub{e_\bot \otimes \omega^\bot}
+e_a\otimes \theta^a
&= \underbrace{e_0\otimes \theta^0}
\sub{e_{\top }\otimes \theta^{\top }}
+\epsilon_a \otimes \omega^a\ ,\cr
X &= X^0 e_0 + X ^a e_a
&= \underbrace{X^0 \epsilon_0}_{\hbox{$\displaystyle X^\bot
e_\bot$}}
+ X_b g^{ba} e_a
&= \underbrace{-M^{-2}X_0e_0}_{\hbox{$\displaystyle X^{\top }
e_{\top }$}}
+ X ^a \epsilon_a\ ,\cr
X^\flat &= X_0 \omega^0 + X _a \omega^a
&= \underbrace{-N^2X^0\omega^0}_{\hbox{$\displaystyle X_\bot
\omega^\bot$}}
+ X_a \theta^a
&= \underbrace{X_0\theta^0}_{\hbox{$\displaystyle X_{\top }
\theta^{\top }$}}
+ X ^b\gamma_{ba}\omega^a\ .\cr}
\ee
where
\be \meqalign{
X^\bot = -X^\alpha n_\alpha = NX^0\ ,\qquad &
X^\top= -X^\alpha m_\alpha =M(X^0-M_aX^a)\ ,\cr
X_\bot = X_\alpha n^\alpha = N^{-1}(X_0-N^aX_a)\ ,\qquad &
X_\top= X_\alpha m^\alpha =M^{-1}X_0\ .\cr}
\ee
Figure \ref{fig:xdecomp}
illustrates the three decompositions of a
vector field $X$ in a suggestive way.\message{Figure 3.5}
\fig{Decompositions of a vector field}{A pictorial representation
of the various decompositions of a vector field.}{xdecomp}
In the slicing point of view, the covariant reference spatial components
parametrize the spatial part of the vector field $X$ or 1-form $X^\flat$,
while in the threading point of view
it is the contravariant reference spatial components which
do this. Similarly the contravariant reference temporal component
parametrizes the temporal part in the slicing point of view, while the
covariant reference temporal component does this in the threading
point of view.
To recover those reference components which are not adapted
to the observer orthogonal decomposition, one needs the
transformation
\be \meqalign{
X^a &= -N^{-1}N^a X^\bot + g^{ab}X_b &= -N^a X^0 + g^{ab} X_b \ ,\cr
X_a &= M M_a X_\top + \gamma_{ab} X^b &= M_a X_0 + \gamma_{ab} X^b \ ,\cr
X^0 &= M^{-1}X^\top + M_a X^a &= -M^{-2} X_0 + M_a X^a\ ,\cr
X_0 &= N X_\bot + N^a X_a &= -N^2 X^0 + N^a X_a\ .\cr
}\ee
This generalizes in an obvious way to higher rank tensors.
The orthogonal projection via computational frame indices is
accomplished in the slicing point of view by choosing all possible
sets of components
with the zero indices up and the spatial indices down, while these
positions are reversed in the threading point of view, as emphasized
by Zel'manov \Cite{1956, 1973}.
The lapse
in each case can be used to rescale the zero-indexed components
to obtain partially-orthonormal components along the given time
direction.
For the sake of an explicit example,
the reference splitting, slicing splitting, and threading splitting
of a $1\choose 1$-tensor field $S$ are parametrized by the
original computational frame components in the following way
\be \eqalign{
S=& S^0{}_0 e_0\otimes\omega^0 + S^0{}_a e_0\otimes\omega^a
+ S^a{}_0 e_a\otimes\omega^0 + S^a{}_b e_a\otimes\omega^b\cr
&\qquad \leftrightarrow
\{S^0{}_0, S^0{}_a\omega^a, S^a{}_0 e_a, S^a{}_b e_a\otimes \omega^b
\}\ ,\cr
S^\flat=& (-N^2)^2 S^{00} \omega^0\otimes\omega^0
+ (-N^2) S^0{}_a \omega^0\otimes\theta^a
+ (-N^2) S_a{}^0 \theta^a\otimes\omega^0 + S_{ab} \theta^a\otimes\theta^b\cr
=& S^{\bot\bot} \omega^\bot\otimes\omega^\bot
+ (-1) S^\bot{}_a \omega^\bot\otimes\theta^a
+ (-1) S_a{}^\bot \theta^a\otimes\omega^\bot
+ S_{ab} \theta^a\otimes\theta^b\cr
&\qquad \leftrightarrow
\{S^{\bot\bot}, -S^\bot{}_a\theta^a, -S_a{}^\bot \theta^a,
S_{ab} \theta^a\otimes \theta^b \}\ ,\cr
S^\sharp=& (-M^{-2})^2 S_{00} e_0\otimes e_0
+ (-M^{-2}) S_0{}^a e_0\otimes\epsilon_a
+ (-M^{-2}) S^a{}_0 \epsilon_a\otimes e_0 +
S^{ab} \epsilon_a\otimes\epsilon_b\cr
=& S_{\top\top} \epsilon_\top\otimes\epsilon_\top
+ (-1) S_\top{}^a e_\top\otimes\epsilon_a
+ (-1) S^a{}_\top \epsilon_a\otimes e_\top
+ S^{ab} \epsilon_a\otimes\epsilon_b\cr
&\qquad \leftrightarrow
\{S_{\top\top}, - S_\top{}^a\epsilon_a, - S^a{}_\top \epsilon_a,
S^{ab} \epsilon_a\otimes \epsilon_b \}\ .\cr
}\ee
Once the tensor is split into a collection of spatial fields in the
slicing and threading points of view, their ``spatial" indices may be
shifted using the spatial metric as follows from
\ref{eq:sis},
while the normalized temporal indices can be shifted by a sign change.
Zel'manov has called the collection of spatial fields which represents
a spacetime tensor as ``kinematically invariant quantities"
(invariant under a change of threading") in the slicing point of view
\Cite{Zel'manov 1973} and as ``chronometrically invariant"
(invariant under a change of slicing") in the threading point of view
\Cite{Zel'manov 1956}.
Note that in the special case of a tensor which
is already a spatial field in a given point of view,
the projected computational frame components are directly
equal to the reference spatial components.
The remaining computational frame components are parametrized by these
reference spatial components in such a way that index shifting the
reference spatial components with the spatial metric matrices exactly
reproduces index shifting of the tensor
with respect to the spacetime metric taking
into account its remaining components.
For a tensor which is not spatial, the projected computational components
differ from the reference spatial components by terms involving the temporal
components and the shift.
\Subsection{The slicing, threading and reference representations}
\label{sec:rr}
The splitting of spacetime into space plus time is done to be able to
work with time-dependent fields on a 3-dimensional space rather than
the original spacetime fields on the spacetime manifold. This may be done
in several ways.
In either point of view one may consider either a
``\emf{slicing representation}"
or a ``\emf{threading representation}" by mapping the family of spatial
fields which represents a given spacetime field onto a family of
time-dependent fields on the slicing or threading computational 3-space
respectively. In the slicing representation, one may \emf{pullback}
the covariant family of spatial fields to $\Sigma$ using $I_t^\ast$.
In the threading representation, one may \emf{pushforward} (down)
the contravariant family of spatial fields to $\four M/{Threading}$
using $\pi_t\,_\ast$. In each case the spatial metric $P(o)^\flat$
maps onto a time-dependent Riemannian metric which may be used to
shift indices back to their original positions.
The slicing representation of the slicing point of view
is used by the extended Fischer-Marsden group
\Cite{Fischer and Marsden 1979, 1980, Gotay et al 1990}.
One may also stay on the spacetime and project the original family
of spatial fields representing a given spacetime tensor field using
the reference spatial projection $\Pscr$ which
reduces $\theta^a$ to $\omega^a$ and $\epsilon_a$ to $e_a$.
This is often what is done when working in adapted coordinates where one
frequently makes many convenient identifications in notation which
obscure the fine distinctions between the two computational 3-spaces
and the slices in spacetime.
For example, in adapted local coordinates, one often sees the reference
spatial field $\Pscr
P(n)^\flat =g_{ab}dx^a\otimes dx^b$ rather than $P(n)^\flat=
g_{ab}(dx^a+N^a dt)\otimes(dx^b +N^b dt)$ referred to as
the slicing spatial metric, and indeed if one interprets $\{x^a\}$
as the restrictions of the spatial coordinates to the slices, then
the induced metric on those slices is the former expression, which
may also be identified with the metric on the slicing computational
3-space if
one instead interprets these coordinates as their pullbacks
$I_t{}^\ast x^a$ to that manifold.
Durrer and Straumann \Cite{1988}
have chosen this option for the slicing point of view, referring
to the reference spatial fields as ``horizontal fields".
Similarly one might choose to use $\Pscr P(m)^\sharp = \gamma^{ab}
\partial/\partial x^a \otimes \partial/\partial x^b$
as a representative for $P(m)^\sharp = \gamma^{ab}
(\partial/\partial x^a+ M_a \partial /\partial t) \otimes
(\partial/\partial x^b + M_b \partial /\partial t) $
in the threading point of view. If one interprets the spatial coordinates
instead as the coordinates on the threading computational 3-space which
they naturally induce, then this is the correct expression for the
inverse metric there.
Note that the restriction to a slice of either the
spacetime metric, or the reference spatial metric, or the spatial metric
yields the same induced metric on the slice
in the slicing point of view, or
the same pullback to the slicing
computational 3-space for a given value of $t$.
Similarly in the threading point of view, the projection onto a slice
of the spacetime inverse metric, the reference inverse metric, or the
inverse spatial metric all yield the inverse spatial metric when
restricted to the slice,
or the same pushforward down to the threading computational 3-space
for a given value of $t$.
Given any spatial tensor field $S$, the reference spatial projection
\be
\Pscr S = S^{a\ldots}_{\ \ b\ldots} e_a \otimes \cdots \otimes
\omega^b \otimes \cdots
\ee
may be used to as a representative of the original spatial field in
the ``\emf{reference representation}" of the given point of view.
This isomorphism gives a representation of the computational 3-space
and its geometry on the spacetime itself.
The reference spatial metric $\Pscr P(o)^\flat$ ($g_{ab}$ or $\gamma_{ab}$)
has the ``inverse" $\Pscr P(o)^\sharp$ ($g^{ab}$ or $\gamma^{ab}$) in the
sense that their contraction yields the reference spatial identity tensor
\be
\Pscr P(o)^\sharp \leftcontract \Pscr P(o)^\flat =
\Pscr P(o) =\Pscr\ .
\ee
Index-shifting of reference spatial fields with the reference spatial
metric then corresponds to the reference projection of index-shifting
of spatial fields with respect to the spacetime metric or equivalently
with respect to the spatial metric.
The reference representation is the only way one can interpret
in terms of fields on spacetime the manipulations one does
when working in classical coordinate notation
and partitioning components according to their
temporal and spatial index structure. The style in which Landau
and Lifshitz discusses the threading point of view typifies this
classical approach. Durrer and Straumann have adopted the
explicit modern reference representation for the slicing point of view.
Using the slicing point of view in the black hole context,
Thorne \Cite{Thorne and MacDonald 1984} originally
referred to the computational 3-space for a stationary nonlinear
reference frame
with its time-independent slicing spatial metric
as ``absolute space",
but later decided to use this term for the observer space, namely the
space of observer worldlines (normal congruence)
rather than threading curves, preferring to think of the
grid of an adapted spatial coordinate system as moving relative to this
space \Cite{Thorne et al 1986}.
%Rather than trying to decide which
%possible 3-manifold to label as ``space",
Gerosh \Cite{1971} while studying solution generation techniques
for stationary spacetime explored the geometry
of the computational 3-space in the threading point of view.
In this case stationary fields may be represented by families of
time-independent fields on the threading computational 3-space,
as discussed below.
\Subsection{Transformation between slicing and threading points of view}
In many applications both $n$ and $m$ are timelike on some region of
spacetime, and so one may consider the relative observer boost between
them as discussed in section \ref{sec:relobsboost}.
%in which case the two local rest spaces orthogonal to them
%are related by a boost $B(m,n)$
%(a pure Lorentz transformation acting in the plane of $n$ and $m$
%actively mapping $n$ onto $m$)
Introduce the following abbreviated
notation to handle this application of those formulas
\be
v^a = \nu(m,n)\ , \qquad
V^a = \Nu(m,n)= - \nu(n,m) \ , \qquad
\gamma_L = \gamma(m,n) = \gamma(n,m) \ .
\ee
Then one has
\be \eqalign{
& m= B(m,n)n=\gamma_L(n +v^ae_a)\ ,\quad \hbox{or} \quad
n=B(n,m)m= \gamma_L(m -V^a\epsilon_a)\ , \cr
& v^a =N^a/N \ ,\quad V^a=\gamma_L^{-1}v^a=M M^a\ , \quad
V_a=\gamma_L v_a\ ,\cr
& v^2 =g_{ab}v^av^b =N^{-2}N^aN_a =M^2M^aM_a= \gamma_{ab}V^aV^b= V^2\ ,\cr
& \gamma_L=N/M= \gamma^{1/2}/g^{1/2} =(1-v^2)^{-1/2}\ .\cr}
\ee
\Message{?? more??}
Then
\be \vec v= v^ae_a= N^{-1}\vec N= v(m,n)
\ee
is the spatial velocity
associated with this boost as seen from the local rest space associated
with the unit normal, while
\be \vec V=V^a\epsilon_a= M \vec M= -v(n,m)
\ee
is the one as seen
from the local rest space orthogonal to the time lines.
%and
%\be \gamma_L =\gamma(m,n)=\gamma(n,m)
%\ee
%is the Lorentz gamma factor associated with the proper time dilation
%and the spatial volume contraction.
If either $m$ or $n$ (but not both) changes causal character by becoming null,
this boost becomes singular, with $v\to1$.
Note also that the inverse relative projection $P(m,n)^{-1} : LRS_m \to LRS_n$
is just equal to the reference spatial projection $\Pscr$, since by
definition the latter map undoes the projection into the threading local
rest space of $m$.\message{more??}
Any remarks in what follows involving relationships
between the slicing and threading perspectives will be understood
to apply only when both $n$ and $m$ exist as timelike unit vectors,
the case of a spacelike slicing and a timelike threading. When this is
the case, the different representations of the observer orthogonal
decomposition of a vector or a tensor as given in Section \ref{sec:refobsrel}
then become a relationship between the two points of view which may be
re-expressed in terms of this Lorentz boost.
Figure \ref{fig:boost}\message{Figure 3.6}
illustrates the 2-plane in which the Lorentz boost takes place, making
use of the unit vectors along the relative velocity vectors
\be
\hat v^a = v^{-1} v^a\ , \qquad
\hat V^a = V^{-1} V^a\ .
\ee
\fig{Lorentz boost 2-plane}{The 2-plane of the Lorentz boost between
the slicing and threading observers.}{boost}
%\Subsection{Transformation of gravitational variables}
Instead of analyzing the relationship between the two points of view
in geometrical terms, one can treat the problem as a transformation of
gravitational variables. Either set of variables $(N,N^a,g_{ab})$ or
$(M,M_a,\gamma_{ab})$ may be used to parametrize the spacetime metric
and express the field equations. The transformation between these two
sets is easily derived by comparing the two parametizations of the
spacetime metric and its inverse. The result is
\be\meqalign{
M &= (N^2-N^c N_c)^{1/2}\ , & & \cr
M_a &= (N^2-N^c N_c)^{-1} N_a &\hfill
M^a &= N^{-2} N^a\ ,\cr
\gamma_{ab} &= g_{ab} + (N^2-N^c N_c)^{-1} N_a N_b\ , \qquad &\hfill
\gamma^{ab} &= g^{ab} - N^{-2} N^a N^b\ , \cr
}\ee
with inverse
\be\meqalign{
N &= (M^{-2}-M^c M_c)^{-1/2}\ , & &\cr
N^a &= (M^{-2}-M^c M_c)^{-1} M^a \qquad &
N_a &= M^{2} M^a\ ,\cr
g_{ab} &= \gamma_{ab} -M^2 M_a M_b\ , \qquad
& g^{ab} &= \gamma^{ab} + (M^{-2}-M^c M_c)^{-1} M^a M^b\ . \cr
}\ee
This transformation can then be pushed up to the connection and
curvature levels. The transformation of the Einstein tensor
is very instructive in considering how the two formulations of
the Einstein equations relate to each other.
%TO FINISH.
\Subsection{So far:}
To summarize, the pair of fields $(e_0,\omega^0)$ with the compatibility
condition $\omega^0(e_0)=1$
represent the parametrized
nonlinear reference frame at the linear level. In each point of view the
causal assumption about the nonlinear reference frame leads one of these
fields to introduce a local notion of time which is represented by a unit
timelike vector field $o$, interpretable as the 4-velocity of a field of
test observers, while the other field represents the gauge freedom remaining
in the nonlinear reference frame holding that family of observers fixed,
a freedom which is natural to call
the spatial gauge freedom. This observer 4-velocity
field $o$ is the unit normal $n=e_\bot$
to the slicing in the slicing point of view,
obtained by normalizing and index-lowering the
sign-reversed slicing 1-form $-\omega^0$,
and it is
the unit tangent $m=e_\top$ to the threading in the threading point of view,
obtained by normalizing the threading vector field $e_0$.
The orthogonal decomposition of the tangent space
into the local time direction of the observer and the local rest space
$LRS_o$ may be used to ``measure" a spacetime field, leading to a collection
of ``spatial" fields of different ranks
obtained by contraction with $-o$ rather than projection
along $o$ in the tensor product of the orthogonal decomposition.
The ``measurement" of the spacetime metric gives only one nontrivial piece,
the ``spatial metric" which is the metric on the local rest space $LRS_o$,
while the decomposition of the spatial gauge field, respectively $e_0$ and
$\omega^0$, leads to the remaining pieces of the spacetime metric. The
spatial projection gives the shift vector field and 1-form respectively,
while the temporal contraction yields the lapse and the negative reciprocal
of the lapse respectively. In each case the lapse function is the normalizing
function (with the appropriate power)
for the parametrized nonlinear reference frame field which determines
the local time direction, and determines the rate of change of the observer
proper time with respect to the time parameter of the parametrized nonlinear
reference frame.
All of the spatial fields may be interpreted as time-dependent
fields on either of the two
computational 3-spaces in the slicing or threading representation.
The reference projections
also allow one to represent these fields on the spacetime in terms of fields
which instead are ``spatial" with respect to the reference decomposition of
the tangent space associated with the nonlinear reference frame, and these
reference fields lead to the same time-dependent fields on the
computational 3-spaces as the
original spatial fields. Finally, in the event that both points of view hold,
one has a unique Lorentz boost which transforms between the two points of view.
The principal difference between the two points of view is that the test
observers are fixed in the computational 3-spaces
in the threading point of view, but move with
respect to those spaces
in the slicing point of view, the position of a given observer
at time $t$ determined by an integral curve of the time-dependent
sign-reversed shift
vector field. This makes the slicing point of view a hybrid type of formalism
based on two distinct congruences, while the threading point of view is
entirely determined by a single congruence.
\Newpage
\Section{Derivatives}
The slicing point of view, with its somewhat ugly but in general
necessary separation of the observer and evolution congruences,
wins as soon as one considers spatial derivatives, which are integrable
in this point of view, i.e., representable in terms of partial derivatives
with respect to spatial coordinates only plus linear transformations.
It is the nonintegrability of the spatial derivatives which sabotages
the somewhat more aesthetically pleasing threading point of view, where
the measurement and evolution are both determined by a single congruence.
Each of the derivative operators which act on spatial tensor fields may
be expressed in terms of corresponding operators in the slicing, threading,
and reference representations of a given point of view. These operators
are defined in the latter point of view by reference spatial projection
of the corresponding spatial operators. The reference representation
is just a way of modeling the computational 3-space on the spacetime
itself.
For the natural differential operators, this leads to a
\emf{reference spatial Lie derivative}
\be
[\Pscr \Lie]\sub{X} S = \Pscr [\Lie\sub{X} S]
\ee
and a \emf{reference spatial exterior derivative}
\be
[\Pscr d] S = \Pscr [d S]\ .
\ee
\Subsection{Evolution}
Once a given tensor field has been decomposed into a collection of spatial
tensor fields in each point of view, corresponding to the notion of
measurement, one can discuss the evolution of these ``measured quantities",
namely how they change along the threading congruence relative to the
structure imposed on spacetime by the nonlinear reference frame.
Given a parametrization of this nonlinear reference frame, the 1-parameter
group of diffeomorphisms of the threading vector field $e_0$
provides a natural way of comparing fields at different ``times" relative
to that reference frame itself, namely by Lie dragging.
``No evolution" corresponds to Lie invariance.
However, if one
evolves spatial fields rather than spacetime fields, one must compose the
Lie dragging with the spatial
projection in order to compare spatial fields since in general,
spatial fields will not remain spatial under Lie dragging. At the differential
level, one therefore needs the spatial Lie derivative
rather than the Lie
derivative along $e_0$
\be
\PLie(o)\sub{e_0}S
\ee
in order to discuss the evolution of a spatial tensor field $S$.
This Lie derivative will be referred to as the\message{useful or not??}
\emf{evolutionary Lie derivative}. It is this derivative which
will correspond to the ordinary time derivative on the
computational 3-space.
In the threading point of view this is just a rescaling of the
observer Lie temporal derivative when acting on spatial fields
since in that case $o = Me_0$
and the observer and evolution congruences coincide.
However, in the slicing point of view the observer congruence
and threading congruence are in general distinct so
the measurement projection is not orthogonal to the evolution congruence,
slightly complicating the situation.
\Subsection{Natural time derivatives}
The time derivative relevant for the evolution is in good shape in both
points of view, representable in terms of an ``ordinary time derivative".
By assumption the computational frame vectors and 1-forms are Lie dragged
along $e_0$ so
\be
\PLie(n)\sub{e_0} e_a = \Lie\sub{e_0} e_a = 0\ , \qquad
\PLie(n)\sub{e_0} \omega^a = \Lie\sub{e_0} \omega^a = 0\ .
\ee
The spatial Lie derivatives along $e_0$ of the remaining projected
computational frame spatial vectors and 1-forms are instead
\be \eqalign{
\Lie\sub{e_0} \theta^a &= (e_0 N^a) \omega^0\ ,\qquad
\Lie(n)\sub{e_0} \theta^a = 0\ ,\cr
\Lie\sub{e_0} \epsilon_a &= (e_0 M_a) e_0\ ,\qquad
\Lie(m)\sub{e_0} \epsilon_a = 0\ ,\cr
}\ee
so the projected computational spatial frame and dual frame are spatially
comoving along $e_0$. This means that the spatial
Lie derivative of a spatial tensor by $e_0$ is obtained simply by
differentiating the projected computational components by $e_0$, which
amounts to a partial derivative with respect to $t$ in adapted coordinates.
In either the slicing or threading representation, this corresponds to the
time-derivative of the time-dependent field on the computational 3-space
to which a given spatial field is mapped, i.e., the derivative of the
1-parameter family of fields with respect to its parameter
\be
S \mapsto S_t\ ,\qquad \PLie(o)\sub{e_0} S \mapsto d/dt\, S_t
\equiv \dot{S}_t\ .
\ee
A dot will be used to indicate the time derivative of a time-dependent
field on the computational 3-space.
The evolution time derivative $\PLie(o)\sub{e_0}$
is simply related to the observer spatially-projected
Lie derivative $\PLie(o)\sub{o}$.
In the threading point of view one has for any spatial field $S$
\be
\PLie(m)\sub{e_0} S = M \PLie(m)\sub{m} S \qquad {\rm or} \qquad
\PLie(m)\sub{m} S = M^{-1} \PLie(m)\sub{e_0} S\ ,
\ee
so one need only rescale the observer Lie temporal time derivative
operator used in the congruence point of view discussion. In the slicing
point of view, one has instead for a spatial field $S$
\be
\PLie(n)\sub{e_0} S = \PLie(n)\sub{Nn} S + \PLie(n)\sub{\vec N} S
=N \PLie(n)\sub{n} S + \PLie(n)\sub{\vec N} S\ ,
\ee
or
\be \eqalign{
\PLie(n)\sub{n} S &= N^{-1}[ \PLie(n)\sub{e_0} S -\PLie(n)\sub{\vec N} S ]
= N^{-1} \PLie(n)\sub{\epsilon_0} S\cr
& \mapsto N_t{}^{-1} [ \dot S_t - \Lie(n)\sub{\vec N_t} S_t ] \ ,\cr
}\ee
so in addition the shift Lie derivative appears, i.e., a spatial derivative
necessarily accompanies the ordinary time derivative in the observer time
derivative if the threading is tilted.
In other words the evolution time derivative is simpler in the threading
point of view since the it coincides with the reference time derivative,
while a correction term is in general necessary in the slicing point
of view.
\Subsection{Natural spatial derivatives}
The situation is reversed for the spatial derivatives in comparison with
the temporal derivatives.
The slicing spatial
derivatives are simpler since the reference spatial projection
is the observer spatial projection in the slicing point of view, but
a correction term is necessary for the spatial derivatives in the
threading point of view. To see this in detail, one must
discuss how the spatial derivatives are represented in
terms of the nonlinear reference frame and the spatial metric,
i.e., in terms of derivative operators which make sense on the computational
3-space with its time-dependent
Riemannian metric and its tensor algebra representation
of the algebra of spatial tensor fields.
For the slicing point of view this is simple and well known.
The spatial exterior derivative is equivalent to the restriction to the
slice of the exterior derivative, while the spatial Lie derivative for
spatial fields is equivalent to the intrinsic
Lie derivative on the slice itself and the spatial connection is
equivalent to the connection of the induced metric on the slice.
For the threading point of view two differences arise.
First, the direction of spatial differentation, whether in the spatial
exterior derivative, the spatial Lie derivative along a spatial vector
field,
or in the spatial covariant derivative, is no
longer along the slicing subspace of the tangent space and the spatial
derivatives decompose into a part along the slicing subspace
and a part along the threading using the reference decomposition.
This decomposition
closely parallels the splitting of the total spatial
covariant derivative in the congruence point of view using the observer
decomposition.
Second, the spatial part of
the spatial connection itself is distinct from the metric connection
associated with the projected metric on the slice, so a difference
tensor arises between the two connections.
The first question is understood by examining the natural spatial
derivatives. Let $\Pd f \equiv \Pscr [df] = \omega^a e_a f$ define the
\emf{reference spatial exterior derivative}
which differentiates a function $f$
along the slicing directions. The same relation
\be
\Pd S = \Pscr [dS]
\ee
defines this operator for any differential form $S$ and one easily sees
that $ \Pd^2 =0$.
For the slicing point of view, the slicing spatial exterior derivative
projects to the reference spatial exterior derivative
\be
\Pscr \circ d(n)= \Pd \ .
\ee
In the threading point of view one has instead for a function $f$
\be
d(m)f = \omega^a \epsilon_a f = \omega^a [e_a + M_a e_0 ] f=
\Pd f + \Overeq M \PLie(m)\sub{e_0} f \ ,
\ee
which is easily generalized to a spatial
differential form $S$ by including the wedge product
\be
\Pscr d(m)S = d(m)S = \{ \Pd + \Overeq M \wedge \PLie(m)\sub{e_0} \} S\ .
\ee
(Note that reference spatial and spatial covariant tensor fields in the
threading point of view coincide.)
In an adapted coordinate frame, the reference spatial derivatives
$ e_a f = \partial f/\partial x^a$ are just ordinary partial derivatives,
but the threading spatial derivatives $\epsilon_a f= (
\partial /\partial x^a + M_a \partial/\partial t) f$ are not,
and are often called ``\emf{transverse partial derivatives}".
On a computational 3-space, the reference spatial
exterior derivative becomes the ordinary exterior derivative
while the threading spatial exterior derivative
acquires a correction term involving the time derivative
\be \eqalign{
S & \mapsto S_t \ ,\cr
\Pd S &\mapsto d S_t\ ,\cr
d(m) S &\mapsto d S_t + \Overeq M_t \wedge\, \dot{S}_t\ .\cr
}\ee
Similar formulas hold for the spatial Lie derivative of spatial
tensor fields with respect to a spatial vector field $X$
For a function $f$ one has
\be\eqalign{
\Lie(n)\sub{X} f &= [\Pscr\Lie]\sub{\Pscr X} f = \Pscr X f \cr
&\qquad \mapsto X_t f_t\ ,\cr
\Lie(m)\sub{X} f &= [\Pscr\Lie]\sub{\Pscr X} f
+\Overeq M (X) \Lie(m)\sub{e_0} f \cr
&\qquad \mapsto X_t f_t + \Overeq M_t (X_t) \dot f_t\ , \cr
}\ee
while for a spatial vector field $Y$ one has
\be\eqalign{
\Lie(n)\sub{X} Y &= [\Pscr\Lie]\sub{\Pscr X} \Pscr Y \cr
&\qquad \mapsto \Lie\sub{X_t} Y_t \ ,\cr
\Lie(m)\sub{X} Y &= [\Pscr\Lie]\sub{\Pscr X} \Pscr Y
+ \Overeq M(X) \Lie(m)\sub{e_0} Y
- \Overeq M(Y) \Lie(m)\sub{e_0} X \cr
&\qquad \mapsto \Lie\sub{X_t} Y_t
+ \Overeq M_t(X_t) \dot Y_t
- \Overeq M_t(Y_t) \dot X_t \ .\cr
}\ee
\Subsection{Gauge transformations of the nonlinear reference frame}
For a given partial splitting of spacetime, one can consider all of the
compatible full splittings, i.e., the equivalence class of nonlinear
reference frames with one of its two components fixed by the partial
splitting.
For a fixed threading, the nonlinear reference frame is only defined
up to a choice of slicing (spatial gauge freedom) and its
parametrization (the remaining temporal gauge freedom). For a fixed slicing,
the nonlinear reference frame is only defined up to a choice of threading
(spatial gauge freedom) and its parametrization (the remaining temporal
gauge freedom). It is therefore of interest to consider what happens
to the gauge variables, namely the lapse and shift, under such gauge
transformations. The spatial metric is gauge invariant of course, as are
all quantities which only depend on the local time direction of the point
of view.
One may use the pair $(e_0,\omega^0)$ to discuss these gauge transformations
on the linear level.
In the threading point of view, the threading
vector field can only be scaled, while the slicing 1-form may be changed
arbitrarily consistent with the transversality condition
\be
e_0 = \Mscr^{-1} e_0^\prime\ , \qquad
\omega^0 = \Mscr \omega^0{}^\prime\ .
\ee
The lapse and shift then behave in the following manner
\be
M^\prime = \Mscr M \ ,\qquad
\overeq M{}^\prime = \Mscr^{-1} ( \overeq M - \overeq \Mscr ) \ .
\ee
The condition $\Mscr=1$ fixes the parametrization in the case that
a preferred parametrization exists.
In the slicing point of view, the slicing 1-form
can only be scaled, while the threading vector field may be changed
arbitrarily consistent with the transversality condition
\be
\omega^0 = \Nscr \omega^0{}^\prime\ , \qquad
e_0 = \Mscr^{-1/2} (e_0^\prime - \Nscr ) \ .
\ee
The lapse and shift then behave in the following manner
\be
N^\prime = \Nscr N \ ,\qquad
\vec N{}^\prime = \Nscr \vec N + \Nscr \ .
\ee
The condition $\Nscr=1$ fixes the parametrization.
Most discussions of these gauge transformations take place in the
context of an adapted coordinate system, in which case the above relations
interpret the corresponding class of coordinate transformations for either
a given threading or slicing.
TO DO.
\message{?? TO DO.} %pig
\Newpage
\Subsection{Observer-adapted frame structure functions
and kinematical quantities}
In order to examine spatial and spacetime covariant derivatives
in relation to the nonlinear reference frame, it is enough to
extend the correspondence (\ref{eq:sctcorr})
between the projected computational frame
in each point of view and the observer-adapted frame of the congruence
point of view associated with the respective observer congruence
to include the structure functions of the frame.
%
Denote the projected computational structure
functions by $\Cscr(o)^\alpha{}_{\beta\gamma}$ and those of the partially
normalized projected computational frame by $\Dscr(o)^\alpha{}_{\beta\gamma}$.
The purely spatial structure functions in all cases are the same as those
of the original computational frame
\be
\Cscr(o)^a{}_{bc} =\Dscr(o)^a{}_{bc} =C^a{}_{bc}\ .
\ee
In the slicing point of view the only other nonzero structure functions are
\be
\Cscr(n)^b{}_{0a} = - \theta^b(\PLie(n)\sub{\vec N} e_a) \ ,
\ee
which describe the failure of the frame to be spatially comoving along
$\epsilon_0$, or equivalently along $n$,
and the corresponding functions for the partially-normalized
frame
\be
\Dscr(n)^b{}_{\bot a} = - N^{-1} \theta^b(\PLie(n)\sub{\vec N} e_a) \ ,
% =H(n)^b{}_a + [\Cscr^b{}_{ca} - \Gamma(n)^b{}_{ca}] N^c\ ,
\ee
plus the functions
\be
\Dscr(n)^\bot{}_{\bot a} = a(n)_a = [d(n) \ln N]_a %= -g(n)_a
\ee
which describe the acceleration of the observer congruence.
It is convenient to re-express the structure functions $\Dscr(n)^a{}_{\bot b}$
in terms of the spatial covariant derivative using the comma-to-semicolon
formula for the Lie derivative
\be\eqalign{\label{eq:sfn}
\Dscr(n)^a{}_{\bot b} &= N^{-1}[ N^a{}_{|b} - \Gamma(n)^a{}_{cb} N^c] \cr
&= N^{-1} N^a{}_{|b} - \Gamma(n)^a{}_{cb} v^c \ ,\cr
}\ee
where $\vec v = N^{-1} \vec N$ is the relative velocity of the
threading curves with respect to the test observers.
In the threading point of view one has
\be \eqalign{
\Cscr(m)^0{}_{0 a} &= e_0 M_a = [ \PLie(m)\sub{e_0} \Overeq M]_a\ ,\cr
\Cscr(m)^0{}_{ab} &= \epsilon_a M_b -\epsilon_b M_a
- M_c C^c{}_{ab} = [d(m)\Overeq M]_{ab} = 2M^{-1} \omega(m)_{ab}\ , \cr
}\ee
and
\be \eqalign{
\Dscr(m)^\top{}_{\top a} &= e_0 M_a + \epsilon_a \ln M
= [d(m) \ln M]_a + [ \PLie(m)\sub{e_0} \Overeq M]_a
= a(m)_a \equiv C_a\ ,\cr
\Dscr(m)^\top{}_{ab} &= M [\epsilon_a M_b -\epsilon_b M_a
- M_c C^c{}_{ab}] = M [d(m)\Overeq M]_{ab} = 2\omega(m)_{ab}
\equiv \tilde\Omega_{ab}\ , \cr
}\ee
the latter of which coincide with the projected computational frame
components of the acceleration and twice the
rotation of the observer congruence.
The symbols $ C_a $ and $\tilde{\Omega}_{ab}$ were used by
Cattaneo \Cite{1958, 1959a,b} in his formalism based on adapted coordinates.
The vanishing of $C(m)^a{}_{0b}$ and $D(m)^a{}_{\top b}$ means that the
projected computational frame is spatially comoving in the threading point
of view. This is not the case in general in the slicing point of view where
instead the frame is spatially comoving not along the observer congruence
but along the threading congruence, i.e., the spatial Lie derivatives
along $e_0$ of the spatial frame and dual frame are zero.
The only kinematical quantity associated with the observer congruence which
does not enter into the structure functions is the expansion tensor, which
arises from the temporal Lie derivative of the spatial metric.
Gathering the above results together with the evaluation of the expansion
tensor leads to the following expressions for the kinematical quantities of the
observer congruence and their representations on the computational 3-space
\be\eqalign{
a(n)^\flat &= d(n) \ln N \phantom{M + \Lie(m)\sub{e_0} \Overeq M M}
\mapsto d \ln N_t \ ,\cr
a(m)^\flat &= d(m) \ln M + \Lie(m)\sub{e_0} \Overeq M
\mapsto d \ln M_t + \dot{\Overeq M}_t \ ,\cr
& \cr
\omega(n)^\flat &=0 \ ,\cr
\omega(m)^\flat &= \half M d(m) \Overeq M \ ,\cr
& \vec\omega(m) \mapsto \half [ M_t \curl \vec M_t
+ \vec M_t \times \dot{\vec M}_t ] \ ,\cr
&\cr
\theta(n)^\flat &= \half N^{-1} \Lie(n)\sub{\epsilon_0} P(n)^\flat
= \half N^{-1} [ \Lie(n)\sub{e_0}
- \Lie(n)\sub{\vec N} ] P(n)^\flat \ ,\cr
& \theta(n)_{ab} \mapsto
(2N_t)^{-1} [ \dot g_{t\, ab} - \Lie\sub{\vec N_t} g_{t\,ab} ]
\ ,\cr
\theta(m)^\flat &= \half M \Lie(m)\sub{e_0} P(m)^\flat \cr
& \theta(m)_{ab} \mapsto (2M_t)^{-1} \dot \gamma_{t\,ab} \ .\cr
}\ee
The expansion tensor components enter into the structure functions if one
passes to an orthonormal spatial frame from the projected computational
frame. Such an orthonormal frame is used by Durrer and Straumann \Cite{1988}
in the slicing point of view.
\Subsection{Spatial covariant derivative}
Given the correspondence with the results of the congruence point of view,
one can immediately evaluate the components of the spatial part of the
spatial connection in the projected computational frame.
It is convenient to introduce the symbols
\be \eqalign{
\Gamma[g]^a{}_{bc} &= \Gamma(\Pscr,n)^a{}_{bc}
= \half g^{ad} ( g_{\{db,c\}_-} + C(n)_{\{dbc\}_-} )\ ,\cr
\Gamma[\gamma]^a{}_{bc} &= \Gamma(\Pscr,m)^a{}_{bc}
= \half \gamma^{ad} ( \gamma_{\{db,c\}_-} +
C(m)_{\{dbc\}_-} ) \ ,\cr
}\ee
where $C(n)^a{}_{bc}$ and $C(m)^a{}_{bc}$ indicates that the indices on
$C^a{}_{bc}$ are shifted with the respective spatial metrics.
The notation $\Gamma(\Pscr,o)^a{}_{bc}$ allows simultaneous reference
to both points of view, while the notation $ \Gamma[g]^a{}_{bc}$
and $ \Gamma[\gamma]^a{}_{bc}$ is more suggestive of the
symmetric connection of the Riemannian spatial metric on a computational
3-space.
Then \Eq(\ref{eq:scc})
%(2.94)\message{(2.94)?}
with $h_{ab} = g_{ab}$ yields
\be
\Gamma(n)^a{}_{bc} = \Gamma[g]^a{}_{bc}\ ,
\ee
while with $h_{ab} = \gamma_{ab}$ it yields
\be \eqalign{
\Gamma(m)^a{}_{bc} &= \Gamma[\gamma]^a{}_{bc}
+ \Delta \Gamma(\Pscr,m)^a{}_{bc}\ ,\cr
\Delta \Gamma(\Pscr,m)^a{}_{bc} &=
\half \gamma^{ad} \PLie(m)\sub{e_0} \gamma_{\{db} \, M_{c\}_-}\cr
&= \gamma^{ad} M \theta(m)_{\{db} \, M_{c\}_-}\ .\cr
}\ee
On a computational 3-space, the difference tensor
becomes
\be
\Delta \Gamma(\Pscr,m)^a{}_{bc} \mapsto
\half \gamma^{ad} \dot{\gamma}_{\{db} \, M_{c\}_-}\ ,
\ee
suppressing the index $t$ representing the time dependence of the
fields on this space.
One can introduce \emf{reference spatial covariant derivatives} associated
with the induced metric on the slices in the slicing point of view and
the projected metric on the slices in the threading point of view
in each case modeling the metric connection of the Riemannian computational
3-space at a given value of the time $t$.
For a reference field $S= S^{a\ldots}_{\ \ b\ldots} e_a\otimes \cdots \omega^a
\otimes \cdots$, define the reference spatial components of its reference
spatial covariant derivative by
\be
\del(\Pscr,o)_c S^{a\ldots}_{\ \ b\ldots} = S^{a\ldots}_{\ \ b\ldots,c}
+ \Gamma(\Pscr,o)^a{}_{cd} S^{d\ldots}_{\ \ b\ldots} + \cdots
- \Gamma(\Pscr,o)^d{}_{cb} S^{a\ldots}_{\ \ d\ldots} + \cdots \ .
\ee
If $S$ is a spatial tensor in a given point of view, then the reference
spatial covariant derivative of its reference spatial projection
$\del(\Pscr,o) \Pscr S$ maps onto the covariant derivative of $S_t$
on the computational 3-space using the metric connection at time $t$
on this space.
For the slicing point of view, this is an isomorphism which then maps the
spatial covariant derivative on the spatial tensor algebra onto the
metric covariant derivative on the computational 3-space at corresponding
times $t$
\be
\Pscr [ \del(n)\sub{X} S ] = \del[g]\sub{\Pscr X} \Pscr S\ .
\ee
For the threading point of view, it is not.
Instead there are two correction terms, each due to the tilting of the local
rest spaces of the observers away from the slices, one involving the
direction of differentiation of $S$ and a second involving
the derivatives occurring
in the connection components themselves
\be \eqalign{
\Pscr [ \del(m)\sub{X} S ] &= \del[\gamma]\sub{\Pscr X} \Pscr S
+ \Overeq M(X) \PLie(m)\sub{e_0} S + \sigma(\Delta \Gamma(X)) S\ ,\cr
\Delta \Gamma(X)^a{}_b &= \Delta \Gamma(\Pscr,m)^a{}_{cb} X^c \ ,\cr
}\ee
where $X$ is a spatial vector field and $\sigma$ is the Lie algebra
tensor representation appropriate to the index structure of $S$.
The first correction term maps onto $\Overeq M_t(X_t) \dot{S}_t$ on the
computational 3-space, while the coefficients in the
second correction term have already been described above.
\Subsection{Spatial vector analysis}
To discuss spatial vector analysis, let $\diff$ stand for any of the
operator symbols $\grad$, $\curl$ and $\div$. Then one may introduce
operators $\diff(\Pscr,o)$, with alternative symbols
$\diff[g]$ for $o=n$ and $\diff[\gamma]$ for $o=m$, for the
corresponding reference spatial operators. These operators
represent on the spacetime the corresponding operators defined
on the computational 3-space in terms of its Riemannian connection
and may be used to express the spatial derivative operators $\diff_o$
in the reference representation.
One also needs a reference spatial dot product and cross product
between two reference spatial vector fields $X$ and $Y$
\be\eqalign{
X \cdot_{(\Pscr,o)} Y &= [\Pscr P(o)]_{ab} X^a Y^b\ ,\cr
X \times_{(\Pscr,o)} Y &= p(o)^{1/2}
[\Pscr P(o)^\sharp]^{ad}\epsilon_{dbc} X^b Y^c e_a\ ,\cr
}\ee
with $ p(o) = \det ([\Pscr P(o)^\flat]_{ab}) $ standing
respectively for $g$ and $\gamma$. Similarly one has
the alternative notation $X \cdot_{[g]} Y$ and so on.
This notation may also be extended to the spatial duality operation,
though admittedly it becomes more and more cumbersome. 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.
For the various gradient operators one has
\be
\grad(\Pscr,o) f = [\Pscr P(o)^\sharp] \rightcontract [\Pscr d] f
= [ P(o)^{ab} e_b f ] e_a\ ,
\ee
or equivalently
\be
\grad[g] f = [ g^{ab} e_b f] e_a \ ,\qquad
\grad[\gamma] f = [ \gamma^{ab} e_b f ] e_a \ .
\ee
For a reference spatial vector field $X=X^a e_a$ one defines
\be
\curl(\Pscr,o) X = p(o)^{-1/2} \epsilon^{abc} \del(\Pscr,o)_b \left(
[\Pscr P(o)^\flat]_{cd} X^d \right) e_a
\ee
and
\be
\div(\Pscr,o) X = \del(\Pscr,o)_a X^a
= p(o)^{-1/2} \partialslash_a [ p(o)^{1/2} X^a]\ ,
\ee
where
\be
\partialslash_a X^a \equiv (e_a - C^b{}_{ab}) X^a \ .
\ee
Given these operators one can then express the spatial derivative
operators $\diff_o$ in the reference representation, with the
corresponding images on the computational 3-space. The results are
\be\eqalign{
\Pscr \grad_n f &= \grad[g] f \mapsto \grad[g_t] f_t \ ,\cr
\Pscr \grad_m f &= \grad[\gamma] f + [ \Lie(m)\sub{e_0} f ]
\, \Pscr \vec M \cr
&\qquad \mapsto \grad[\gamma_t] f_t + \dot f_t \vec M_t\ ,\cr
}\ee
for the gradient,
\be\eqalign{
\Pscr \curl_n X &= \curl[g] \, \Pscr X \mapsto \curl[g_t] X_t \ ,\cr
\Pscr \curl_m X &= \curl[\gamma] \, \Pscr X + \Pscr \vec M \times_{[\gamma]}
[\Pscr\Lie]\sub{e_0} \Pscr X \cr
&\qquad \mapsto
\curl[g_t] X_t + \vec M_t \times_{[\gamma_t]} \dot X_t \cr
}\ee
for the curl, and
\be\eqalign{
\Pscr \div_n X &= \div[g] \, \Pscr X \mapsto \div[g_t] X_t \ ,\cr
\Pscr \div_m X &= \div[\gamma] \, \Pscr X + \Pscr \vec M \rightcontract
p(o)^{-1/2}[\Pscr\Lie]\sub{e_0}
\left( p(o)^{1/2} \Pscr X \right) \cr
&\qquad \mapsto
\div[\gamma_t] \, X_t + \vec M_t \rightcontract
p(o)_t{}^{-1/2} \left( p(o)_t{}^{1/2} X_t \right)\dot{\,} \cr
}\ee
for the divergence, where in each occurrence $X$ is assumed to be
a spatial vector field of the appropriate type.
It is also of interest to relate the spatial divergences of spatial
vector fields to the spacetime divergences. Here one finds
\be\eqalign{
[\del(n)_a + a(n)_a ] X^a &= N^{-1} \div_n \, N X^a \cr
&= X^\alpha{}_{;\alpha}\ ,\cr
[\del(m)_a + a(m)_a ] X^a &= M^{-1} \div_n \, M X^a
+ [\Lie(m)\sub{e_0} M_a] X^a \cr
&= X^\alpha{}_{;\alpha}\ .\cr
}\ee
The spatial derivative operators that appear in the text of Landau and
Lifshitz \Cite{1975} are just the reference spatial derivative operators
appropriate for the threading point of view.
Hanni \Cite{1977} has noted the complementary definitions which
correspond to the slicing point of view.
\Subsection{Partially-observer-adapted frames: connection components}
Given the correspondence with the results of the congruence point of view,
one can immediately express the splitting of the tensor representing the
connection components in the projected computational frame. It is also
of interest to split the tensor representing the connection components
with respect to the computational frame itself. Many of the spatial
fields from these two families coincide, since the transformation matrix
between the two frames, whose derivative is responsible for the difference
tensor, is of a very simple form.
To examine the relationship between the two frames, a more efficient
notation is needed.
Let
\be
e(o)_\alpha = A(o)^\beta{}_\alpha e_\beta
\ee
denote the projected computational frame vectors in each point of view.
The transformation matrix has a very simple form
\be\meqalign{
& A(n)^0{}_0 = 1\ , \quad & A(n)^0{}_a = 0\ , \quad
& A(n)^a{}_0 = -N^a \ , \quad & A(n)^a{}_b = \delta^a{}_b \ , \cr
& A(n)^0{}_0 = 1\ , \quad & A(n)^0{}_a = M_a\ , \quad
& A(n)^a{}_0 = 0 \ , \quad & A(n)^a{}_b = \delta^a{}_b \ .\cr
}\ee
Let $\four\bfomega$ and $\four\bfomega(o)$ denote the tensor-valued
connection 1-forms in the original computational frame and in the projected
computational frame. They are related by the well known
inhomogeneous transformation law
\be
\four\bfomega(o)^\alpha{}_\beta =
A(o)^{-1\, \alpha}{}_\gamma \four\bfomega^\gamma{}_\delta
A(o)^\delta{}_\beta
+ A(o)^{-1\, \alpha}{}_\gamma d A(o)^\gamma{}_\beta\ ,
\ee
or
\be
\four\bfomega(o) = \four \bfomega + \Delta \four \bfomega(o)\ .
\ee
The difference term is rather simple, having only the following components
nonzero in the projected computational frame
\be\meqalign{
[A(n)^{-1} d A(n)]^a{}_\bot &= - d N^a
&= - \epsilon_0 N^a \omega^0 - e_b N^a \theta^b\ ,\cr
[A(m)^{-1} d A(m)]^\top{}_a &= \phantom{-} d M_a
&=\phantom{-} e_0 M_a \theta^0 + \epsilon_b M_a \omega^b\ .\cr
}\ee
It leads to correction terms for the projected computational frame
components of the connection coefficients with respect to the computational
frame with respect to those of the projected one
\be
\four \Gamma(o)^\alpha{}_{\beta\gamma} =
A(o)^{-1\, \alpha}{}_\gamma \four \Gamma^\alpha{}_{\beta\gamma}
A(o)^\delta{}_\beta +
\Delta \four \Gamma(o)^\alpha{}_{\beta\gamma} \ ,
\ee
where
\be
\four \Gamma^\alpha{}_{\beta\gamma} =
\omega^\alpha( \four \del\sub{e_\beta} e_\gamma )
\ee
are the computational frame components of the spacetime connection.
The projected computational components of the nonzero correction terms
\be\eqalign{
& \Delta \four \Gamma(n)^a{}_{\bot\bot}\ , \qquad
\Delta \four \Gamma(n)^a{}_{\bot b}\ , \cr
& \Delta \four \Gamma(m)^\top{}_{\bot a}\ , \qquad
\Delta \four \Gamma(m)^\top{}_{b a}\ , \cr
}\ee
may be read off from the above derivatives of the shift components.
In particular the following component expressions for
the kinematical quantities hold in either frame
\be\meqalign{
a(n)_a &= \four \Gamma^{\bot\bot}{}_a\ ,\qquad &
a(m)^a &= \four \Gamma^a{}_{\top\top}\ ,\cr
k(n)_{ab} &= -\four \Gamma^\bot{}_{ab}\ ,\qquad &
k(m)^{ab} &=- \four \Gamma^{ab}{}_\top\ .\cr
}\ee
\Subsection{Total spatial covariant derivatives}
The Lie and Fermi-Walker total spatial covariant derivatives have
been introduced in the congruence point of view so they are available
for the observer congruence
in both the slicing and threading points of view with the identification
$u=o$ and $U = \gamma(U,o) [ o + \nu(U,o) ]$.
The discussion of the congruence point of view
is relevant to the threading point of view,
while that of the hypersurface point
of view is relevant to the slicing point of view.
Because two congruences are involved,
the entire qualifier $(U,o)$ cannot be
suppressed but only abbreviated to $(o)$.
When differentiating a spatial
tensor field along the worldline with 4-velocity $U$,
the Lie total spatial covariant derivative has the following expressions
in the two points of view
\be\eqalign{
D\lie(U,o) S / d\tau_o &= [ \Lie(o)\sub{o} + \del(o)\sub{\nu(o)} ] S \cr
&= \left\{
\eqalign{ & M^{-1} \Lie(m)\sub{e_0} S + \del(m)\sub{\nu(m)} S\cr
& N^{-1} [\Lie(n)\sub{e_0} - \Lie(n)\sub{\vec N} ] S
+ \del(n)\sub{\nu(n)} S\cr} \right. \cr
&\mapsto \left\{
\eqalign{ & M_t{}^{-1} \dot S_t + \del(m)\sub{\nu(m)_t} S_t\cr
& N_t{}^{-1} [\dot S_t - \Lie(n)\sub{\vec N_t} S_t ]
+ \del(n)\sub{\nu(n)_t} S_t \ .\cr}\right. \cr
}\ee
Explicit component expressions for these operators may be obtained in
the projected computational frame using \Eq(\ref{eq:tscdc})
once the correspondence
\be
\vbox{\tabskip=10pt
\halign{#&&\hfil $#$ \hfil \cr
& n & \leftarrow & u=o & \rightarrow & m \cr
& \Dscr(n)^a{}_{\bot b} & \leftarrow & C^a{}_{\top b}
& \rightarrow & 0 \cr
}}\ee
with the congruence observer-adapted frame
is made. The Lie total spatial covariant derivative of a spatial vector
then has the expression
\be \eqalign{
D\lie (U,m) X^a / d\tau_m &=
d X^a / d\tau_m + \Gamma(m)^a{}_{bc} \nu(m)^b X^c \ ,\cr
D\lie (U,n) X^a / d\tau_n &=
d X^a / d\tau_n + \Gamma(n)^a{}_{bc} [\nu(n)^b - v^b] X^c
+ N^{-1}N^a{}_{|b} X^b\ .\cr
}\ee
The projected computational frame is a spatially-comoving observer-adapted
frame in the threading point of view, and the entire discussion of the
congruence point of view applies directly to the threading point of view.
The simplifying assumption of a spatially-comoving frame
convenient for many calculations is also valid. However, in the slicing
point of view, the projected computational frame is instead spatially
comoving along $e_0$ rather than $\epsilon_0$, or equivalently $n$,
and this leads to the two additional shift terms in the expression.
However, in the slicing point of view
it is more appropriate to use the Lie derivative
$\Lie(n)\sub{e_0}$ along the threading congruence
than the temporal Lie derivative
$\Lie(n)\sub{n}$ along the observer congruence,
in order to measure evolution with respect to the
nonlinear reference frame. Making this change
leads to still another Lie total spatial covariant derivative
which will be designated at the slicing point of view operator,
while the original one will be referred to as the hypersurface operator.
This slicing Lie total spatial covariant derivative is an operator which
uses the reference decomposition $U= U^0 e_0 + \Pscr U$ of the
4-velocity vector rather than the slicing decomposition as in the
hypersurface operator, constructed with the spatial Lie derivative
along the $e_0$ and the spatial covariant derivative along $P(n)U$
\be\eqalign{
D\lie(U,\Pscr,n)/d\tau_U &=
U^0 \Lie(n)\sub{e_0} + \del(n)\sub{\Pscr U}\ ,\cr
D\lie(U,\Pscr,n)/d\tau_n &=
N^{-1} \Lie(n)\sub{e_0} + \del(n)\sub{\nu - \vec v}\ .\cr
% &= \del\lie(U,\Pscr,n) \ .\cr
}\ee
This leads to a difference term with respect
to the corresponding hypersurface point of view operator
\be\eqalign{
D\lie(U,\Pscr,n)/d\tau_n - D\lie(U,n)/d\tau_n &=
[N^{-1}\Lie(n)\sub{e_0} + \del(n)\sub{\nu-\vec v}]
- [\Lie(n)\sub{n} + \del(n)\sub{\nu}] \cr
&= N^{-1} [ \Lie(n)\sub{\vec N} - \del(n)\sub{\vec N}] \ .\cr
% & = \del\lie(U,\Pscr,n) - \del\lie(U,n) \ .\cr
}\ee
The difference operator
represented by the last equation is a linear
transformation operator, not a derivative operator. The comma-to-semicolon
formula for the Lie derivative, spatially projected, expresses this
linear transformation in terms of the covariant derivative of the shift
$\vec N$. If $S$ is a tensor of type $\sigma$ then
\be
\Lie(n)\sub{\vec N} S - \del(n)\sub{\vec N} S = - \sigma(\del \vec N) S \ .
\ee
Thus for a spatial vector field $X$ or its covariant form,
the difference operator is
\be \eqalign{
D\lie(U,\Pscr,n) X^a/d\tau_n &=
D\lie(U,n) X^a/d\tau_n - N^{-1} N^a{}_{|b} X^b \ , \cr
D\lie(U,\Pscr,n) X_a/d\tau_n &=
D\lie(U,n) X_a/d\tau_n + N^{-1} N^b{}_{|a} X_b\ ,\cr
}\ee
which leads to the formulas
\be\eqalign{
D\lie(U,\Pscr,n) X^a/d\tau_n &= d X^a/d\tau_n
+ \Gamma(n)^a{}_{bc} [ \nu(n)^b - v^b] X^c \ ,\cr
D\lie(U,\Pscr,n) X_a/d\tau_n &= d X_a/d\tau_n
- \Gamma(n)^c{}_{ba} [ \nu(n)^b - v^b] X_c \ . \cr
}\ee
The change from the hypersurface to the slicing point of view
introduces the shift derivative term in the total
spatial covariant derivative in order to change to the spatial
velocity $\vec\nu - \vec v$
relative to the threading curves. This term
must therefore appear explicitly
in the spatial gravitational force defined using the new operator.
The relationship of these various Lie operators is nicely represented
in Figure \ref{fig:tscdsugg}.\message{Figure 3.7}
%\fig{Total spatial covariant derivatives}{A suggestive representation
%of the various total spatial covariant derivatives, normalized to
%correspond to the proper time of the appropriate observers.}{tscdsugg}
\def\dsub#1{_{\hbox{$\displaystyle #1$}}}
\def\del{\nabla}
\def\UNIT{0.8cm}
\begin{figure}\typeout{figure gemfig3-7: total cov der 3 ways}
\label{fig:tscdsugg}
$$ %\kern-2cm
\vbox{
\beginpicture
% \twelvept
\setcoordinatesystem units <\UNIT,\UNIT> point at 0 100 %% to put baseline at top
%\begincomment
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\put {\mathput{\phantom{\del\dsub{\vec\nu(m)}}}} [B] at 2.3 6.3
% to get vertical spacing on partial plot
% lines:a:
\setsolid \setlinear
\plot 0 0 4 6.05 /
\plot 0 0 1.75 5.3 /
\plot 1.6 4.8 3.65 5.5 / % nu
\setdashes
\plot 1.75 5.3 4 6.05 / % % gamma nu
\putrule from 0 6.05 to 4 6.05 % top horizontal leg
\putrule from 0 4.8 to 1.6 4.8 % left mid horizontal leg
\setsolid
\putrule from 0 4.5 to 3 4.5 % bot horizontal leg
\putrule from 0 0 to 0 6.05 % vertical leg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% text:a:
\put {\mathput{n}} [rt] at -.5 4.3
\put {\mathput{\gamma(m,n)n}} [rB] at -.5 4.8
\put {\mathput{\gamma(u,n)n}} [rB] at -.5 6.05
\put {\mathput{m}} [rB] at 1.3 5
% \put {\mathput{\gamma(u,m)m}} [B] at 2 6.5
\put {\mathput{\gamma(u,m)m}} [lt] at .25 5.85
\put {\mathput{u}} [lB] at 4 6.5
\put {\mathput{\vec\nu(m)}} [lB] at 3.8 4.8
\put {\mathput{e_0}} [lB] at 1.9 4.6
\put {\mathput{N^{-1} e_0}} [lB] at 0.5 3.0
\put {\mathput{\vec\nu(n)-\vec v}} [lB] at 1.5 4
\put {\mathput{\vec v}} [t] at .7 4.25
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% arrowheads:a:
\arrow <.3cm> [.1,.4] from 0 4.2 to 0 4.5 %n:
\arrow <.3cm> [.1,.4] from 0 4.5 to 0 4.8 %gmn:
\arrow <.3cm> [.1,.4] from 0 5.75 to 0 6.05 %gun:
\arrow <.3cm> [.1,.4] from 1.475 4.5 to 1.6 4.8 % m:
\arrow <.3cm> [.1,.4] from 1.65 5 to 1.75 5.3 % gum:
\arrow <.3cm> [.1,.4] from 3.35 5.4 to 3.65 5.5 % num:
\arrow <.3cm> [.1,.4] from 3.8 5.75 to 4 6.05 % u:
\arrow <.3cm> [.1,.4] from 1.175 4.5 to 1.475 4.5 % v:
\arrow <.3cm> [.1,.4] from 2.7 4.5 to 3 4.5 % nu-v:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%lines:b:
\setcoordinatesystem units <\UNIT,\UNIT> point at -6 100
\plot 0 0 3 4.5 / % x
\putrule from 0 0 to 0 4.5 %vertical leg
\putrule from 0 4.5 to 3 4.5 %horizontal leg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% arrowheads:b:
\arrow <.3cm> [.1,.4] from 0 4.2 to 0 4.5 %vertical leg
\arrow <.3cm> [.1,.4] from 2.7 4.5 to 3 4.5 %horizontal leg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% text:b:
\put {\mathput{\Lie(n)\dsub{n}}} [rB] at -.5 1.5
\put {\mathput{\del(n) \dsub{\vec \nu(n)}}} [B] at 1.5 5
\put {\mathput{D_{(\bot)}(u)/d\tau_n}} [lB] at 0 -1
\put {\mathput{\gamma(U,n)^{-1}U=n+\vec\nu(n) }} [lB] at -1.5 -2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%lines:c:
\setcoordinatesystem units <\UNIT,\UNIT> point at -11 100
\plot 0 0 3 4.5 / % x
\plot 0 0 1.475 4.5 / % y
\putrule from 1.475 4.5 to 3 4.5 %short horizontal leg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% arrowheads:c:
\arrow <.3cm> [.1,.4] from 2.7 4.5 to 3 4.5 % horizontal leg
\arrow <.3cm> [.1,.4] from 1.375 4.2 to 1.475 4.5 %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% text:c:
\put {\mathput{N^{-1}\Lie(n)\dsub{e_0}}} [rB] at 0 1.5
\put {\mathput{\del(n)\dsub{\vec\nu(n) -\vec v}}} [B] at 2.2 5
\put {\mathput{D_{(sl)}(u)/d\tau_n}} [lB] at 0 -1
\put {\mathput{\gamma(U,n)^{-1}U=N^{-1}e_0 +[\vec\nu(n)-\vec\nu] }} [lB] at -1.5 -2.8
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%lines:d:
\setcoordinatesystem units <\UNIT,\UNIT> point at -15 100
\plot 1.6 4.8 3.65 5.5 / % nu
\plot 0 0 1.6 4.8 / % x
\plot 0 0 3.65 5.5 / % y
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% text:d:
\put {\mathput{\Lie(m)\dsub{m}}} [rB] at 0 1.5
\put {\mathput{\del(m)\dsub{\vec\nu(m)}}} [B] at 2.7 6
\put {\mathput{D_{(th)}(u)/d\tau_m}} [lB] at 0 -1
\put {\mathput{\gamma(U,m)^{-1}U=m+\vec\nu(m) }} [lB] at -1.5 -2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% arrowheads:d:
\arrow <.3cm> [.1,.4] from 3.35 5.4 to 3.65 5.5 % num:
\arrow <.3cm> [.1,.4] from 1.475 4.5 to 1.6 4.8 % m:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\endpicture}$$
\caption{
A suggestive representation
of the various total Lie spatial covariant derivatives for the hypersurface,
slicing and threading points of view, normalized to
correspond to the proper time of the appropriate observers.
Each covariant derivative is the sum of a temporal Lie derivative and a spatial
covariant derivative along the temporal and spatial parts of the relevant
multiple of the 4-velocity $U$ represented in the triangle,
each vector decomposition given under the symbol for the corresponding covariant
derivative.
}
\end{figure}
As with the previous Lie total spatial covariant derivative, index shifting
with the spatial metric generates Lie derivative terms
\be \eqalign{
D\lie(U,\Pscr,n) g_{ab} / d \tau_n &= N^{-1} \Lie(n)\sub{e_0} g_{ab}
= 2 \theta(n)_{ab} + 2 N^{-1} N_{(a|b)} \ , \cr
D\lie(U,\Pscr,n) g^{ab} / d \tau_n &= N^{-1} \Lie(n)\sub{e_0} g^{ab}
= -2 \theta(n)^{ab} - 2 N^{-1} N^{(a|b)} \ . \cr
}\ee
These terms must be taken into account in comparing index-shifted formulas.
\message{argument notation here perhaps not right}%
\message{warning: Ca0b terms?}
The relative velocity may be represented in terms of the time derivatives
of the spatial coordinates
\be\eqalign{
\nu(m)^a &= d x^a / d \tau_m = M^{-1} d x^a / d t \cr
& \mapsto M_t{}^{-1} \dot x{}^a_t \ , \cr
\nu(n)^a &= d x^a / d \tau_n + v^a = N^{-1} [ d x^a / d t + N^a ] \cr
& \mapsto N_t{}^{-1} [\dot x{}^a_t + N^a_t] \ . \cr
}\ee
This leads to analogous expressions for the slicing and threading
Lie total spatial covariant derivatives
\be\eqalign{
D\lie(U,m) X^a/d\tau_m &= M^{-1} [ d X^a/d t
+ \Gamma(m)^a{}_{bc} d x^b / d t \, X^c \ ,\cr
D\lie(U,\Pscr,n) X_a/d\tau_n &=N^{-1} [ d X_a/d t
- \Gamma(n)^c{}_{ba} d x^b / d t \, X_c \ . \cr
}\ee
This leads to an analogous gravitomagnetic force in the slicing point of view
even though the gravitomagnetic force in the hypersurface
point of view vanishes.
\Subsection{Spatial gravitational forces}
The discussion of spatial gravitational forces depends on the choice
of derivative operator one uses to express the acceleration equation.
The discussion for the congruence point of view applies to the observer
congruence in both the slicing and threading points of view.
However, the change from the hypersurface to the
slicing Lie total spatial covariant derivative leads
to another representation of spatial gravitational forces.
The difference term in this derivative operator compared with the hypersurface
operator in the decomposition of the acceleration equation leads
to a Lie gravitomagnetic force in the slicing point of view which is absent
in the hypersurface point of view due to the vanishing of the rotation.
The various Lie gravitomagnetic tensors are
\be \imeqalign{
H\lief(m)_{ab} &= 2 M M_{[b || a]} \ , \qquad &
H\lies(m)_{ab} &= 2 M M_{[b || a]}
- 2 M^{-1} \Lie(m)\sub{e_0} \gamma_{ab} \ , \cr
H\lief(n)_{ab} &= 0 \ ,\qquad &
H\lies(n)_{ab} &= - N^{-1} [ \Lie(n)\sub{e_0} g_{ab}
- 2 \del(n)_{(a} N_{b)} ] \ ,\cr
H\lief(\Pscr,n)_{ab} &= N^{-1} N_{b|a}\ , \qquad &
H\lies(\Pscr,n)_{ab} &= N^{-1} N_{b|a}
- N^{-1} \Lie(n)\sub{e_0} g_{ab} \ ,\cr
}\ee
and the gravitomagnetic vectors are
\be\imeqalign{
\vec H\lie(m) &= M \curl_m \vec M \ , \qquad &
\ H\lie(m)^a &= M \eta(m)^{abc} \del(m)_b M_c\ ,\cr
\vec H\lie(n) &= 0\ , &
H\lie(n)^a &= 0\ , \cr
\vec H\lie(\Pscr,n) &= N^{-1} \curl_n \vec N \ ,\qquad &
H\lie(\Pscr,n)^a &= N^{-1} \eta(n)^{abc} \del(n)_b N_c\ .\cr
}\ee
With these definitions
the newly defined Lie total spatial gravitational forces are then
\be\eqalign{
\vec F\g\lief(\Pscr,n) &= \gamma(U,n) [ \vec g(n)
+ \nu(n) \cdot_n \bivec H\lief(\Pscr,n) ] \cr
&= \gamma(U,n) [ \vec g(n)
+ \half \nu(n) \times_n \vec H\lie(\Pscr,n) ]
+ \nu(n) \cdot_n \SYM \bivec H\lief(\Pscr,n) ] \ ,\cr
\vec F\g\lies(\Pscr,n) &= \gamma(U,n) [ \vec g(n)
+ \nu(n) \cdot_n \bivec H\lies(\Pscr,n)] \cr
&= \gamma(U,n) [ \vec g(n)
+ \half \nu(n) \times_n \vec H\lie(\Pscr,n) ]
+ \nu(n) \cdot_n \SYM \bivec H\lies(\Pscr,n) ] \ .\cr
}\ee
Note that the same factor of one half which appears in the
gravitomagnetic vector force term in the Fermi-Walker spatial gravitational
force in the congruence point of view also appears
in the Lie spatial gravitational forces in the slicing point of view.
The energy equation (\ref{eq:energy1}) may also
be expressed in the slicing and threading points of view. The total spatial
covariant derivatives all agree for a scalar but the spatial gravitational
forces differ. Using the covariant Lie spatial gravitational forces one has
the result
\be\eqalign{
D (U,m) E(U,m) /d\tau_m
&=[ \vec F\g\lief(m) + \vec F(m) ] \cdot_m \nu(m)
- [ 2 M]^{-1} \Lie(m)\sub{e_0} \gamma_{ab} \nu(m)^a \nu(m)^b \ ,\cr
D (U,\Pscr, n) E(U,n) /d\tau_n
&=[ \vec F\g\lief(\Pscr,n) + \vec F(n) ] \cdot_n \vec\nu(n)
- [ 2 N]^{-1} \Lie(n)\sub{e_0} g_{ab} \nu(n)^a \nu(n)^b \ .\cr
}\ee
The gravitoelectric and gravitomagnetic force fields have been discussed in
the black hole case by Thorne et al \Cite{1986} using the hybrid
total spatial covariant derivative operator in the slicing point of view.
Both Zel'manov \Cite{1956} and Cattaneo \Cite{1958} have discussed them
from the threading point of view, while Landau and Lifshitz \Cite{1975}
discuss only the stationary case in the threading point of view.
M\"oller \Cite{1952, 1972} discusses them both from his
parametrization-dependent description of the threading point of view
as well as for the threading point of view. All of these threading point
of view discussions introduce the Lie spatial gravitational forces.
Massa \Cite{1974b} has re-expressed the Cattaneo approach in a
somewhat more modern framework, using the spatial co-rotating Fermi-Walker
total spatial covariant derivative.
\message{edit here??}
\message{!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!}
\Subsection{Second-order acceleration equation}
INCORPORATE MATERIAL FROM MFGREV.TEX:
In the context of a nonlinear reference frame which enables
one to represent the spacetime geometry in terms of time-dependent
fields on a computational 3-space, one can re-express the
first order spatial acceleration equation for the spatial momentum
as a second order
equation describing the evolution of the spatial coordinates along the
worldline under consideration.
Assume a projected computational frame obtained from adapted coordinates.
The rates of change of the spatial coordinates
with respect to the coordinate time $t$ then define the
reference spatial components of the coordinate spatial velocity
\be
\Uscr^a = d x^a / d t =
\left\{\meqalign{ & M \nu(m)^a &= \Gamma(U)^{-1} p(n)^a \cr
& N \nu(n)^a - N^a &= \Gamma(U)^{-1} p(m)^a - N^a\cr}\right.
\ .
\ee
This may be interpreted as a spatial vector in each point of view
\be
\Uscr(n) = \Uscr^a e_a \ ,\qquad
\Uscr(m) = \Uscr^a \epsilon_a \ .
\ee
The projected computational frame components of the
appropriate Lie total spatial covariant derivative of this spatial
vector
then yield the corresponding
second derivatives of the spatial coordinates
\be
\left(\matrix{ D\lie(U,m)^2 x^a / dt^2 \cr
D\lie(U,\Pscr,n)^2 x^a / dt^2 \cr
}\right) =
\left(\matrix{ D\lie(U,m) \Uscr(m)^a / dt \cr
D\lie(U,\Pscr,n) \Uscr(n)^a / dt \cr
}\right) = \frac{d^2 x^a}{dt^2}
+ \Gamma(o)^a{}_{bc} \frac{dx^b}{dt} \frac{dx^c}{dt} \ .
\ee
These derivatives may then be related either to the relative
acceleration equation and logarithmic derivatives of the lapse or
to the spatial momentum equation and logarithmic derivatives of the
coordinate gamma factor.
For example, in the threading point of view one has
\be\eqalign{
D\lie(U,m) \Uscr(m)^a / dt
&= M^2 \gamma(U,m)^{-1} D\lie(U,m) p(m)^a / d\tau_m
- \Uscr^a D(U,m) \ln \Gamma(U) / d t \cr
&= M^2 D\lie(U,m) \nu(m)^a / d\tau_m
+ \Uscr(m)^a D(U,m) \ln M / d t \ .\cr
}\ee
Since the relative acceleration has already been evaluated
in equation (\ref{eq:relacctwo}),
the second expression is easier to evaluate. It only leads to an
extra term along the relative velocity of the form
\be
\Uscr(m)^a \{ \Lie(m)\sub{e_0} \ln M
+ [-F\g\lies(m){}_b - F(m)_b + a(m)_b
- \Lie(m)\sub{e_0} M_b] \Uscr(m)^b \}
\ .
\ee\message{??}%
%which reduces to $\Uscr^a [2 a(m)_b \Uscr^b$ in a stationary nonlinear
%reference frame in a stationary spacetime.
The rest is just the
rescaling of the total spatial force
\be
M^2 [ \vec g(m) + \Uscr(m) \times_m \vec H(m) + \vec F(m) ]^a \ .
\ee
Thus apart from the term along the direction of motion due to the
change of parametrization of the worldline, the terms in the second-order
acceleration equation directly correspond to the spatial force terms.
In the slicing point of view, something more interesting happens because
of the additional shift term in the coordinate relative velocity
\be\eqalign{
D\lie(U,\Pscr,n) \Uscr(n)^a / dt
&= N^2 D\lie(U,\Pscr,n) \nu(n)^a / d\tau_n \cr \qquad
& - D\lie(U,\Pscr,n) N^a / d t
+ \Uscr(n)^a D(U,n) \ln N / d t \cr
&= N^2 D\lie(U,\Pscr,n) \nu(n)^a / d\tau_n
-N [ \Lie(n)\sub{e_0} N^a + \del(n)_b N^a \Uscr(n)^b ] \cr
& \qquad
+ \Uscr(n)^a [ \Lie(n)\sub{e_0} N + a(n)_b \Uscr(n)^b ]\cr
&= N^2 [ \vec g(n) - N^{-1} \Lie\sub{e_0} \vec N
+ \{ \Uscr(n) \times_n \vec H(n) + \vec F(n) \} \cr
&\qquad
- \half \{ v \times \vec H(n) \}]^a
+ \Uscr(n)^a [ \ldots ] \ .\cr
}\ee\message{??}%
The additional shift spatial covariant derivative term combines with
the existing term in the spatial gravitational force to form twice its
antisymmetric part, eliminating the contribution of the
symmetric part of the shift tensor gravitomagnetic force term and doubling
the contribution of the gravitomagnetic vector force to yield a term
exactly analogous to the threading point of view expression,
modulo a term quadratic in the shift vector field.
A shift Lie derivative term also adds to the gravitoelectric field to form
an expression analogous to the one in the threading point of view.
These changes to the slicing point of view discussion are not surprising since
these expressions must result from a transformation of the threading
point of view expressions, and in the nonrelativistic limit the lapse and
shift of the two points of view coincide. Thorne et al \Cite{1986}
discuss these changes
in the slicing point of view second-order acceleration equation for
black hole spacetimes in the weak field slow motion limit.
Forward \Cite{1961}
briefly discusses a reference decomposition of the second-order
acceleration equation before linearizing to go to the same limit for
an isolated body.
\message{comp 3-spacE??}
\message{reference decomp ala forward??}
\message{mixed d/dt Lie??}
%The reference version of these equations was considered by Forward \Cite{1961}
%before linearizing it in his discussion of the analogy between
%electromagnetism and linearized general relativity. The linearization of
%this equation has also been considered by Thorne in several contexts.
\Subsection{The spin transport equation}
The gravitomagnetic force fields govern the evolution of the spin vector
of a gyro carried by the observers of the observer congruence. In
the present context the spin transport equation along the observer
congruence can be expressed as follows
\be\eqalign{
D\lie(m,m) S / d\tau_m &= -\half \vec H(m) \times_m S
+ \theta(m) \rightcontract S
= H\lies(m) \rightcontract S \ ,\cr
D\lie(n,\Pscr,n) S / d\tau_n &= -\half \vec H(n) \times_n S
+ \theta(n) \rightcontract S
= H\lies(\Pscr,n) \rightcontract S \ .\cr
}\ee
Thus the gravitomagnetic tensor force $H\lies$ controls the evolution
of the spin vector in the slicing and threading points of view,
as it does in the congruence and hypersurface points of view.
The splitting of the general spin transport equation is
not so interesting since the physical information most useful is
described by the hypersurface point of view discussion as far as
it relates to the observations of the observer carrying the gyro itself.
This topic will therefore be omitted.
\message{edit here?? above and below}
\message{!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!}
\Subsection{Transformation of spatial gravitational fields}
In the case of a spacelike
slicing and a timelike threading, both the slicing and threading
points of view hold simultaneously and one can consider transforming
between the two. The unique boost discussed above relates the two orthogonal
decompositions and may be used to transform between the two
families of spatial tensors
which represent a given spacetime field.
However, in the case of the spatial
gravitational fields, their definitions in each point of view determine
the transformation which relates them to each other.
The gravitoelectric and gravitomagnetic vector fields turn out to be related
in a way which is closely connected to the Lorentz boost of the electric
and magnetic fields.
In comparing the two points of view it is helpful to recall that
$\vec v= N^{-1}\vec N$ and $\vec V= M \vec M$ define the relative
velocity of the threading observers with respect to the normal observers
as measured by the normal and threading observers respectively. This
explains the reciprocally related
factors of the lapse and dual lapse which appear
in the gravitomagnetic fields and in many other contexts.
A simple
calculation yields the following formulas for the transformation of the
the gravitomagnetic vector field and the gravitoelectric 1-form
\be \eqalign{
P(m)\half\vec H(n)
& = \half\vec H(m) + \vec V \times_m \vec g(m)
+\vec V \times_m
\left[\half M^{2} \Lie(m) \subb{e_0}(M^{-2} \Overeq M)\right]^\sharp\ ,\cr
\Pscr\, \Overeq {\,g}(n)
&= \gamma_L{}^2 [\vec g(m) - H(m)\rightcontract \vec V]^\flat
+ \vec V\,^\flat [\Lie\subb{m}\ln (M\gamma_L)]
+\gamma_L{}^2 \Lie(m) \subb{e_0} \Overeq M\cr
&= \gamma_L{}^2 [\vec g(m) -\vec V \times_m \half \vec H(m)
-\SYM H(m)\rightcontract \vec V]^\flat + \ldots
\ .\cr}
\ee\message{inconsistent with choice of direction of transformation in
appendix; check??}%
King and Ellis\Cite{1973}
represent the threading kinematical quantities
in terms of those of the slicing, which is relevant to the inverse
transformation
\be \eqalign{
\Pscr \half\vec H(m)
& = \gamma_L{}^2 \left[ \half\vec H(n) +(\half \vec H(n) \cdot_n \vec v)\vec v
+ \vec v \times_n (\vec g(n) + \SYM H(n)\rightcontract \vec v) \right]
\cr
&\qquad + \vec v \times_n
\left[\half \Lie(n) \subb{e_0}(M^{-2} \Overeq N)\right]^\sharp\ ,\cr
P(n)\, \Overeq {\,g}(m)
&= \gamma_L{}^2 [\vec g(n) + H(n) \rightcontract \vec v]^\flat
- \gamma_L \vec v\,^\flat \Lie\subb{e_0}\ln (M)
- \Lie(n) \subb{e_0}(M^{-2}\Overeq N)\cr
&= \gamma_L{}^2 [\vec g(n) +\vec v \times_n \half \vec H(n)
+\SYM H(n) \rightcontract \vec v]^\flat + \ldots
\ .\cr}
\ee
The relations
\be \meqalign{
& P(m) (Y^a e_a) = Y^a \epsilon_a\ , \qquad
& \Pscr (\sigma_a \theta^a) = \sigma_a \omega^a\ ,\cr
& \Pscr (Y^a \epsilon_a) = Y^a e_a\ , \qquad
& P(n) (\sigma_a \omega^a) = \sigma_a \theta^a\ ,\cr}
\ee
enable one to convert the above index-free equations for the transformation
of the gravitomagnetic vector field and the gravitoelectric 1-form back into
component form
\be \meqalign{
& \half H(n)^a = \half H(m)^a +\ldots\ ,\qquad &
g(n)_a = \gamma_L{}^2 [g(m)_a +\ldots\ ,\cr
& \half H(m)^a = \gamma_L{}^2 [\half H(n)^a +\ldots\ ,\qquad &
g(m)_a = \gamma_L{}^2 [g(n)_a +\ldots\ ,\cr}
\ee
Since the gravitomagnetic vector field
and the gravitoelectric 1-form only involve
the exterior derivative, their
transformation laws are relatively simple and closely analogous to
the electromagnetic case, especially in the stationary case when all the
Lie derivative terms vanish.
The transformation formulas for the gravitomagnetic vector field
may be directly compared with the transformation of the magnetic field
under the boost between the two points of view, but the
formulas for the gravitoelectric 1-form must be compared with
the boost of the electric 1-form field.
The symmetric part of the gravitomagnetic
tensor on the other hand involves the spatial connection and has a much
more complicated transformation law.
In order to compare the above equations with the familiar orthonormal component
form of the Lorentz transformation of the electric and magnetic fields,
one must bridge the gap between the four-dimensional spatial notation and
the usual orthonormal frame component notation which is so familiar. This
is done in appendix B. However, one thing is clear independent of these
details. The gravitomagnetic vector field transforms with an additional
factor of one half with respect to the magnetic field. This is contrary
to the usual analogy used in weak field slow motion arguments between
the transformation laws for the gravitoelectric and gravitomagnetic fields
and those for the electric and magnetic fields under a boost.
In fact in those contexts one has a transformation between two different
nonlinear reference frames associated with nearly inertial coordinate
systems, not a transformation between two sets of observers associated
with the same nonorthogonal nonlinear reference frame. Such a transformation
law has not really been carefully studied.
\Newpage
\Section{Spatial curvature}
%This correspondence may be pushed to the curvature level in order to
%relate
Having expressed the spatial part of the spatial connection in terms of
the metric connection on the computational 3-space or its equivalent
reference representation representative on the spacetime itself,
one can now relate
the spatial curvature tensor to the curvature tensor of the
Riemannian computational 3-space or its reference representative
representative.
This explains in a more direct way the unusual symmetries
that occur and the additional kinematical terms that litter the classical
equations involving the spatial curvature in the congruence point of view
for a rotating congruence.
In the slicing point of view there is instead a direct isomorphism, which
has made it so useful in computations in so many problems over the years.
The complications in the threading point of view have instead
apparently discouraged
anyone from even exploring the geometry in detail.
The symmetry-obeying spatial curvature is the most
closely connected with the Riemann curvature tensor of the spatial metric
on a computational 3-space. The two curvature tensors
coincide in the slicing point of view
but differ in the threading point of view
\be \eqalign{
R\sym(n)_{abcd} =& R[g]_{abcd} \ ,\cr
R\sym(m)_{abcd} =& R[\gamma]_{abcd} + \Delta R(\Pscr,m)_{abcd} \ ,\cr
}\ee
where the difference term has the following representation on
the computational 3-space
\be \eqalign{
\Delta R(\Pscr,m)_{abcd} \mapsto
& \half \left\{
M_b M_c \ddot\gamma_{ad} - M_b M_d \ddot\gamma_{ac}
+ M_a M_d \ddot\gamma_{bc} - M_a M_c \ddot\gamma_{bd}
\right\} \cr
& \quad+ \half \left\{ (
2 M_{(b} \del[\gamma]_{c)} + \del[\gamma]_{(b} M_{c)}
+ M_{(b} \dot M_{c)} ) \dot\gamma_{ad} \right. \cr
& \qquad\quad \left. - (c\leftrightarrow d) + (a\leftrightarrow b)
-(c\leftrightarrow d) \,\right\} \cr
&\quad - \fraction14 g^{mn} \left\{
\dot\gamma_{\{mc} M_{a\}_-} \dot\gamma_{\{nd} M_{b\}_-}
- \dot\gamma_{\{md} M_{a\}_-} \dot\gamma_{\{nc} M_{b\}_-} \right\}\ . \cr
}\ee
The three terms in curly brackets in the threading point of view
expression each have all the usual symmetries of a curvature tensor,
leading to the same properties for the symmetry-obeying spatial curvature.
\Section{Initial value problem?}
\Subsection{Hypersurface and slicing points of view}
The geometry of the
initial value problem for the gravitational variables in the
hypersurface point of view is well known, often referred to as the
Cauchy problem.
The question one asks is,
can an initial spacelike hypersurface be imbedded in some spacetime
with certain sources present? Of the gravitational variables
only the intrinsic metric of the
hypersurface and its extrinsic curvature are required to state and
resolve this question. One must solve the gravitational constraints,
which consist of the three components of the supermomentum contraint
$\four E^\bot{}_a=0$ and the scalar super-Hamiltonian contraint
$\four E^\bot{}_\bot=0$. In these contraints only hypersurface derivatives
appear, so they represent a set of tensor equations within the hypersurface
itself.
These may be expressed entirely in terms of the
spatial metric $g_{ab}$ and the extrinsic curvature tensor $K_{ab}$
(and their spatial derivatives)
at an initial time on the computational 3-space of an adapted
nonlinear reference frame. The resolution of these equations via
conformal methods has a long history starting with Lichnerowicz
and Choquet-Bruhat
and ending with the work of York and his contemporaries. A useful
summary of this history may be found in a survey article by
Choquet-Bruhat and York\Cite{1980}.
Once an initial data set is found, then one must choose a nonlinear
reference frame to evolve this data and determine the spacetime which
contains the initial hypersurface. This amounts to a choice of the
lapse and shift variables, and this may be done arbitrarily, but
certain useful conditions have been found which lead to preferred
choices of nonlinear reference frames. Smarr and York discuss this
question in detail \Cite{Smarr and York 1978a,b??, York 1979}.
\Subsection{Thin sandwich problem}
In the slicing point of view, one may reinterpret the initial value
problem in terms of the gravitational variables and their time derivatives
on the computational 3-space at a given moment of time. The same equations
become constraints on the variables $(g_ab, \dot g_{ab}, N, N^a)$.
One method of solving these contraints is to specify the metric and its
time derivative arbitrarily and attempt to find values of the lapse and
shift which solve them. This problem, known as the thin sandwich problem,
is much more complicated and
seems to have no definitive answer, as discussed by Belasco and Ohanian
\Cite{1969}.
\Subsection{Congruence and threading points of view}
The initial value problem for the congruence point of view is fundamentally
different from the hypersurface point of view. Instead of initiating
a slicing of a spacetime
by considering a problem confined to an initial hypersurface
involving only hypersurface derivatives, one is trying to initiate
a timelike congruence in a spacetime. The basic gravitational variable is
a unit vector field defined on an initial hypersurface which no longer
requires a spacelike causality assumption, together with a metric on the
its orthogonal complement within the tangent space to the hypersurface
at each point.
%Perhaps nontimelike is the
%more appropriate condition, from what is known about
The fundamental difficulty with the corresponding gravitational
contraint equations
in the congruence point of view
is that they involve spatial derivatives of the gravitational variables
at the initial hypersurface,
but the spatial derivatives do not coincide with the hypersurface derivatives.
Since the congruence itself is the object one is trying to get started,
the lapse and shift variables $(M,M_a)$
(or variables equivalent to them)
must be specified in order to consider the initial value problem. In fact
there
is little point in not passing immediately to the threading point of view
and discussing the contraints in their representation on the
computational 3-space. In this sense the problem is much closer related
to the thin sandwich problem than the hypersurface initial value problem.
To evaluate spatial derivatives of spatial fields in their representation
on the computational 3-space, one must specify not only the field but
its time derivative. Both the first and second time derivatives of the
spatial metric $(\gamma_{ab})$ are necessary to consider the threading
constraints. First time derivatives appear in the components of the spatial
connection, while second time derivatives occur in the spatial curvature and
in spatial derivatives of the expansion tensor.
The expansion tensor is now so
closely related to the first time derivative of the spatial metric
$\theta_{ab} = (2M)^{-1} \dot\gamma_{ab}$ that there seems little point
to using a first order formalism.??
How can one deal with $\ddot \gamma_{ab}$ creeping into the initial value
problem? Does it make sense to try to solve the threading evolution
equations for this quantity and then replace it in the constraints?
This is equivalent to taking new linear combinations of the Einstein
equations related to the hypersurface directions.
Should one therefore abandon the threading splitting of the
Einstein equations for the initial value problem?
To investigate these questions one can look at two extremes:
stationary spacetimes where the time derivatives drop out, and Bianchi
type I cosmological models, where the slicing spatial derivatives drop out.
A third possibility is linearization, but this seems to eliminate some
of these ugly questions which start at second order.
Finally if one actually solves the initial value problem, to what extent
can one specify $(M,M_a)$ (consistent with the causality assumption) to
evolve $\gamma_{ab}$? How does the spatial gauge freedom enter into
the evolution in this description? And if one satisfies a hybrid
initial value problem, what does one do with the actual threading
constraints and the evolution?
\Newpage
\Subsection{Perfect fluids}
TO DO:\message{develop perfect fluid section??}
The threading point of view
lapse and shift play an important role in the dynamics of
an isoentropic perfect fluid in Taub's adapted comoving coordinate
system \Cite{Taub 1969, Taub and MacCallum 1972}.
Taub chooses the threading point of view adapted to
the fluid in the following way
\be
e_0=\mu^{-1}u\ ,\qquad M=\mu^{-1}\ , \qquad M_a=v_a=\mu u_a\ ,
\ee
where $u$ is the unit four-velocity of the fluid, $\mu$ is the chemical
potential of the fluid in the notation of
Misner, Thorne and Wheeler \Cite{1973},
and in this section only,
$v_a$ are the spatial components of the circulation 1-form
$v=\mu u^\flat$. (The time component has the value $v_0=-1$.)
Note also the relation $\gamma_L=u^\bot$.
This identifies the threading congruence with the fluid flowlines, but
chooses the parametrization to simplify the fluid equations of motion.
The choice of initial slice for the slicing is still arbitrary.
This choice of parametrized threading of the perfect
fluid spacetime makes the fluid equations of motion very simple
since the normal and tangential projections of the
conservation equations $T_\alpha{}^\beta{}_{;\,\beta}=0$ are just
\be
\four\div nu =0=\Lie_{\hbox{$e_0$}} v\ ,
\ee
where in this section only,
$n$ refers to the baryon number density in the
notation of Misner, Thorne and Wheeler \Cite{1973}.
The tangential equation just says that the circulation components are
time-independent in this computational frame, and the normal equation
says that the spatial scalar density
\be
\ell\equiv ng^{1/2}u^\bot=\gamma_L g^{1/2} n=Ng^{1/2} \mu n
=M\gamma^{1/2} \mu n
\ee
is time-independent as well.
(Note that $\mu n=\rho +p$, where $\rho$ is the energy density and $p$
the pressure.)
The Taub approach to isoentropic perfect fluids fits nicely into
threading point of view and allows an elegant geometrization in terms of
a principal fiber bundle in a way similar to
the stationary case described below.
The extension of Taub's work to the slicing point of view
by Bao, Marsden and Walton \Cite{1985}
might also benefit somewhat from
a comparison with the present framework; the alternative fluid gauge
variables of Walton are in fact related to a reference decomposition
of the fluid circulation vector.
\endinput
One can introduce a ``\emf{reference point of view}" analogous to the
slicing and threading points of view based on the reference decomposition
for the measurement process in place of the orthogonal decomposition
associated with an observer congruence. The evolution is simpler, defined
by comparison with Lie dragging along by $e_0$.
This is represented differentially by
the Lie derivative by $e_0$, which commutes with the reference projections,
so the reference spatially projected Lie derivative
\be
[\Pscr \Lie]\sub{e_0} S \equiv \Pscr [\Lie\sub{e_0} S]
\ee
acts on the reference spatial fields which represent the orthogonal
decomposition of a given spacetime tensor field.
However, unless at least one of the two causality conditions on the
nonlinear reference frame is satisfied, the use of the terms ``temporal"
and ``spatial" in this context has no meaning. This point of view
corresponds to what usually occurs when one works in an adapted coordinate
system in the old-fashioned component style of discussion.
In the terminology of the congruence point of view, the projected
computational frame is an observer-adapted frame which is spatially
comoving along the threading vector field
\be \eqalign{
\hbox{slicing:}\qquad &
\PLie(n)\sub{e_0} e_a = 0 = \PLie(n)\sub{e_0} \theta^a\ ,\cr
\hbox{threading:}\qquad &
\PLie(m)\sub{e_0} \epsilon_a = 0 = \PLie(m)\sub{e_0} \omega^a\ ,\cr
}\ee
a trivial consequence of the fact that these fields are invariant
under dragging along the threading congruence.
In the threading point of view
this implies that the spatially projected Lie derivatives $\PLie(m)\sub{e_0}
= M \PLie(m)\sub{m}$ of these spatial fields also vanish so the
projected computational frame is
just a spatially-comoving observer-adapted frame already used
extensively for evaluating the spacetime splittings of many interesting
fields and operators. These results may be taken over as is by expressing
the frame vectors and 1-forms with the present notation.
In the slicing point of view, it is instead a
more general class of frame not yet considered, so many results must
be rederived for this more general frame.
\endinput
The result is the following in the slicing point of view
\be \eqalign{
D(U,\Pscr,n)^2 x^a / d t^2
&= d^2 x^a / d t^2 + \Gamma(n)^a{}_{bc} d x^b / d t \,\, d x^c / d t \cr
&= N^2 [ \vec g(n) + \vec \nu(n) \times_n \vec H(n) ]^a \cr
& \qquad + [ \Lie(n)\sub{e_0} N \, \nu(n)^a - \Lie(n)\sub{e_0} N^a
+ N g^{ac} \{ \Lie(n)\sub{e_0} g_{cb}\} \nu(n)^b ]
+ N^a{}_{|b} N^b \cr
& \qquad
+ N^2 [ -2\vec g(n)\cdot_n \vec\nu(n)
+ \vec v \cdot_n \vec g(n)
+ \vec\nu(n) \cdot_n \bivec \theta(n) \cdot_n \vec\nu(n) ]
\nu(n)^a \ , \cr
}\ee
where
\be
\nu(n)^a = N^{-1} [ d x^a / d t + N^a ] \ .
\ee
In the threading point of view one has a simpler result since no relative
velocity of the observer enters the problem
\be\eqalign{
D(U,m)^2 x^a / d t^2
&= d^2 x^a / d t^2 + \Gamma(m)^a{}_{bc} d x^b / d t \,\, d x^c / d t \cr
&= M^2 [ \vec g(m) + \vec \nu(m) \times_m \vec H(m) ]^a \cr
& \qquad + \Lie(m)\sub{e_0} M \, \nu(m)^a +
M^2 \{ \nu(m) \times_m [ \vec M \times_m \Lie(m)\sub{e_0} \vec M ]\}^a
- 2 M^2 \theta(m)^a{}_b \nu(m)^b \cr
&\qquad + M^2 [ -2\vec g(m)\cdot_m \vec\nu(m)
- \{\Lie(m)\sub{e_0} \overeq M \} \rightcontract \vec\nu(m)
+ \vec\nu(m) \cdot_m \bivec \theta(m) \cdot_m \vec\nu(m) ]
\nu(m)^a \ , \cr
}\ee
where
\be
\nu(m)^a = M^{-1} d x^a / d t \ .
\ee
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Subsection{Rotating coordinates in flat spacetime}
No discussion of spatial gravitational forces would be complete without
relating them to the familiar notions of centrifugal and Coriolis forces
which arise in everyday experience. It is these forces which have been
generalized by the splitting discussion of
the acceleration equation, so it
is important to understand exactly how they fit into the general picture
as a special case.
The point of view in which they are discussed in classical mechanics is
neither the slicing or threading point of view but rather the reference
point of view.
%They appear not directly in
%the guise of either the
%slicing or threading spatial gravitational forces, but rather those which
%appear in the reference point of view decomposition of the geodesic equation.
Consider adapted coordinates $\{t,x^a\}$ for a nonlinear reference frame
in an arbitrary spacetime. The reference decomposition of the geodesic
equation considered as a (contravariant) vector field equation
is just its coordinate decomposition.
The calculation of the spatial reference projection of that equation
may be found in Forward's discussion \Cite{1961} of linearized
general relativity based on M\o ller's first edition discussion \Cite{1952} of
spatial gravitational forces using the coordinate time as the parameter for
the geodesic, but modified to use the spatial coordinate decomposition of the
geodesic equation rather than the threading decomposition. Linearization allows
one to sidestep the some of the complications which arise in the general case,
but the logarithmic time derivative of the
coordinate gamma factor $\Gamma$ %defined by \Eq(\ref{eq:coordgamma})
leads to an additional acceleration-dependent term proportional to
the velocity. (Compare with \Eq(6.25) of Misner, Thorne and Wheeler
\Cite{1973}.)
However, in flat spacetime with a coordinate system $\{t,x^a\}$ consisting of a
rigidly rotating system of orthonormal Cartesian spatial coordinates and
the usual time coordinate $t$ of the related nonrotating inertial coordinates,
the complications wash out and lead to the usual formulas
\be \eqalign{
\ddot t & = 0\ ,\cr
\ddot {\vec x} &= \vec g_{(ref)} + \dot{\vec x} \times \vec H_{(ref)} \ .\cr
}\ee
Here the dot is the usual time coordinate derivative and the usual
vector notation is employed. The spatial
forces are
\be \eqalign{
& \vec g_{(ref)} = -\vec\Omega \times (\vec\Omega \times \vec x)\ ,\cr
& \vec H_{(ref)} = 2\vec\Omega\ ,\cr
}\ee
where $\vec\Omega$ is the constant vector
describing the angular velocity of the rotating system,
identified with a constant vector field.
The reference gravitoelectric field is exactly the centrifugal force
(per unit mass) and the reference gravitomagnetic field vector is
just twice the global rotation of the coordinate system, leading to
the Coriolis force (per unit mass).
To see how these relate to the slicing and threading fields, one must
evaluate them for the nonlinear reference frame for which these coordinates are
adapted. The slicing is a flat slicing of Minkowski spacetime but
the threading is an inhomogeneous tilted threading relative to that slicing,
and it is timelike only within a certain cylinder in space called the light
cylinder
inside of which the
velocity of rotation is less than the speed of light. Thus the slicing
point of view holds everywhere,
%although the physical interpretation of
%``Space" breaks down outside the light cylinder,
while the threading
point of view is limited to the interior of the light cylinder.
Since the spatial metric is Euclidean in the slicing point of view, one can
use the customary vector notation unambiguously in expressing the line element
(instead of metric to allow the dot product notation)
\be \eqalign{
& ds^2 = - dt^2 + ( d\vec x + \vec\Omega \times \vec x dt) \cdot
( d\vec x + \vec\Omega \times \vec x dt)\ ,\cr
& N=1\ ,\qquad \vec N = \vec\Omega \times \vec x \ ,\qquad
g_{ab}=\delta_{ab}\ .\cr
}\ee
The differentials $ d\vec x + \vec\Omega \times \vec x dt$ are the rotated
differentials of the nonrotating Cartesian coordinates.
The slicing spatial gravitational fields in this notation are
\be
\vec g(n)=0 = \SYM \bivec H(n)\ ,\qquad
\vec H(n) = 2 \vec\Omega = \vec H_{(ref)}
\ee
and the expansion tensor vanishes $\theta(n)=0$.
The Lorentz boost parameters are
\be
\vec v= \vec N\ , \qquad \gamma_L =(1-v^2)^{-1/2}\ ,
\ee
leading to the threading quantities
\be \eqalign{
& M=\gamma_L{}^{-1}\ , \qquad
\overeq M = \gamma_L{}^2 \delta_{ab} v^a dx^b\ , \qquad
\gamma_{ab} = \delta_{ab} +\gamma_L{}^{-2} M_a M_b\ ,\cr
& g(m)_a = \gamma_L{}^2 \delta _{ab}g(m)_{(ref)}^b\ ,\qquad
H(m)^a = \gamma_L{}^2 H_{(ref)}^a\ .\cr
}\ee
Thus it is the gravitoelectric force in the threading point of view, apart from
a time reparametrization and a projection,
which corresponds to the centrifugal force.
The slicing gravitomagnetic vector field directly equals the reference
gravitomagnetic vector field, which in turn is twice the global angular velocity
of the coordinate system, and leads to the Coriolis force.
The threading gravitomagnetic vector field
differs from twice the constant global rotation vector
by the time reparametrization and a projection
in such a way that it equals the inhomogeneous
local rotation of the threading congruence.
Since the slicing spatial velocity differs from the reference spatial
velocity by the velocity of the threading curves
\be
\nu(n)^a = d x^a / d t + N^a \ ,
\ee
the slicing second-order acceleration equation takes the form
\be \eqalign{
D(U,\Pscr,n)^2 x^a / d t^2
&= d^2 x^a / d t^2 \cr
&= [ \vec\nu(n) \times_n \vec H(n) ]^a
+ N^a{}_{|b} N^b \cr
&= [ d\vec x /dt \,\times_n \vec H(n) ]^a
+ 2 N_{[b|}{}^a{}_] N^b+ N^a{}_{|b} N^b \cr
&= [ d\vec x /dt \,\times_n \vec H(n) ]^a
+ N_{b|}{}^a N^b \ .\cr
}\ee
The resultant shift derivative term corresponds to the reference
gravitoelectric field, i.e., the centrifugal force term.
For the threading point of view where there is no relative velocity term
\be
\nu(m)^a = M^{-1} d x^a / d t \ ,
\ee
the second-order acceleration equation is instead
\be\eqalign{
D(U,m)^2 x^a / d t^2
&= d^2 x^a / d t^2 + \Gamma(m)^a{}_{bc} d x^b / d t \,\, d x^c / d t \cr
&= M^2 [ -2\vec g(m)\cdot_m \vec\nu(m) \, \vec\nu(m)
+ \vec g(m) + \vec \nu(m) \times_m \vec H(m) ]^a \ .\cr
}\ee
Here the quadratic term in the spatial velocity involving the gravitoelectric
field cancels the connection term, leaving a direct correspondence between
the threading and reference spatial gravitational forces.
The spatial curvature in the threading point of view is easily evaluated from
the splitting of the vanishing spacetime curvature tensor.
Since the expansion tensor is zero, one has in the projected
computational frame
\be\eqalign{
& \four R^{ab}{}_{cd} = 0 \rightarrow \cr
& \qquad R\lie(m)^{ab}{}_{cd} = R\sym(m)^{ab}{}_{cd} =
- 2 \omega(m)^a{}_{[c} \omega(m)^b{}_{d]}
- 2 \omega(m)^{ab} \omega(m)_{cd} \ .\cr
}\ee
The double-sided spatial dual gives the equivalent Einstein tensor
\be
G\lie(m)^{ab}
G\sym(m)^{ab} = 3 \omega(m)^a \omega(m)^b
= \gamma_L{}^4 \Omega^a \Omega^b \ .
\ee\message{sign??}%
%pig
\message{!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!}
Given any timelike curve parametrized by the time function $t$, one can
rescale the tangent vector by the reciprocal lapse function
to correspond to a new parametrization by
the proper time $\tau_n$ and $\tau_m$ respectively measured by the
appropriate test observers.
Using a sloppy classical notation one has
the coordinate velocity $\Uscr$ and the unit 4-velocity $U$ defined by
\be \label{eq:coordgamma}
\Uscr^\alpha = dx^\alpha/dt\ ,\qquad
U^\alpha= dx^\alpha/d\tau_U = \Gamma(U) \Uscr^\alpha\ ,\qquad
\Gamma(U) =|\Uscr^\beta \Uscr_\beta|^{-1/2} = dt/d\tau_U\ ,
\ee
which in turn may be rescaled to represent the rate of change of
position with respect to observer proper time
\be
\eqalign{
\hbox{slicing:}\qquad
& d\tau_n/dt = N\ ,\qquad
dx^\alpha/d\tau_n = N^{-1} \Uscr^\alpha\ ,\cr
\hbox{threading:}\qquad
& d\tau_m/dt = M\ ,\qquad
dx^\alpha/d\tau_m = M^{-1} \Uscr^\alpha\ ,\cr
}\ee
where $\tau_U$ indicates the reparametrization of the curve by its own
proper time and $U$ is the unit tangent to the curve.
The quantity $\Gamma(U)$ is the ``coordinate gamma factor"
introduced by M\o ller \Cite{1952, 1972}.
When both points of view hold, the case of a spacelike
threading and a timelike threading, then
$d\tau_n/d\tau_m =N/M =\gamma(m,n)$ relates the two observer proper times
along this worldline and
defines the
Lorentz gamma factor of the boost in the plane of $n$ and $m$ which
maps $n$ onto $m$.
Defining the common notation for the observer lapse
\be
L(n) = N\ , \qquad L(m) = M \ ,
\ee
one can represent the gamma factor of a 4-velocity relative to the
observer congruence in terms of the coordinate gamma factor and the
lapse
\be
\gamma(U,o) = \frac{d\tau_o}{d\tau_U} =
\frac{d\tau_o}{dt} \frac{dt}{d\tau_U} = L(o) \Gamma(U) \ .
\ee
The coordinate gamma factor converts a proper time derivative along the
timelike worldline to a coordinate time derivative while
the lapse converts the observer proper time derivative to a coordinate
time derivative\Message{??problems here; see article}
\be
\frac{d}{d t} = ?? \frac{d}{d\tau_o} = \Gamma(U)^{-1} \frac{d}{d\tau_U} \ .
\ee
M\o ller's approach to relative kinematics is based on the coordinate time
parametrization of timelike worldlines.