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\Chapter{The congruence point of view and the measurement process}

Although special relativity may be studied with all of the
tools of general relativity, the existence
of a special class of ``inertial observers" in Minkowski spacetime
enables one to
simplify considerably the analysis of its spacetime geometry.
With each such observer in Minkowski spacetime one can associate
an entire  family of such observers of zero relative velocity filling the
spacetime and one may synchronize the proper times of all of them
so that they describe a global time function for the family.
This family constitutes the global reference frame
of the given observer. 
The 4-velocity field of this family of observers is a
covariant constant timelike unit vector field on Minkowski space.

Two inertial observers in relative motion lead to two distinct
such global reference systems.
Although the worldlines of two particular inertial observers in relative
motion may be transformed into each other by an active Poincar\'e
transformation of Minkowski space, their associated reference
frames may be transformed into each other with
a pure Lorentz tranformation or ``boost" of Minkowski spacetime
into itself about some arbitrary point.  This in turn may be identified
with the boost
which maps the 4-velocity associated with the
first reference frame onto that of the second 
in the tangent space at each point of
Minkowski spacetime. It is this 
globally constant
boost (the mixed tensor field which represents it is covariant constant)
which allows one transform
the measurements of physical quantities at some spacetime point
from one family of observers to the other.

However, in a general spacetime in general relativity, no such
preferred families of inertial observers exist, and one must rely on
arbitrary families of observers to establish a reference frame with
which to measure physical quantities. The global boost of special
relativity relating two families of inertial observers
gives way to a point by point boost
between the 4-velocities of the observers 
in the tangent space at each spacetime point associated with two families of
observers covering the spacetime. Thus all of the algebra associated
with a ``single" boost of Minkowski space between two inertial reference
frames must be repeated at each point independently in general relativity
in order to relate the two reference frames associated with two families
of observers. Furthermore, the differential operators associated with
a single family of observers necessarily reflect the state of motion
of those observers, leading to the description of noninertial effects
due to their acceleration and relative rotation.
The analysis of these questions is the natural generalization
of special relativity kinematics to general relativity.

\Section{Algebra}

Suppose $u$ is a future-pointing timelike unit vector field on our spacetime 
$\four M$
\beq\meqalign{
   -1 &= \four g_{\alpha\beta} u^\alpha u^\beta
      &= \four g^{\alpha\beta} u_\alpha u_\beta
      &= u_\alpha u^\alpha \ ,\cr
   -1 &= \four \gm (u,u) &= \four \gm^{-1}(u^\flat,u^\flat)
      &= u^\flat (u) \ .\cr
}\eeq 
It may be interpreted as the 4-velocity field of a family of
{\it test observers\/} whose worldlines are the integral curves of $u$, each of
which may be parametrized by the proper time $\tau_u$ which is defined up
to an additive constant on each curve. This defines an observer congruence on
the spacetime which may be used to induce a pointwise orthogonal decomposition 
of the spacetime tensor algebra. (Sachs and Wu \Cite{1977} call $u$ itself a
``reference frame" and its proper time parametrized integral curves the 
``observers".)

%%%%%%%%%
\Subsection{Observer orthogonal decomposition}

The orthogonal decomposition of the spacetime tensor algebra 
begins with the orthogonal direct sum of each tangent space into the
``{\it local time axis\/}" along $u$ and the ``{\it local rest space\/}" 
$LRS_u$ orthogonal
to $u$.  This may be accomplished using the
orthogonal projection operators $T(u)$ 
for the temporal projection and $P(u)$ for the spatial projection.
Each of them can be identified with a $1\choose1$-tensor field, defined by
\beq\imeqalign{
  T(u)^\alpha{}_\beta  &= -u^\alpha u_\beta \ ,\qquad&
            T(u) &= -u\otimes u^\flat\ ,\cr
  P(u)^\alpha{}_\beta  &= \delta^\alpha{}_\beta  + u^\alpha u_\beta \ ,\qquad &
            P(u) &= \four id + u\otimes u^\flat\ ,\cr
}\eeq 
which are related to the identity tensor, the
metric tensor and its inverse tensor in the following way
\beq\imeqalign{
   \delta^\alpha{}_\beta &= 
     T(u)^\alpha{}_\beta + P(u)^\alpha{}_\beta\ ,\qquad&
     \four id &= T(u) +P(u)\ ,\cr
   \four g_{\alpha\beta} &= T(u)_{\alpha\beta} + P(u)_{\alpha\beta} \ ,\qquad&
     \four \gm &= T(u)^\flat + P(u)^\flat \ ,\cr
  \four g^{\alpha\beta} &= T(u)^{\alpha\beta} + P(u)^{\alpha\beta} \ ,\qquad&
     \four \gm^{-1} &= T(u)^\sharp + P(u)^\sharp \ .\cr
}\eeq 
Here $T$ suggests ``time" or ``tangential" (to the congruence), while $P$ 
suggests
``perpendicular" or ``projection". The latter makes sense since often attention
is focused on the spatial projection tensor $P(u)$ which is conventionally 
denoted by the kernel symbol $h$, while the temporal projection $T(u)$
is usually suppressed in favor of contraction with the timelike unit vector
field or 1-form.

$1\choose1$-tensor fields may be interpreted as linear transformations of
each tangent space into itself
\beq
    X^\alpha \to  A^\alpha{}_\beta X^\beta\ , \qquad
    X \to  A \rightcontract X 
\eeq
and the right contraction corresponds to matrix multiplication of the
component matrices in the index-notation, both in this action and in their
successive action by composition
\beq
     A^\alpha{}_\gamma B^\gamma{}_\beta 
         = [ A\rightcontract B ]^\alpha{}_\beta \ .
\eeq
It is convenient to extend the ``square" notation for the self-composition
of a linear map with the corresponding self-right-contraction of the
tensor field
\beq
       [ A^2 ]^\alpha{}_\beta = A^\alpha{}_\gamma A^\gamma{}_\beta \ ,\qquad
             A^2 = A \rightcontract A \ .
\eeq
With these notational conventions, the orthogonal projection tensor fields
then satisfy the usual relations for an orthogonal decomposition
\beq\imeqalign{
    P(u)^\alpha{}_\gamma P(u)^\gamma{}_\beta &= P(u)^\alpha{}_\beta\ ,\qquad &
      P(u)^2 &= P(u)\ ,\cr
    T(u)^\alpha{}_\gamma T(u)^\gamma{}_\beta &= T(u)^\alpha{}_\beta\ ,\qquad &
      T(u)^2 &= T(u)\ ,\cr
    P(u)^\alpha{}_\gamma T(u)^\gamma{}_\beta 
        &= T(u)^\alpha{}_\gamma P(u)^\gamma{}_\beta = 0\ ,\qquad &
      P(u)\rightcontract T(u) &= T(u) \rightcontract P(u) = 0\ ,\cr
}\eeq
which follow from the unit character of $u$ and its orthogonality to the
spatial projection tensor
\beq\eqalign{
      P(u)^\alpha{}_\beta u^\beta &= 0 = u_\alpha P(u)^\alpha{}_\beta \ ,\cr
      P(u) \rightcontract u  &= 0 = u^\flat \rightcontract P(u) \ .\cr      
}\eeq

It is convenient to extend the projection operators to act on a tensor $S$
by understanding $P(u)S$ to mean the projection of $S$ on each index by the
projection $P(u)$, i.e., contraction of $S$ on each index by the appropriate
index of $P(u)$
\beq\eqalign{
  [P(u)S]^{\alpha\ldots}_{\ \ \beta\ldots} &=
     P(u)^\alpha{}_\gamma \cdots  P(u)^\delta{}_\beta \cdots 
        S^{\gamma\ldots}_{\ \ \delta\ldots} \ ,\cr
  [T(u)S]^{\alpha\ldots}_{\ \ \beta\ldots} &=
     T(u)^\alpha{}_\gamma \cdots  T(u)^\delta{}_\beta \cdots 
        S^{\gamma\ldots}_{\ \ \delta\ldots} \ .\cr
}\eeq 
These give the purely spatial and purely temporal parts of $S$. 

The orthogonal
decomposition of $S$ is accomplished by extending the orthogonal decomposition
to each of its indices, and expanding out the result
\beq\eqalign{
S^{\alpha\ldots}_{\ \ \beta\ldots} 
  &= \delta^\alpha{}_\gamma \cdots \delta^\delta{}_\beta \cdots 
      S^{\gamma\ldots}_{\ \ \delta\ldots} \cr 
  &= [T(u)^\alpha{}_\gamma +P(u)^\alpha{}_\gamma] \cdots 
   [T(u)^\delta{}_\beta +P(u)^\delta{}_\beta] \cdots 
      S^{\gamma\ldots}_{\ \ \delta\ldots} \cr 
  &= [T(u)S]^{\alpha\ldots}_{\ \ \beta\ldots} 
    + ({\rm mixed\ terms}) + [P(u)S]^{\alpha\ldots}_{\ \ \beta\ldots} 
   \ .\cr
}\eeq 
The coefficent tensors of those factors of $u$ with a free index in this formula
define a family of tensors of all ranks between zero and the rank of $S$.
These are the tensors which result from replacing the projection along $u$
with contraction by $-u^\delta$ or $-u_\gamma$ as appropriate. They represent
the spatial projections of all possible contractions of $S$ with any number of
factors of $-u$ or $-u^\flat$ as appropriate, and can be easily used
to reconstruct $S$ by factoring in the appropriate missing factors of 
$u^\flat$ and $u$ respectively. 
The reduction of a tensor $S$ to such a family of tensors will
be called the {\it measurement\/} of $S$ by the observer congruence.

A tensor which has been spatially projected on all of its free indices is
called a {\it spatial tensor\/} and
gives zero upon contraction of any index with
$u$. All of the members of the family of tensors which result from the
measurement of a single spacetime tensor are spatial tensors.
For example, the measurement of a $1\choose1$-tensor $S$
leads to the following result in the index notation
\beq\eqalign{
& \hskip 1truein S^\alpha{}_\beta \qquad \leftrightarrow\cr
& \{ 
  \underbrace{S^\gamma{}_\delta u^\delta u_\gamma}_{\rm scalar} ,
  \underbrace{-P(u)^\alpha{}_\gamma S^\gamma{}_\delta u^\delta}_{\rm vector} ,
  \underbrace{- u_\gamma S^\gamma{}_\delta P(u)^\delta{}_\beta}_{1-{\rm form}} ,
  \underbrace{ P(u)^\alpha{}_\gamma S^\gamma{}_\delta  P(u)^\delta{}_\beta}
          _{{1\choose1}-{\rm tensor}}
  \} \ .\cr
}\eeq 


$P(u)$ and its index-shifted forms are all spatial tensors which result from
the purely spatial part of the measurement of the identity tensor and
the metric and its inverse. The remaining parts of these measurements
are trivial
\beq\eqalign{
 \four id & \quad\leftrightarrow\quad \{ -1, 0, 0, P(u)\}\ ,\cr
 \four \gm & \quad\leftrightarrow\quad \{ -1, 0, P(u)^\flat\}\ ,\cr
 \four \gm^{-1} & \quad\leftrightarrow\quad \{ -1, 0, P(u)^\sharp\}\ .\cr
}\eeq 
For the symmetric metric tensor and its inverse
the two rank-one mixed parts are equal and vanish,
while for the mixed identity tensor, the two distinct rank one parts are
the zero vector field and the zero 1-form. In each case the rank zero or
spatial scalar part is the constant $-1=u_\alpha u^\alpha$.

When tensors have symmetries among their contravariant indices or among
their covariant indices, the measurement process may be collapsed
to list only the distinct spatial fields which result from the measurement,
as done above for the metric and its inverse.
The simplest case occurs for $p$-forms, which give only two distinct fields
since at most one contraction with $u$ is possible before spatially projecting
the result. No contraction yields
the purely spatial part or ``magnetic part" $p$-form
and one contraction yields the single mixed part or ``electric part"
$(p-1)$-form.
This splitting may be expressed in the form
\beq\imeqalign{
  [ S\E(u) ]_{\alpha_1\ldots \alpha_{p-1}} &= -u^\alpha S_{\alpha \alpha_1
      \ldots\alpha_{p-1}}\ , \qquad &
    S\E(u) &= - u \leftcontract S \ ,\cr
  [ S\M(u) ]_{\alpha_1\ldots \alpha_p} 
             &=[P(u) S]_{\alpha_1\ldots \alpha_p}\ , \qquad &
    S\M(u) &= P(u) S \ ,\cr
}\eeq 
while the orthogonal decomposition of $S$ has the form
\beq\eqalign{\label{eq:si}
   S_{\alpha_1 \ldots \alpha_p} 
         &= p! u_{[\alpha_1} S\E(u)_{\alpha_2\ldots \alpha_p]} 
              + S\M(u)_{\alpha_1 \ldots \alpha_p} \ ,\cr
   S &= u^\flat \wedge S\E(u) + S\M(u) \ .\cr
}\eeq 
This is equivalent to the following decomposition of the identity
operator $\ALT$ (the antisymmetrizer) on differential forms 
\beq
  \ALT =u^\flat\wedge(-u\leftcontract\,) + \ALT\circ P(u)\ ,
\eeq 
or in index notation, of the generalized Kronecker delta tensor $\delta^{(p)}$
\beq\eqalign{
  \delta^{\alpha_1\ldots\alpha_p}_{\ \ \beta_1\ldots\beta_p}
   &= p! \delta^{\alpha_1}{}_{[\beta_1} \cdots \delta^{\alpha_p}{}_{\beta_p]} 
           \cr
   &= - p! u^{[\alpha_1} u_{[\beta_1} \delta^{\alpha_2}{}_{\beta_2} \cdots
       \delta^{\alpha_p]}{}_{\beta_p]}
       +  \delta(u)^{\alpha_1\ldots\alpha_p}_{\ \ \beta_1\ldots\beta_p} \ ,\cr
}\eeq
where the spatial generalized Kronecker delta tensor 
$\delta(u)^{(p)} = P(u)\delta^{(p)}$ is defined by
\beq
    \delta(u)^{\alpha_1\ldots\alpha_p}_{\ \ \beta_1\ldots\beta_p}
     = p!  P(u)^{[\alpha_1}{}_{[\beta_1} \cdots
               P(u)^{\alpha_p]}{}_{\beta_p]}\ .
\eeq 
Notice that the contraction $u\leftcontract S$ is automatically spatial
due to the antisymmetry of $S$ (since $u\leftcontract\,(u\leftcontract S)=0$
or equivalently
$ u^\alpha u^\beta S_{\beta\alpha \gamma_3\ldots \gamma_p} = 0$)
and so requires no subsequent spatial
projection.

The (unit oriented) volume 4-form $\four\eta$ may be split in this way
but only its electric part survives since it is easily shown that
the magnetic part $P(u)\four \eta$ vanishes. The electric part
reversed in sign
\beq
    \eta(u) =u\leftcontract \four\eta \ ,\qquad \eta(u)_{\alpha\beta\gamma}
            = u^\delta \,\four \eta_{\delta\alpha\beta\gamma}
\eeq 
defines the (unit oriented) volume 3-form for the local rest space $LRS_u$.
In an (oriented, time-oriented) 
orthonormal frame adapted to $u$ and $LRS_u$, then
$\eta_{0123} = 1 = -\eta^{0123}$ and
$\eta(u)_{123}=1=\eta(u)^{123}$.
This spatial volume 3-form defines the spatial duality operation $\dualp{u}$
for antisymmetric spatial tensors and
the usual cross product of two vectors
$X$ and $Y$ in the local rest space
\beq
  X \times_u Y = \dualp{u} [X \wedge Y] \ ,\qquad
  [X \times_u Y]^\alpha = 
          \eta(u)^\alpha{}_{\beta\gamma} X^\beta  Y^\gamma\ .
\eeq 
One can also introduce the spatial ``dot product" between two spatial vectors
in order to mimic Euclidean vector analysis
\beq
  X \cdot_u Y = P(u)^\flat(X,Y) =P(u)_{\alpha\beta}X^\alpha Y^\beta \ ,
\eeq 
This just gives the spacetime inner product of the two spatial vectors.
Note that one can replace the spacetime metric tensor in any contraction
with spatial tensors by the appropriate valence spatial projection
which acts as the metric on the local rest space and its tensor algebra.

The spatial volume 3-form $\eta(u)$ is used to define the spatial duality
operation ``$\,^{\ast_{(u)}}\,$" for spatial differential forms
\beq
   \dualp{u} S_{\alpha_1\cdots \alpha_{3-p}} = {1\over p!}
               S_{\beta_1\cdots \beta_p} \eta(u)^{\beta_1\cdots \beta_p}{}
                   _{\alpha_1\cdots \alpha_{3-p}} \ ,
\eeq
It satisfies $\dualp{u}\, \dualp{u}  S = S$.
Using this spatial operation, one can
split the spacetime duality operation ``$\,^\ast\,$"
on differential forms.  Assuming the right dual convention of Misner,
Thorne and Wheeler \Cite{1973} this is given by
\beq
   \dual S_{\alpha_1\cdots \alpha_{4-p}} = {1\over p!}
               S_{\beta_1\cdots \beta_p} \eta^{\beta_1\cdots \beta_p}{}
                   _{\alpha_1\cdots \alpha_{4-p}} \ ,
\eeq
This operation instead satisfies $\dual\, \dual S= (-1)^{p-1} S$ for a
$p$-form $S$.
Using the definition of the electric and magnetic parts of a differential
form, one finds for a $p$-form $S$
\beq\eqalign{
    [\dual S]\E(u) &= (-1)^{p-1}\, \dualp{u} S\M(u) \ ,\cr
    [\dual S]\M(u) &=              \dualp{u} S\E(u) \ .\cr
}\eeq

In order to have unique index-free symbols for the three valence forms of
a 2-form, let $F$ ($F^\alpha{}_\beta$)
be the mixed valence form of a 2-form $F^\flat$ 
($F_{\alpha\beta}$) and let $F^\sharp$ 
($F^{\alpha\beta}$) be the 2-vector or contravariant form of the field.
Then the above relationship with $S= F^\flat$ is
\beq\label{eq:splitdualtwo}
    [\dual F^\flat]\E(u) = - \dualp{u} F^\flat{}\M(u) \ ,\qquad
    [\dual F^\flat]\M(u) =   \dualp{u} F^\flat{}\E(u) \ .
\eeq

\Typeout{Add electric and magnetic parts of 2-form, curvature 4-form.}

%%%%%%%%%%%%%
\Subsection{Observer-adapted frames}

Components with respect to a frame adapted to the observer orthogonal
decomposition can be quite useful in the splitting game.
An {\it observer-adapted frame} $\{e_\alpha\}$ with dual frame 
$\{\omega^\alpha\}$ will be any frame for which
$e_0$ is along $u$ and the ``spatial frame" $\{e_a\}$ spans the local
rest space $LRS_u$
\beq\meqalign{
  & u= \L^{-1} e_0 \equiv e_\top\ , & u^\flat (e_a)=0\ ,\cr
  & u^\flat= - L \omega^0 \equiv -\omega^\top\ ,\qquad &  \omega^a(u)=0\ .\cr
}\eeq 
The orientability and time-orientability assumptions imply
$\L>0$ and $\eta(u)_{123}>0$.
Thus for a time-oriented oriented orthonormal
observer-adapted frame, one has $L=1$ or $e_0 =u$.
A frame which only has $L=1$ will be called {\it partially normalized}.

The splitting of a tensor field $S$ amounts to a partitioning of the
components in an observer-adapted frame
according to whether or not individual indices are zero or not.
The purely spatial part corresponds to those components which have only
``spatial indices" 1,2,3,
while the purely temporal part corresponds to the single component with
every index equal to the ``temporal index" 0.
The remaining components parametrize the mixed parts of the tensor.
Spatial tensors have only the spatially-indexed components nonzero.
Thus any tensor equation in index-form involving only spatial tensor fields,
when expressed in an observer-adapted frame, reduces to an equation
whose only nonzero components correspond to replacing Greek indices with
Latin indices.

For a $1\choose1$-tensor $S$ one has
\beq\eqalign{
S & \qquad \leftrightarrow \qquad
  \{ S^0{}_0,   S^a{}_0,  S^0{}_a, S^a{}_b \}\cr
 & \qquad \leftrightarrow \qquad
  \{ S^\top{}_\top,   S^a{}_\top, S^\top{}_a, S^a{}_b \} \ ,\cr
}\eeq 
where the index ``$\top$" (pronounced ``tan")
suggests ``tangential" to the congruence 
(or even ``time" in this context) and
corresponds to the orthonormal temporal component obtained by scaling
the zero-indexed frame  component by the normalization factor $\L$
(evaluation on contravariant arguments on $e_\top$ and covariant
arguments on $\omega^\top$).
This family of spatial fields differs by a sign from the
contraction/projection algorithm described above which picks off the
coefficients of $u$ and $u^\flat$ rather than of $e_\top = u$ and 
$\omega^\top = -u^\flat$
in the expansion of a tensor. One can represent this decomposition in
index-free notation by using an index ``$\bot$" (pronounced ``perp")
standing for ``perpendicular"
to the congruence in place of the spatial index
\beq
S  \qquad \leftrightarrow \qquad
    \{ S^\top{}_\top,   S^\bot{}_\top, S^\top{}_\bot, S^\bot{}_\bot \} \ .
\eeq 
The symbol $\top$ was selected for being the reflection of the symbol
$\bot$ to a dual position.

Similarly a $p$-form $S$ splits in component form in the following way
\beq\eqalign{
\{ S_{\alpha_1 \ldots \alpha_p} \} & \qquad \leftrightarrow \qquad 
    \{ S_{0 a_2 \ldots a_p}  , S_{ a_1 \ldots a_p} \} \cr
 &\qquad \leftrightarrow \qquad 
    \{ S_{\top a_2 \ldots a_p}  , S_{ a_1 \ldots a_p} \} \cr
 &\qquad \leftrightarrow \qquad 
    \{ S_{\top \bot \ldots \bot}  , S_{ \bot \ldots \bot} \}=
    \{ -S\E(u), S\M(u) \}\ . \cr
}\eeq 

The components of the spacetime metric and its inverse in such a frame
may be represented in the conventional  form
\beq\meqalign{
& \four g_{00} = -\L^2\ ,\qquad & \four g_{0a} = 0\ ,\qquad 
     \four g_{ab} = P(u)_{ab} =h_{ab}\ ,\cr
& \four g^{00} = -\L^{-2}\ ,\qquad & \four g^{0a} = 0\ ,\qquad 
     \four g^{ab} = P(u)^{ab} =h^{ab}\ ,\cr
}\eeq 
or equivalently
\beq\eqalign{
\four \gm &= -\L^2 \omega^0\otimes \omega^0 + h_{ab} \omega^a\otimes \omega^b\cr
    &= - \omega^\top\otimes \omega^\top + h_{ab} \omega^a\otimes \omega^b
\ ,\cr
\four \gm^{-1} &= - \L^{-2} e_0\otimes e_0 + h^{ab} e_a\otimes e_b\cr
    &= - e_\top\otimes e_\top + h^{ab} e_a\otimes e_b\ .\cr
}\eeq 
The kernel symbol $P(u)$ is here being
replaced by $h$ as often used in specific
formalisms where the frame is understood to be adapted to a given congruence.
The spacetime metric determinant factor has the expression
$ \four g^{1/2} =  L h^{1/2}$, while the oriented spatial volume 3-form
has components $\eta(u)_{abc} =\four\eta_{\top abc}= h^{1/2}\epsilon_{abc}$.
The three valence forms of the spatial projection are
\beq 
P(u)^\flat = h_{ab} \omega^a\otimes \omega^b\ ,\qquad
P(u)^\sharp = h^{ab} e_a\otimes e_b\ ,\qquad
P(u)= h^a{}_b e_a\otimes \omega^b = e_a\otimes \omega^a \ .
\eeq 
Note that the cross product and dot product have the explicit expressions
\beq\eqalign{
  [X \times_u Y]^a &= \eta(u)^a{}_{bc} X^b Y^c
  = h^{1/2} h^{ad} \epsilon_{dbc} X^b Y^c \ ,\cr
  X \cdot_u Y &= h_{ab} X^a Y^b\ .\cr
}\eeq 

It is also quite useful to consider frames which are only adapted to
one of the two subspaces of the orthogonal decomposition of the
tangent spaces associated with a given family of test observers.
An ``{\it observer-partially-adapted frame}" $\{e_\alpha\}$ will be a frame
for which either 1) $e_0$ is along $u$ or 2) $\{e_a\}$ spans $LRS_u$
(equivalently $\omega^0$ is along $u^\flat$), i.e.,
that spacetime frame contains either a subframe for the distribution
of local time directions or for the distribution of local rest spaces.
Both conditions together characterize the observer-adapted frames.
Those observer-partially-adapted frames which are not observer-adapted will
be considered below in the context of a nonlinear reference frame.

%\Subsection{Electric and magnetic fields}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Typeout{Edit here: (electric and magnetic fields)}

For an electromagnetic 2-form $F^\flat$, the electric and magnetic
parts defined above correspond directly to the electric field and the
spatial dual of the magnetic field.
It is convenient to let $F$ (in index notation $F^\alpha{}_\beta$)
be the mixed form of the 2-form and
$E(u) = F\rightcontract u$ and $B(u)$ be the vector fields in the
index-free notation, so one has
\beq\eqalign{
     E(u)^\flat &= F^\flat{}\E(u) = - u \leftcontract F^\flat 
                  = F^\flat \rightcontract u \ ,\cr
     B(u)^\flat &= \dualp{u} F^\flat{}\M(u) \ , \qquad
            F^\flat{}\M(u) = \dualp{u} B(u)^\flat \ ,\cr
}\eeq
or equivalently
\beq\eqalign{
     E(u)_\alpha &= F_{\alpha\beta} u^\beta \ ,\cr
    B(u)^\alpha &= \half \eta(u)^{\alpha\beta\gamma} F_{\beta\gamma}
        = -\half \eta^{\alpha \beta\gamma\delta} F_{\beta\gamma} u_\delta\ .\cr
}\eeq
In an observer-adapted frame, these definitions take the index-form
\beq\eqalign{
    E(u)_a &= - F_{\top a} = F_{a\top} \ ,\cr
    B(u)_a &= \half \eta(u)_a{}^{bc} F_{bc} \ , \qquad
       F_{ab} = \eta(u)_{ab}{}^c B(u)_c \ .\cr
}\eeq
One may re-express the 2-form and its spacetime dual
in terms of the electric and magnetic 1-forms in the following way
\beq\eqalign{
   F^\flat &=  u^\flat \wedge E(u)^\flat + \dualp{u} B(u)^\flat \ ,\quad
   F_{\alpha\beta} =  2 u_{[\alpha} E(u)_{\beta]} 
        + \eta_{\alpha\beta}{}^{\gamma\delta} u_\gamma B(u)_\delta\ ,\cr
   \dual F^\flat &=  -u^\flat \wedge B(u)^\flat + \dualp{u} E(u)^\flat \ ,\quad
   \dual F_{\alpha\beta} =  -2 u_{[\alpha} B(u)_{\beta]} 
        + \eta_{\alpha\beta}{}^{\gamma\delta} u_\gamma E(u)_\delta\ .\cr
}\eeq

\Newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Typeout{Revise this section according to articles:}

\Subsection{Relative kinematics: algebra}\label{sec:relobsboost}

Suppose $U$ is another unit timelike vector field representing a different
family of test observers. One may then examine how the measurements 
of the two families of observers are related to each other.
This involves as well
the measurement of the various projection and boost
maps between their local rest spaces which are necessary to interpret
the ``transformation" from the spatial quantities and operators
of one set to those of the other.

The relative velocities of one set of observers with respect to the other
are defined by measuring the other's 4-velocity
\beq\eqalign{\label{eq:boost}
   U &= \gamma(U,u) [ u + \nu(U,u) ] \ ,\cr
   u &= \gamma(u,U) [ U + \nu(u,U) ] \ .\cr
}\eeq
The relative velocity $\nu(U,u)$ of $U$ with respect to $u$ 
is spatial with respect to $u$ and vice versa.
Both have the same
magnitude $|| \nu(U,u) ||= [ \nu(U,u)_\alpha \nu(U,u)^\alpha ]^{1/2}$, 
while the common gamma factor is related to that magnitude by
\beq
  -\four g(U,u) = \gamma(U,u) = \gamma(u,U) = [ 1 - ||\nu(U,u)||^2 ]^{-1/2}
    \ .
\eeq
Let $\hat\nu(U,u) = ||\nu(U,u)||^{-1} \nu(U,u)$ 
be the unit vector giving the direction of the relative
velocity $\nu(U,u)$, when nonzero.
Introduce also the energy and spatial momentum per unit mass relative to $u$
\beq
    \tilde E(U,u) = \gamma(U,u) \ , \qquad \tilde p(U,u) = \gamma(U,u) \nu(U,u)
  \ .
\eeq
Multiplication of these quantities by the nonzero
rest mass $m$ of a test particle whose worldline coincides with one of the
curves of the congruence of $U$
yields its energy $E(U,u)$ and spatial momentum $p(U,u)$ as
seen by the observers with 4-velocity $u$. These in turn
are the result of the measurement by $u$ of the 4-momentum $P=mU$
of the test particle.

In addition to the natural parametrization of the worldlines of $U$ by
the proper time $\tau_U$, one may introduce a new parametrization $\tau_{(U,u)}$
by
\beq\label{eq:repar}
     d \tau_{(U,u)} / d\tau_U = \gamma(U,u) \ .
\eeq
This corresponds to the sequence of
proper times of the  family of observers from the
$u$ congruence which cross paths with a given worldline of the $U$ congruence.
It is convenient to abbreviate $\gamma(U,u)$ by $\gamma$ when its meaning
is clear from the context in order to simplify the appearance of equations
which involve this factor.

When the relative velocity is nonzero, $U$ and $u$ define a 2-dimensional
subspace of each tangent space called the {\it relative observer plane}.
Its orthogonal complement is the subspace $LRS_u \cap LRS_U$ common
to both local rest spaces representing in each the directions orthogonal
to the direction of relative motion.
Equations (\ref{eq:boost}) describe a unique active ``relative observer boost"
$B(U,u)$ which acts as the identity on $LRS_u \cap LRS_U$ 
such that in the relative observer plane
\beq
   B(U,u) u = U\ , \qquad      B(U,u) \nu(U,u) = - \nu(u,U) \ . 
\eeq
The inverse
boost $B(u,U)$ ``brings $U$ to rest" relative to $u$.
It will be convenient to use the same symbol for a linear map of the tangent
space into itself and the corresponding $1\choose1$-tensor acting by
contraction. The right contraction between two such maps will
represent their composition. When the contraction symbol is suppressed,
the linear map will be implied.


The boost $B(U,u)$ restricts to an invertible map
$B\lrs(U,u) \equiv P(U)\circ B(U,u)\circ P(u) : LRS_u \to LRS_U$  
between the
local rest spaces which acts as the identity on their common
subspace.
Similarly the projection $P(U)$ restricts to an invertible
{\it relative projection} map $P(U,u)=P(U)\circ P(u):LRS_u \to LRS_U$  
with inverse $P(U,u)^{-1} : LRS_U \to LRS_u$
and vice versa, and these maps also act as the
identity on the common subspace of the local rest spaces.
The boosts and projections between the local rest spaces differ only by
a gamma factor along the direction of motion.
It is exactly the inverse projection map 
which describes Lorentz contraction of lengths along the direction of motion.
Figure \ref{fig:relobsplane}
illustrates these maps and the relative velocities
on the relative observer plane.

% emfig21.tex->emfig22.tex
%
%\fig{The relative observer plane and associated maps}{%
%The relative observer plane, the relative velocities and the associated
%relative observer maps.}{relobsplane}

\begin{figure}\typeout{figure gemfig21: relobsplane}
\label{fig:relobsplane}

$$ \vbox{
\beginpicture
  \setcoordinatesystem units <1cm,1cm> point at 0 100  %% to put baseline at top
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% rules:

    \putrule from 0 0   to 0 5.0 % vertical leg   u
    \putrule from 0 0   to 8.0 0  % hor leg       
    \putrule from 0 5.0 to 2.0 5.0 % short hor leg  \nu(U,u)
    \putrule from 7.5 0 to 7.5 3.0 % short ver leg  P(u,U)

% linear plots:
  \setlinear

    \plot 0 0 2.18  5.45  /  %U
    \plot 0 0 7.5 3.0  /     % -\hat\nu(u,U)  
    \plot 0 4.55 2.18 5.45 / %  \nu(u,U) 
    \plot 6.3 0 7.5 3.0 /    %  P(U,u)^{-1}

\setdashes

    \plot 0 0 5 5 /  %null

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% quadratic plots:  

\setsolid
\setquadratic

  \plot 6.875 0  6.95 0.9  7.05 1.5  7.15 1.9  7.5 3.0 /

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% arrowheads:

\setlinear

    \arrow <.3cm> [.1,.4]    from  0 4.7 to 0 5.0           % u
    \arrow <.3cm> [.1,.4]    from  2.06 5.15 to 2.18 5.45   % U
    \arrow <.3cm> [.1,.4]    from  1.7 5.0 to 2.0 5.0       % \nu(U,u)
    \arrow <.3cm> [.1,.4]    from  0.3 4.7 to 0 4.55        % \nu(u,U)
 
    \arrow <.3cm> [.1,.4]    from  4.7 0 to 5.0 0           %  \hat\nu(U,u)
    \arrow <.3cm> [.1,.4]    from  5.15 2.06 to  5.45 2.18  % -\hat\nu(u,U)
    \arrow <.3cm> [.1,.4]    from  7.2 2.88 to  7.5 3.0     %  X

    \arrow <.3cm> [.1,.4]    from  6.0 0 to 6.3 0           %  P^{-1} X
    \arrow <.3cm> [.1,.4]    from  6.575 0 to 6.875 0       %  B X
    \arrow <.3cm> [.1,.4]    from  7.2 0 to 7.5 0           %  B X
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% text:a:
  \put {\mathput{u}}                          [rB]   at  -.5 3.0
  \put {\mathput{U}}                          [lB]   at  1.7 3.0
  \put {\mathput{\nu(U,u)}}                   [rB]   at  0.2 5.4
  \put {\mathput{\nu(u,U)}}                   [lB]   at  1.5 5.8

  \put {\mathput{X}}                          [rB]   at  6.7 3.1
  \put {\mathput{\hat\nu(U,u)}}               [rB]   at  4.5 0.4
  \put {\mathput{-\hat\nu(u,U)}}              [rB]   at  4.8 2.4


%  \put {\mathput{P(U,u)X}}                               [lt]   at  8   -.3
%  \put {\mathput{B(U,u)= \gamma^{-1} P(u,U)X}}           [lt]   at  6.7 -1.0
%  \put {\mathput{P(u,U)^{-1}X = \gamma^{-1} B(u,U)X}}    [lt]   at  5.9 -1.7

  \put {\mathput{P(u,U)X}}                               [rt]   at  7   -1.7
  \put {\mathput{B(u,U)X = \gamma^{-1} P(u,U)X}}         [rt]   at  6.5 -1.0
  \put {\mathput{P(U,u)^{-1}X = \gamma^{-1} B(u,U)X}}    [rt]   at  5.8 -.3

\setquadratic
  \plot 5.90 -0.45 6.05 -0.30 6.15 -0.10 /
  \plot 6.60 -1.15 6.68 -0.80 6.725 -0.10 /
  \plot 7.10 -1.85 7.25 -1.50 7.35 -0.10 /

  \plot 0.30 5.50  0.45 5.40 0.50 5.10 /
  \plot 1.40 5.90  1.25 5.70 1.40 5.25 /

\endpicture}$$
% \noindent JANTZEN, CARINI, BINI: The Many Faces of Gravitoelectromagnetism.
%  page 5027.  Figure 1.
\caption{
The relative observer plane, the relative velocities and the associated
relative observer maps.
}
\end{figure}

If $Y\in LRS_u$, then the orthogonality condition $0=u_\alpha Y^\alpha$ implies
that $Y$ has the form
\beq\label{eq:otherspdec}
   Y= [ \nu(u,U) \cdot_U P(U,u) Y ] U + P(U,u) Y \ .
\eeq
If $X= P(U,u)Y \in LRS_U$ is the field seen by $U$, then $Y=P(U,u)^{-1} X$
and
\beq\eqalign{
    P(U,u)^{-1} X &= [ \nu(u,U) \cdot_U X ] U + X \cr
           &= [ P(U) + U \otimes \nu(u,U)^\flat ] \rightcontract X \ , \cr
}\eeq
which gives a useful expression for the inverse projection
in terms of the tensor which represents it.

Any map between the local rest spaces may be ``measured" by one of 
the observers, i.e., expressed
entirely in terms of quantities which are spatial with respect to
that observer.
For example, the mixed tensor representing the relative observer projection
\beq\eqalign{
   P(U,u) &= P(U) \rightcontract P(u) \cr
           &= P(U) \rightcontract P(U,u)
           = P(U,u) \rightcontract P(u) \cr
}\eeq
(which expands to $P(u) + \gamma  U \otimes \nu(U,u)$),
corresponding to the linear map $P(U,u) : LRS_u \to LRS_U$, is spatial
with respect to $u$ in its covariant index and with respect to $U$
in its contravariant index, i.e., is a ``connecting tensor" in the
terminology of Schouten \Cite{1954}.
It has associated with it two tensors
\beq\eqalign{
   P(U) &= P(U,u) \rightcontract P(U,u)^{-1} \ , \cr
   P(u) &= P(U,u)^{-1} \rightcontract P(U,u) \ , \cr
}\eeq
which are spatial with respect to $U$ and $u$ respectively and correspond
to identity transformations of each local rest space into itself.
In the same way any linear map $M(U,u) : LRS_u \to LRS_U$ is represented
by such a connecting tensor and has associated with it two tensors
$M_U(U,u)$ and $M_u(U,u)$ which are spatial with respect to $U$ and $u$
respectively and act as linear transformations of the respective
local rest spaces into themselves
\beq\eqalign{
    M(U,u) 
    &= M_U(U,u) \rightcontract P(U,u) 
            = P(U,u) \rightcontract M_u(U,u) \ , \cr
    M_U(U,u) &= M(U,u) \rightcontract P(U,u)^{-1} \ ,\cr
    M_u(U,u) &= P(U,u)^{-1} \rightcontract M(U,u) \ .\cr
}\eeq
These latter tensors enable one to express the map in terms of the spatial
projections of just one of the observers.

The relative boosts and projections all involve the further decomposition
of each local rest space into the subspaces parallel and perpendicular
to the direction of relative motion. Introducing the notation
\beq
       LRS_u = LRS\parp_u \oplus LRS\perpp_u 
\eeq
for these subspaces,
then $LRS\perpp_u = LRS\perpp_U = LRS_u \cap LRS_U$ is the common subspace
of the two local rest spaces and
and each of the above maps decompose into maps 
\beq\eqalign{ 
   M\parp(U,u) : LRS\parp_u \to LRS\parp_u \ ,\cr
   M\perpp(U,u) : LRS\perpp_u \to LRS\perpp_u \ ,\cr
}\eeq
where the latter map is the identity map for both the boosts and projections.

The individual projections parallel and perpendicular to the direction of
relative motion between the local rest spaces and within each local rest space
have the representations
\beq\imeqalign{
     P\parp(U,u) &= -\gammauu 
                    \hat\nu(u,U) \otimes \hat\nu(U,u)^\flat \ ,\qquad &
     P\perpp(U,u) &= P(U,u) - P\parp(U,u) \ ,\cr
     P_U\parp(U,u) &= \hat\nu(u,U) \otimes \hat\nu(u,U)^\flat \ ,\qquad &
     P_U\perpp(U,u) &= P(U) - P_U\parp(U,u) \ ,\cr
     P_u\parp(U,u) &= \hat\nu(U,u) \otimes \hat\nu(U,u)^\flat \ ,\qquad &
     P_u\perpp(U,u) &= P(u) - P_u\parp(U,u) \ ,\cr
}\eeq
where $ P(U,u) \hat\nu(u,U) = - \gammauu  \hat\nu(U,u)$ explains the $\gamma$
factor in the first relation.
The vector identity
\beq
  A \times_u [B\times_u C] = (A\cdot_u C) B - (A\cdot_u B) C
\eeq
may be used to express this decomposition of a spatial vector $X$ in the following way
\beq
  P_u\parp(U,u) X =  \hat\nu(U,u) [\hat\nu(U,u)\cdot_u X]\ ,\
  P_u\perpp(U,u) X =  \hat\nu(U,u) \times_u [\hat\nu(U,u)\times_u X]\ .
\eeq


These parallel and perpendicular projections in turn may be used to similarly 
decompose the boost $B\lrs(U,u)$ and
the inverse projection $P(u,U)^{-1}$, for which one has the obvious
relations (see Figure \ref{fig:relobsplane})
\beq\eqalign{\label{eq:ipbp}
     P\parp(u,U)^{-1} &= \gammauu ^{-1} B\lrs\parp (U,u)
              = \gammauu ^{-2} P\parp (U,u) \ ,\cr
     P\perpp(u,U)^{-1} &= B\lrs\perpp(U,u) = P\perpp(U,u) \cr
}\eeq
which may be used to reconstruct the spatial tensors associated with the
boost and inverse projection.

For example, for the inverse boost $B\lrs(u,U)$ one has
\beq\eqalign{
    B\lrs{}_u(u,U) &= P(u) 
     - \gammauu  (\gammauu  + 1)^{-1} \nu(U,u) \otimes \nu(U,u)^\flat
         \ ,\cr
    B\lrs{}_U(u,U) &= P(U) 
     - \gammauu  (\gammauu  + 1)^{-1} \nu(u,U) \otimes \nu(u,U)^\flat
         \ ,\cr
}\eeq
which follows from the expansion of
\beq\eqalign{  
    B\lrs{}_u(u,U) &= B\lrs{}\parp{}_u(u,U) + B\lrs\perpp(u,U)_u \cr
             &= P_u\parp(u,U) + \gammauu ^{-1} P_u\perpp(u,U) \ .\cr
}\eeq
Thus if $S\in LRS_U$, then its inverse boost is
\beq\eqalign{\label{eq:boostback} 
    B\lrs(u,U) S
    &= [ P(u) - \gamma(\gamma + 1)^{-1} \nu(U,u) \otimes \nu(U,u){}^\flat ] 
              \rightcontract P(u,U)S \ .\cr
}\eeq 

The map 
\beq
P(u,U,u) = P(u,U) P(U,u) = P(u) P(U) P(u) 
\eeq
is an isomorphism of $LRS_u$ into itself which
turns up in manipulations with these maps. It and its inverse 
(in the sense $P(u,U,u)^{-1} P(u,U,u) = P(u)$)
can be expressed in the following equivalent ways
\beq\eqalign{\label{eq:ppuu}
     P(u,U,u) 
   &=  P_u\perpp(u) + \gammauu ^2 P\parp_u(u,U)\ ,\cr
   &= P(u) + \gammauu ^2 \nu(U,u) \otimes \nu(U,u)^\flat \ ,\cr
     P(u,U,u)^{-1} &= P(U,u)^{-1} P(u,U)^{-1} = P_u(U,u)^{-1}\cr
   &= P_u\perpp(U,u) + \gammauu ^{-2} P\parp_u(U,u) \cr
   &= P(u) - \nu(U,u) \otimes \nu(U,u)^\flat \ .\cr
}\eeq
The inverse map represents a ``double Lorentz contraction"
along the direction of relative motion.

The map $P(U,u)^{-1}$ appears in the transformation law for the electric and magnetic fields.
Using the fact that
\beq
   [ P(U) \four F ] \rightcontract \nu(u,U) = \nu(u,U) \times_U B(U) \ ,
\eeq
one finds
\beq\eqalign{
  P(U,u) E(u)
       &= \gammauu  P(U) \{ \four F \rightcontract [ U + \nu(u,U) ] \} \cr
      &= \gammauu  [ E(U) + \nu(u,U) \times_U B(U) ] \ , \cr
}\eeq
and similarly
\beq
 P(U,u) B(u) = \gammauu  [ B(U) - \nu(u,U) \times_U E(U) ] \ .
\eeq
Equivalently one may write
\beq\eqalign{
 E(u) &= \gammauu  P(U,u)^{-1} [ E(U) + \nu(u,U) \times_U B(U) ] \ ,\cr
 B(u) &= \gammauu  P(U,u)^{-1} [ B(U) - \nu(u,U) \times_U E(U) ] \ .\cr
}\eeq

The transformation of the electric and magnetic fields takes a more
familiar form if one re-expresses it in terms of the parallel/perpendicular
decomposition of the boost using equation (\ref{eq:ipbp})
\beq\eqalign{ 
    E\parp(u) &= B\parp\lrs(u,U) E\parp(U) \ ,\cr
    E\perpp(u) &= \gammauu  B\perpp\lrs(u,U) 
          [ E\perpp(U) - \nu(u,U) \times_U B\perpp(U) ]  ,\cr
}\eeq
with analogous expressions for the magnetic field. When expressed in a
pair of orthonormal frames adapted to the two local rest spaces and related
by the boost, these reduce to the familiar component expressions in a
direct way.

The map $\gamma P(u,U)^{-1}$ appears rather than the projection $P(U,u)$
in these transformations due to the spatial duality operation.
Suppose $F(u)$ is a spatial 2-vector with respect to $u$. Then
$F(U) = P(U,u)F(u)$ is the purely spatial part of this 2-vector as seen by
$U$. Its spatial dual with respect to $U$ is
\beq\eqalign{
      \vec F(U) &= \dualp{U} F(U) = [ \dualp{U} P(U,u) \dualp{u} ] \vec F(u) \cr
                &= \gamma P(u,U)^{-1} \vec F(u) \ .\cr
}\eeq
This is easily verified by an index calculation
\beq\eqalign{
   F(U)^\alpha &= U_\delta \eta^{\delta\alpha}{}_{\beta\gamma} 
           u^\epsilon \eta_\epsilon{}^{\beta\gamma}{}_\sigma F(u)^\sigma\cr
       & = -  U_\delta u^\epsilon \delta^{\delta\alpha}_{\, \epsilon\sigma}
         F(u)^\sigma \cr
    & = -  U_\delta u^\delta F(u)^\alpha + u^\alpha U_\sigma F(u)^\sigma \cr
    & = \gamma [ F(u)^\alpha + u^\alpha \nu(U,u)_\sigma F(u)^\sigma ]
       = [ \gamma P(u,U)^{-1} \vec F(u) ] ^\alpha \ .\cr
}\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Typeout{TO DO:}
\Subsection{Splitting along parametrized spacetime curves and test particle worldlines}
%%%%%%%%%%%%%%%
\label{sec:splitU}

The gravitational field is felt through its action on test particles, whose paths in
spacetime (world lines) are either timelike (nonzero rest mass) or null (zero rest mass)
curves. However, other gravitational effects manifest themselves along spacelike 
curves---synchonization of clocks, for example. It is therefore necessary to consider
curves of all three causality types when viewing them in terms of a given observer family.

Given any parametrized curve $c$ in spacetime, with parameter $\lambda$,
its tangent vector $V(\lambda)=c^\prime(\lambda)$ is
defined in local coordinates by
\beq
        V^\alpha (\tau_U) = d x^\alpha (c^\prime (\lambda) ) 
               = d [ x^\alpha \circ c(\lambda) ] / d\lambda\ .
\eeq 
This tangent vector can be decomposed into its parts
parallel and perpendicular to the observer 4-velocity
\beq
     V(\lambda)^\alpha 
         = V(\lambda)^{(||_u)} \, u^\alpha + [P(u) V(\lambda)]^\alpha
\ ,
\eeq
where its temporal and spatial parts are respectively 
\beq
V(\lambda)^{(||_u)} = -u_\gamma V(\lambda)^\gamma\ ,\ 
[P(u)V(\lambda)]^\alpha = P(u)^\alpha{}_\gamma V(\lambda)^\gamma
\ .
\eeq

Dividing the spatial part by the temporal
part defines the relative velocity of the path in spacetime
repesented by the curve as seen by the observer family, 
regardless of its causality type.
This relative velocity and (when nonzero) the unit vector
defining its direction are defined by 
\begin{equation}
     \nu(V(\lambda),u)^\alpha 
         = [ P(u)V(\lambda) ]^\alpha / V(\lambda)^{(||_u)} \ ,
\end{equation}
and 
\beq
    \hat\nu(V(\lambda),u)^\alpha 
        = \sgn V(\lambda)^{(||_u)} 
         [ P(u)V(\lambda) ]^\alpha / ||P(u)V(\lambda)|| \ ,
\eeq
while the relative speed is the magnitude of the relative
velocity
\beq
    ||\nu(V(\lambda),u)||
        = ||P(u)V(\lambda)|| \ .
\eeq
For a null curve, the relative velocity is itself a unit vector
and the relative speed is 1. For a curve whose tangent vector
is spatial with respect to the observer family, this speed
goes to infinity.

The relative velocity, direction and speed are invariant 
under a change of parametrization which scales the tangent vector
by the rate of change of one parameter with respect to the other
by the chain rule
\beq
 \lambda \to \lambda(\lambda^\prime)\ ,\
 \frac{d}{d\lambda} \to 
           \frac{d}{d\lambda^\prime} 
    = \frac{d\lambda}{d\lambda^\prime} \frac{d}{d\lambda}\ .
\eeq

Suppose one has a timelike worldline $c$
parametrized by its proper time $\lambda=\tau_U$ 
with unit timelike tangent vector $U = c^\prime$.
In local coordinates one has
\beq
        U^\alpha (\tau_U) = d x^\alpha (c^\prime (\tau_U) ) 
               = d [ x^\alpha \circ c(\tau_U) ] / d\tau_U \ .
\eeq
This worldine may represent the path of a test particle of mass $m$ and
charge $q$ in spacetime.
Measurements made by a test observer following the same worldline
can be related to
the sequence of measurements made by the family of
different test observers of the observer
congruence whose worldlines intersect the given worldline.
All of the algebra developed above for two families of test observers 
may be restricted to the given worldline to relate the test particle
quantities to the single observer congruence.
In particular multiplying the energy and spatial momentum per unit mass
by the rest mass yields the the energy and spatial momentum
of the test particle
\beq\eqalign{
    E(P,u) &= m \tilde E(U,u) = m \gamma(U,u) \ ,\cr
    p(P,u) &= m \tilde p(U,u) = m \gamma(U,u) \nu(U,u) \ ,\cr
}\eeq
which themselves result from the measurement of
the 4-momentum $P = m U$
\beq
       P = E(P,u) u + p(P,u) = E(P,u) [ u + \nu(P,u) ] \ .
\eeq

One may also extend to a null worldline 
those quantities which do not rely on the timelike nature of $P$ 
in order to study a test particle with zero rest mass $m=0$. 
In this case
one must work directly with the 4-momentum $P$ and the energy and
spatial momentum.
\beq\eqalign{
    m{}^2 =
         - P_\alpha P^\alpha 
          &= E(P,u)^2 - || p(P,u)||^2 \cr
%          &= E(P,u)^2 [1 - \nu(P,u)^2 ] \cr
}\eeq
In the null case $m=0$, 
the relative velocity is a unit vector
$\nu(P,u) = \hat\nu(P,u)$, corresponding to the unit speed of light.
To handle simultaneously the cases of zero and nonzero rest
mass, introduce the ``rest mass per unit mass" parameter $\tilde m$
which has the value 0 for the first case and 1 for the second.

\Newpage
\Subsection{Addition of velocities and the aberration map}

Another map of physical interest between the local rest spaces of two 
observers is the
nonlinear aberration map between their unit spheres
which describes the relationship between the apparent
directions of a light ray as seen by the two observers.
This in turn is a special case of the addition of velocity formula
which relates the relative velocities of a given test  particle
with 4-momentum $P$ as seen by two different observers
with 4-velocities $u$ and $U$
\beq
    P = E(P,u) [ u + \nu(P,u) ]  = E(P,U) [ U + \nu(P,U) ] \ .
\eeq

One easily finds
\beq\eqalign{
    E(P,U) &= \gamma(U,u) [ 1 - \nu(U,u) \cdot_u \nu(P,u) ] E(P,u) \ ,\cr
    \nu(P,U) &= \gamma(U,u)^{-1} [ 1 - \nu(U,u) \cdot_u \nu(P,u) ]^{-1}
                             P(U,u) [ \nu(P,u) - \nu(U,u) ] \ .\cr
}\eeq
Re-expressing the relative projection in terms of the boost leads
to the formulas
\beq\eqalign{
    \nu(P,U)\perpp &= \gamma(U,u)^{-1} [ 1 - \nu(U,u) \cdot_u \nu(P,u) ]^{-1}
                         \nu(P,u)\perpp \ ,\cr
    \nu(P,U)\parp &= [ 1 - \nu(U,u) \cdot_u \nu(P,u) ]^{-1}
                        B\lrs(u,U) [ \nu(P,u)\parp - \nu(U,u) ] \ .\cr
}\eeq

The null case describes the 4-momentum of a photon or the tangent vector
to the path of a light ray, depending on the choice of language.
The relative velocity is then a unit vector giving the direction of the
photon or light ray.
Dividing the energy and spatial momentum
by  Planck's constant yields
the frequency $ \omega(P,u) = E(P,u) / \hbar$
and wave vector $k(P,u)= p(P,u)/ \hbar = \omega(P,u)\nu(P,u)$ of the photon.
The relationship between the observed frequencies
for the two different observer
congruences determines the relativistic Doppler effect 
\beq
   \omega(P,u) = \gammauu [ 1 - \nu(U,u) \cdot_u \nu(P,u) ]
         \omega(P,u)  \ .
\eeq
The above relationship between the unit relative velocites 
is the aberration map.
For the special case
\beq
             \nu(P,u)\parp = 0 \ ,
           \qquad  || \nu(P,u)\perpp || = 1
\eeq
of a light ray orthogonal to the relative motion as seen by $u$, one has
the familiar result
\beq
       \tan \theta_U = {|| \nu(P,U)\parp || \over || \nu(P,U)\perpp || }
      = \gammauu ||\nu(U,u) ||
\eeq
for the angle away from the perpendicular direction as seen by $U$, using
the fact that boosts preserve lengths.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Newpage
\Section{Derivatives}

The orthogonal decomposition associated with a choice of observer congruence
has been used to split each spacetime tensor field 
into a family of spatial fields
which represent its measurement by the observers.
The action of a spacetime differential operator on a spacetime tensor field
leads to a new tensor field which may be measured.  This leads to a
representation of the differential operator itself
in terms of two independent derivative
operators which act on the collection of spatial fields which represent
the spacetime tensor field being differentiated.
The spatial projection of the derivative along $u$ (when appropriate)
leads to a temporal
differential operator, while the spatial projection of the differential
operator itself yields a spatial version of the differential operator.
This splitting process may be applied to the natural derivatives, namely
Lie and exterior derivatives, and to the covariant derivative
and variations of it like the Fermi-Walker derivative.
%The latter derivative depends on the observer congruence

While the operators so defined preserve the spatial character of
tensor fields they differentiate, they may also be applied to nonspatial
tensor fields and this is actually useful in certain cases.
However, because of the spatial projection, any terms containing
undifferented factors of the observer 4-velocity will vanish, breaking
the appropriate product rule for tensor (or wedge) products
of nonspatial fields, while differentiated
factors lead to the appearance of kinematical factors.

When a second observer congruence is
present, one must also introduce relative derivative operators which preserve
the spatial character of fields which are spatial with respect to the
original observer congruence.
This is necessary in order to measure derivatives along the second
observer congruence of 
the quantities which result from the measurement by the original observer 
congruence of fields associated with the second congruence.
These operators may be restricted to the case of a single timelike
worldline of a test particle of nonzero rest mass and suitably
extended to the case of zero rest mass in order to study test particles
of either type.
This leads to the subject of apparent spatial gravitational forces
acting on test particles
due to the motion of the family of test observers, as well as their
consequences for the behavior of test gyroscopes or spinning test
particles.

\Subsection{Natural derivatives}

Two natural derivatives are important for various considerations, the
Lie derivative, which for vector fields becomes the Lie bracket, and the
exterior derivative. Each of these may be split as described above.
The {\it  spatial  Lie bracket} is defined by
for arbitrary vector fields $X$ and $Y$ by
\beq
   [X,Y](u) = P(u) [X,Y] =
   \Lie(u)\sub{X} Y   \ ,
\eeq
and in general the 
{\it spatial  Lie derivative} of an arbitrary
tensor $S$ is defined in the obvious way
\beq
   \Lie(u)_X S = P(u) [\Lie_X S]\ .
\eeq
Similarly the {\it spatial exterior derivative} 
of an arbitrary differential form $S$ is defined by
\beq
  d(u) S = P(u) d S\ .
\eeq 

Each of these spatial 
derivative operators can be applied to nonspatial fields $S$
but are perhaps most useful when applied to spatial fields. 
However,  when applied to nonspatial fields 
both operators violate the various product rules that usually hold
since the spatial projection eliminates terms with temporal factors.

For example,
when $S$ is spatial and 
the vector field $X = f u$ is a temporal field,
the spatial Lie derivative along $X$ has the special property of being
linear in the temporal component $X^\top =f$ 
due to the orthogonality properties
\beq\label{eq:liescale}
  \Lie(u)\sub{fu} S = f \Lie(u)\sub{u} S \ , \qquad S\ {\rm spatial}\ .
\eeq 
This property also holds for the spatial Lie derivative of the metric itself
\beq
  \Lie(u)\sub{fu} \four\gm = f \Lie(u)\sub{u} \four\gm 
\eeq
since it differs from the spatial field $P(u)^\flat$ only by the term
$ u^\flat \otimes u^\flat$ which has zero spatial Lie derivative along $u$.

For the same reason, the spatial exterior derivative of a temporal 1-form
$ X^\flat = f u^\flat$ 
is linear in the temporal component $X_\top = -f$
\beq
   d(u) [fu^\flat] =P(u) [ df\wedge u^\flat + f du^\flat]
                   = f d(u) u^\flat\ .
\eeq

The derivative $\Lie(u)\sub{u}$ provides a natural time derivative
which will be referred to both as the {\it temporal Lie derivative}
and as the {\it Lie temporal derivative}, 
while the derivative $\Lie(u)\sub{X}$ along a spatial vector field $X$
provides a natural spatial derivative.
These two operators are necessary to describe
the splitting of the Lie derivative. On the other hand, 
the temporal Lie derivative and the spatial exterior derivative together
are necessary to describe the splitting of the exterior derivative.

It is convenient to introduce the alternate notation
$\del\lie(u) = \Lie(u)\sub{u}$
for the Lie temporal derivative associated with $u$. This will be needed
for efficient handling of three different preferred temporal derivatives.

%%%%%%%%%%%%%%
\Subsection{Covariant derivatives}

Each spacetime tensor field $S$ can be split into its representative
family of spatial fields by the measurement process. Its covariant derivative
$\four \del S$ can be split in exactly the same way and the result can be
represented in terms of two projected differential operators acting either on
members of
the family of spatial fields representing $S$ or on $u$ itself, generating
the kinematical quantities associated with the observer congruence.

The two projected differential operators are the {\it spatial covariant 
derivative}, for differentiating along spatial directions,
and the spatial Fermi derivative, or 
{\it spatial Fermi-Walker derivative},
for differentiating along the local time direction of the observer congruence.
They are defined for an arbitrary tensor field $S$ by
\beq\eqalign{
 \del(u) S &= P(u) [\four\del S]\ ,\cr
 \del\fw (u) S &= P(u) [\four\del\sub{u} S]\ .\cr
}\eeq 
In index notation it is convenient to introduce the double vertical bar symbol
in place of the semicolon for the alternative notation for the spatial
covariant derivative
\beq\eqalign{
 [\four \del S]^{\alpha\ldots}_{\ \ \beta\ldots \gamma} &=
   \four \del_\gamma S^{\alpha\ldots}_{\ \ \beta\ldots} = 
  S^{\alpha\ldots}_{\ \ \beta\ldots ;\gamma}\ ,\cr
 [\del(u) S]^{\alpha\ldots}_{\ \ \beta\ldots \gamma} &=
   \del(u)_\gamma S^{\alpha\ldots}_{\ \ \beta\ldots} = 
  S^{\alpha\ldots}_{\ \ \beta\ldots ||\gamma}\ .\cr
}\eeq 

For a spatial tensor $S$, the spatial Fermi-Walker derivative coincides with 
the Fermi-Walker deriva\-tive \Cite{Walker 1932} 
which in turn reduces to the Fermi derivative \Cite{Fermi 1922} 
defined only for spatial tensors. For a nonspatial tensor,
the spatial Fermi-Walker derivative and the Fermi-Walker derivative 
yield distinct results. One might also refer to the spatial Fermi-Walker
derivative as the {\it Fermi-Walker temporal derivative}  in analogy with
the Lie temporal derivative, a term which is more suggestive
of the direction of differentiation.

Similarly the spatial covariant derivative may be applied to a nonspatial
tensor, although it is most useful when applied to spatial tensors. One can
speak of its spatial part $\del(u)\circ P(u)$ and its temporal part
$\del(u)\circ T(u)$.
In the literature the spatial covariant derivative is consistently defined 
to act only on spatial tensor fields, so only the geometry of the spatial
part of spatial covariant derivative $\del(u)$ as defined here has been
explored.
This restricted derivative was mentioned
by M\o ller \Cite{1952} and studied by Cattaneo \Cite{1958,1959} who
called it the ``transverse covariant derivative", while its associated 
curvature
was mentioned by Zel'manov \Cite{1956} and more deeply explored by
Cattaneo-Gasperini \Cite{1961} and Ferrarese \Cite{1963,1965}.
Ellis has recently used the terminology ``orthogonal covariant derivative"
\Cite{Ellis, Hwang and Bruni, 1990}.

The spacetime metric and the volume 4-form are obviously covariant constant
with respect to the spatial covariant derivative.
It is straightforward to show that the spatial projection and each of its
index-shifted forms are all covariant constant with respect to the spatial
covariant derivative, as is the spatial volume 3-form
\beq
    \del(u) \four g = \del(u) P(u) = \del(u) P(u)^\flat = 0\ , \ \hbox{etc.}
\eeq
Thus index shifting of spatial fields and the spatial duality operation
commute with the spatial covariant derivative.

Fermi-Walker transport of a spatial tensor along the observer congruence
is defined by requiring its  spatial Fermi-Walker derivative to vanish.
Similarly spatial parallel transport of a spatial tensor along a curve 
everywhere orthogonal to the observer congruence is defined by requiring
that its spatial covariant derivative along the curve vanish. Under both
transports spatial tensors remain spatial by undergoing a unique 1-parameter
family of boosts relative to parallel transport in order to preserve
the orthogonality with $u$.
Since parallel transport itself preserves inner products,
so do these boosted transports, which means that the spatial projections
and volume 3-form are also covariant constant with respect to the
spatial Fermi-Walker derivative
\beq
  \del\fw(u) P(u) = \del\fw(u) P(u)^\flat = \del\fw(u) \eta(u) = 0
        \ ,\ \hbox{etc.}
\eeq
This also easily follows from the spatial projection of the vanishing of
the covariant derivatives of the spacetime metric and 4-volume form.
Thus index-shifting and the
spatial duality operations commute with the spatial Fermi-Walker
derivative, exactly as for the spatial covariant derivative.
Fermi-Walker transport of a spatial vector along the observer
congruence has the physical interpretation of describing the behavior
of the spin vector
of a test torque-free gyroscope (gyro) carried by an observer.


%%%%%%%%%%%%%%
\Subsection{Kinematical quantities}

The first nontrivial
covariant derivative to split is $\four\del u$, or $u^\alpha{}_{;\beta}$
in index-notation,
namely the covariant
derivative of the observer congruence 4-velocity vector field.
This mixed tensor may
be identified with a linear transformation of the tangent space, which
is connected with its physical interpretation.
The time projection on the contravariant index vanishes due to the unit nature
of $u$
\beq
     u_\alpha  u^\alpha{}_{;\beta} =\half (u_\alpha u^\alpha)_{;\beta} =0\ ,
\eeq 
so only two of the four terms in the splitting survive.
In index-free form one has
\beq\meqalign{
 \four\del u &= - a(u)\otimes u^\flat  +\del(u) u\ ,\cr
& a(u) = \four\del_u u =\del\fw (u) u\ ,\cr
& \del(u) u = -k(u) = \theta(u) -\omega(u)\ ,\cr
}\eeq 
while the index-lowered form is usually given when expressed in the index form
\beq\eqalign{
 & [\four\del u^\flat]_{\alpha\beta} = \four\del_\beta u_\alpha
      = u_{\alpha;\beta}
  = \theta(u)_{\alpha\beta} -\omega(u)_{\alpha\beta} - a(u)_\alpha u_\beta
          \ ,\cr
  &\qquad k(u)_{\alpha\beta} = - P(u)^\gamma{}_\alpha P(u)^\delta{}_\beta 
              u_{\gamma;\delta} = -\theta(u)_{\alpha\beta} 
                               + \omega(u)_{\alpha\beta}\ ,\cr
  &\qquad \theta(u)_{\alpha\beta} =P(u)^\gamma{}_\alpha P(u)^\delta{}_\beta 
              u_{(\gamma;\delta)}\ ,\cr
  &\qquad \omega(u)_{\alpha\beta} = -P(u)^\gamma{}_\alpha P(u)^\delta{}_\beta 
              u_{[\gamma;\delta]}\ ,\cr
  &\qquad a(u)_\alpha =u_{\alpha;\beta} u^\beta\ ,\cr
}\eeq 
The expansion tensor $\theta(u)^\flat$ and 
the rotation or vorticity tensor $\omega(u)^\flat$ are just the symmetric and
sign-reversed antisymmetric parts of the spatial covariant derivative of 
$u^\flat$.
The expansion tensor is then decomposed into its pure-trace and tracefree 
parts to define the expansion scalar
\beq
  \Theta(u)=\Tr \theta(u)=\theta(u)^\alpha{}_\alpha
\eeq
and the shear tensor
\beq
 \sigma(u)=\theta(u)-\fraction13\Theta(u)P(u)\ ,\qquad
 \sigma(u)_{\alpha\beta}=\theta(u)_{\alpha\beta}
      -\fraction13\theta(u)^\gamma{}_\gamma P(u)_{\alpha\beta}\ .
\eeq 
The expansion scalar, shear tensor and rotation tensor,
together with the acceleration vector, are referred to
as the kinematical quantities of the vector field $u$ or of its
associated congruence of integral curves.

The acceleration vector field is spatial due to the unit character of
$u$
\beq
        u_\alpha a(u)^\alpha = u_\alpha u^\alpha{}_{;\beta} u^\beta
            = \half [ u_\alpha u^\alpha{}]_{;\beta} u^\beta = 0\ .
\eeq
If it is identically zero the observer worldlines are geodesics and the
observers are inertial observers. However, in contrast with the flat
spacetime case, families of inertial observers are often less convenient
to use than noninertial observers unless they happen to admit
additional special properties shared by the usual translation invariant
families of
inertial observers in the flat case. This is due to the attractive
nature of gravitation which causes the inertial observers to 
collapse together unless other conditions oppose this behavior.

The remaining kinematical properties reside in the spatial tensor $k(u)$.
This mixed tensor may be thought of as a linear transformation
of the local rest space into itself, and the decomposition of this linear
transformation into its irreducible parts has a very physical interpretation.
This interpretation involves a comparison with Lie dragging along $u$,
for which the operator $\Lie(u)\sub{u}$ is relevant.
One may express 
the kinematical quantities of $u$ entirely in terms of natural
derivatives  acting on $u^\flat$ or its associated covariant or contravariant
spatial projections
\beq\eqalign{
 du^\flat &= -u^\flat \wedge a(u)^\flat + 2 \omega(u)^\flat \ ,\cr
  a(u)^\flat &= -[du^\flat]\E = u\leftcontract du^\flat = 
       \Lie\sub{u} u^\flat = \Lie(u)\sub{u} u^\flat \ ,\cr
  \omega(u)^\flat &= \half [du^\flat]\M =  \half d(u) u^\flat \ ,\cr
       &\cr
  \Lie\sub{u} P(u)^\flat &= \Lie(u)\sub{u} P(u)^\flat = 
     \Lie(u)\sub{u} \four\gm =  2\theta(u)^\flat\ ,\cr
  \Lie\sub{u} P(u)^\sharp &= \Lie(u)\sub{u} P(u)^\sharp =
       \Lie(u)\sub{u} \four\gm^{-1} = -2\theta(u)^\sharp\ ,\cr
  \Lie\sub{u} P(u) &= u \otimes a(u)^\flat\ , 
      \qquad a(u)^\flat = -u\leftcontract \, [\Lie\sub{u} P(u)]^\flat\ ,
      \qquad   \Lie(u)\sub{u} P(u) =0 \ .\cr
}\eeq
 
The negative sign in the antisymmetrized covariant derivative used to define
the rotation tensor arises because of the convention of adding the additional
exterior derivative index to the left while the additional covariant
derivative index adds to the right. For a 1-form, the two differ by a sign.
The factor of two arises since the antisymmetrized covariant derivative
(for a symmetric connection) is half the exterior derivative, modulo the
sign difference.
The definition of the rotation tensor by Ehlers \Cite{1961, 1993} and Ellis 
\Cite{1971} has the opposite sign, while Misner, Thorne and Wheeler \Cite{1973}
use different signs in the two sections in which kinematical quantities
are discussed.

Since the  spatial  Lie derivative by $u$ of the spatial
metric yields (twice) the expansion tensor,
this derivative operator does not commute
with index-shifting on spatial fields when the expansion tensor is nonzero.
Similarly the result
\beq
  \Lie\sub{u} \eta(u) =\Lie(u)\sub{u} \eta(u)
  = u\leftcontract \Lie\sub{u} \four \eta
  = \Theta(u) \eta(u) 
\eeq 
shows that it does not commute with the spatial duality operation either
as long as the expansion scalar is nonzero. For example,
for a spatial differential form $S$ one has the relation
\beq\label{eq:liederspadual}
   [ \dualp{u} \Lie(u)\sub{u} \dualp{u} S]^\sharp
          = [\Lie(u)\sub{u} + \Theta(u)] S^\sharp \ .
\eeq\message{check??}

However, the spatial projection tensor $P(u)$ does have zero 
 spatial  Lie derivative 
along $u$, so the spatial projection operator
and the  spatial  Lie derivative along $u$ do commute
\beq
    [ \Lie(u)\sub{u}, P(u)] = 0\ ,
\eeq 
which means
\beq
     \Lie(u)\sub{u} P(u) S = P(u) \Lie\sub{u} S =  \Lie(u)\sub{u} S \ .
\eeq 
Thus if a tensor is Lie dragged along $u$, its Lie derivative by $u$ is
zero which implies that the  spatial  Lie derivative of its
spatial projection is also zero. It is natural to define 
{\it spatial  Lie transport} 
of a spatial field along $u$ by requiring that its
 spatial  Lie derivative along $u$ be zero. Then this result
means that the spatial projection of Lie transport along $u$ of a tensor
yields the
 spatial  Lie transport along $u$ of the spatial projection
of that tensor.

The rotation tensor characterizes the failure of the Lie bracket of two
spatial vector fields to again be spatial.  The following short calculation
for a pair of spatial vector fields $X$ and $Y$
using the invariant definition of the exterior derivative of a 1-form
\beq\eqalign{
  2\omega^\flat(u)(X,Y) &= du^\flat (X,Y) = X u^\flat (Y) - Y u^\flat (X)
     - u^\flat([X,Y])  = - u^\flat([X,Y]) \ ,\cr
  T(u) [X,Y] &=  - u^\flat([X,Y]) u = 2\omega(u)^\flat(X,Y) u \ ,\cr
}\eeq 
shows that the nonspatial part of this Lie bracket vanishes only if the
rotation vanishes. The spatial part defines the  spatial 
Lie bracket of the two spatial vector fields. When the rotation vanishes
this corresponds to the Lie bracket defined within the hypersurfaces
orthogonal to the congruence.

The covariant rotation tensor is just half the spatial exterior derivative of 
$u^\flat$,
thus agreeing with its classical 3-dimensional definition in the Euclidean 
case \Cite{Synge and Schild 1949}.
Its index-raised dual defines the spatial rotation or vorticity vector
\beq\eqalign{
  \vec \omega(u) &= \dualp{u} \omega(u) = 
         \half [\dual (u^\flat \wedge du^\flat)]^\sharp\ ,\cr
  \omega(u)^\alpha &= \half u_\beta\, \four\eta^{\beta\alpha\gamma\delta}
        \four\del_\gamma u_\delta =
 \half \four\eta^{\alpha\beta\gamma\delta} u_\beta u_{\gamma;\delta} \ .\cr
}\eeq 
The sign of this formula is affected both by the choice of index range
(0,1,2,3 or 1,2,3,4) and the signature, each of which varies in the
literature.  For the index range 1,2,3,4, the spatial volume 3-form
is defined by the right contraction $\four\eta \rightcontract u$ rather
than the left contraction, while for the opposite signature ($+---$ or
$---+$) the covariant spatial components of a vector field change sign
relative to the contravariant spatial components, forcing another sign
change to correspond to the classical spatial definition.  
For example, Ellis \Cite{1973} in one of his review articles uses the
convention $\four\eta_{0123}=-1$ in an orthonormal frame with signature
$-+++$ and defines the covariant rotation tensor as minus the spatial
exterior derivative of $u^\flat$ for a total of two sign changes, leading
to the same formula as given here.

A stationary spacetime is characterized by the existence of a timelike
Killing vector field $\xi$ which Lie derives the metric
\beq
   \Lie\sub{\xi} \four\gm_{\alpha\beta} = 2 \xi_{(\alpha;\beta)} = 0 \ .
\eeq
The 4-velocity vector field of a congruence of ``{\it Killing observers}"
can be defined by normalizing the Killing vector field
\beq
           u = \ell^{-1} \xi \ , \qquad \ell = | \four\gm(\xi,\xi) |^{1/2} \ .
\eeq
The normalization factor is also stationary, i.e., invariant under this
symmetry, since
\beq
     \Lie\sub{\xi} [ \four\gm_{\alpha\beta} \xi^\alpha \xi^\beta ] 
    = [ \Lie\sub{\xi}  \four\gm_{\alpha\beta} ]\xi^\alpha \xi^\beta  =0
\eeq
implies $\Lie\sub{\xi} \ell =0$.

For such an observer congruence, a ``{\it stationary observer congruence}",
the expansion tensor vanishes
\beq
  \theta(u) = \half \Lie(u)\sub{u} \four\gm 
       = \half \ell^{-1} \Lie(u)\sub{\xi} \four\gm
       = \half \ell^{-1} P(u) \Lie\sub{\xi} \four\gm =0 \ .
\eeq
Similarly the acceleration admits a spatial potential which is the natural
logarithm of the normalization factor $\ell$
\beq\eqalign{\label{eq:statacc}
  a(u)^\flat &= \Lie(u)\sub{u} u^\flat 
    =P(u) \Lie\sub{\ell^{-1}\xi} [\ell^{-1} \xi^\flat ] = \ldots \cr
    &= d(u) \ln \ell = d \ln \ell\ , \cr
}\eeq
where the last equality follows from the stationarity of $\ell$.
Only in the event that $u$ itself is a Killing vector field must the
normalization factor be constant, leading to vanishing acceleration.

%%%%%%%%%%%%
\Subsection{Splitting the exterior derivative}

The rotation and acceleration appear in the splitting of the spacetime exterior
derivative, which may be expressed in terms of the spatial exterior derivative
and the temporal Lie derivative in the same way that the
spacetime covariant derivative can be expressed in terms of the spatial 
covariant derivative and the spatial Fermi-Walker derivative, together with 
the kinematical quantities.
To carry out the splitting of the exterior derivative, one needs several 
identities. Wedging the splitting identity (\ref{eq:si})
by $u^\flat$ yields the useful preliminary identity
\beq
   u^\flat \wedge S = u^\flat \wedge P(u) S\ .
\eeq 
The Lie derivative of a differential form is related to the exterior 
derivative by the well known identity
\beq
   \Lie\sub{u} S = d(u\leftcontract S) + u \leftcontract dS\ .
\eeq 
For a spatial differential form $S$, the contraction of $S$ by $u$ vanishes
(and $u\leftcontract dS$ is automatically spatial) so
\beq
   u\leftcontract dS = \Lie\sub{u} S=  \Lie(u)\sub{u} S\ ,
    \qquad u\leftcontract S =0\ ,
\eeq 
and hence applying the splitting identity (\ref{eq:si}) one obtains
\beq
 dS = -u^\flat \wedge \Lie(u)\sub{u} S + d(u) S \ ,\qquad 
     u\leftcontract S=0\ .
\eeq 
Using these identities it is straightforward to evaluate the electric and
magnetic parts of the exterior derivative of an arbitrary differential
form
\beq\eqalign{\label{eq:splitextder}
[dS]\E &= -[d(u) + a(u)^\flat\wedge ]S\E 
             - \Lie(u)\sub{u} S\M \ ,\cr
[dS]\M &= d(u) S\M + 2 \omega(u)^\flat \wedge S\E\ .\cr
}\eeq 

The measurement of the identity $0= d^2 u^\flat$ leads to identities
for the spatial exterior derivatives of the acceleration and rotation
forms. Just substituting 
$S=d u^\flat = - u^\flat \wedge a(u)^\flat + 2 \omega(u)^\flat$
into the splitting identity for the
exterior derivative leads to
\beq\eqalign{
0= (d^2 u^\flat)\E &= [ d(u) + a(u)^\flat \wedge ] a(u)^\flat 
        - 2 \Lie(u)\sub{u} \omega(u)^\flat \ ,\cr
0= (d^2 u^\flat)\M &= 2 [ d(u) \omega(u)^\flat 
          -  \omega(u)^\flat \wedge a(u)^\flat ] \ ,\cr
}\eeq
leading to the relations
\beq\eqalign{
  d(u) a(u)^\flat &=  2 \Lie(u)\sub{u} \omega(u)^\flat \ ,\cr
 d(u) \omega(u)^\flat &= \omega(u)^\flat \wedge a(u)^\flat \ .\cr
}\eeq

For an arbitrary differential form $S$, the splitting of $d^2 S$ is 
obtained by iterating the splitting of the exterior derivative, with
the result
\beq\eqalign{
0= (d^2 S)\E &=  \{ d(u)^2 - 2\omega(u)^\flat \wedge \Lie(u)\sub{u} \} S\E \cr
 &\qquad
  + \{ d(u) a(u)^\flat - 2 \Lie(u)\sub{u} \omega(u)^\flat \} \wedge S\E \cr
 &\qquad
  + \{ -[\Lie(u)\sub{u} , d(u)] + a(u)^\flat \wedge \Lie(u)\sub{u} \} S\M
       \ ,\cr
0= (d^2 S)\M &=   \{ d(u)^2 - 2\omega(u)^\flat \wedge \Lie(u)\sub{u} \} S\M \cr
 &\qquad    +  2 \{ d(u) \omega(u)^\flat - \omega(u)^\flat \wedge a(u)^\flat\} 
       \wedge S\E \ .\cr
}\eeq
Since $S\E$ and $S\M$ are independent, the terms involving each must
separately vanish.  Two of these terms vanish identically due to the identity
for $u^\flat$ itself ($S\E=1$ and $S\M=0$), while a third gives an identity
for the commutator of the two natural derivatives acting on a spatial
differential form
\beq
    [\Lie(u)\sub{u} , d(u)] = a(u)^\flat \wedge \Lie(u)\sub{u}  S
        \ ,\qquad u\leftcontract S =0\ .
\eeq

This leaves the result that the measurement of $d^2$ acting on a differential
form leads to the operator $d(u)^2 - 2\omega(u)^\flat \wedge \Lie(u)\sub{u}$
acting on each of the two splitting fields which represent that differential
form.  The vanishing of this operator then gives an identity for $d(u)^2$
acting on any spatial differential form
\beq
d(u)^2 S = 2 \omega(u)^\flat \wedge \Lie(u)\sub{u} S
    \ , \qquad  u\leftcontract S =0\ .
\eeq 
Using this result, one can evaluate $d(u)^2$ acting on an arbitrary
differential form, leading to the result
\beq
d(u)^2 S = 2 \omega(u)^\flat \wedge [\Lie(u)\sub{u} S - 
        d(u) (u\leftcontract S) ]\ .
\eeq 

\Subsection{Splitting the differential form divergence operator}

One may merge the splitting of the exterior derivative $d$ and of the
duality operation $^\ast$ to obtain the splitting of the divergence operator
$\delta$, defined for a differential form $S$ on spacetime by
\beq
          \delta S =  \dual d \dual S \ .
\eeq
This takes a more familiar form in index notation using the covariant
derivative (assuming zero torsion)
\beq
         [\delta S]_{\alpha_1 \ldots \alpha_{p-1}}
          = - \four \del_\beta S^\beta{}_{\alpha_1 \ldots \alpha_{p-1}} \ .
\eeq
The spatial divergence operator has an additional sign;
for a spatial $p$-form $S$ one has
\beq\eqalign{
          \delta(u) S &=  (-1)^p \dualp{u} d(u) \dualp{u} S \ ,\cr
          [\delta(u) S]_{\alpha_1 \ldots \alpha_{p-1}} 
        &=-\del(u)_\beta S^\beta{}_{\alpha_1 \ldots \alpha_{p-1}}   \ .\cr
}\eeq

By manipulating the definitions and identities one finds the result
\beq\eqalign{
[\delta  S]\E &= - \delta(u) S\E
      - 2 \omega(u)^\sharp\leftcontractp{2} S\M \ ,\cr
[\delta S]\M &= [\delta(u) - ?? a(u)\leftcontract ] S\M 
             - \dualp{u}\Lie(u)\sub{u} \dualp{u} S\E \ .\cr
}\eeq\message{check?? (-1)^p inconsistent}%
%where $\leftcontractp{p}$
In applying the last formula, the identity 
(\ref{eq:liederspadual}) is useful in re-expressing
the term
$ \dualp{u}\Lie(u)\sub{u} \dualp{u} S$,
especially for a spatial 1-form $S$.


%%%%%%%%%%%%%%%
\Subsection{Spatial vector analysis}

For functions and spatial vector fields, one can introduce the covariant
analogs of the gradient, curl and divergence operations in order to
make contact with traditional vector analysis which takes place in
Euclidean space. These spatial operations are defined in an obvious
way in terms of the spatial covariant derivative. Let $\vec\del(u)$
denote the covariant derivative operator with its differentiating index
raised, i.e., $[\vec\del(u)f]^\alpha = \del(u)^\alpha f$.
These operators are related to the spatial exterior derivative in the
usual manner
\beq\eqalign{
&  \grad_u f =\vec\del(u) f = [d(u)f]^\sharp\ ,\cr
& \curl_u X =\vec\del(u) \times_u X = [\dualp{u}(d(u)X^\flat)]^\sharp\ ,\cr
& \div_u X = \vec\del(u) \cdot_u X = \dualp{u} [d(u) \dualp{u}X^\flat]\ , \cr
}\eeq 
where $X$ is assumed to be a spatial vector field. However,
nothing prevents their application to nonspatial fields.
The spatial curl of an arbitrary vector field can be written in index form as
\beq
    [ \curl_u X ]^\alpha 
        = \eta(u)^{\alpha\beta\gamma} \four\del_\beta X_\gamma \ .
\eeq

\Message{change this dyad stuff??}
Since many second rank tensors also enter into vector analysis in the
form of linear transformations of vectors, one can follow the 
leftright arrow notation of Thorne et al \Cite{1986} 
for contravariant second rank spatial tensors 
\beq
 [\bivec S \cdot_u \vec X]^\alpha =S^{\alpha\beta} P(u)_{\beta\gamma}
  X^\gamma \ .
\eeq 
In the old-fashioned language such tensors are called ``dyads".
Thus the spatial vorticity vector of $u$ is just the spatial dual of half 
the contravariant form of the
spatial exterior derivative of $u$, which defines the vorticity dyad
\beq\eqalign{   
   \bivec\omega(u) &= \omega(u)^\sharp \ ,\cr
  \vec\omega(u)&=\dualp{u} \bivec\omega(u) = \half \curl_u u \ .\cr
}\eeq 
Allowing the spatial curl operator to act on the nonspatial
vector $u$, one sees that the vorticity vector is half its curl.
Expressing this in an orthonormal frame in terms of which $u$ has
a nonrelativistic relative spatial velocity leads to the classical
expression of nonrelativistic mechanics \Cite{Synge and Schild 1949}.
With these conventions the linear transformation corresponding to
the rotation tensor becomes
\beq
    \omega(u) \rightcontract X = - \vec\omega(u) \times_u X \ ,\qquad
    \omega(u)^\alpha{}_\beta X^\beta 
       = -\eta^\alpha{}_{\beta\gamma} \omega(u)^\beta X^\gamma
\eeq
when expressed in terms of the rotation vector.

The spatial exterior derivative of the rotation 2-form itself as given
above may also be rewritten in the 3-vector notation
\beq\label{eq:divvor}
   \div_u \vec \omega(u) = a(u) \cdot_u \vec \omega(u) \ , \qquad
   \del(u)_\alpha \omega(u)^\alpha = a(u)_\alpha \omega(u)^\alpha \ .
\eeq 
The identity for the exterior derivative of the acceleration 1-form
becomes the following when rewritten
\beq\eqalign{\label{eq:curlacc}
  \half \curl_u a(u) &= \dualp{u} [ \Lie(u)\sub{u} \omega(u)^\flat] =
 \Lie(u)\sub{u} \vec \omega(u) + \Theta(u) \vec \omega(u) \cr
   &= \del\fw(u) \vec \omega(u)  
    - [\theta(u) - P(u) \Theta(u)]\rightcontract \vec \omega(u) \ ,\cr
  \half [\curl_u a(u)]^\alpha
      &= [ \del\fw(u) + \fraction23 \Theta(u)] \vec \omega(u)^\alpha
             - \sigma(u)^\alpha{}_\beta \omega(u)^\beta \ ,\cr
}\eeq 
where the expansion scalar appears from the commutation of the 
spatial Lie derivative and the spatial duality operation.

\Newpage
%%%%%%%%%%
\Subsection{Ordinary and Co-rotating Fermi-Walker derivatives}

Suppose one refers to a tensor as being of ``of type $\sigma$",
where $\sigma$ is the $gl(4,R)$-representation corresponding to the
$GL(4,R)$-representation under which the tensor
transforms under a change of
frame. Then the representation function $\sigma$ is defined by
\beq
  [\sigma(\bfA) S]^{\alpha\ldots}_{\ \ \beta\ldots} =
    A^\alpha{}_\gamma S^{\gamma\ldots}_{\ \ \beta\ldots} + \cdots
   -A^\gamma{}_\beta S^{\alpha\ldots}_{\ \ \gamma\ldots} - \cdots\ .
\eeq 
With this notation,
the Fermi-Walker derivative along the observer congruence may be defined by
\beq
   \four \del\fw (u) = \four \del\sub{u} - \sigma( u \wedge a(u)^\flat )\ ,
\eeq
where the argument of $\sigma$ is the antisymmetric tensor (mixed form)
\beq
      [ u \wedge a(u)^\flat ]^\alpha{}_\beta 
         =  u^\alpha a(u)_\beta - a(u)^\alpha u_\beta \ .
\eeq
Such an operator satisfies the obvious product laws with respect
to tensor products and contractions thereof. It is easily verified
the $\four\gm$, $\four\eta$,
$u$, $P(u)$, $T(u)$, and all index-shifted variations
of these tensor fields have an identically vanishing Fermi-Walker
derivative. This means that the Fermi-Walker derivative commutes
with the measurement process:
the measurement of the Fermi-Walker derivative
of a spacetime tensor yields the Fermi-Walker derivative acting
on each member of the family of spatial tensors which represent that tensor.

A tensor field is said to undergo Fermi-Walker transport along $u$ if
its Fermi-Walker derivative is zero, in analogy with parallel transport
along $u$, to which it reduces for a geodesic congruence with vanishing
acceleration.
Thus $\four\gm$, $\four\eta$, $u$, 
$P(u)$, and $T(u)$ all undergo
such Fermi-Walker
transport, while only $\four\gm$ and $\four\eta$
are parallel transported along $u$.
The effect of the difference term in the two derivatives
is to generate an active boost in the velocity-acceleration plane
which maps the parallel transport of $u$ along $u$
onto $u$ itself. %which is Fermi-Walker transported along $u$. 
This same boost maps the parallel transport of any tensor field
along $u$ onto the Fermi-Walker transported tensor field along $u$. Both
transports preserve inner products and commute with index-shifting
and  duality operations, but the Fermi-Walker derivative preserves
the orthogonal projection and measurement operations associated with $u$
as well. The Fermi-Walker derivative of a spatial tensor field is also
spatial with respect to $u$.

In fact, the addition to the covariant derivative
of a term $-\sigma(A)$, where $A$
is a $1\choose1$-tensor field,
will generate a continuous Lorentz transformation
relative to parallel transport along the curve or congruence
as long as $A$ is antisymmetric with respect to the metric, i.e., is
the mixed form of a 2-form. Such a derivative will automatically
commute with index shifting and its corresponding transport
will preserve inner products, as does the
Fermi-Walker derivative.
If $A$ is also spatial with respect to $u$ then it will generate a
continuous rotation of the local rest space of $u$ while leaving
$u$ itself uneffected, and both $u$ and $u^\flat$ will still have vanishing
derivative with respect to the new derivative (as will the metric and
the spatial and temporal projection tensors).

For a congruence of timelike curves, 
one may define a {\it co-rotating Fermi-Walker derivative} 
which further adapts the covariant derivative to the congruence by
adding a term to the Fermi-Walker derivative
which leads to a co-rotation of the local rest spaces
relative to the congruence under the corresponding transport
\beq\eqalign{
     \four \del\cfw (u) &= \four \del\fw(u) + \sigma(\omega(u))\cr
        &= \four \del\sub{u} - \sigma(u\wedge a(u)^\flat - \omega(u))\ .\cr
}\eeq
A vector which is transported by the co-rotating Fermi-Walker transport
has the same angular velocity as the relative velocity vector 
to be defined below,
but does not exhibit the effects of the expansion and shear that
the latter does.

The spatial Fermi-Walker derivative has been introduced by spatially projecting
the covariant derivative $\four\del\sub{u}$ and not the Fermi-Walker derivative
$\four\del\fw(u)$, in order to have a temporal derivative one can apply to $u$
itself (since $\four\del\fw(u)u=0$ while $\del\fw(u)u = a(u)$)
\beq
   P(u) \four\del\fw(u) = \del\fw(u) - P(u) \sigma( u\wedge a(u)^\flat ) \ .
\eeq
The difference term only contributes to the derivative of nonspatial fields.
In the same spirit one can define the
{\it spatial co-rotating Fermi-Walker derivative} 
along $u$ or equivalently
the {\it co-rotating Fermi-Walker temporal derivative} along $u$ 
\beq
      \del\cfw(u) 
               = \del\fw(u) + \sigma(\omega(u))  \ .
\eeq
This spatial derivative is related to its corresponding spacetime derivative
in the same way as the ordinary Fermi-Walker derivatives are related
\beq
   P(u) \four\del\cfw(u) = \del\cfw(u) - P(u) \sigma( u\wedge a(u)^\flat ) \ .
\eeq
On spatial fields the spacetime and spatial ordinary and co-rotating
Fermi-Walker derivatives agree, and so the measurement of the
spacetime derivative leads to the action of the corresponding spatial
derivative on each member of the family of spatial fields which represent
a given spacetime field.

With the present definitions, the derivatives of a multiple of $u$ itself are
\beq
      \del\cfw(u) [ f u ] = \del\fw(u)[ f u ] = f a(u)  \ ,
\eeq
which determines their action on the nonspatial parts of a tensor field.
The alternative would be to define all the spatial operators by projection
of the corresponding
spacetime operators and introduce a new temporal derivative to
handle nonspatial tensor fields
(identical to the present definition of $\del\fw(u)$).

\Newpage
%%%%%%%%%%
\Subsection{Relation between Lie and 
  Fermi-Walker temporal derivatives}

The interpretation of the kinematical quanitites 
depends on the relationship between the Lie temporal derivative and the
Fermi-Walker temporal derivatives. This in turn depends on the
more general relationship between the Lie and
covariant derivatives themselves.

The formulas for the components of the
covariant and Lie derivatives in a coordinate frame $\{e_\alpha\}$
characterized by 
$C^\alpha{}_{\beta\gamma} = \omega^\alpha([e_\beta,e_\gamma])=0$
are
\beq\label{eq:ld}
\eqalign{ 
   [\four\del\sub{u} S]^{\alpha\ldots}_{\ \ \beta\ldots} &=
   S^{\alpha\ldots}_{\ \ \beta\ldots,\gamma} u^\gamma +
        [\sigma(\four\Gamma(u)) S]^{\alpha\ldots}_{\ \ \beta\ldots}\ ,\cr
   [\Lie\sub{u} S]^{\alpha\ldots}_{\ \ \beta\ldots} &=
   S^{\alpha\ldots}_{\ \ \beta\ldots,\gamma} u^\gamma -
        [\sigma(\partial u) S]^{\alpha\ldots}_{\ \ \beta\ldots}\ ,\cr
  &= S^{\alpha\ldots}_{\ \ \beta\ldots;\gamma} u^\gamma -
        [\sigma(\four\del u) S]^{\alpha\ldots}_{\ \ \beta\ldots}\ ,\cr
}\eeq 
where the matrix arguments of $\sigma$ are the suggestive abbreviations
$\four\Gamma(u)^\alpha{}_\beta = \Gamma^\alpha{}_{\gamma\beta}u^\gamma$,
$(\partial u)^\alpha{}_\beta = u^\alpha{}_{,\beta}$, and
$(\four\del u)^\alpha{}_\beta = u^\alpha{}_{;\beta}$.
%while the comma indicates ordinary differentiation of a component function
%along a frame vector field.
The last expression for the Lie derivative
is true for an arbitrary frame and is the result of the ``comma to semicolon
rule" valid for any symmetric connection in a coordinate frame
\beq\label{eq:LieFWK}
    \Lie\sub{u} S= \four\del\sub{u} S - \sigma(\four\del u) S \ .
\eeq

Taking the spatial projection of this equation yields
\beq
 \Lie(u)\sub{u} S = \del\fw (u) S - P(u)\sigma(\four\del u) S
\eeq 
for an arbitrary tensor but
\beq
 \Lie(u)\sub{u} S = \del\fw (u) S + \sigma(k(u)) S
\eeq 
for a spatial tensor. The kinematical tensor $k(u)$ thus acts as the linear
transformation of the local rest space which describes the difference between
 spatial  Lie transport  and Fermi-Walker transport of spatial tensors
along $u$. 
These two transports and their corresponding differential
operators embody two possible choices for describing ``evolution",
which might be called ``Lie" and ``Fermi-Walker" evolution respectively
from their origins in derivative operators which correspond to those
labels. In the absence of further structure,
evolution can only be defined independently
along each observer worldline and then only
relative to a transport of the local rest space along each such worldline which
defines ``no evolution".
Note that for $k(u)$ itself the difference between the spatial Fermi-Walker
derivative and the  spatial  Lie derivative along $u$ vanishes
so 
\beq
   \Lie(u)\sub{u} k(u) = \del\fw (u) k(u)\ .
\eeq 

The Lie derivative along $u$ has the following representation
in terms of the ordinary and co-rotating Fermi-Walker derivatives
\beq\eqalign{\label{eq:liecfwder}
  \Lie\sub{u} = \four \del\sub{u} - \sigma(\four \del u)
              &= \four \del\sub{u} 
                + \sigma(\omega(u) - \theta(u) + a(u)\otimes u^\flat ) \cr
              &= \four \del\fw(u) 
                + \sigma(\omega(u) - \theta(u) + u\otimes a(u)^\flat ) \cr
              &= \four \del\cfw(u) 
                + \sigma( - \theta(u) + u\otimes a(u)^\flat )\ . \cr
}\eeq
In the difference term relative to the co-rotating Fermi-Walker
derivative, the expansion term in the Lie derivative
generates a deformation of the local rest
space of $u$ while the acceleration term tilts the local rest space
relative to $u$ along the acceleration direction,
breaking the orthogonality. Only in the case that
$u$ is a Killing vector field do both of these terms vanish
\beq
        \Lie\sub{u} \four \gm = 0 \rightarrow \theta(u) =0 = a(u) \ ,
\eeq
in which case the Lie and co-rotating Fermi-Walker derivatives coincide.
However, a unit timelike Killing vector field is perhaps only interesting
in flat spacetime.

Spatially projecting the representation of the Lie derivative along $u$
in terms $\del\cfw(u)$
leads to the following relation between that operator and
the Lie temporal derivative
\beq\label{eq:cfwlie}
      \del\cfw(u) 
               = \Lie(u)\sub{u} + \sigma(\theta(u)) 
                                - P(u) \sigma( a(u) \otimes u^\flat) \ .
\eeq
The last term only contributes for nonspatial fields.
Thus
when acting on spatial fields the three temporal derivatives are related by
\beq\eqalign{
      \del\cfw(u) 
               &= \del\fw(u) + \sigma(\omega(u))  \cr
               &= \Lie(u)\sub{u} + \sigma(\theta(u)) \ .\cr
}\eeq

The co-rotating Fermi-Walker temporal derivative is therefore an obvious
compromise between the Fermi-Walker temporal derivative (retaining the latter's
compatibility with spatial orthonormality) and the Lie temporal derivative
(retaining the latter's co-rotation feature).
If one wants to use an orthonormal spatial frame, the best one
can do to adapt it to the congruence is to transport it along the
congruence by co-rotating Fermi-Walker transport,
since under Lie transport, the frame will not remain
orthonormal or spatial. Only when $u$ is a Killing
vector field is Lie transport compatible with orthonormality, since
in this case it coincides with the co-rotating Fermi-Walker transport.

An immediate consequence of the ``comma to semicolon" formula 
for the Lie derivative is the following ``product rule"
\beq
  \Lie\sub{fu} S = f \Lie\sub{u} S - 
      \sigma(u\otimes df) S\ ,
\eeq 
with spatial projection
\beq
  \Lie(u)\sub{fu} S = f \Lie(u)\sub{u} S - 
      P(u) \sigma(u\otimes df) S\ .
\eeq 
%Note that only the terms where $u$ is contracted with a covariant index
%of $S$ survive the spatial projection. 
%
For a stationary spacetime with a timelike Killing vector field $\xi$ and
its corresponding normalized 4-velocity $u= \ell^{-1} \xi$, invariance
under the symmetry group is associated with the Lie derivative by $\xi$
and its spatial projection, not the  corresponding operators for $u$.
The previous identities together with $d \ln\ell = a(u)^\flat$ then imply
\beq\eqalign{ \label{eq:uscaledlieder}
 \Lie\sub{u} S &=
  \ell^{-1} \Lie\sub{\xi} S +
      \sigma(u\otimes a(u)) S\ , \cr
 \Lie(u)\sub{u} S &=
  \ell^{-1} \Lie(u)\sub{\xi} S +
      P(u) \sigma(u\otimes a(u)) S\ . \cr
}\eeq 
These lead to the following relationships between
the co-rotating Fermi-Walker derivative and
co-rotating Fermi-Walker temporal derivative along $u$
and the rescalings of the respective Killing vector field operators
\beq\eqalign{
   \four \del\cfw(u) &= \ell^{-1} \Lie\sub{\xi}\ ,\cr
   \del\cfw(u) &= \Lie(u)\sub{u} - P(u)\sigma(a(u) \otimes u^\flat)  \cr
               &= \ell^{-1} \Lie(u)\sub{\xi}
                         - P(u)\sigma(a(u) \wedge u^\flat)  \ .\cr
}\eeq
Without stationarity, the co-rotating Fermi-Walker operators are the
best that one can do to extend the corresponding Lie operators in the
stationary case.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Both the ordinary and co-rotating Fermi-Walker derivatives along $u$
of a spatial tensor field reduce to the respective spatial derivatives
of that tensor field.  
Thus the spacetime transport of a spatial tensor field along $u$ for
either of these two types of derivatives reduces to the corresponding
spatial transport.

The  spatial  Lie transport of spatial fields is rather simple. Lie transport
a spatial tensor along the observer congruence using the 1-parameter group
of diffeomorphisms generated by $u$ and then spatially project it. 
If $e_\alpha$ is any frame with $e_0$ along $u$ and $\{e_a\}$ Lie dragged
along $u$, then  
$\{e_0,P(u)e_a\}$ is an observer-adapted frame and 
$\{P(u)e_a\}$ is a spatial frame which undergoes  spatial  Lie
transport.  Any spacetime tensor which does not evolve in the Lie
sense  is represented by a family of spatial tensors which have components
in such a spatial frame which are constant along the observer worldlines.
Such tensors are ``anchored" in the observer congruence. The 
spatial Lie derivative along $u$ measures the rate at which a field is evolving
in this sense.

The Fermi-Walker evolution instead describes how a field changes with respect
to what the observers see using an orthonormal spatial frame which is
{\it locally nonrotating}, as determined by a set of torque-free gyros carried
by the observer, rather than by comparing the field to the nearby observers
seen by that observer. Fermi-Walker transport of the spatial frame along
the observer congruence defines mathematically what locally nonrotating
means. A spatial field which has constant components with respect to
such a frame exhibits no evolution in the covariant sense. A spacetime
tensor field which is Fermi-Walker transported along the observer
congruence is represented by a family of spatial tensors which undergo
spatial Fermi-Walker transport.

The interpretation of the kinematical quantities is associated with
the comparison of the two kinds of evolution. These quantities describe
the (limiting) relative motion of the nearby observers as seen by each observer
moving through spacetime and as compared to a locally nonrotating 
orthonormal spatial
frame. Physically an observer can assign a relative position vector 
in his local rest space to each neighboring observer at each moment of
his proper time by using light signals, obtaining the relative distance
from half the light travel time and the direction as an average direction.
This relative position vector in the flat local rest space will undergo
a time-dependent linear transformation compared to the locally nonrotating
orthonormal spatial frame. The rate of change of this transformation
is described by the kinematical tensor $-k(u)$.

Suppose $X$ is a tangent vector which is Lie dragged along a given observer
worldline. If $X$ is sufficiently small,
it has the interpretation of being a ``{\it connecting vector}" whose tip
lies on the worldline of some other fixed  nearby observer worldline, 
identifying nearby points in the spacetime manifold with points in the
tangent space. If it is initially spatial, it will not remain spatial
since its tip moves at unit speed relative to the nearby observer's proper
time, while its initial point moves at unit speed relative to the
given observer.
Its spatial projection $P(u)X$ after an interval
$\Delta\tau_u$ of the given
observer's proper time represents the observed position
vector (``{\it relative position vector}") of the nearby observer 
at the same value of the elapsed proper time
measured by that nearby observer, not by the given observer. The component
along $u$, namely $-u_\alpha X^\alpha$, represents the additional lapse
of the given observer's proper time necessary to reach the event at which
$P(u)X$ is the relative position vector. Neglecting this additional
synchronization question helps define the spacetime neighborhood in which this
discussion \Cite{Ehlers 1961, 1993, Ellis 1971}
makes sense. 

Let $Y=P(u)X$ be this relative position vector, which undergoes 
 spatial Lie transport. Its spatial Fermi-Walker derivative 
(``{\it relative velocity vector}") is therefore
\beq\label{eq:relvelvectorfw}
\del\fw (u) Y =  -k(u)\rightcontract Y = 
  [\theta(u)-\omega(u)] \rightcontract Y = \vec\omega(u) \times_u Y
       +\theta(u) \rightcontract Y\ .
\eeq 
This shows that nearby observers appear to be  rotating with angular
velocity $\vec\omega(u)$ and shearing and expanding by $\sigma(u)$ and
$\Theta(u)$ relative to locally nonrotating orthonormal axes. 
The co-rotating Fermi-Walker temporal derivative describes the deviation
from pure rotation
\beq\label{eq:relvelvectorcfw}
\del\cfw (u) Y =  \theta(u)\rightcontract Y \ .
\eeq 
Thus the relative position vector fails to undergo
spatial co-rotating Fermi-Walker transport along $u$ only because of the
effects of the expansion of the congruence.

\message{shear and Lie; check}
One can further decompose the relative position vector into a relative distance 
$||Y||$ and a relative direction vector $\hat Y$
\beq
     Y = ||Y|| \hat Y \ .
\eeq
This 
leads to a corresponding decomposition of the spatial Fermi-Walker derivative
\beq\eqalign{
\del\fw (u) \hat Y  &=  
    \vec\omega(u) \times_u \hat Y +
       [\sigma(u) - \sigma(\hat Y,\hat Y) P(u)]
           \rightcontract \hat Y\ ,\cr
\del\fw (u) \ln ||Y||  &=  
        {1\over3} \Theta(u) + \sigma(\hat Y,\hat Y) \ .\cr
}\eeq\message{check shear relative velocity eq??}

%by Ehlers \Cite{1961} and Ellis \Cite{1971}.

The corresponding
``{\it relative acceleration}" is obtained by taking one more spatial
Fermi-Walker derivative
\beq\eqalign{
\del\fw (u)^2 Y 
  &= - [\del\fw (u) k(u)]\rightcontract Y 
     - k(u)\rightcontract \del\fw (u) Y \cr
  &= - [\del\fw (u) k(u) - k(u)\rightcontract k(u)] \rightcontract Y\ ,\cr
}\eeq 
but since $k(u) = -\del(u) u$ and $\del\fw (u) u =a(u)$ this becomes
\beq\label{eq:ra}
\del\fw (u)^2 Y 
  =  \{ [\del\fw (u), \del(u)] u  + k(u)\rightcontract k(u)
   +\del(u) a(u) \} \rightcontract Y\ .
\eeq 
To finish this evaluation, one needs to consider the commutator of the spatial
covariant derivative and the spatial Fermi-Walker derivative, which
involves the curvature tensor through the Ricci identity. This will be
postponed until the splitting of the curvature tensor is discussed.

On the other hand, a spatial vector $Z$ which is not evolving relative
to the locally nonrotating orthonormal spatial frame does evolve
with respect to the observer congruence at a rate described by the
kinematical tensor itself. If $\del\fw(u) Z = 0$ then
\beq
   \Lie(u)\sub{u} Z = k(u) \rightcontract Z
   =   -\vec\omega(u) \times_u Z
       -\theta(u) \rightcontract Z\ .
\eeq 
This would describe the evolution of the spin vector of a gyro carried
by the observers relative to the observer congruence itself.
Equivalently, the rotation tensor alone describes the evolution with respect
to a co-rotating Fermi-Walker spatial frame
\beq
   \del\cfw(u)\sub{u} Z = \omega(u) \rightcontract Z
   =   -\vec\omega(u) \times_u Z \ .
\eeq

%%%%%%%%%%%%%
\Newpage
\Typeout{Update this section according to articles:}

\Subsection{Total spatial covariant derivatives}
\label{sec:tscd}

Suppose one has a parametrized timelike or null curve $c$ with
parameter $\lambda$ and
tangent vector $c'$ in spacetime, thought of as the worldline
of a test particle of nonzero rest mass $m\neq0$ ($\tilde m = 1$)
or of zero rest mass $m=0$ ($\tilde m=0$)
respectively.
It is then of interest to
split the absolute or intrinsic derivative, here called the
total covariant derivative operator $\four D/d \lambda=
\four\del\sub{c'(\lambda)}$ along this curve. This operator
is discussed in section 
\ref{app:tcd} of the appendix. Before splitting this operator,
it is useful to explore first its spatial projection and several
related operators which are useful in representing the splitting.
To identify this operator more explicitly, introduce the expanded
notation $\four D(c'(\lambda)) / d\lambda$.

In the case that $c$ is a proper-time-parametrized
worldline belonging to another observer
congruence with 4-velocity $U$, with $c'(\lambda) = U\circ c(\lambda)$,
then this is just the splitting of the operator $\four\del\sub{U}$
with respect to $u$.
However, if one uses the %affine 
parameter $\lambda = m^{-1} \tau_U$,
then the tangent vector $c' = P = m U$ equals the 4-momentum of the
test particle, and one can handle the case of zero rest mass in
a parallel fashion. Recall the notation of section (\ref{sec:splitU}).
%assuming an affine parametrization also in that case.

In both of these cases a new parametrization of the curve $c$
can be defined which corresponds
to the continuous sequence of proper times of the observers whose
worldlines intersect this curve
\beq
      d\tau_{(P,u)} / d \lambda = E(P,u) \ .
\eeq
Here the subscript notation on $\tau$ indicates the proper time
with respect to $u$ along $P$.
For the case $\tilde m =1$ of a timelike worldline, substituting $m=1$
replaces $P$ everywhere by the 4-velocity $U$, $E(P,u)$ by the
gamma factor $\gamma(U,u)$, and the affine parameter $\lambda$ by
the proper time $\tau_U$. This particular relation then connects
the proper times
\beq
      d\tau_{(U,u)} / d \tau_U = \gamma(U,u) \ ,
\eeq
corresponding to the Lorentz dilation of the test particle's proper
time in relative motion with respect to the observer congruence.

It is convenient to follow the custom of not distinguishing the operator
$\four D(c'(\lambda)) /d\lambda$ 
which acts on tensors defined only along the curve $c$
from the operator $\four\del\sub{c'(\lambda)}$ which acts on tensor fields
defined in a neighborhood of $c$. When necessary they will be understood
to apply to any smooth extension of a tensor along $c$ to a tensor field,
since only the values along the curve enter into these derivatives.

From the total covariant derivative one may extract
a derivative operator along the curve (equivalently, along $P$)
which preserves the spatial character of a
tensor and can therefore be used in the representation of the splitting
of the total covariant derivative of a spacetime 
tensor in terms of its representative spatial fields.
The obvious way to do this is simply by spatial projection of the total
covariant derivative
\beq\eqalign{
  D\fw(P,u) / d \lambda &= P(u) \four D(P) / d \lambda
      = P(u) \four\del\sub{P} \cr
    &= E(P,u) [ \del\fw(u) + \del\sub{\nu(P,u)} ] \ , \cr
}\eeq
and hence changing the parametrization to observer proper time
\beq\eqalign{
   D\fw(P,u) / d \tau_{(P,u)} 
          &= E(P,u)^{-1}  D\fw(P,u) / d \lambda \cr
         &= \del\fw(u) + \del\sub{\nu(P,u)}  \ . \cr
}\eeq
The qualifying notation $(P,u)$ indicates the derivative along $P$
as seen by $u$.

However, two other useful possibilities exist.
$P$ itself can be split into spatial and temporal parts. Clearly one
can use the spatial covariant derivative along the spatial part of $P$
but one has
the choice of the ordinary or co-rotating spatial Fermi-Walker derivative 
or the spatial Lie derivative along the temporal part.
This leads to three different possibilities.
One then has the option to rescale the derivative to correspond to
the rate of change with respect to the observer proper time.

Suppose $S$ is a spatial tensor defined only along the worldline with tangent
$U$.  In order to discuss derivative operators along this worldline in terms 
of the three temporal derivatives 
$\{ \del\tem(u) \}_{{\rm tem} \,=\, {\rm fw,cfw,lie}}$  
and the spatial covariant derivative
$\del(u)$ already introduced
for tensor fields, assume that $S$ may be smoothly extended to a spatial
field on $\four M$.
The three possible {\it total spatial covariant derivatives} 
that one can define
for such spatial tensors along the worldline
can then be represented in terms of the following
operators on the corresponding extended spatial tensor fields
\beq\eqalign{
   D\tem (P,u)/d \lambda &=
     E(P,u) [      \del\tem (u)    + \del(u)\sub{\nu(P,u)} ] \cr
  &=   E(P,u) \del\tem (u)  + \del(u)\sub{p(P,u)}\ , \qquad
 {\scriptstyle {\rm tem} \,=\, {\rm fw, cfw, lie} } \ ,\cr
}\eeq 
which may then be rescaled 
to correspond to a derivative with respect to the observer proper time
\beq\eqalign{\label{eq:tscdc}
   D\tem (P,u)/d\tau_{(P,u)} &= E(P,u)^{-1}    D\tem (P,u)/d \lambda \cr
   &=  \del\tem (u)    + \del(u)\sub{\nu(P,u)}
     \qquad
 {\scriptstyle {\rm tem} \,=\, {\rm fw, cfw, lie} } \ .\cr
}\eeq 

These three operators will be called respectively
the Lie, Fermi-Walker, and
co-rotating Fermi-Walker
total spatial covariant derivatives along $P$ with respect to $u$.
All of them, like the Lie and Fermi-Walker
temporal derivatives and the spatial covariant derivative, may be
applied to nonspatial tensor fields but are most useful for spatial
tensor fields. When applied to functions, all of these operators
reduce to the ordinary parameter derivative along the worldline if the
function is only defined on the worldline or the ordinary derivative
by $P$ or $E(P,u)^{-1} P$ if the function is defined in a neighborhood
of the worldline
\beq
   D\tem (P,u) f / d\tau_{(P,u)}     \equiv    d f / d\tau_{(P,u)}  = P f
       \ .
\eeq

Since the spatial metric is covariant constant with respect to the
spatial covariant derivative and has zero 
ordinary and co-rotating spatial Fermi-Walker derivative
along $u$, the ordinary and co-rotating Fermi-Walker 
total spatial covariant derivatives of the spatial metric vanish,
so that each of these derivatives commute with index-shifting on spatial
fields.
However, the Lie total spatial covariant derivative of the spatial metric,
which coincides with its spatial Lie derivative,
is not zero as long as the expansion tensor is nonvanishing
\beq
 D\lie (P,u) P(u)^\flat  /d\tau_{(P,u)} = \Lie(u)\sub{u} P(u)^\flat = 
   2\theta(u)^\flat \ .
\eeq 
Thus the Lie
total spatial covariant derivative does not commute with
index-shifting on spatial tensor fields and extra expansion
terms must be taken into account when shifting indices.

The co-rotating Fermi-Walker total spatial covariant derivative
is a  compromise between the Fermi-Walker and Lie
total spatial covariant derivatives.
It has the 
Fermi-Walker property of commuting with the shifting of indices but
its associated transport has the same rotational properties as the Lie 
total spatial
covariant derivative. One simply absorbs the expansion tensor into
the latter derivative or equivalently the rotation tensor into the former
to obtain this result
\beq
    D\fw(P,u) /d\tau_{(P,u)} + \sigma(\omega(u)) = 
    D\cfw(P,u) /d\tau_{(P,u)} =
    D\lie(P,u) /d\tau_{(P,u)} + \sigma(\theta(u)) \ .
\eeq
This new total covariant derivative
was introduced by Massa \Cite{1974b, 1990}.

The difference between the rotation and expansion tensors
parametrizes the difference between the Lie and Fermi-Walker
total spatial covariant derivatives 
\beq
          D\lie (P,u) /d\tau_{(P,u)} - D\fw (P,u) /d\tau_{(P,u)} 
       = \sigma(k(u)) \ .
\eeq 
The Lie total spatial covariant derivative was used by
Zel'manov \Cite{1956} and Cattaneo \Cite{1958}.

Every timelike worldline with 4-velocity $U$ has one spatial field
defined along it, namely its relative velocity vector $\nu(U,u)$
as seen by $u$. The temporal derivative of this
relative velocity defines the 
relative acceleration of $U$.
Introduce therefore the
ordinary and co-rotating and the Lie {\it relative accelerations}
of $U$ with respect to $u$ by
\beq\eqalign{
       a\tem(U,u) &= D\tem(U,u) \nu(U,u) / d\tau_{(U,u)} \ ,\cr
 &\qquad {\scriptstyle {\rm tem} \,=\, {\rm fw, cfw, lie} } \ .\cr
}\eeq
These are related to each other in the same way as the corresponding
derivative operators
\beq\eqalign{
      a\cfw(U,u) &= a\lie(U,u) + \theta(u) \rightcontract \nu(U,u) \cr
                 &= a\fw(U,u) + \omega(u) \rightcontract \nu(U,u) \cr
                 &= a\fw(U,u) - \vec\omega(u) \times_u \nu(U,u) \ .\cr
}\eeq

The relative acceleration may in turn be related to the temporal derivative
of the spatial momentum (per unit mass) $\tilde p(U,u) = \gamma \nu(U,u)$
using the product rule and explicitly differentiating the expression for the
gamma factor
\beq\eqalign{
      \gamma &= [ 1 - ||\nu(U,u)||^2 ]^{-1/2} \ ,\cr
    d \ln \gamma / d\tau_{(U,u)} 
       &=  \gamma^2 \nu(U,u) \cdot_u [ a\tem(U,u) 
           + \half D\tem(U,u) P(u)^\flat / d\tau_{(U,u)} 
                  \rightcontract \nu(U,u) ] \ ,\cr
       &\qquad {\scriptstyle {\rm tem} \,=\, {\rm fw, cfw, lie} } \ \cr
}\eeq
to obtain
\beq\eqalign{\label{eq:dpdtacc}
   D\tem(U,u) \tilde p(U,u) / d\tau_{(U,u)}
        &=  \gamma [ D\tem(U,u) \tilde \nu(U,u) / d\tau_{(U,u)}
              + d \ln \gamma / d\tau_{(U,u)} \, \nu(U,u) ]\cr
        &= \gamma [ P(u,U) \rightcontract P(U,u) \rightcontract a\tem(U,u) \cr
        &\qquad\qquad  + \half \gamma^2 D\tem(U,u) P(u)^\flat / d\tau_{(U,u)}
              (\nu(U,u),\nu(U,u)) \, \nu(U,u) ] \ .\cr
}\eeq

Only in the Lie case is the derivative of the spatial metric nonzero,
producing the expansion tensor 
$\theta(u)^\flat = \half D\lie(U,u) P(u)^\flat / d\tau_{(U,u)}$.
In all cases this relationship may be inverted 
to give the relative
acceleration as a function of the temporal derivative of the spatial
momentum
either from the previous equation using equation (\ref{eq:ppuu}) 
or by directly differentiating $\gamma^{-1} \tilde p(U,u)$.
The result is
\beq\eqalign{\label{eq:accdpdt}
       a\tem(U,u)
          &= \gamma^{-1} P_u(U,u)^{-1} 
   D\tem(U,u) \tilde p(U,u) / d\tau_{(U,u)} \cr
           & \qquad - \half \gamma^2 D\tem(U,u) P(u)^\flat / d\tau_{(U,u)}
              (\nu(U,u),\nu(U,u)) \, \nu(U,u)  \ .\cr
}\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Newpage
\Subsection{Splitting the total covariant derivative}
\label{sec:stcd}

Consider the splitting of the total
covariant derivative along an arbitrary timelike or null 
parametrized curve in spacetime
representing the worldline of a test particle of nonzero or zero
rest mass  $m$ respectively with 4-momentum $P$ and 
associated parameter $\lambda$. Although one could derive formulas
for the splitting of this operator acting on arbitrary tensors defined
along the curve,
the main application will be for a vector defined along the curve.
Let 
\beq
    Y = Y^\top(Y,u) + \vec Y(Y,u) \ , \qquad  \vec Y(Y,u) \equiv P(u) Y
\eeq
be such a vector.

Recalling the splitting of $P$ 
\beq
    P = E(P,u) [ u + \nu(P,u) ] \ ,
\eeq
and the Fermi-Walker total spatial covariant
derivative of $u$
\beq
    D\fw(P,u) u / d\tau_{(P,u)}
       = [ \del\fw(u) + \del(u)\sub{\nu(P,u)} ] u
       = a(u) - k(u) \rightcontract \nu(P,u) \ ,
\eeq
one finds
\beq\eqalign{
      P(u) \four D(P) Y / d\tau_{(P,u)}
       &= E(P,u)^{-1} P(u) \four D(P) Y / d\tau_{(P,U)}
        = D\fw(P,u) Y / d\tau_{(P,U)} \cr
       &= Y^\top(Y,U)   D\fw(P,u) u / d\tau_{(P,U)}
           + D\fw(P,u) \vec Y(Y,u) / d\tau_{(P,U)} \cr
}\eeq
for the spatial projection. For the temporal projection recall that
\beq\eqalign{
     \four \del\sub{P} u^\alpha &= u^\alpha{}_{;\beta} P^\beta
           =  - a(u)^\alpha u_\beta P^\beta - k(u)^\alpha{}_\beta P^\beta \cr
        &= E(P,u) [ a(u)^\alpha - k(u)^\alpha{}_\beta \nu(P,u)^\beta ] \cr
        &= E(P,u) D\fw(P,u) u / d\tau_{(P,u)} \cr
}\eeq
and
\beq\eqalign{
     [ \four \del\sub{P} Y ]^\top
        &= - u^\alpha \four\del\sub{P} Y_\alpha
         = \four\del\sub{P} [- u^\alpha Y_\alpha ] 
              + Y_\alpha \four\del\sub{P} u^\alpha \cr
       &= P Y^\top(Y,u) +  E(P,u) \vec Y(Y,u)^\flat
            \rightcontract D\fw(P,u) u / d\tau_{(P,u)} \cr
       &= E(P,u) [ d Y^\top(Y,U) / d\tau_{(P,u)} 
           + \vec Y(Y,u)^\flat
            \rightcontract D\fw(P,u) u / d\tau_{(P,u)} ] \cr
}\eeq
so
\beq\eqalign{
   [\four D(P) Y / d\tau_{(P,u)} ]^\top
      &= E(P,u)^{-1} [\four D(P) Y / d\lambda ]^\top \cr
      &= d Y^\top(Y,U) / d\tau_{(P,u)} 
           + \vec Y(Y,u) \cdot_u D\fw(P,u) u / d\tau_{(P,u)} \ . \cr
}\eeq

In each case the splitting of the total covariant derivative leads
to the Fermi-Walker total spatial covariant derivative of the
splitting fields plus a term involving the Fermi-Walker total
spatial covariant derivative of the observer 4-velocity, which
introduces kinematical terms. The same result applies to the
covector $Y^\flat$ since index shifting commutes with these operators.
This allows one to generate a formula for the splitting of the
total covariant derivative of any tensor.

\Newpage
\Section{Observer-adapted frame derivatives}

The previous section does not rely on the introduction of
an explicit spatial frame to
perform calculations.  However, spatial frames can be quite useful in 
making calculations much more concrete and automatic.
This is especially true in splitting spacetime fields involving derivatives,
where the use of an observer-adapted spacetime frame makes calculations
almost routine.

\Subsection{Natural frame derivatives}

It is convenient to have several notations for the frame derivatives of
functions (including component functions)
\beq
  e_\alpha f = \partial_\alpha f = f_{,\alpha}\ .
\eeq 
The commutators of these derivatives define the
structure functions $C^\alpha{}_{\beta\gamma}$ of the
frame $\{e_\alpha\}$
\beq
       [e_\alpha,e_\beta] = C^\gamma{}_{\alpha\beta} e_\gamma\ ,\qquad
       d \omega^\alpha = -\half C^\alpha{}_{\beta\gamma} 
           \omega^\beta \wedge \omega^\gamma\ ,
\eeq 
or
\beq
       \Lie\sub{e_\alpha} e_\beta = C^\gamma{}_{\alpha\beta} e_\gamma\ ,\qquad
            \Lie\sub{e_\alpha} \omega^\beta =- C^\beta{}_{\alpha\gamma}
                         \omega^\gamma\ .
\eeq 
These functions satisfy the Jacobi identity
\beq\label{eq:ji}
\eqalign{
 0&= [e_\alpha ,[e_\beta,e_\gamma]] + [e_\beta ,[e_\gamma,e_\alpha]] +
         [e_\gamma ,[e_\alpha,e_\beta]] \cr
 &= 2(\partial_{[\alpha} C^\delta{}_{\beta\gamma]} - C^\delta{}_{\epsilon [\alpha}
       C^\epsilon{}_{\beta\gamma]} )e_\delta\ ,\cr
}\eeq 
equivalent to $d^2 \omega^\alpha =0$.

The structure functions for an observer-adapted frame
\beq\meqalign{
& [e_a,e_b] = C^c{}_{ab} e_c + C^0{}_{ab} e_0\ , \qquad
    & d\omega^a = -\half C^a{}_{bc} \omega^b\wedge\omega^c 
            -C^a{}_{0b}\omega^0\wedge\omega^b\ ,\cr
& [e_0,e_a] = C^b{}_{0a} e_b + C^0{}_{0a} e_0\ , \qquad
    & d\omega^0 = -\half C^0{}_{ab} \omega^a\wedge\omega^b
            -C^0{}_{0a}\omega^0\wedge\omega^a\ ,\cr
}\eeq 
may be interpreted in terms of the geometry of the orthogonal decomposition,
using only noncovariant derivatives.  The purely spatial structure functions
govern the spatial Lie bracket and spatial exterior derivative for
purely spatial fields
\beq\eqalign{
  [e_a,e_b](u) &= \Lie(u)\sub{e_a} e_b = C^c{}_{ab} e_c\ ,\cr
  d(u) \omega^c  &= -\half C^c{}_{ab} \omega^a \wedge \omega^b\ ,\cr
}\eeq 
leading to the expressions
\beq\eqalign{
  [X,Y](u) &= 
   (X^b \partial_b Y^a - Y^b \partial_b X^a + C^c{}_{ab} X^a Y^b) e_a\ ,\cr
  d(u) \sigma  &= \half (\partial_a \sigma_b - \partial_b \sigma_a -
   \sigma_c C^c{}_{ab}) \omega^a \wedge \omega^b\ ,\cr
}\eeq 
for a spatial vector fields and 1-forms.

The structure functions $C^0{}_{ab}$ 
are proportional to the components of the rotation 2-form
\beq\eqalign{
    &\omega(u)^\flat = \half d(u) u^\flat = -\half  L d(u) \omega^0 
       = \fraction14  L C^0{}_{ab}\omega^a\wedge\omega^b\ ,\cr
    &\omega(u)_{ab} = \half  L C^0{}_{ab}\ .\cr
}\eeq 
These structure functions, or equivalently the rotation,
directly measure the failure of the
spatial frame vector fields to be hypersurface-forming since in the latter
case their Lie brackets would again lie in the distribution determining the
hypersurfaces, i.e., would again be spatial vector fields
\beq
    [X,Y] = [X,Y](u) + 2\omega(u)^\flat (X,Y) u
\eeq 

Similarly
\beq\eqalign{
  \Lie(u)\sub{u} e_a &= \L^{-1} \Lie(u)\sub{e_0} e_a =
         \L^{-1} C^b{}_{0a} e_b\ ,\cr
  \Lie(u)\sub{u} \omega^a &= u\leftcontract d\omega^a 
      =   -\L^{-1} C^a{}_{0b} \omega^b\ ,\cr
}\eeq 
show that the structure functions $C^b{}_{0a}$ directly describe the
 spatial  Lie derivatives along $u$
of the spatial frame and spatial
dual frame and which appear in the component expression for the
expansion tensor
\beq\eqalign{
  \theta(u)_{ab} &= \half [\Lie\sub{u} P(u)]_{ab} =
        (2\L)^{-1}[ h_{ab,0} - 2 C_{(a|0|b)}]\ ,\cr
\Theta(u) &= \theta(u)^c{}_c = - \Tr k(u) = 
     \Lie(u)\sub{u} \ln h^{1/2} - \L^{-1} C^c{}_{0c}\cr
}\eeq 
as follows from its Lie derivative definition.
For an orthonormal spatial frame, as used by Durrer and Straumann \Cite{1988a,b,
1989} in the slicing point of view,  for example,
then $h_{ab} = \delta_{ab}$
and $h_{ab,0}=0$, so that the symmetric combination of these structure
functions alone determine the expansion tensor.

The structure functions
$C^0{}_{0a}$ enter into the expression for the
acceleration of the congruence
\beq
  a(u)^\flat = \Lie(u)\sub{u} u^\flat =  
     d(u) \ln  L + C^0{}_{0a}\omega^a\ , 
\eeq 
and directly give the 
spatial acceleration components in a partially-normalized observer-adapted
frame (for which $\L=1$).

For a given observer congruence,
one may impose conditions on the structure functions
which do not directly represent kinematical quantities, provided that they
are compatible with the Jacobi identities.
The structure functions $C^a{}_{0b}$ determine how the spatial frame
is transported along the observer congruence. Introducing the
quantities
\beq\label{eq:ctem}
         \del\tem(u) e_a = C\tem(u)^b{}_a e_b \ ,
\eeq
one finds
\beq\eqalign{\label{eq:ctemformula}
  C\lie(u)^b{}_a &=  L^{-1} C^b{}_{0a}  \ ,\cr
  C\cfw(u)^b{}_a &=  L^{-1} C^b{}_{0a} + \theta(u)^b{}_a  \ ,\cr
  C\fw(u)^b{}_a &=  L^{-1} C^b{}_{0a} + \theta(u)^b{}_a 
                                  - \omega(u)^b{}_a \ .\cr
}\eeq
Setting the matrix $C\tem(u)^a{}_b$ to zero for each of the three choices
in turn
respectively defines the spatial frame's spatial Lie transport, 
its co-rotating Fermi-Walker transport, and its Fermi-Walker transport 
along $u$. 
An observer-adapted frame for which one of these three conditions hold
will be referred to respectively as a spatially-comoving observer-adapted
frame, a spatially co-rotating
observer-adapted frame
or a locally nonrotating observer-adapted frame.

For the
spatially-comoving observer-adapted frame, 
letting $(\alpha,\beta,\gamma,\delta)=(0,b,c,d)$ in  \ref{eq:ji}
reduces it to 
$C^d{}_{bc,0} =0$ under the condition $C^a{}_{0b}=0$,
showing that the spatial structure functions are constant along the
congruence for this class of frame.
That such a frame always exists is clear since one can always take
any partially-observer-adapted frame for which $e_0$ is along $u$
and the spatial frame is Lie dragged along $e_0$ (a {\it comoving 
partially-observer-adapted frame})
and simply spatially project that spatial frame. Similarly one can always
choose a {\it spatially-holonomic observer-adapted frame} by
taking comoving coordinates for the congruence and spatially projecting
the coordinate frame vector fields for the three coordinates which
parametrize the congruence. Such a frame is then also spatially comoving
and has $C^a{}_{bc}=0$, two conditions which make it the simplest choice
of frame in which to perform certain calculations. In a 
spatially-comoving observer-adapted frame, 
the  spatial  Lie derivative of a spatial tensor
with respect to either $e_0$ or $u$
just reduces to the ordinary derivative by these vector fields of the 
components.

Introduce the spatial tensor $C(u)$ 
\beq
          C(u)^a{}_{bc} =  C^a{}_{bc}
\eeq
whose spatial components in the observer-adapted frame
equal the corresponding structure functions.
One may then freely shift the indices on $C(u)$ unambiguously with
the spatial metric. This tensor represents the freedom left in the
choice of the spatial distribution of the spatial frame, once
the evolution along the observer congruence has been fixed.
\Message{clean up here??}

\Subsection{Splitting the connection coefficients}

So far, a temporal and a spatial covariant derivative operator
have been extracted from the spacetime covariant derivative, together with
the kinematical quantities of the observer congruence. Each of the
derivative operators, namely the spatial Fermi-Walker derivative and
the spatial covariant derivative, may act on any spacetime tensor fields,
but preserve the spatial property of spatial fields.
These operators, together with the spatial kinematical fields, are enough
to express the measurement of the
spacetime covariant derivative of a spacetime tensor in terms of spatial
operators acting on the spatial fields which represent that tensor.

The simplest way to obtain such measurement formulas is to use a
spatially-comoving spatially-holono\-mic observer-adapted frame and
simply evaluate the standard formulas (see Appendix A)
for the components of the
connection and of the covariant derivative of those spacetime tensors
which are of interest.  Such a frame is used as a fundamental tool in
the approaches of both Zel'manov \Cite{1956} and
Cattaneo \Cite{1958,1960} to the threading point of view.

The components of the spacetime metric connection with respect to a frame
$\{e_\alpha\}$ will here be defined with the ordering of the covariant 
components following the ``del convention" of Hawking and Ellis \Cite{1977}
\beq
       \four \Gamma^\alpha{}_{\beta\gamma} =
   \omega^\alpha{}( \four\del\sub{e_\beta} e_\gamma) 
\eeq 
rather than the ``semi-colon convention"
$       \four \Gamma^\alpha{}_{\beta\gamma} =
   \omega^\alpha{}( e_{\beta;\delta} e_\gamma^\delta)$
of Misner, Thorne and Wheeler \Cite{1973}.
Using the notation
\beq
      A_{\{\delta\beta\gamma\}_-} = A_{\delta\beta\gamma} -
         A_{\beta\gamma\delta} + A_{\gamma\delta\beta} \ .
\eeq 
and shifting indices on the structure functions 
with the frame component matrix of the spacetime metric,
the connection components can be written
\beq
       \four \Gamma^\alpha{}_{\beta\gamma} =
   \half \four\gm^{\alpha\delta}( \four g_{\{\delta\beta,\gamma\}_-}
       + C_{\{\delta\beta\gamma\}_-} ) \ .
\eeq 

Consider the
$1\choose2$-tensor
\beq
   \four \Gamma^\alpha{}_{\beta\gamma}
    e_\alpha \otimes \omega^\beta \otimes \omega^\gamma\ 
\eeq 
which represents the connection coefficients in a particular choice of frame
and of course changes to a different tensor under a change of frame, according
to the inhomogeneous transformation law for the connection coefficients.
This tensor can be split as any other, but the results will be frame dependent
like the tensor itself. However, if the frame is at least a 
partially-observer-adapted frame, then
some of the splitting information is frame independent 
and one can interpret some of the
fields which result from the splitting in a geometric way, 
similar to the interpretation of the structure functions of an
observer-adapted frame. The more general partially-observer-adapted
frames will be considered below in the discussion of a nonlinear
reference frame. Here attention will be restricted to the class
of observer-adapted frames.

%%%%%%%%%%%%
\Subsection{Observer-adapted connection components}

Measurement of the connection  coefficient ``tensor" in the
observer-adapted frame leads to the splitting
\beq\eqalign{
  \four\Gamma^\alpha{}_{\beta\gamma} &\quad\leftrightarrow\quad
  \big\{ \four\Gamma^0{}_{00};
       \four\Gamma^0{}_{a0}, \four\Gamma^0{}_{0a}, \four\Gamma^a{}_{00}; 
       \four\Gamma^0{}_{ab}, \four\Gamma^a{}_{0b}, \four\Gamma^a{}_{b0}; 
       \four\Gamma^a{}_{bc} \big\}\cr
&\quad\leftrightarrow\quad
  \big\{ \four\Gamma^\top{}_{\top\top};
       \four\Gamma^\top{}_{a\top}, \four\Gamma^\top{}_{\top a}, 
          \four\Gamma^a{}_{\top\top}; 
       \four\Gamma^\top{}_{ab}, \four\Gamma^a{}_{\top b}, 
           \four\Gamma^a{}_{b\top}; 
       \four\Gamma^a{}_{bc} \big\}\cr
}\eeq 
where the 0 indices have been rescaled by $\L$ to correspond to components
along $e_\top$ and $\omega^\top$.
These fields are also connected closely with the geometry of the congruence
\beq\eqalign{
&\four\Gamma ^\top {}_{\top \top } = \Lie(u)\sub{u} (\ln \L)\ ,\cr
&\four\Gamma ^a {}_{\top \top } =   a(u)^a\ ,\cr
&\four\Gamma ^\top {}_{a \top } = [d(u) (\ln \L)]_a \ ,\cr
   %=a(u)_a - C^0{}_{0a}\ ,\cr
&\four\Gamma ^\top {}_{\top a } = a(u)_a\ ,\cr
&\four\Gamma ^\top {}_{ab} = k(u)_{ba} \ ,\cr \typeout{sign??}
&\four\Gamma ^a {}_{\top b } = -k(u)^a{}_b +  C\lie(u)^a{}_b \ ,\cr  
&\four\Gamma ^a {}_{b \top } = -k(u)^a{}_b \ ,\cr  
&\four\Gamma ^a {}_{b c } = \Gamma(u)^a{}_{bc} \ .\cr 
}\eeq 

The spatial
components of the spacetime connection equal the components of the spatial
part of the spatial connection $\del(u) \circ P(u)$
\beq\eqalign{ \label{eq:scc}
   \del(u)\sub{e_a} e_b &= \Gamma(u)^c{}_{ab} \ ,\cr
   \Gamma(u)^a{}_{bc} 
      &= \half h^{ad}( h_{ \{ db,c \}_- } + C(u)_{ \{dbc\}_- }) \ ,\cr
}\eeq 
For the hypersurface point of view, one makes the replacements $u \to n$,
%\message{hyp LMN?} 
$ L \to N$, 
$\top \to \bot$ for ``perpendicular" to $n$, $\omega(u) \to 0$, and
$k(u) \to K(n)$, the extrinsic curvature tensor.
The spatial part of the
spatial connection is then equivalent to the connection of the induced
metric on each slice.

Only when $\L=1$ do the above formulas give the components of the connection
in the partially-normalized frame.  This distinction only matters for the
explicitly $\L$-dependent fields $\four \Gamma^\top{}_{\top\top}$
and $\four \Gamma^\top{}_{a \top}$,
which are associated with the threading parametrization. These two fields
are not components of $\four\del e_\top$
as conventional notation would imply, but rather of $\L^{-1}\four\del e_0$,
unless $\L=1$ when $e_0=e_\top$,
in which case they vanish. Otherwise
they serve the
purpose of converting temporal and spatial derivatives of 0-indexed
components to derivatives of the corresponding $\top$-indexed components.
The purely spatial components determine the spatial part of the spatial
connection while the remaining connection components dependent only on
the kinematical quantities, which determine the temporal part of the
spatial connection.
The contribution $C\lie(u)^a{}_b$ to the field $\four\Gamma^a{}_{\top b}$
comes from the  spatial  Lie derivatives of the spatial frame
vectors and depends on their choice.

Using a 
partially-normalized  spatially-comoving spatially-holonomic
observer-adapted frame 
\beq
   \L=1\ , \qquad C^a{}_{0b} = 0 = C^a{}_{bc}\ , \quad 
\eeq 
eliminates the ``gauge terms" which depend on the choice of $e_0$ and the
spatial frame,
making it the most convenient frame to use in
splitting the covariant derivative of a tensor field, leading to a
family of spatial fields which arise from the original family representing
the split of the tensor itself by a linear operation involving the
spatial covariant derivative, the  spatial  Lie derivative along
$u$ and the kinematical quantities of the congruence. This was already
done for the geodesic operator without the aid of such a frame.
Furthermore, the  spatial  Lie derivative may be replaced by the
spatial Fermi-Walker derivative plus a kinematical term to yield
alternative expressions for the measured fields.
The resulting formulas, in either case, are valid in any observer-adapted
frame, the above ``gauge terms" entering into the expressions for 
the observer-adapted components
of the spatial Lie and covariant derivatives.

%%%%%%%%%%%
\Subsection{Splitting covariant derivatives}

It is quite useful to explicitly consider splitting the covariant derivatives 
of  vector fields, 1-forms and $1\choose1$-tensor fields. A vector field
$X^\alpha$ splits into the family $\{ X^\top, X^a\}$ and its covariant
derivative $X^\alpha{}_{;\beta}$ splits into the family
\beq\eqalign{
X^\top{}_{; \top} &= \Lie(u)\sub{u} X^\top  + a(u)_d X^d \ ,\cr
X^a   {}_{; \top} &= \Lie(u)\sub{u} X^a -k(u)^a{}_d X^d +a(u)^a X^\top \ ,\cr
X^\top{}_{;    a} &= \del(u)_a X^\top -k(u)_{da} X^d   \ ,\cr
X^a   {}_{;    b} &=  \del(u)_b X^a - k(u)^a{}_b X^\top  \ ,\cr
}\eeq 
with trace
\beq
  X^\alpha{}_{;\alpha} = \four\div X = \{ \Lie(u)\sub{u} + \Theta(u) \} X^\top
        + \{ \del(u)_c + a(u)_c \} X^c\ .
\eeq 
In these expressions and in similar ones below, the symbols 
$\Lie(u)\sub{u} X^{\top}$ and  $\Lie(u)\sub{u} X^a$ indicate respectively
the spatial Lie derivative of a scalar $X^\top$ and the 
spatial components of the spatial Lie derivative of the spatial vector
$X^a$. The same applies to the spatial covariant derivative.
In other words scalar projections along $e_\top$ are intended
before the action of spatial derivative operators.

If $X$ and $Y$ are spatial vector fields, then these formulas imply the result
\beq
  \four\del_X Y - \del(u)_X Y = -k(u)(Y,X)\, e_\top 
         = [\theta(u)(Y,X)-\omega(u)(Y,X)] e_\top\ .
\eeq
Thus $k(u)$ describes the difference between spatial and spacetime differentiation
in the local rest space of the observer. When the vorticity $\omega(u)$ is zero, 
then $k(u)$ reduces to the extrinsic curvature tensor $-\theta(u)$ of the 
hypersurfaces orthogonal to the observer family, namely the sign-reversed second
fundamental form of those hypersurfaces. As one parallel transports $Y$ along $X$
in the local rest space using the spatial connection, $-k(u)(Y,X)$ describes the
spacetime rate of change of the component of the transported field along the direction 
of the observer 4-velocity necessary to keep $Y$ spatial compared to the spacetime parallel transported field.

It is worth noting that for any tensor 
$S^{\alpha\ldots}_{\ \ \beta\ldots}$ %expressed in a spatially-comoving frame, 
one has
\beq
  \{ \Lie(u)\sub{u} + \Theta(u) \} S 
    = h^{-1/2} \Lie(u)\sub{u}(h^{1/2} S) - \L^{-1} C^c{}_{0c} S\ ,
\eeq 
which gives an alternative expression for the temporal part of the
spacetime divergence, the second term of which vanishes in a spatially-comoving
frame.
The spatial part on the other hand is the spacetime
divergence of the spatial part of the vector field
\beq
 [P(u)X]^\alpha{}_{;\alpha} = \{ \del(u)_c + a(u)_c \} X^c\ ,
\eeq 
leading to the addition of the acceleration term to the spatial divergence.
Such formulas can also be obtained without using an adapted frame by relying
only on the properties of the spatial projection operator,
as described by Ehlers \Cite{1961, 1993} and Ellis \Cite{1971}. For example
\beq\eqalign{ 
\four \del_\alpha X^\alpha &=
    \four \del_\alpha (P(u)^\alpha{}_\beta X^\beta)
   =P(u)^\alpha{}_\beta \four \del_\alpha X^\beta
     + X^\beta \four\del_\alpha P(u)^\alpha{}_\beta\cr
   &= \del(u)_\beta X^\beta + X^\gamma P(u)^\beta{}_\gamma \four\del_\alpha
         (u^\alpha u_\beta)
    = [\del(u)_\beta + a(u)_\beta] X^\beta\cr
}\eeq 
illustrates this technique for a spatial vector field $X$.

For a 1-form $X_\alpha$ represented by the spatial fields $\{X_\top,X_c\}$,
the covariant derivative $X_{\alpha;\beta}$
has the following splitting
\beq\eqalign{
X_{\top ; \top} &= \Lie(u)\sub{u} X_\top - X_d a(u)^d \ ,\cr
X_{a    ; \top} &= \Lie(u)\sub{u} X_a + X_d k(u)^d{}_a -  % sign!!
            X_\top a(u)_a \ ,\cr
X_{\top ;  a  } &=  \del(u)_a X_\top + X_d k(u)^d{}_a \ ,\cr
X_{a    ;    b} &=  \del(u)_b X_a + X_\top k(u)_{ba}\ .\cr
}\eeq 
Twice 
the symmetrized covariant derivative of the 1-form equals the Lie derivative
of the spacetime metric with respect to its corresponding vector field
\beq\eqalign{
[\Lie\sub{X} \four\gm]_{\top \top} &=  
    2\Lie(u)\sub{u} X_\top  - 2X_d a(u)^d \ ,\cr
[\Lie\sub{X} \four\gm]_{\top    a} &= 
    \{ \del(u)_a - a(u)_a \} X_\top + \Lie(u)\sub{u} X_a 
      + 2 X_b k(u)^b{}_a \ ,\cr&= 
    \{ \del(u)_a - a(u)_a \} X_\top + h_{ab}\Lie(u)\sub{u} X^b
        + 2X_b \omega(u)^b{}_a    \ ,\cr
[\Lie\sub{X} \four\gm]_{a       b} &=  
    \Lie(u)\sub{P(u)X} P(u)_{ab} - 2 X_\top \theta(u)_{ab}
   = \Lie(u)\sub{X} P(u)_{ab} \ ,\cr
}\eeq 
where
\beq
   \Lie(u)\sub{P(u)X} P(u)_{ab} = 2\del(u)_{(a} X_{b)}\ ,
\eeq 
and
\beq
\Lie(u)\sub{u} X_a - 2 X_b \theta(u)^b{}_a 
    = h_{ab} \Lie(u)\sub{u} X^b\ .
\eeq 

Finally consider splitting the covariant derivative of the $1\choose1$-tensor
$S^\alpha{}_\beta$
\beq\eqalign{
S^\top{}_{\top ; \top } &= \Lie(u)\sub{u} S^\top{}_\top  + a(u)_d S^d{}_\top
  - S^\top{}_d a(u)^d  \ ,\cr
%
S^a   {}_{\top ; \top } &= \Lie(u)\sub{u} S^a{}_\top - k(u)^a{}_d S^d{}_\top
   -S^a{}_d a(u)^d  \ ,\cr
S^\top{}_{a    ; \top } &= \Lie(u)\sub{u} S^\top{}_a +S^\top{}_d k(u)^d{}_a  
   - a(u)_a S^\top{}_\top +a(u)_d S^d{}_a  \ ,\cr
S^\top{}_{\top ; a    } &= \del(u)_a S^\top{}_\top - k(u)_{ad} S^d{}_\top
   +S^\top{}_d k(u)^d{}_a    \ ,\cr
%
S^a   {}_{b    ; \top } &= \Lie(u)\sub{u} S^a{}_b - k(u)^a{}_d S^d{}_b
  + S^a{}_d k(u)^d{}_b  +a(u)^a S^\top{}_b -S^a{}_\top a(u)_b \ ,\cr
S^a   {}_{\top ; b    } &= \del(u)_b S^a{}_\top   + S^a{}_d k(u)^d{}_b  
- S^\top{}_\top k(u)^a{}_b  \ ,\cr
S^\top{}_{a    ; b    } &= \del(u)_b S^\top{}_a   + S^\top{}_\top k(u)_{ba}  
 - k(u)_{bd} S^d{}_a  \ ,\cr
S^a{}_{b       ; c    } &= \del(u)_c S^a{}_b - k(u)^a{}_c S^\top{}_b + 
  S^a{}_\top  k(u)_{bc}  \ ,\cr
%
}\eeq 
with trace
\beq\eqalign{
S^\alpha{}_{\top ; \alpha} &=  
[\Lie(u)\sub{u} + \Theta(u)] S^\top{}_\top  + [\del(u)_c +a(u)_c ] S^c{}_\top 
+ S^c{}_d k(u)^d{}_c - a(u)^c S^\top{}_c \ ,\cr
S^\alpha{}_{   a ; \alpha} &=  [\Lie(u)\sub{u} + \Theta(u)] S^\top{}_a
+ S^\top{}_c k(u)^c{}_a +  S^c{}_\top k(u)_{ca} -a(u)_a S^\top{}_\top\cr
  &\qquad + [\del(u)_c +a(u)_c ] S^c{}_a \ .\cr
}\eeq 
Notice that the temporal derivative which occurs in the divergence
is the $\Theta(u)$-augmented Lie derivative, while the spatial divergence
occurs in each case accompanied by an acceleration term to form
the spacetime divergence of the spatial field $S^a{}_\top$ and 
the spatial projection of the spacetime divergence of the spatial field 
$S^a{}_b$.

These formulas have interesting special cases for symmetric and antisymmetric
tensors. For a symmetric tensor one has
\beq\eqalign{
S^\alpha{}_{\top ; \alpha} &=  
[\Lie(u)\sub{u} + \Theta(u)] S^\top{}_\top  
  + [\del(u)_c +2a(u)_c ] S^c{}_\top \message{check}
  -S^c{}_d \theta(u)^d{}_c \ ,\cr
S^\alpha{}_{   a ; \alpha} &=  [\Lie(u)\sub{u} + \Theta(u)] S^\top{}_a
  -a(u)_a S^\top{}_\top
  + [\del(u)_c +a(u)_c ] S^c{}_a \ .\cr
}\eeq 
while for an antisymmetric tensor one has
\beq\eqalign{\label{eq:atdsplit}
S^\alpha{}_{\top ; \alpha} &= \del(u)_c S^c{}_\top - 
  \eta(u)^{cab}\omega(u)_c S_{ab}
  = \del(u)_c S\E{}^c 
   - 2\vec\omega(u) \cdot_u [\dualp{u} S\M ]^\sharp  \ ,\cr
S^\alpha{}_{   a ; \alpha} &=  h_{ac} [\Lie(u)\sub{u} + 
  \Theta(u)] S\E{}^c
  + [\del(u)_c +a(u)_c] S\M{}^c{}_a \cr
    &= h_{ab}  [\Lie(u)\sub{u} +   \Theta(u) ] S\E{}^{\sharp\, b}
  + \{ [ \vec\del(u) +a(u) ] \times_u [ \dualp{u} S\M ]^\sharp \}_a
\ .\cr
}\eeq 
\message{check?? previous too}

Note that analogous to the relation for a spatial vector field,
the spatial projection of the spacetime divergence 
and the spatial divergence of a $1\choose1$-tensor
differ by an acceleration term
\beq \label{eq:sd}
\eqalign{
X^\gamma{}_{;\gamma} &= [\del(u)_c +a(u)_c ]X^c 
  =[\del(u)_\gamma +a(u)_\gamma ]X^\gamma \ ,\cr
X_a{}^\gamma{}_{;\gamma} &= [\del(u)_c +a(u)_c ]X_a{}^c\ ,
  \qquad {\rm or} \qquad 
P(u)^\delta{}_\alpha X_\delta{}^\gamma{}_{;\gamma} 
  = [\del(u)_\gamma +a(u)_\gamma ]X_\alpha{}^\gamma\ ,\cr
}\eeq 
while for a spatial 2-form one has
\beq
X_a{}^\gamma{}_{;\gamma} = 
  \{ [\vec\del(u) +a(u) ]\times_u \dualp{u} X{}^\sharp \}_a\ ,
\eeq 
leading to the acceleration-augmented curl of its spatial dual which
represents the spatial projection of the spacetime divergence.

In the case of a stationary spacetime, where one may define $u$ by
normalizing a timelike
Killing vector field $ u = \ell^{-1} \xi$, then the
Killing equation $\Lie\sub{\xi} \four\gm =0$ may be expressed in terms of
$u$ using the splitting of $\Lie\sub{X} \four\gm$ given above with
$X=\xi$, $X^\top=\ell\neq 0$ and $X^a = 0$.
This immediately yields three conditions on $u$ (the spatial scalar, spatial
vector and spatial tensor equations which represent the splitting 
of the Killing equation)
\beq
   \Lie(u)\sub{u} \ell = 0 \ ,\qquad
  a(u)_a = \del(u)_a \ln \ell\ ,\qquad
  \theta(u)_{ab} =0 \ .
\eeq
The normalization factor must be a stationary function, the logarithm of which
serves as an acceleration potential (modulo sign conventions),
and the expansion tensor must vanish.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Subsection{Observer-adapted components 
of total spatial covariant derivatives}
\label{sec:extscd}

Given the definitions (\ref{eq:ctem}) and (\ref{eq:scc}),
the total spatial covariant derivative of the spatial frame along the worldline
of a test particle with 4-momentum $P$ is given by
\beq\eqalign{
  D\tem (U,u) e_a / d\tau_{(P,u)} &=  [ \del\tem(u) + \del(u)\sub{\nu} ] e_a \cr
     &= [ C\tem(u)^b{}_a + \Gamma(u)^b{}_{ca} \nu(P,u)^c ] e_b \cr
      &= [ C\tem(u)^b{}_a + \bfomega(u)(\nu(P,u))^b{}_a ] e_b \ ,\cr
  &\qquad {\scriptstyle {\rm tem} \,=\, {\rm fw, cfw, lie} } \ .\cr
}\eeq 
where $(\bfomega(u)(\nu(P,u))^b{}_a)$ is the matrix of spatial connection
1-forms 
\beq
   \bfomega(u)^b{}_a = \Gamma(u)^b{}_{ca} \omega^c 
\eeq
evaluated on the relative velocity vector. 
The spatial connection 1-form is discussed below in
section (\ref{sec:sc}).

From the appropriate product rule,
one then has for a spatial vector field the expression
\beq
  D\tem (P,u) X^a / d\tau_{(P,u)} = 
       d X^a / d\tau_{(P,u)} +  C\tem(u)^a{}_b X^b
         + \Gamma(u)^a{}_{bc} \nu(P,u)^b X^c \ ,
\eeq
or in general for a spatial tensor field
\beq\eqalign{
&  D\tem (P,u) S^{a\ldots}_{\ \ b\ldots} / d\tau_{(P,u)}  \cr
&\qquad\qquad = d S^{a\ldots}_{\ \ b\ldots}  / d\tau_{(P,u)} 
          +  [ \sigma([C\tem(u) + \bfomega(u)(\nu(P,u) ]\rightcontract X) S ]
                   ^{a\ldots}_{\ \ b\ldots} \ .\cr
}\eeq
When the spatial frame undergoes the corresponding transport along the
observer congruence, the term involving $C\tem(u)^a{}_b=0$ 
vanishes and the total
spatial covariant derivative of that type
reduces to a simpler expression.
One may find the Lie total spatial covariant derivative in this form
mentioned by
M\o ller \Cite{1952} and in subsequent work by Zel'manov \Cite{1956} and
Cattaneo \Cite{1958}, all of whom use spatially-comoving spatially-holonomic
spatial frames.

In general
only the ordinary and co-rotating Fermi-Walker transport conditions 
for the spatial frame along $u$
are compatible with the condition
of orthonormality. \message{when is Lie transport comp with ON condition??}%
An orthonormal spatial frame of this latter type
has an arbitrary distribution of orientation
on a given hypersurface transversal to the observer congruence, but its
subsequent behavior away from that hypersurface 
along the observer congruence is completely fixed by
the transport condition.
For a given choice of such a frame, one may examine the relative rotation
of a spatial frame which is transported along an arbitrary timelike worldline
by the corresponding transport, which accomplished
by requiring that the corresponding
total covariant derivative of the frame along the worldline vanish.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Suppose  $\{e_a\}$ is any spatial frame defined on some fixed hypersurface
$\Sigma$ transversal to the observer congruence. One can extend it to a
spatial frame on the spacetime by transporting it along the congruence
by one of the three spatial transport conditions
\beq
       \del\tem(u) e_a = 0 
  \quad\leftrightarrow\quad C\tem(u)^a{}_b = 0 \ .
\eeq
One can also spatially transport it along an arbitrary timelike or null
worldline from its value at the intersection of that worldline with the
hypersurface to define a spatial frame along that single worldline
\beq
         D\tem(P,u) \Escr(P,u,e)_a = 0 \ .
\eeq
The two such frames along the worldline
are linearly related
\Typeout{Change symbol for rotation matrix from kernel letter R}
\beq
      \Escr\tem(P,u,e)_a = R\tem(P,u,e)^b{}_a e_b  \ .
\eeq

If $\{e_a\}$ is an orthonormal spatial frame on $\Sigma$, then in general
only in the ordinary and co-rotating Fermi-Walker cases  will it remain
orthonormal as it is extended along the observer congruence
to a spatial frame on spacetime, as is the
case for its corresponding spatial transport along the single given worldline.
In these cases the matrix $\bfR\fw(P,u,e)$ or $\bfR\cfw(P,u,e)$ respectively
will be an orthogonal matrix describing the relative rotation of the
orthonormal spatial frame transported along the worldline with respect to
the given orthonormal spatial frame on spacetime.

Suppose $X=X^a e_a$  undergoes the corresponding transport along the given
worldline
\beq
    0 = D\tem(P,u) X^a / d\tau_{(P,u)}
       = d X^a / d\tau_{(P,u)} + \bfomega(\nu(P,u))^a{}_b X^b
\eeq
or
\beq
    d X^a / d\tau_{(P,u)} = -\bfomega(u)(\nu(P,u))^a{}_b X^b \ .
\eeq 

Since $\{e_a\}$ is an orthonormal spatial frame, its connection
1-form is antisymmetric in its tensor-valued indices
(see section (\ref{sec:ced}) of the appendix),
so its spatial dual defines a vector-valued 1-form whose value on the
relative velocity defines the relative angular velocity of the two
orthonormal frames $\{e_a\}$ and $\{\Escr(P,u)_a\}$ along the worldline
\beq
        \zeta\sc(P,u,e) = \dualp{u} \bfomega(u)(\nu(P,u))^\sharp \ ,
\eeq
or
\beq\eqalign{
   \zeta\sc(P,u,e)^a 
       &= \half \eta(u)^{abc} \bfomega(u)(\nu(P,u))_{bc}\ , \cr
   \bfomega(u)(\nu(P,u))^a{}_b 
       &= \eta(u)^a{}_{bc} \zeta\sc(P,u,e)^c \ ,\cr
       &\qquad {\scriptstyle {\rm tem} \,=\, {\rm fw, cfw} } \ .\cr
}\eeq
The transport condition for $X$ then takes the more familiar form
\beq
    d X^a / d\tau_{(P,u)} = \eta(u)^a{}_{bc} \zeta\sc(P,u,e)^b X^c
                           = [\zeta(P,u,e) \times_u X]^a \ .
\eeq
Letting $X=\Escr\tem(P,u,e)_a$ leads to the relationship between the
relative rotation and the relative angular velocity
\beq
         d R\tem(P,u,e)^a{}_c / d\tau_{(P,u)} \, R\tem(P,u,e)^{-1\, c}{}_b 
            = \eta(u)^a{}_{cb}\zeta\sc(P,u,e)^c \ .
\eeq
This determines the finite rotation of the frame $\{ \Escr\tem(P,u,e)_a\}$ 
defined along the worldline with 4-momentum $P$ with
respect to the frame $\{e_a\}$ restricted to this worldline.
Note that relative rotation along the worldline
of the co-rotating Fermi-Walker transported
spatial frame with respect to the Fermi-Walker transported spatial frame
is related to the vorticity vector $\vec\omega(u)$ in the same way
that $\bfR\tem(P,u)$ is related to the relative angular velocity
$\zeta\tem(P,u)$ since
\beq
     D\cfw(P,u) X^a / d\tau_{(P,u)} = 
     D\fw(P,u) X^a / d\tau_{(P,u)} + \omega(u)^a{}_b X^b
\eeq
and $\vec\omega(u)$ is the spatial dual of $\omega(u)^\sharp$.

The interpretation of the rotation $\bfR\tem(P,u)$
is clear. The initial spatial 
distribution of the orientation of $e_a$ on the hypersurface $\Sigma$ is
completely arbitrary and this arbitrariness is propagated along
the observer congruence by the choice of either ordinary or co-rotating
spatial Fermi-Walker transport.
This rotation removes the arbitrariness along
the given worldline. Of course different paths between the same pair
of points in spacetime will in general lead to different results.
This is a manifestation of spatial curvature for the given observer
congruence.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Newpage
\Section{Relative kinematics: applications}

There are two interesting applications of 
the splitting of the total
covariant derivative
along an arbitrary timelike or null 
parametrized curve in spacetime
representing the worldline of a test particle of nonzero or zero
rest mass  $m$ respectively with 4-momentum $P$ and 
associated parameter $\lambda$. 
The first 
\beq
   \four D(P) P / d \lambda = m f(P) \ ,  \qquad
        f(P)_\alpha P^\alpha = 0 \ ,
\eeq
is the
``acceleration equals force per unit mass"
equation for a timelike worldline with 4-velocity $ U = m^{-1}P$
\beq
    \four D(U) P / d \tau_U = f(U) 
      \quad\leftrightarrow\quad 
    \four D(U) U / d \tau_U = \tilde f(U) \ ,
\eeq
or the affinely parametrized geodesic equation for a null
worldline ($m=0$)
\beq
      \four D(P) P / d \lambda = 0 \ .
\eeq
The former
reduces to the geodesic equation
in the absense of an applied  force, which when nonzero 
must be orthogonal to $U$
in order to maintain the unit character of the 4-velocity
\beq
     0 = \four D(U) [ U^\alpha U_\alpha] / d\tau_U 
       = 2 U^\alpha \four D U_\alpha / \tau_U 
       = 2 U^\alpha \tilde f(U)_\alpha \ .
\eeq
These two variations of original equation will both be referred to
as the acceleration equation.

The second is the equation for the
Fermi-Walker transport of a vector along an arbitrary timelike
worldline to which it is orthogonal,
which describes the behavior of the spin vector 
of a torque-free test gyro carried along that worldline or the spin of
a spinning test particle following that worldline
\beq
 \del\fw (U) S = 0\ ,   \qquad U_\alpha S^\alpha=0 \ .
\eeq 
This will be referred to simply as the
spin transport equation.

\Subsection{Splitting the acceleration equation}

Letting $Y=P$ in subsection (\ref{sec:stcd}) leads to 
\beq\eqalign{\label{eq:delppsplit}
      P(u) \four D(P) P / d\tau_{(P,u)}
       &= E(P,u) D\fw(P,u) u / d\tau_{(P,u)}
           + D\fw(P,u) p(Y,u) / d\tau_{(P,u)} \ ,\cr
   [\four D(P) P / d\tau_{(P,u)} ]^\top
      &= d E(P,u) / d\tau_{(P,u)} 
           + p(Y,u) \cdot_u D\fw(P,u) u / d\tau_{(P,u)} \ . \cr
}\eeq

Consider first the timelike case, where one can use the unit 4-velocity $U$
instead of the 4-momentum $P$
and 
\beq
      \four D(U) U / d\tau_U \equiv a(U)
\eeq
defines the 4-acceleration.
The acceleration vector and hence the force vector must be spatial
with respect to $U$, so their temporal components are determined by
the orthogonality condition
\beq\imeqalign{
     a(U) &= \gamma [ \Ascr(U,u) u + A(U,u) ] \ , \qquad&
         \Ascr(U,u) &= A(U,u) \cdot_u \nu(U,u) \ , \cr
     f(U) &= \gamma [ \wp(U,u) u +  F(U,u) ]\ ,\qquad &
          \wp(U,u) &=  F(U,u) \cdot_u \nu(U,u) \ .\cr
}\eeq
Here $\wp(U,u)$ is the power and $F(U,u)$ the spatial force relative
to the observer congruence as defined in special relativity
\Cite{Anderson 1967}. These are defined respectively so that
the appropriate time derivatives of the spatial momentum and energy
as seen by the observer equal respectively the spatial force and
power.

Dividing out the mass parameter in the above equations (\ref{eq:delppsplit})
describes the splitting of the acceleration vector
\beq\eqalign{\label{eq:accsplit}
     A(U,u)      
       &= \gamma D\fw(U,u) u / d\tau_{(U,u)}
           + D\fw(U,u) \tilde p(U,u) / d\tau_{(U,u)} \ ,\cr
    \Ascr(U,u) 
      &= d \tilde E(U,u) / d\tau_{(U,u)} 
           + \tilde p(U,u) \cdot_u D\fw(U,u) u / d\tau_{(U,u)} \ . \cr
}\eeq
Having split both the acceleration and the force, equating their
spatial and temporal parts leads to the splitting of the acceleration
equation. With this in mind, the following preliminary definition proves
useful
\beq\eqalign{
    \tilde F{}\fw\g(U,u) &= - \gamma D\fw(U,u) u / d\tau_{(U,u)} \cr
       &= \gamma [ -a(u) + k(u) \rightcontract \nu(P,u)] \ ,\cr
}\eeq
so that
\beq\eqalign{\label{eq:auuf}
     A(U,u)      
       &= D\fw(U,u) \tilde p(U,u) / d\tau_{(U,u)} 
              - \tilde F{}\fw\g(U,u)\ ,\cr
       &\equiv D\tem(U,u) \tilde p(U,u) / d\tau_{(U,u)} 
              - \tilde F{}\tem\g(U,u)\ ,\cr
         &\qquad {\scriptstyle {\rm tem} \,=\, {\rm fw, cfw, lie} } \ .\cr
}\eeq

Equating $A(U,u)$ to the spatial force $\tilde F(U,u)$ per unit mass
then leads to the spatial projection of the acceleration equation
\beq\eqalign{\label{eq:derspamomentum}
     D\tem(U,u) \tilde p(U,u) / d\tau_{(U,u)}
         &=  \tilde F{}\tem\g(U,u) + \tilde F(U,u) \ ,\cr
         &\qquad {\scriptstyle {\rm tem} \,=\, {\rm fw, cfw, lie} } \ .\cr
}\eeq
This defines the
ordinary and co-rotating and the Lie {\it spatial gravitational forces}
per unit mass, related to each other in the same way as the corresponding
total spatial covariant derivatives
\beq\eqalign{
      \tilde F{}\cfw(U,u) &= 
      \tilde F{}\fw(U,u) + \omega(u) \rightcontract \tilde p(U,u) \cr
  &= \tilde F{}\lie(U,u) + \theta(u) \rightcontract \tilde p(U,u) \ .\cr
}\eeq
Since the Lie total spatial covariant derivative does not
commute with index lowering, it is convenient to introduce also
the Lie covariant spatial gravitational force
\beq
     [ D\tem(U,u) \tilde p(U,u)^\flat / d\tau_{(U,u)} ]^\sharp
         =  \tilde F\lief\g(U,u) + \tilde F(U,u) \ .
\eeq
The four distinct spatial gravitational forces have the explicit 
form
\beq\eqalign{
  \tilde F\g\fw(U,u) &= \gamma [ - a(u) +   \nu(U,u) \times_u \vec\omega(u) 
                      - \theta(u) \rightcontract  \nu(U,u) ] \ ,\cr
  \tilde F\g\cfw(U,u) &= \gamma [ - a(u) + 2  \nu(U,u) \times_u \vec\omega(u)  
                      - \theta(u) \rightcontract  \nu(U,u) ] \ ,\cr
  \tilde F\g\lie(U,u) &= \gamma [ - a(u) + 2  \nu(U,u) \times_u \vec\omega(u)
                      - 2\theta(u) \rightcontract  \nu(U,u) ] \ ,\cr
 \tilde F\g\lief(U,u) &= \gamma [ - a(u) + 2  \nu(U,u) \times_u \vec\omega(u) ]
                                \ .\cr
}\eeq

Similarly the temporal projection of the acceleration equation
becomes the equation specifying the rate of change of energy
\beq\eqalign{\label{eq:derenergy1}
    d \tilde E(U,u) / d\tau_{(U,u)} 
           &= \nu(U,u) \cdot_u [ \tilde F\fw\g(U,u) + \tilde F(U,u) ] \cr
           &= - \gamma [ a(u) \cdot_u \nu(U,u)
%?? minus: ??
               - \theta(u)^\flat(\nu(U,u),\nu(U,u)) ] + \wp(U,u) \ .\cr
}\eeq\Message{Minus??}%
This can be written in terms of all the variously defined
spatial gravitational forces as
\beq\eqalign{\label{eq:derenergy2}
    d \tilde E(U,u) / d\tau_{(U,u)} 
          &= \nu(U,u) \cdot_u [ \tilde F{}\tem\g(U,u) + \tilde F(U,u) ] \cr
       &\qquad + \epsilon\tem \gamma\theta(u)^\flat(\nu(U,u),\nu(U,u))\ ,\cr
   \epsilon\tem &\equiv (0, 0, 1, -1) \ ,
       \qquad {\scriptstyle {\rm tem} \,=\, {\rm fw, cfw, lie, lie\flat} } 
           \ ,\cr
}\eeq

The relative acceleration defined in section \ref{sec:tscd}
can be expressed in terms of the spatial projection of the acceleration
using equations (\ref{eq:accdpdt}) and (\ref{eq:auuf})
\beq\eqalign{
      a\tem(U,u) 
       &= \gamma^{-1} P_u(U,u)^{-1} [ \tilde F{}\tem\g(U,u) + A(U,u) ] \cr
           & \qquad - \half \gamma^2 D\tem(U,u) P(u)^\flat / d\tau_{(U,u)}
              (\nu(U,u),\nu(U,u)) \, \nu(U,u)  \ .\cr
}\eeq
Using the acceleration equation this becomes
\beq\eqalign{\label{eq:relacctwo}
      a\tem(U,u) &\equiv D\tem(U,u) \nu(U,u) / d\tau_{(U,u)} \cr
       &= \gamma^{-1} P_u(U,u)^{-1} [ \tilde F{}\tem\g(U,u) + F(U,u) ] \cr
           & \qquad - \half \gamma^2 D\tem(U,u) P(u)^\flat / d\tau_{(U,u)}
              (\nu(U,u),\nu(U,u)) \, \nu(U,u)  \ .\cr
}\eeq
The derivative of the spatial metric is nonzero only in the Lie case
where it reduces to an expansion tensor term. However, this term 
in the relative acceleration is along
the relative velocity and so does not contribute to the part orthogonal
to the relative motion
\beq
   a\tem(U,u) \times_u \nu(U,u) 
     = \gamma^{-1} [ \tilde F{}\tem\g(U,u) + \tilde F(U,u) ] 
           \times_u \nu(U,u) \ .
\eeq

\Message{comment or not??}
\begincomment
The terms proportional to the relative velocity which drop out of
the relationship
are the familiar ones associated with changing the parametrization
of the curve in the second order acceleration equation, namely the
acceleration equation expressed in terms of local coordinates.
\endcomment

\Subsection{Analogy with electromagnetism: gravitoelectromagnetism}

Comparison of the spatial projection of the acceleration equation
with the Lorentz force per unit mass and charge
exerted by an electromagnetic field
expressed in terms of the electric and magnetic fields leads to analogous
electric and magnetic spatial gravitational forces.
To appreciate the analogy, for which the term
``{\it gravitoelectromagnetism}" seems an appropriate description,
one must split the Lorentz force
associated with the electromagnetic 2-form $\four F^\flat$.
The prefix $\four$ will
subsequently be used on the electromagnetic field
to avoid confusion with the kernel letter for spatial forces.

The Lorentz 4-force
on a charged particle of charge $q$, mass $m$, and
4-velocity $U$ is
\beq\eqalign{
        [f\em(U)]^\alpha &= q \four F^\alpha{}_\beta U^\beta
          = q \gamma [ \four F^\alpha{}_\beta u^\beta 
                 + \four F^\alpha{}_\beta \nu(U,u)^\beta ] \ ,\cr
  f\em(U)  &= q \, \four F \rightcontract U 
    = q\,  \gamma [ \four F \rightcontract u 
           + \four F\rightcontract \nu(U,u)\,]\ .\cr
}\eeq 
This splits into the  spatial force and power 
\beq\eqalign{
  F\em(U,u)
     &= q[ E(u) + \nu(U,u) \times_u B(u)] \ ,\cr
   \wp\em(U,u)
     &= q E(u) \cdot_u \nu(U,u)    \ ,\cr
}\eeq 
with similar equations for the per unit mass quantities.

These expressions may be compared with the spatial gravitational forces which
appear in the splitting of the acceleration in terms of either the 
ordinary or co-rotating Fermi-Walker or Lie
total spatial covariant derivative of the covariant spatial
momentum. 
In each case the spatial gravitational force and power have the following form
\beq\eqalign{
   F\tem\g(U,u) 
        &= m \gamma  [ \vec g(u) + H\tem(u) \rightcontract \nu(U,u) ] \ ,\cr
   \wp\g(U,u)
        &= m \gamma [ \vec g(u) \cdot_u \nu(U,u)    
                 - \theta(u)(\nu(U,u),\nu(U,u)) ] \ .\cr
}\eeq
Apart from an additional overall gamma factor not present in the
electromagnetic case, one has both electric-like and magnetic-like
spatial gravitational forces which Thorne \Cite{Thorne et al 1986}
has called the {\it gravitoelectric} and {\it gravitomagnetic} forces 
respectively in slightly different contexts.
The same terminology will be extended to each of the various points of
view and choices of temporal derivatives.

Independent of the choice of time derivative one has a unique
gravitoelectric  vector field
\beq
       \vec g(u) =-a(u)
\eeq 
due to the acceleration of the observer congruence. The gravitomagnetic
tensor field depends on the choice of time derivative
and in the Lie case the contravariant or covariant form of the
momentum equation
\beq\eqalign{
    H\fw (u) &
         = \omega(u) - \theta(u) = k(u) \ ,\cr
    H\cfw (u) &= 
          2\omega(u) - \theta(u)\ ,\cr
    H\lies (u) &= 
          2\omega(u) - 2\theta(u) = 2 k(u)\ ,\cr
    H\lief (u) &= 2 \omega(u) \ .\cr
}\eeq 
The antisymmetric part of the gravitomagnetic tensor, apart from a
factor of two, defines a unique gravitomagnetic vector field
\beq
    \vec H(u)= 2 \vec\omega(u)
\eeq 
which is just twice the vorticity vector of the observer congruence.
Using these definitions the spatial gravitational forces per unit mass
may be written
\beq\eqalign{
\tilde F\g\fw(U,u)& = \gamma [ \vec g(u) + \half\nu(U,u) \times_u \vec H(u)
     - \theta(u) \rightcontract \Vec\nu\,]\ ,\cr
\tilde F\g\cfw(U,u)& = \gamma  [ \vec g(u) + \nu(U,u) \times_u \vec H(u)
     - \theta(u) \rightcontract \nu(U,u)\,]\ ,\cr
\tilde F\g\lie(U,u)& = \gamma  [ \vec g(u) 
    +  \nu(U,u) \times_u \vec H(u)
     - 2\theta(u) \rightcontract \nu(U,u)\,]\ ,\cr
\tilde F\g\lief(U,u)& = \gamma  [ \vec g(u) 
    +  \nu(U,u) \times_u \vec H(u)]\ .\cr
}\eeq

The gravitomagnetic symmetric tensor field $\SYM H\tem(u)$,
which has no analog in electromagnetism, arises from the temporal
derivative of the spatial metric, which is the new ingredient in 
gravitoelectromagnetism that makes it fundamentally different
from electromagnetism.
The spatial
derivatives of the spatial metric also
enter the total spatial covariant derivative of the spatial
momentum as a ``space curvature" force term
when expressed in terms of an observer-adapted frame
\beq\eqalign{\label{eq:lietscd}
&   D\tem(U,u) \tilde p(U,u)^a /  d \tau_{(U,u)} \cr&\qquad=
        d \tilde p(U,u)^a / d \tau_{(U,u}) 
             +  C\tem(u)^a{}_b \tilde p(U,u)^b \cr
     &\qquad\quad + \Gamma(u)^a{}_{bc} \tilde p(U,u)^c \nu(U,u)^b \ .\cr
}\eeq
This ``space curvature" force term
is quadratic in the spatial velocity. One can thus think of the spatial
metric as a potential for  these two different spatial forces,
both associated with the spatial geometry, namely the relative
distances and directions of nearby observers in the observer
congruence.
The matrix $C\tem(u)^a{}_b$ given by equation (\ref{eq:ctemformula})
depends on how the spatial frame is transported along the congruence
and may be conveniently be chosen to vanish for one of the three temporal
derivatives.

To handle the case of a lightlike test particle with zero rest mass,
one must work with the null 4-momentum $P$ instead of a timelike 
4-velocity $U$.
The spatial gravitational force and power then take the form
\beq\eqalign{
   F\tem\g(P,u) 
        &= E(P,u)  [ \vec g(u) + H\tem(u) \rightcontract \nu(P,u) ] \ ,\cr
   \wp\g(P,u)
        &= E(P,u) [ \vec g(u) \cdot_u \nu(P,u)  
             + \SYM H\tem(u)^\flat(\nu(U,u),\nu(U,u))   \ ,\cr
}\eeq
and measurement of the null geodesic equation yields
\beq\eqalign{
     D\tem(P,u) p(P,u) / d\tau_{(P,u)}  &=  F{}\tem\g(P,u) \ ,\cr
     d E(P,u) / d\tau_{(P,u)} &= \wp\g(P,u) \ ,\cr
   &\qquad {\scriptstyle {\rm tem} \,=\, {\rm fw, cfw, lie, lie\flat} } \ .\cr
}\eeq

The gravitoelectric force roughly
corresponds to the centrifugal force experienced
in uniformly rotating coordinates in flat spacetime, while the gravitomagnetic
force roughly
corresponds to the Coriolis force, but the details 
of the correspondence will be discussed below
in the context of a nonlinear reference frame.

\Subsection{Maxwell-like equations}

The analogy between the gravitoelectromagnetic vector fields and the
electromagnetic ones
shows that the exterior derivative of the observer
velocity 1-form corresponds to the electromagnetic 2-form, which is
itself locally the exterior derivative of a 4-potential 1-form
\beq\eqalign{\label{eq:obsgpot}
     d \four A &= u^\flat \wedge E(u)^\flat + \dualp{u} B(u)^\flat 
                                            = \four F^\flat \ ,\cr
     d u^\flat &= u^\flat \wedge \vec g(u)^\flat 
         + \dualp{u} \vec H(u)^\flat \ .\cr
}\eeq
The observer 4-velocity thus acts as the 4-potential for the
gravitoelectromagnetic vector fields. Similarly in direct
analogy with the Lorentz force,
the right contraction of the 4-velocity $U$ of a test particle
with the
mixed form of the 2-form $du^\flat$ also produces the part of
the spatial gravitational force per unit mass $\tilde F\tem\g(U,u)$
which involves only the gravitoelectromagnetic vector fields for
${\scriptstyle\rm tem = cfw,lie,lie\flat}$, although not for the
Fermi-Walker case which involves an additional factor of $\half$.

The splitting of the
identity 
$ d\four F^\flat = d^2 \four A^\flat =0$ leads to half of Maxwell's equations
\beq\meqalign{\label{eq:meq}
    \dualp{u} [ d\four F^\flat]\M
            &= \div_u B(u) + \vec H(u) \cdot_u E(u) &= 0 \ , \cr
    \dualp{u} [ d\four F^\flat]\E{}^\sharp
            &= -\curl_u E(u) + \vec g(u) \times_u E(u)   \cr
            &\qquad 
             - [ \Lie(u)\sub{u} + \Theta(u) ] B(u) &= 0 \ ,\cr
}\eeq
which follow from equation (\ref{eq:splitextder}) with $S= \four F^\flat$ and
making use of the identity (\ref{eq:liederspadual}) with $S=B(u)$.
Replacing $\four A$ by $\four u$ reduces these to the corresponding
gravitoelectromagnetic equations
\beq\eqalign{\label{eq:gemeqsourcefree}
    & [ \div_u + \vec g(u) \cdot_u ] \vec H(u) = 0 \ , \cr
    & \curl_u \vec g(u) + [ \Lie(u)\sub{u} + \Theta(u) ] \vec H(u) =0 \ ,\cr
}\eeq
which are just equations (\ref{eq:divvor}) and (\ref{eq:curlacc}) rewritten
in terms of the gravitoelectromagnetic vector fields.

Splitting the remaining half of Maxwell's equations
\beq
    \dual d \dual \four F = 4\pi \four J
\eeq
using equation (\ref{eq:splitextder}) with $S=\dual \four F^\flat$
and the identities (\ref{eq:splitdualtwo}), which become
\beq
      [\dual \four F^\flat ]\E(u) = - B(u)^\flat\ ,\qquad
      [\dual \four F^\flat ]\M(u) = \dualp{u} E(u)^\flat
\eeq 
when rewritten in terms of the electric and magnetic fields,
leads to
\beq\meqalign{\label{eq:meqsources}
    \dualp{u} [ d\dual \four F^\flat]\M{}^\sharp
    &= \div_u E(u) - \vec H(u) \cdot_u B(u) &= 4\pi \rho(u) \ , \cr
    \dualp{u} [ d\dual \four F^\flat]\E{}^\sharp
    &= \curl_u B(u) - \vec g(u) \times_u B(u) 
       &\cr&\qquad 
             -  [ \Lie(u)\sub{u} + \Theta(u) ] \vec E(u) &= 4\pi J(u) \ ,\cr
}\eeq
where $  \four J = \rho(u) u + J(u) $
is the splitting of the 4-current.

The remaining Maxwell-like equations for the gravitoelectromagnetic
vector fields arise from the Einstein equations.
In order to state them one must first introduce appropriate spatial
curvature tensors associated with
the spatial part of the spatial connection of the observer congruence $u$,
and then split the spacetime curvature.
This will be done below.
Forward \Cite{1961}
systematically developed the analogy between general relativity
and electromagnetism in the post-Newtonian limit.

%pig
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Newpage
\Subsection{Splitting the spin transport equation}

The kinematical tensor $k(u)$ has already been seen to relate
spatial Fermi-Walker transport along the observer congruence
to the  spatial  Lie transport.
When applied to a spatial vector $S$ this may be interpreted as the
equation describing the evolution of the spin of a torque-free gyro
carried by an observer. The following are equivalent ways of stating this
transport
\beq\eqalign{\label{eq:restspinp}
  \del\fw (u) S &= 0 \ ,\cr
  \del\cfw (u) S &= \omega(u)\rightcontract  S\cr
          &=-\vec\omega(u)\times_u  S   \ ,\cr
   \del\lie (u) S &= k(u)\rightcontract  S\cr
          &=-\vec\omega(u)\times_u  S - \theta(u)\rightcontract  S\ .\cr
}\eeq 
The spin vector must rotate with the angular velocity 
of {\it gravitomagnetic precession}
\beq
   \zeta\gmag(u) = - \vec\omega(u) = -\half \vec H(u)
\eeq
equal to minus the
vorticity relative to a spatially Lie dragged spatial frame
or a co-rotating Fermi-Walker propagated spatial frame
in order to undo the local rotation of such a frame.
It must also change
in a way that undoes the expansion and shearing of the spatially
Lie dragged frame vectors
relative to a Fermi-Walker or co-rotating Fermi-Walker propagated spatial 
frame.

One can also split the equation describing the 
transport of the spin vector
of a gyro along an arbitrary worldline with 4-velocity $U$.
Since the spin vector is spatial with respect to $U$, it has the form
\beq
   S = [\nu(U,u) \cdot_u \vec S ] u + \vec S \ ,
\eeq 
where is the spatial
projection $\vec S = P(u,U)S$ of the spin.
Since Fermi-Walker
transport does not change the magnitude $||S||$ of the 4-vector,
the magnitude of the spatial projection of the spin will change
as the inner product $\nu(U,u) \cdot_u \vec S$ changes
\beq
  || \vec S ||^2 = ||S||^2 + [ \nu(U,u) \cdot_u \vec S ]^2 \ ,
\eeq
leading to a change in the observed frequency of rotation associated
with the spin vector.

Since the temporal component of the spin
is determined by its spatial projection,
one only needs an evolution equation for the spatial projection,
which is provided by the spatial projection of the Fermi-Walker 
transport equation.
Because of the spatial condition on $S$ relative to $U$, the Fermi-Walker
transport condition reduces to
\beq
  0= \four\del\fw (U) S = \del\fw (U) S = 
     \four\del\sub{U} S -  a(U)_\beta S^\beta \, U
\eeq
and its spatial projection is
\beq\eqalign{
    0 &=\gamma^{-1} P(u) \del\fw (U) S \cr
      &= \gamma^{-1} P(u) [\four\del\sub{U} S
           - a(U)_\beta S^\beta \, U ]\cr
      &= D\fw(U,u) [ ( \nu(U,u)\cdot_u \vec S ) u + \vec S] / d\tau_{(U,u)}
            - a(U)_\beta S^\beta \, \nu(U,u) \cr
      &=  ( \nu(U,u)\cdot_u \vec S ) D\fw(U,u) u / d\tau_{(U,u)}
           + D\fw(U,u) \vec S / d\tau_{(U,u)}
            - a(U)_\beta S^\beta \, \nu(U,u) \ .\cr
}\eeq
Solving this for the Fermi-Walker total spatial covariant derivative 
and observing that
\beq\eqalign{
     a(u)_\beta S^\beta &= 
       - \gamma [\nu(U,u)\cdot_u A(U,u)][\nu(U,u)\cdot_u \vec S] 
       + \gamma A(U,u)\cdot_u \vec S \cr
      &= \gamma
        A(U,u)^\flat \rightcontract P_u(U,u)^{-1} \rightcontract \vec S \cr
}\eeq
using equation (\ref{eq:ppuu}),
one gets the result
\beq\eqalign{\label{eq:dervecS}
D\fw(U,u) \vec S /d\tau_{(U,u)} &= 
  \gamma^{-1}[ \nu(U,u) \cdot_u \vec S ] \tilde F{}\g\fw(U,u)
   - \gamma [ A(U,u)^\flat \rightcontract P_u(U,u)^{-1} \rightcontract \vec S ]
             \nu(U,u) \ ,\cr
}\eeq 
which is easily re-espressed in terms of
the co-rotating Fermi-Walker derivative
\beq\eqalign{
D\cfw(U,u) \vec S /d\tau_{(U,u)} &= 
                -\vec\omega(u) \times_u \vec S
  + \gamma^{-1}[ \nu(U,u) \cdot_u \vec S ] \tilde F{}\g\fw(U,u) \cr
     &\qquad  - \gamma
         [ A(U,u)^\flat \rightcontract P_u(U,u)^{-1} \rightcontract \vec S ]
             \nu(U,u) \ .\cr
}\eeq 

These equations may be decomposed into separate evolution equations for the
length $||\vec S||$ and direction $\hat S = ||\vec S||^{-1} \vec S$ of the
spatial projection of the spin vector. The results are
\beq\eqalign{ 
  D(U,u)  \ln || \vec S || /d\tau_{(U,u)} 
     &= \gamma^{-1} [ \nu(U,u) \cdot_u \hat S ] \,
                [ \tilde F{}\g\fw(U,u)  \cdot_u \hat S] \cr
 &\qquad
  -  \gamma [A(U,u)^\flat \rightcontract P_u(U,u)^{-1} \rightcontract \vec S ]
       \, [\nu(U,u)  \cdot_u \hat S ] \ ,\cr
}\eeq
and
\beq
  D\tem(U,u) \hat{S} /d\tau_{(U,u)} 
             = \Omega\tem(\hat S,U,u) \times_u \hat{S} \ ,
\eeq
where
\beq\eqalign{ 
   \Omega\fw (\hat S,U,u)
      &=
     -  \gamma^{-1} [ \nu(U,u) \cdot_u \hat{S} ]  \, 
          \tilde F{}\g\fw(U,u) \times_u \hat{S} \cr
      & \qquad
  +  \gamma [A(U,u)^\flat \rightcontract P_u(U,u)^{-1} \rightcontract \hat{S} ]
      \, \nu(U,u) \times_u \hat{S} \ ,\cr
   \Omega\cfw  (\hat S,U,u) &=     \Omega\fw (\hat S,U,u)
                                            - \vec\omega(u) \ .\cr
}\eeq
These formulas describe the precession of the spin vector as seen by
the family of different observers of the observer congruence along the
gyro's worldline. Note that the angular velocity of the direction of the
spatial projection of the spin vector
depends on the direction of that spin vector.

These formulas describe how the spin vector $S$ changes as seen by 
the 1-parameter family of observers belonging to the observer congruence
which intersect the worldline of the gyro.
The observed
spin vector $\vec S =P(u)S$ does not undergo a rotation with
respect to the Fermi-Walker propagated axes of the observer
but a more complicated motion required to keep $S$ orthogonal to $U$.
This is superimposed upon the  gravitomagnetic precession when
compared to co-rotating Fermi-Walker propagated axes, and complicated by
the expansion and shear effects when compared to spatial Lie dragged axes.


\Newpage
%%%%%%%%%%
\Subsection{Relative Fermi-Walker transport and gyro precession}

Another question one can consider is how a single observer with 4-velocity
$U$ following the
gyro worldline would see the orientation of the gyro change with respect
to a spatially co-rotating observer-adapted frame anchored in
the given observer congruence $u$.
The observer following the gyro will see these axes to be in relative
motion and not even appear orthonormal. However, a basic assumption of
special relativity implicit when one discusses pure Lorentz transformations
in Minkowski space
is that the orientation of a
set of orthonormal axes in relative motion is defined to be the orientation
they would have if they were not in relative motion. In other words
one ``brings them to rest" by the inverse relative observer boost in
order to define their orientation with respect to a given orthonormal spatial
frame in the local rest space of an observer.

Suppose $\{e_a\}$ is a spatially co-rotating observer-adapted spatial frame
along the observer congruence $u$. Then $B\lrs(U,u) e_a$ are the axes
``momentarily at rest" with respect to the gyro, and the orientation of the
spin vector $S$ with respect to them is well-defined and represents the
orientation of $S$ with respect to the moving axes $e_a$.
However, since the boost is an isometry between the two local rest spaces,
the orientation of $S$ with respect to $B(U,u) e_a$ is the same as the
orientation with respect to $e_a$ of the boosted spin vector
\beq
       \Sscr \equiv B(u,U) S
\eeq
in the local rest space of the observer congruence. Again since the boost
is an isometry, the lengths of $\Sscr$ and $S$ are the same, and since
the latter is constant under Fermi-Walker transport, so is the former.
In other words $\Sscr$ will rotate with respect to the orthonormal
spatial frame $e_a$, and its angular velocity with respect to the sequence
of observers from the observer congruence along the gyro worldline will
be the same as the angular velocity of $S$ with respect to the boosted
spatial frame $B(U,u) e_a$, apart from a proper time renormalization.
In order to calculate these angular velocities, one must consider
the evolution of $\Sscr$ along the gyro worldline.

This boosted spin vector has the explicit expression
\beq
   \Sscr   = \vec S
         - \gamma^{-1}(\gamma + 1)^{-1} 
                 [\tilde p(U,u) \cdot_u \vec S] \tilde p(U,u)  \ ,
\eeq
which is obtained by rewriting equation (\ref{eq:boostback}) in terms
of the spatial momentum per unit mass.
Since the Fermi-Walker total spatial covariant derivatives of $\vec S$,
$\tilde p(U,u)$, and $\gamma= \tilde E(U,u)$ are given by equations
(\ref{eq:dervecS}), (\ref{eq:derspamomentum}), and 
(\ref{eq:derenergy1}), it is straightforward
to evaluate the corresponding derivative of $\Sscr$.

After some algebra one finds a result which may be expressed in the
following equivalent forms
\beq\eqalign{
&   D\tem(U,u) \Sscr / d\tau_{(U,u)} 
        = \zeta\tem(U,u) \times_u \Sscr \ ,\cr
&  \qquad {\scriptstyle \rm tem = fw, cfw} \ ,\cr
}\eeq
where the Fermi-Walker and co-rotating Fermi-Walker
``relative angular velocities" are related to each other by
by the gravitomagnetic precession 
\beq\eqalign{
         \zeta\cfw(U,u) &= -\vec\omega(u) + \zeta\fw(U,u) \cr
                &= -\half \vec H(u) + \zeta\fw(U,u)  \cr
                &= \zeta\gmag(u) + \zeta\fw(U,u)  \cr
}\eeq
and the former is defined by
\beq\eqalign{
   \zeta\fw(U,u) 
      &=  - \gamma   (\gamma   +1)^{-1}      \nu(U,u) \times_u \tilde F(U,u)
    + (\gamma   + 1 )^{-1} \, \nu(U,u) \times_u \tilde F\g\fw(U,u) \cr
   &= \zeta\thom(U,u) + \zeta\geo(U,u) \ .\cr
}\eeq
This may also be expressed in terms of the relative acceleration
\beq\eqalign{
   \zeta\fw(U,u) &=
   - \gamma  ^2 (\gamma   + 1 )^{-1}  \nu(U,u) \times_u  a\cfw(U,u) \cr
   &\qquad + \nu(U,u) \times_u [ \tilde F\g\cfw(U,u)
   - \gamma (\gamma   + 1 )^{-1} \nu(U,u) \times_u \vec\omega(u) ] \ ,\cr
}\eeq
a formula first obtained by
Massa and Zordan \Cite{1975}.
These angular velocities, in contrast with the result for 
$\Omega\cfw(\hat S,U,u)$,
depend only on the relative boost between the local rest spaces.

For a gyro at rest relative to the observer congruence, only the
gravitomagnetic precession term is nonzero. 
This is also known as the
frame-dragging precession or the Lense-Thirring precession, since the
effect was first studied by Lense and Thirring \Cite{1918}
in the context of weak fields and slow motion.

The Thomas precession term 
\beq
  \zeta\thom(U,u) = 
   - \gamma (\gamma + 1)^{-1} \nu(U,u) \times_u \tilde F(U,u)
\eeq
in the relative angular velocity is due to the applied force
acting on the gyro.
In Minkowski spacetime choosing $u$ to be a unit timelike rotation-free
Killing vector field corresponding to a time translation,
only this term is nonzero and has the equivalent expression
\beq\eqalign{
    \zeta\thom(U,u) 
       &= - \gamma^2 (\gamma +1)^{-1} \nu(U,u) \times_u a\cfw(U,u) \cr
       &= - ||\nu(U,u)||^{-2} (\gamma -1) \nu(U,u) \times_u a\cfw(U,u)\cr
}\eeq
in terms of the relative acceleration
\beq
  a\cfw(U,u) = a\fw(U,u) = a\lie(U,u) 
        = \gamma^{-1} P_u(U,u)^{-1} \tilde F(U,u)
\eeq
which in this case represents the usual spatial acceleration of special
relativity. 

The acceleration of the gyro in Minkowski space causes it
to precess as discussed by Thomas \Cite{1927}. 
In the limit $||\nu(U,u)|| \ll 1$ and $\gamma \to 1$ of
nonrelativistic relative motion, 
the Thomas precession under these conditions reduces to
\beq
       \zeta\thom(U,u) \to -\half \nu(U,u) \times_u a\cfw(U,u) 
                          = -\half \nu(U,u) \times_u \tilde F(U,u) \ .
\eeq
For circular motion with angular velocity $\vec\Omega$,
the precession angular velocity is $[\gamma-1] \vec\Omega$, as
described in exercise 6.9 of Misner, Thorne and Wheeler \Cite{1973}.
The first term in their equation (6.28) is exactly the boosted
spin vector $\Sscr$.

The second term in the Fermi-Walker angular velocity
\beq\eqalign{
   \zeta\geo(U,u)
       &= (\gamma + 1)^{-1} \nu(U,u) \times_u  \tilde F\g\fw(U,u) \cr
       &= \gamma (\gamma + 1)^{-1} \nu(U,u) \times_u
     [ \vec g(u) +\half  \nu(U,u) \times_u \vec H(u) 
           - \theta(u)\rightcontract \nu(U,u) ] \cr
}\eeq
is due to the spatial
gravitational force alone, which is present even in the absence of
an applied force, i.e., for geodesic motion of the gyro. For this
reason it is often called the geodetic precession, in addition to
the commonly used alternatives de Sitter precession or Fokker
precession or even Fokker-de Sitter precession,
named after the original investigators of this phenomenon
\Cite{de Sitter 1916, Fokker 1920, Pirani 1956}.

In the nonrelativistic limit $\gamma \to 1$, neglecting terms
of second order in the velocity,
this term has the limiting expression
\beq
    \zeta\so(U,u) = \half \nu(U,u) \times_u \vec g(u) \ .
\eeq
Thorne \Cite{1989} describes this nonrelativistic term as an
``induced gravitomagnetic precession" or ``spin-orbit" precession
since it corresponds to the gravitomagnetic precession due to
an additional ``induced" gravitomagnetic field
$\vec H(u)\ind = - \nu(U,u) \times_u \vec g(u)$ induced by the motion
of the gyro in the gravitoelectric field in analogy with the induced
magnetic field due to motion in an electric field.
\Typeout{Update with bcjw.tex.}

These relative angular velocities must still be interpreted in terms
of the original discussion of the angular velocity of the relative
orientation of the gyro spin and the spatially co-rotating 
observer-adapted spatial frame.
These angular velocities do not depend on the choice of   spatial
frame but only involve the relative observer boost between the
gyro and the observer congruence.
They describe the angular velocities of the boosted spin vector $\Sscr$
with respect to a spatial frame  which is transported along the gyro
worldline by the corresponding spatial transport.
This transport seeks to eliminate the arbitrariness in the spatial
distribution of the orientation of any given spatially co-rotating
observer-adapted spatial frame, at least along the given gyro worldline.

However, if the gyro returns to a given observer worldline belonging
to the observer congruence, the spatial frame transported along its
worldline will in general differ from the value of the spatially co-rotating
observer-adapted spatial frame there. This is a manifestation of
``space curvature" which occurs as long as the spatial metric is not
flat.
If one is really interested in measuring the relative orientation of
$\Sscr$ with respect to a particular spatially co-rotating
observer-adapted orthonormal spatial frame, 
one must remove the additional angular
velocity of the co-rotating Fermi-Walker transported orthonormal
spatial frame
with respect to it, leading to a frame-dependent result
for the ordinary derivative of the components of $\Sscr$ with respect
to the former frame
\beq\eqalign{
   d \Sscr^a / d\tau_{(U,u)}  &=
       \epsilon_{abc} [\zeta\cfw(U,u) + \zeta\sc(U,u,e)]{}^b \Sscr^c \ .\cr
}\eeq
Thorne \Cite{Thorne et al 1986} has referred to the relative
angular velocity $\zeta\sc(U,u)$ in the precession formula
as the space curvature term.
Of course now that ordinary derivatives are being used one has to
choose the observer-adapted frame carefully in order that this make some
physical sense. 

Finally, normalizing the proper time to correspond to that of the gyro
instead of that of the observer congruence,
the angular velocity of the given spatially co-rotating 
observer-adapted orthonormal spatial frame $\{e_a\}$
as seen by the gyro has components
\beq
      \zeta\gyro(U,u,e)^a = \gamma [ \zeta\cfw(U,u)^a + \zeta\sc(U,u,e)^a ]
\eeq
with respect to the boosted frame $B\lrs(U,u)e_a$.
In other words, if $S^a$ are the components of the spin vector with respect
to this boosted frame, then
\beq
    d S^a / d\tau_U = \epsilon_{abc} \zeta\gyro(U,u,e)^b S^c \ .
\eeq

The precession formula will have a physical meaning in a given spacetime
first only if 
the observer congruence itself has some physical meaning and
second if the spatially co-rotating observer-adapted orthonormal
spatial frame has some physical meaning.
The obvious candidate satisfying the first condition is an observer
congruence associated with a timelike Killing vector field in a
stationary spacetime. Since the shear of such an observer congruence
vanishes, the co-rotating Fermi-Walker transport along $u$ reduces
to spatial Lie transport, and so the spatially co-rotating
observer-adapted orthonormal spatial frame is also a stationary
frame.

This spatial frame should also be as
``Cartesian-like" as possible, i.e., the distribution of 
its orientation should be as aligned as possible, in order that the
angular velocity have some meaning at each point along the gyro
worldline. 
For spacetimes which are asymptotically flat at spacelike infinity,
Nester \Cite{1991a, 1991b}
has shown that a preferred spatial orthonormal frame exists
on a given asymptotically flat spacelike hypersurface which asymptotically
approaches a given inertial frame at spacelike infinity
and in some sense has ``Cartesian-like properties."
Given a preferred slicing by spacelike hypersurfaces, one could then boost
these spatial frames into the local rest space of a preferred
observer congruence to define a preferred class of observer-adapted
orthonormal frames which might be used in interpreting the angular velocity.
Stationary axially symmetric spacetimes have a preferred slicing
orthogonal to the zero-angular-momentum observers, and so must possess
such preferred frames. Whether or not they are compatible with
the spatially-comoving condition is not clear.
\Message{MORE HERE ON NESTER??}

Alternatively, one can consider worldlines which return
to a given observer worldline and attempt to
calculate the total rotation of the
spin between the initial and final intersection points 
(corresponding to a closed loop in the space of observers)
to avoid this problem. 
This is the case for circular orbits often considered
in gyro precession calculations, where the net rotation per
revolution is of interest. 


%%%%%%%%%%%%
\Newpage
\Subsection{The Schiff Precession Formula}

The classic spin precession formula of Schiff \Cite{1960}
describes how the spin vector precesses relative to the ``distant stars"
as seen by an observer carrying the gyro in the gravitational field of an
isolated body \Cite{Weinberg 1972, Misner, Thorne and Wheeler 1973}.
This is evaluated locally by using a spatial frame which is somehow
anchored in the
asymptotically flat part of the spacetime around such an isolated body.

The discussion is cleaner for a stationary spacetime of this type
with a stationary frame adapted to the asymptotically nonrotating stationary
observers, the so-called {\it static observers}.
Since spatial Lie transport along $u$ coincides with co-rotating Fermi-Walker
transport, the spatial projection of any frame which is comoving with respect
to $u$ will yield a spatial frame which is
spatially comoving and which undergoes co-rotating Fermi-Walker
transport along $u$. Call this a {\it static spatial frame}. 
Let $\{u,e_a\}$
be a spatially-comoving observer-adapted frame, whose spatial frame is
therefore static.
One can choose this frame to be orthonormal. For a black hole
spacetime in Boyer-Lindquist coordinates, for example, the spatial coordinate 
differentials are orthogonal to each other and to the asymptotically-%
nonrotating timelike Killing vector field, and so one may normalize them
to obtain such a frame.

Light rays from ``distant stars"
will arrive at a given static observer from fixed directions
relative to the static spatial frame. In this way the stationary
symmetry anchors the static
observer frame to the asymptotically flat part of the spacetime.
It can then be used for a local comparison of the spin vector of a
gyroscope with the ``distant stars." 

How can the observer carrying the
gyro make this comparison?
The static observer's spatial frame is moving with respect to the
gyro. The boost $E_a = B(U,u)e_a$ 
of the static observer's spatial frame to the rest frame
of the gyro is the spatial frame the gyro would see if the static
observer were not in relative motion. 
The angular velocity $\zeta\gyro(U,u,e)$ evaluated above describes the rotation
of the spin vector with respect this boosted frame.

This angular velocity depends crucially on the spatial distribution
of the orientation of the static spatial frame $\{e_a\}$, i.e.,
on how the orientation depends on the location of the observer in the
observer congruence. If one has a stationary axially symmetric spacetime
with an axially symmetric static spatial frame like
the one described above for a black hole
arising from the normalized Boyer-Linquist spatial coordinate differentials,
then the static frame will itself rotate through an angle of $2\pi$ for
each loop around a circular orbit of the axial symmetry subgroup.
Clearly one does not want to measure the gyro precession with respect
to such a frame, but rather one with Cartesianlike properties, if such a
frame exists.

The actual
Schiff formula corresponds to the value of the angular velocity
$\zeta\gyro(U,u,e)$
in the slow motion weak field limit of general relativity for
the linearized gravitational field of an isolated distribution of matter.
It has been generalized to the PPN theory as discussed by Misner, Thorne
and Wheeler \Cite{1973} or Thorne \Cite{1989}.
The key difference  between it and the
limiting expression for $\zeta\cfw(U,u)$ 
\beq
    \zeta\cfw(U,u) \to
        -\half \vec H(u) -\half \nu(U,u) \times_u \tilde F(U,u)
          + \half \nu(U,u) \times_u \vec g(u)
\eeq
is the
fact that in this limit within general relativity, 
the space curvature precession has twice the
value of the spin orbit precession $\zeta\so(U,u) \to 
\half \nu(U,u) \times_u \vec g(u)$
leading to a total coefficient of
$\frac32$. The sum of these two terms is conventionally
referred to as the geodetic precession.
This will be discussed below in chapter ?? on the 
post-Newtonian limit.

This same result generalizes in a simple way to the 
parametrized post-Newtonian (PPN) context
following the same notation as  Misner, Thorne and Wheeler \Cite{1973}.
The PPN spatial coordinates are orthogonal to lowest order and may
be normalized to define an orthonormal spatial frame which may be completed
to an orthonormal frame by the addition of the normalization of the
time coordinate derivative vector field which is orthogonal to the
spatial coordinate differentials to lowest order. This is exactly
analogous to the black hole case described above.
However, all of the linearized discussions of spin precession are
somewhat clouded by the mixing of the linearization
process itself with the features of the spin precession.

Without the assumption of stationarity, the entire discussion no longer
has a very solid foundation. In the PPN context, the PPN coordinate grid
is anchored to the asymptotically flat part of the spacetime but
one has an entire class of gauge transformations which allow the coordinate
grid to change. 
\message{MORE HERE SCHIFF END??}

The measurement of the 
precession of a torque-free gyro in a free-fall earth orbit is the
goal  of the long awaited Stanford Gravity-Probe B experiment 
\Cite{Everitt 1979}.
Of course to get explicit formulas for the the actual cumulative precession
measured, one must consider the
actual geometry of the earth-satellite system, as well as the aberration
effects involved in the star-tracking telescope. 
It is this experiment, the proposed LAGEOS experiment 
\Cite{Ciufolini 1990}, 
and others with similar goals \Cite{Nordvedt 1988, Mashhoon, Paik, and Will
1989}
which have provided much of the motivation for talking
about ``gravitomagnetism." 
%However,
%a key point in the analysis is that the difference between the boosted
%spin vector $\Sscr$ and the one $\vec S$ seen by the stationary observers 
% is orientational in nature, depending on the location in the orbit and
% the momentary direction of the spin, so that cumulative effects become
% much greater after a large number of orbits.
\message{confused??}
\Typeout{Emphasize secular effects so that these details are irrelevant
to the actual experiment.}
%%%%%%%%%%%%%%%%%%%%%%%
\Newpage
\Subsection{The relative angular velocity as a boost derivative}

The formula for the co-rotating relative angular velocity 
$\zeta\cfw(U,u)$ has been
obtained in a two-step process, first evaluating the evolution of
the spatial projection of the spin vector and then of the transformation
which converts it to the boosted spin vector.
This quantity may be obtained instead directly from the evolution of the
relative boost between the local rest spaces along the worldline.

Starting from the Fermi-Walker transport equation for the spin vector $S$
\beq
  \four D(U) S / d\tau_U = \sigma(U\wedge a(U)) S \ ,
\eeq
one can calculate the spatial projection of the total covariant derivative
of the boosted spin vector $\Sscr = B\lrs(u,U) S$
\beq\eqalign{
  D\fw(U,u) \Sscr / d\tau_{(U,u)}
     &= P(u) \four D(U) \Sscr / d\tau_u \cr
     &= P(u) \four D(U) B\lrs(u,U) / d\tau_{(U,u)} \, B\lrs(U,u) \Sscr \cr
     &\qquad  + P(u) B\lrs(u,U) \four D(U) S / d\tau_{(U,u)} \ ,\cr
}\eeq
suppressing the contraction notation of the linear transformations involved.
The second term vanishes 
\beq\eqalign{
  P(u) B\lrs(u,U) \four D S / d \tau_{(U,u)} 
      &= \gamma(U,u) P(u) B\lrs(u,U) \sigma(U\wedge a(U)) B\lrs(U,u) \Sscr \cr
      &= \gamma(U,u) P(u)  \sigma(u \wedge B\lrs(u,U) a(U)) \Sscr =0 \cr
}\eeq
since the factor of $u$ is killed by the spatial projection
or by contraction with the spatial field $\Sscr$.

The first term defines the boost angular velocity tensor
\beq
     W(U,u) = P(u) [ \four D(U) B\lrs(u,U) / d\tau_{(U,u)} \, B\lrs(U,u) ] 
             = - \dualp{u} \zeta\fw(U,u) \ ,
\eeq
which is an antisymmetric spatial tensor. Its sign-reversed
spatial dual therefore defines the {\it relative
angular velocity} vector $\zeta\fw(U,u)$.
The antisymmetry of $W(U,u)$ follows
since this represents the spatial part of the covariant Lie algebra
derivative of the 1-parameter family of Lorentz transformations.

%One need only
%differentiate the Lorentz transformation condition satisfied by the
%boost
%\beq
%       B\lrs(u,U)^\gamma{}_\alpha g_{\gamma\delta} B\lrs(u,U)^\delta{}_\beta 
%         = P(U)_{\alpha\beta}
%\eeq
%along the worldine
%and contract the free indices by the inverse boost to obtain
%\beq
%  P(u)  \four D B(u,U)^\gamma{}_\delta / d\tau_u \, 
%    B(U,u)^\delta{}_{(\beta} g_{\alpha)\gamma} = 0 \ .
%\eeq
%Spatial projection of this condition gives the corresponding condition
%on the angular velocity tensor $W(U,u)$.

This leads to the result
\beq
     D\fw(U,u) \Sscr / d\tau_{(U,u)} = \zeta\fw(U,u) \times_u \Sscr
\eeq
describing the precession of the boosted spin vector. The relative
angular velocity differs from the co-rotating angular velocity
by the vorticity of the observer congruence
\beq
       \zeta\cfw(U,u) = \zeta\fw(U,u) - \vec\omega(u) \ ,
\eeq
i.e., by the gravitomagnetic precession.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Newpage
\Subsection{Relative kinematics: transformation of spatial gravitational 
fields}

%pig

The spatial gravitational force fields are simply related to the kinematical
quantities associated with the observer congruence. If one has two
distinct observer congruences with unit tangents $u$ and $U$, one
can describe the transformation law between the spatial gravitational
fields observed by each. One need only express the quantities and operators
in the expression for the spatial gravitational fields of one in terms of those
of the other to obtain such laws, as in the above derivation of the 
transformation law for the electric and magnetic fields.

The acceleration and kinematical field transform as follows
\beq\eqalign{
    a(U) &= \gamma^2 P(u,U)^{-1} [ a(u) - k(u) \rightcontract \nu(U,u) ] 
              \cr&\quad 
               + \gamma^2 P(U,u) a\fw(U,u)  \ ,\cr
   k(U) &= \gamma P(U,u) [ k(u) - a(u) \otimes \nu(U,u)^\flat ] 
               \cr&\quad 
             - \gamma \del(U) \nu(U,u) \ .\cr
}\eeq    
The expansion tensor and the 
rotation tensor and vector then transform as
\beq\eqalign{
    \theta(U)^\flat &= \gamma^2 P(U,u) 
           [ \theta(u)^\flat + \half \{ a(u) \otimes \nu(U,u) +
                                          \nu(U,u) \otimes a(u) ]
               \ ,\cr
    \omega(U)^\flat &= \gamma^2 P(U,u) 
           [ \omega(u)^\flat - \half a(u)\wedge \nu(U,u) ]
               \cr&\quad 
           + \half \gamma d(U) \nu(U,u)^\flat \ ,\cr
   \vec\omega(U) &= \gamma^2 P(u,U)^{-1}
           [ \vec\omega(u) + \half\nu(U,u)\times_u a(u) ]
               \cr&\quad 
           + \half \gamma \curl_U \nu(U,u) \ ,\cr
}\eeq    

Converting to the gravitoelectromagnetic symbols leads to
\beq\eqalign{
   \vec H(U) &= \gamma^2 P(u,U)^{-1}
           [ \vec H(u) - \nu(U,u)\times_u \vec g(u) ]
               \cr&\quad 
           + \gamma \curl_U \nu(U,u) \ ,\cr
    \vec g(U) &= \gamma^2 P(u,U)^{-1} 
         [ \vec g(u) + \half \nu(U,u) \times_u \vec H(u) 
               \cr&\quad 
           - \theta(u) \rightcontract \nu(U,u) ]
               - \gamma^2 P(U,u) a\fw(U,u)  \cr
       &=  \gamma^2 P(u,U)^{-1} 
           [ \vec g(u) + \nu(U,u) \times_u \vec H(u) 
               \cr&\quad 
           - \theta(u) \rightcontract \nu(U,u) ]
               - \gamma^2 P(U,u) a\cfw(U,u)  \ ,\cr
}\eeq    
where the expressions in square brackets in the gravitoelectric field
transformation laws are just $\gamma^{-1} \tilde F{}\g\fw(U,u)$ and
$\gamma^{-1} \tilde F{}\g\cfw(U,u)$ respectively,
analogous to the Lorentz force and its magnetic analog which appear in the
transformation law for the electric and magnetic fields.
The terms explicitly involving the gravitoelectromagnetic
vector fields
in the transformation law for the 
gravitomagnetic vector field and in the second form of the one for the
gravitoelectric vector field
are exactly analogous to the
corresponding transformation laws for the magnetic and electric fields,
apart from the extra gamma factor
also present in the force law itself.

Apart from the expansion term,
the remaining part of the transformation law which breaks this
correspondence, namely the
relative acceleration and the relative velocity curl, 
can be further expanded. The relative acceleration, for example,
can be re-expressed using equation (\ref{eq:relacctwo}).
For the relative curl, one can apply the following useful formula
\beq\eqalign{
   \curl_U X 
      &= \gamma P(u,U)^{-1} \{ \curl_u X 
               \cr&\quad 
        + \nu(U,u) \times_u [\Lie(u)\sub{u} X^\flat ]^\sharp \} \ ,\cr
}\eeq    
valid when $X$ is spatial with respect to $u$.

For an observer-adapted co-rotating Fermi-Walker orthonormal frame,
the gravitoelectric and gravitomagnetic fields are related
to the 2-form which results from evaluating the
tensor-valued connection 1-form on $u$ in a way similar to the way
the electric and magnetic fields are related to the electromagnetic
2-form \Cite{jancar91}.
The homogeneous part of the transformation law for  the
connection then leads to the terms in the transformation law for the
gravitoelectric and gravitomagnetic vector fields which are
analogous to those for the electric and magnetic fields.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Newpage
\
Section{The hypersurface point of view}

The zero vorticity case $\omega(u)=0$ of the congruence point of view
essentially defines the hypersurface point of view, described in this way
by Zel'manov \Cite{1973}, Ehlers \Cite{1961, 1993} and Ellis \Cite{1971}.
In this special case one
has an integrable family of local rest spaces $LRS_u$ which define a 
spacelike slicing of
spacetime with $u$ equal to the unit normal $n$ for this slicing. 
The spatial metric $P(n)^\flat$ when restricted to a slice equals the induced
Riemannian metric of the slice, while the
the kinematical tensor $k(n)^\flat=-\theta(n)^\flat$ 
restricts to the extrinsic curvature tensor $K^\flat$ of the slice.  
There is a natural isomorphism between spatial tensor fields on a given
slice and
the tensor algebra of the slice as a manifold itself, under which
the spatial covariant derivative operator for spatial tensor fields corresponds
to the covariant derivative of the induced metric connection.

Because the vorticity vanishes,
the covariant normal  admits a family of integrating factors,
so  the acceleration or equivalently the gravitoelectric field admits a
family of potentials 
\beq
  n^\flat = -N dt \ , \qquad a(n)=\grad_n \ln N = -\vec g(n)\ .
\eeq 
Thus $-\ln N$ serves as an ``acceleration potential" \Cite{Ehlers 1961, 1993} 
while
$\ln N$ serves as the gravitoelectric potential. The integrating factor $N$
is only defined to within a factor whose spatial derivative vanishes,
corresponding to a reparametrization of the slicing time function $t$.
Since the vorticity vanishes, the gravitomagnetic vector field is zero.

Of course given explicitly the congruence on spacetime in terms of some
local coordinates, one would have to integrate partial differential equations
(for a time function)
to obtain the slices, and conversely given a slicing in terms of local
coordinates one would have to integrate ordinary differential equations
(for the orthogonal trajectories)
in order to obtain the normal congruence. This means that in practice
one might have to distinguish the hypersurface and corresponding congruence
points of view, but as far as the spacetime 
geometry of the splitting is concerned, they are the same.

In the hypersurface point of view, it is natural to interchange the
symbols $\top$ and $\bot$. Since the slicing is considered primary
rather than the implicit normal congruence,
it is logical to let $\perp$ stand for perpendicular to the slicing
and $\top$ for tangential to the slicing, compared to the symbols $\top$
and $\bot$ for tangential and perpendicular to the congruence in the
corresponding congruence point of view adapted to the normal congruence.
Thus the decomposition of a spacetime $1\choose1$-tensor field $S$
corresponding to the ordered family of spatial fields listed above
for the equivalent normal congruence point of view becomes instead
\beq\eqalign{
S & \qquad \leftrightarrow \qquad
  \{ S^0{}_0,   S^a{}_0,  S^0{}_a, S^a{}_b \}\cr
 & \qquad \leftrightarrow \qquad
  \{ S^\bot{}_\bot,   S^a{}_\bot, S^\bot{}_a, S^a{}_b \} \ ,\cr
 & \qquad \leftrightarrow \qquad
    \{ S^\bot{}_\bot,   S^\top{}_\bot, S^\bot{}_\top, S^\top{}_\top \} \cr
}\eeq 
in the dual hypersurface point of view.
The commonly used single vertical bar notation for the spatial covariant
derivative index notation will also be adopted in place of the
double vertical bar 
\beq
  [\del(n) S]^{\alpha\ldots}_{\ \ \beta\ldots} = 
   \del(n)_\gamma S^{\alpha\ldots}_{\ \ \beta\ldots} 
   = S^{\alpha\ldots}_{\ \ \beta\ldots | \gamma}\ .
\eeq 

\endinput

\Subsection{Relation between Lie and 
  Fermi-Walker temporal derivatives}

Suppose one refers to a tensor as being of ``of type $\sigma$",
where $\sigma$ is the $gl(4,R)$-representation corresponding to the
$GL(4,R)$-representation under which the tensor
transforms under a change of
frame
\beq
  [\sigma(\bfA) S]^{\alpha\ldots}_{\ \ \beta\ldots} =
    A^\alpha{}_\gamma S^{\gamma\ldots}_{\ \ \beta\ldots} + \cdots
   -A^\gamma{}_\beta S^{\alpha\ldots}_{\ \ \gamma\ldots} - \cdots\ .
\eeq 
This allows a compact notation for the formula for the components of the
covariant and Lie derivatives in a coordinate frame $\{e_\alpha\}$
characterized by 
$C^\alpha{}_{\beta\gamma} = \omega^\alpha([e_\beta,e_\gamma])=0$
\beq\label{eq:ld}
\eqalign{ 
   [\four\del\sub{u} S]^{\alpha\ldots}_{\ \ \beta\ldots} &=
   S^{\alpha\ldots}_{\ \ \beta\ldots,\gamma} u^\gamma +
        [\sigma(\four\Gamma(u)) S]^{\alpha\ldots}_{\ \ \beta\ldots}\ ,\cr
   [\Lie\sub{u} S]^{\alpha\ldots}_{\ \ \beta\ldots} &=
   S^{\alpha\ldots}_{\ \ \beta\ldots,\gamma} u^\gamma -
        [\sigma(\partial u) S]^{\alpha\ldots}_{\ \ \beta\ldots}\ ,\cr
  &= S^{\alpha\ldots}_{\ \ \beta\ldots;\gamma} u^\gamma -
        [\sigma(\four\del u) S]^{\alpha\ldots}_{\ \ \beta\ldots}\ ,\cr
}\eeq 
where the matrix arguments of $\sigma$ are the suggestive abbreviations
$\four\Gamma(u)^\alpha{}_\beta = \Gamma^\alpha{}_{\gamma\beta}u^\gamma$,
$(\partial u)^\alpha{}_\beta = u^\alpha{}_{,\beta}$, and
$(\four\del u)^\alpha{}_\beta = u^\alpha{}_{;\beta}$.
%while the comma indicates ordinary differentiation of a component function
%along a frame vector field.
The last expression for the Lie derivative
is true for an arbitrary frame and is the result of the ``comma to semicolon
rule" valid for any symmetric connection in a coordinate frame
\beq
    \Lie\sub{u} S= \four\del\sub{u} S - \sigma(\four\del u) S \ .
\eeq

Taking the spatial projection of this equation yields
\beq
 \Lie(u)\sub{u} S = \del\fw (u) S - P(u)\sigma(\four\del u) S
\eeq 
for an arbitrary tensor but
\beq
 \Lie(u)\sub{u} S = \del\fw (u) S + \sigma(k(u)) S
\eeq 
for a spatial tensor. The kinematical tensor $k(u)$ thus acts as the linear
transformation of the local rest space which describes the difference between
 spatial  Lie transport  and Fermi-Walker transport of spatial tensors
along $u$. 
These two transports and their corresponding differential
operators embody two possible choices for describing ``evolution",
which might be called ``Lie" and ``Fermi-Walker" evolution respectively
from their origins in derivative operators which correspond to those
labels. In the absence of further structure,
evolution can only be defined independently
along each observer worldline and then only
relative to a transport of the local rest space along each such worldline which
defines ``no evolution".
Note that for $k(u)$ itself the difference between the spatial Fermi-Walker
derivative and the  spatial  Lie derivative along $u$ vanishes
so 
\beq
   \Lie(u)\sub{u} k(u) = \del\fw (u) k(u)\ .
\eeq 

The  spatial  Lie transport is rather simple. Lie transport
a spatial tensor along the observer congruence using the 1-parameter group
of diffeomorphisms generated by $u$ and then spatially project it. 
If $e_\alpha$ is any frame with $e_0$ along $u$ and $\{e_a\}$ Lie dragged
along $u$, then  
$\{e_0,P(u)e_a\}$ is an observer-adapted frame and 
$\{P(u)e_a\}$ is a spatial frame which undergoes  spatial  Lie
transport.  Any spacetime tensor which does not evolve in the Lie
sense  is represented by a family of spatial tensors which have components
in such a spatial frame which are constant along the observer worldlines.
Such tensors are ``anchored" in the observer congruence. The 
 spatial Lie derivative along $u$ measures the rate at which a field is evolving
in this noncovariant sense. 

The Fermi-Walker evolution instead describes how a field changes with respect
to what the observers see using an orthonormal spatial frame which is
{\it locally nonrotating}, as determined by a set of torque-free gyros carried
by the observer, rather than by comparing the field to the nearby observers
seen by that observer. Fermi-Walker transport of the spatial frame along
the observer congruence defines mathematically what locally nonrotating
means. A spatial field which has constant components with respect to
such a frame exhibits no evolution in the covariant sense. A spacetime
tensor field which is Fermi-Walker transported along the observer
congruence is represented by a family of spatial tensors which undergo
spatial Fermi-Walker transport.
The {\it Fermi-Walker derivative} along $u$ is defined by
\beq\eqalign{
   [\four\del\fw (u) S]^{\alpha\ldots}_{\ \ \beta\ldots} &=
 S^{\alpha\ldots}_{\ \ \beta\ldots;\gamma} u^\gamma -
        [\sigma(A) S]^{\alpha\ldots}_{\ \ \beta\ldots}\ ,\cr
   A^\alpha{}_\beta &= [u \wedge a(u)^\flat]^\alpha{}_\beta
    =  u^\alpha a(u)_\beta   - a(u)^\alpha u_\beta \ .\cr
}\eeq 
Such an operator satisfies the obvious product laws with respect
to tensor products and contractions thereof. It is easily verified
the $u$, $\four\gm$, $P(u)$, $T(u)$, and all index-shifted variations
of these tensor fields have an identically vanishing Fermi-Walker
derivative. This means that the Fermi-Walker derivative commutes
with the measurement process:
the measurement of the Fermi-Walker derivative
of a spacetime tensor yields the spatial Fermi-Walker derivative acting
on each member of the family of spatial tensors which represent that tensor.

The interpretation of the kinematical quantities is associated with
the comparison of the two kinds of evolution. These quantities describe
the (limiting) relative motion of the nearby observers as seen by each observer
moving through spacetime and as compared to a locally nonrotating 
orthonormal spatial
frame. Physically an observer can assign a relative position vector 
in his local rest space to each neighboring observer at each moment of
his proper time by using light signals, obtaining the relative distance
from half the light travel time and the direction as an average direction.
This relative position vector in the flat local rest space will undergo
a time-dependent linear transformation compared to the locally nonrotating
orthonormal spatial frame. The rate of change of this transformation
is described by the kinematical tensor $-k(u)$.

Suppose $X$ is a tangent vector which is Lie dragged along a given observer
worldline. If $X$ is sufficiently small,
it has the interpretation of being a ``{\it connecting vector}" whose tip
lies on the worldline of some other fixed  nearby observer worldline, 
identifying nearby points in the spacetime manifold with points in the
tangent space. If it is initially spatial, it will not remain spatial
since its tip moves at unit speed relative to the nearby observer's proper
time, while its initial point moves at unit speed relative to the
given observer.
Its spatial projection $P(u)X$ after an interval
$\Delta\tau_u$ of the given
observer's proper time represents the observed position
vector (``{\it relative position vector}") of the nearby observer 
at the same value of the elapsed proper time
measured by that nearby observer, not by the given observer. The component
along $u$, namely $-u_\alpha X^\alpha$, represents the additional lapse
of the given observer's proper time necessary to reach the event at which
$P(u)X$ is the relative position vector. Neglecting this additional
synchronization question helps define the spacetime neighborhood in which this
discussion \Cite{Ehlers 1961, 1993, Ellis 1971}
makes sense. 

Let $Y=P(u)X$ be this relative position vector, which undergoes 
 spatial Lie transport. Its spatial Fermi-Walker derivative 
(``{\it relative velocity vector}") is therefore
\beq\label{eq:relvelvectorfw}
\del\fw (u) Y =  -k(u)\rightcontract Y = 
  [\theta(u)-\omega(u)] \rightcontract Y = \vec\omega(u) \times_u Y
       +\theta(u) \rightcontract Y\ .
\eeq 
This shows that nearby observers appear to be  rotating with angular
velocity $\vec\omega(u)$ and shearing and expanding by $\sigma(u)$ and
$\Theta(u)$ relative to locally nonrotating orthonormal axes. 

\message{shear and Lie; check}
One can further decompose the relative position vector into a relative distance 
$||Y||$ and a relative direction vector $\hat Y$
\beq
     Y = ||Y|| \hat Y \ .
\eeq
This 
leads to a corresponding decomposition of the spatial Fermi-Walker derivative
\beq\eqalign{
\del\fw (u) \hat Y  &=  
    \vec\omega(u) \times_u \hat Y +
       [\sigma(u) - \sigma(\hat Y,\hat Y) P(u)]
           \rightcontract \hat Y\ ,\cr
\del\fw (u) \ln ||Y||  &=  
        {1\over3} \Theta(u) + \sigma(\hat Y,\hat Y) \ .\cr
}\eeq\message{check shear relative velocity eq??}

%by Ehlers \Cite{1961} and Ellis \Cite{1971}.

The corresponding
``{\it relative acceleration}" is obtained by taking one more spatial
Fermi-Walker derivative
\beq\eqalign{
\del\fw (u)^2 Y 
  &= - [\del\fw (u) k(u)]\rightcontract Y 
     - k(u)\rightcontract \del\fw (u) Y \cr
  &= - [\del\fw (u) k(u) - k(u)\rightcontract k(u)] \rightcontract Y\ ,\cr
}\eeq 
but since $k(u) = -\del(u) u$ and $\del\fw (u) u =a(u)$ this becomes
\beq\label{eq:ra}
\del\fw (u)^2 Y 
  =  \{ [\del\fw (u), \del(u)] u  + k(u)\rightcontract k(u)
   +\del(u) a(u) \} \rightcontract Y\ .
\eeq 
To finish this evaluation, one needs to consider the commutator of the spatial
covariant derivative and the spatial Fermi-Walker derivative, which
involves the curvature tensor through the Ricci identity. This will be
postponed until the splitting of the curvature tensor is considered.


On the other hand, a spatial vector $Y$ which is not evolving relative
to the locally nonrotating orthonormal spatial frame does evolve
with respect to the observer congruence at a rate described by the
kinematical tensor itself
\beq
   \Lie(u)\sub{u} Y = k(u) \rightcontract Y
   =   -\vec\omega(u) \times_u Y
       -\theta(u) \rightcontract Y\ .
\eeq 
This would describe the evolution of the spin vector of a gyro carried
by the observers relative to the observer congruence itself.

One can also take the spatial projection of the noncovariant formula
for the Lie derivative in (\ref{eq:ld}) with $u$ replaced by
$fu$ to obtain the generalization of (\ref{eq:liescale}) 
to an arbitrary tensor $S$
\beq
  \Lie(u)\sub{fu} S = f \Lie(u)\sub{u} S - 
      P(u) \sigma(u\otimes d(u)f) S\ ,
\eeq 
or equivalently
\beq\label{eq:uscaledlieder}
 \Lie(u)\sub{u} S =
  f^{-1} \Lie(u)\sub{fu} S +
      P(u) \sigma(u\otimes d(u) \ln f) S\ .
\eeq 
Note that only the terms where $u$ is contracted with a covariant index
of $S$ survive the spatial projection.

\Newpage
%%%%%%%%%%%%%%%
\Subsection{Co-rotating Fermi-Walker derivative}

The Fermi-Walker derivative along the observer congruence is defined by
\beq
   \four \del\fw (u) = \four \del\sub{u} - \sigma( u \wedge a(u)^\flat)\ ,
\eeq
where the argument of $\sigma$ is the antisymmetric tensor (mixed form)
\beq
      [ u \wedge a(u)^\flat ]^\alpha{}_\beta 
         =  u^\alpha a(u)_\beta - a(u)^\alpha u_\beta 
\eeq
which
generates an active boost in the velocity-acceleration plane
which maps the parallel transport of $u$ along $u$
onto $u$ itself, which is Fermi-Walker transported along $u$. 
This same boost maps the parallel transport of any tensor field
along $u$ onto the Fermi-Walker transported tensor field along $u$. Both
transports preserve inner products and commute with index-shifting
and  duality operations
since the metric, volume element, and $u$ itself all have zero
Fermi-Walker derivative
\beq
       0 = \four\del\fw(u) \four \gm
         = \four\del\fw(u) \four \eta
         = \four\del\fw(u) u \ .
\eeq

In fact, the addition to the covariant derivative
of a term $-\sigma(A)$, where $A$
is a $1\choose1$-tensor field,
will generate a continuous Lorentz transformation
relative to parallel transport along the curve or congruence
as long as $A$ is antisymmetric with respect to the metric, i.e., is
the mixed form of a 2-form. Such a derivative will automatically
commute with index shifting and its corresponding transport
will preserve inner products, as does the
Fermi-Walker derivative.
If $A$ is also spatial with respect to $u$ then it will generate a
continuous rotation of the local rest space of $u$ while leaving
$u$ itself uneffected, and both $u$ and $u^\flat$ will still have vanishing
derivative with respect to the new derivative.

For a congruence of timelike curves, 
one may define a {\it co-rotating Fermi-Walker derivative} 
which further adapts the covariant derivative to the congruence by
adding a term to the Fermi-Walker derivative
which leads to a co-rotation of the local rest spaces
relative to the congruence under the corresponding transport
\beq\eqalign{
     \four \del\cfw (u) &= \four \del\fw(u) + \sigma(\omega(u))\cr
        &= \four \del\sub{u} - \sigma(u\wedge a(u)- \omega(u))\ .\cr
}\eeq
A vector which is transported by the co-rotating Fermi-Walker transport
has the same angular velocity as the relative velocity vector,
but does not exhibit the effects of the expansion and shear that
the latter does.

The Lie derivative along $u$ has the following representation
in terms of the ordinary and co-rotating Fermi-Walker derivatives
\beq\eqalign{\label{eq:liecfwder}
  \Lie\sub{u} = \four \del\sub{u} - \sigma(\four \del u)
              &= \four \del(u) 
                + \sigma(\omega(u) - \theta(u) + a(u)\otimes u^\flat ) \cr
              &= \four \del\fw(u) 
                + \sigma(\omega(u) - \theta(u) + u\otimes a(u)^\flat ) \cr
              &= \four \del\cfw(u) 
                + \sigma( - \theta(u) + u\otimes a(u)^\flat )\ . \cr
}\eeq
In the difference term relative to the co-rotating Fermi-Walker
derivative, the expansion term generates a deformation of the local rest
space of $u$ while the acceleration term tilts the local rest space
relative to $u$ along the acceleration direction,
breaking the orthogonality. Only in the case that
$u$ is a Killing vector field do both of these terms vanish
\beq
        \Lie\sub{u} \four \gm = 0 \rightarrow \theta(u) =0 = a(u) \ ,
\eeq
in which case the Lie and co-rotating Fermi-Walker derivatives coincide.
However, a unit timelike Killing vector field is perhaps only interesting
in flat spacetime.

If one wants to use an orthonormal spatial frame, the best one
can do to adapt it to the congruence is to transport it along the
congruence by co-rotating Fermi-Walker transport,
since under Lie transport, the frame will not remain
orthonormal or spatial. Only when $u$ is a Killing
vector field is Lie transport compatible with orthonormality, since
in this case it coincides with the co-rotating Fermi-Walker transport.

The spatial Fermi-Walker derivative has been introduced by spatially projecting
the covariant derivative $\four\del\sub{u}$ and not the Fermi-Walker derivative
$\four\del\fw(u)$ in order to have a temporal derivative one can apply to $u$
itself (since $\four\del\fw(u)u=0$ while $\del\fw(u)u = a(u)$)
\beq
   P(u) \four\del\fw(u) = \del\fw(u) - P(u) \sigma( u\wedge a(u)^\flat ) \ .
\eeq
The difference term only contributes to the derivative of nonspatial fields.
In the same spirit one can define the
{\it spatial co-rotating Fermi-Walker derivative} 
along $u$ or equivalently
the {\it co-rotating Fermi-Walker temporal derivative} along $u$ 
\beq
      \del\cfw(u) 
               = \del\fw(u) + \sigma(\omega(u))  \ .
\eeq
This spatial derivative is related to its corresponding spacetime derivative
in the same way as the ordinary Fermi-Walker derivatives are related
\beq
   P(u) \four\del\cfw(u) = \del\cfw(u) - P(u) \sigma( u\wedge a(u)^\flat )
\eeq
which leads to the following relation to the Lie temporal derivative
\beq
      \del\cfw(u) 
               = \Lie(u)\sub{u} + \sigma(\theta(u)) 
                                - P(u) \sigma( a(u) \otimes u^\flat) \ ,
\eeq
where the last term only contributes for nonspatial fields.
It guarantees that
\beq
      \del\cfw(u)u = \del\fw(u)u = a(u)  \ .
\eeq

The alternative would be to define all the spatial operators by projection
of the corresponding
spacetime operators and introduce a new temporal derivative to
handle nonspatial fields (identical to the present definition of $\del\fw(u)$).
When acting on spatial fields these three temporal derivatives are related by
\beq\eqalign{
      \del\cfw(u) 
               &= \del\fw(u) + \sigma(\omega(u))  \cr
               &= \Lie(u)\sub{u} + \sigma(\theta(u)) \ .\cr
}\eeq

The co-rotating Fermi-Walker temporal derivative
reduces to the Lie temporal derivative in the stationary case 
in which $u$ is proportional to a Killing vector field leading to 
a vanishing expansion tensor field. It reduces to the Fermi-Walker
temporal derivative when the observer congruence is nonrotating.
The alternate notation $\del\lie(u)$ for the Lie temporal derivative
allows one to handle all three of these temporal derivatives
uniformly, writing
\beq
     \{ \del\tem(u) \}_{{\rm tem} \,=\, {\rm lie}, {\rm fw}, {\rm cfw} }
       = \{ \Lie(u)\sub{u}, \del\fw(u), \del\cfw(u) \} \ .
\eeq

Both the ordinary and co-rotating Fermi-Walker derivatives along $u$
of a spatial tensor field reduce to the respective spatial derivatives
of that tensor field.  This is equivalent to the fact that these
derivative operators commute with spatial projection.
Thus the spacetime transport of a spatial tensor field along $u$ for
either of these two types of derivatives reduces to the corresponding
spatial transport.
In particular, for a relative position vector field $Y$ which undergoes
spatial Lie transport along $u$ as in equation (\ref{eq:relvelvectorfw}),
the spatial co-rotating Fermi-Walker derivative satisfies
\beq\label{eq:relvelvectorcfw}
\del\cfw (u) Y =  \theta(u)\rightcontract Y \ .
\eeq 
Thus the relative position vector fails to undergo
spatial co-rotating Fermi-Walker transport along $u$ only because of the
effects of the expansion of the congruence.

For a stationary spacetime with a timelike Killing vector field $\xi$ and
its corresponding normalized 4-velocity $u= \ell^{-1} \xi$, invariance
under the symmetry group is associated with the Lie derivative by $\xi$
and its spatial projection, not the  corresponding operators for $u$.
For such a spacetime
the co-rotating Fermi-Walker derivative and
the co-rotating Fermi-Walker temporal derivative along $u$
reduce to the
rescalings of the respective Killing vector field operators
\beq
   \four \del\cfw(u) = \ell^{-1} \Lie\sub{\xi}\ ,\qquad
   \del\cfw(u) = \Lie(u)\sub{u}  = \ell^{-1} \Lie(u)\sub{\xi}\ .
\eeq
The first equation follows from equation (\ref{eq:liecfwder}) 
with vanishing expansion and equation (\ref{eq:statacc}) which
represents the acceleration as $a(u) = d \ln \ell$,
together with equation (\ref{eq:uscaledlieder}) substituting $f=\ell$.
The second equation is just the spatial projection of the first.
Without stationarity, the co-rotating Fermi-Walker operators are the
best that one can do to extend the corresponding Lie operators in the
stationary case.
