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% \filename{em5.tex} \version{25-jul-1997<-16-apr-1992->5-mar-1993}
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\Chapter{Stationary spacetimes}
\Section{Stationary nonlinear reference frame}
For stationary spacetimes the parametrized nonlinear reference frame
can be adapted to the isometry group by
choosing the parametrized threading to coincide with the flow of a
timelike Killing vector field $e_0$ associated with the stationarity.
It is natural to call this a ``{\it stationary nonlinear reference frame}".
One is still free to choose some initial hypersurface for the slicing
as well as consider affine transformations of the threading parametrization.
For such a nonlinear reference frame, the tensor algebra of stationary spatial
tensor fields on spacetime, including the covariant derivative,
spatial Lie derivative, and exterior derivative operators
on such fields, is isomorphic to the tensor algebra of fields on the
computational space with its time-independent Riemannian metric and its
associated symmetric covariant derivative, Lie derivative, and
exterior derivative. It is exactly this case which motivated Landau and
Lifshitz to use the reference representation of the computational space
geometry for the threading point of view.
For a stationary nonlinear reference frame, the Lie derivatives
$\Lie\sub{e_0}$ of all stationary fields vanish, so the spatial Lie
derivatives $\Lie(m)\sub{m}$ of all stationary spatial fields also
vanish. The expansion tensor $\theta(m)$ of the threading congruence
is therefore zero
\beq
\theta(m) = 0 \ ,\qquad \theta(n)_{ab} = - N^{-1} N_{(a|b)} \ ,
\eeq
leading
to a direct correspondence between the components of the spatial
part of the spatial connection in the threading point of view and
the components of the connection of the projected spatial metric
on each slice or on the threading computational 3-space.
The Lie and symmetry obeying
spatial curvature tensors in this point of view also coincide
and in turn correspond directly to the Riemann
tensor of the quotient spacetime or of the time-independent
Riemannian geometry of the computational 3-space.
REWRITE:
The spatial covariant derivative of a stationary spatial field
when projected to the quotient space agrees with the
covariant derivative with respect to the spatial
metric on the quotient space. %i.e., the Landau-Lifshitz connection.
One can also raise and lower indices on spatial quantities without generating
additional Lie derivative terms in the total spatially covariant derivative,
and the Lie derivative terms drop out
in the Fermi-Walker derivative
and in the transformation formulas for the gravitoelectric and
gravitomagnetic fields.
The latter formulas for the gravitoelectric
and gravitomagnetic vector fields
are then almost conformally related to the Lorentz boost formulas for the
electric and magnetic parts of a 2-form, except for the symmetric
part of the gravitomagnetic tensor in the slicing point of view.
The $1\choose1$-tensor-valued connection 1-form with respect to
the threading projected computational frame, evaluated on $m$
\beq
- \four \bfomega(m)(m)^\flat = \four \Gamma_{\alpha \top \beta}
\omega^\alpha \otimes \omega^\beta =
m^\flat \wedge \overeq g(m) + \half H(m)^\flat
\eeq
defines a 2-form whose electric part is the
gravitoelectric 1-form and whose magnetic part is the spatial dual of
half the gravitomagnetic vector field in the threading point of view.
In the slicing point of view the connection 1-form evaluated on $n$
has a symmetric part like the gravitomagnetic tenosr but is not exactly
the analogous expression in terms of these two fields.
However,
the transformation between the two sets of connection components
has the form
\beq
\bfomega(n)(n) = \gamma_L{}^{-1} \bfomega(m)(m) + \cdots
\eeq\message{??check;develop}
which explains why it so closely resembles a Lorentz transformation
with an overall gamma factor missing as discussed in Appendix B.
??MORE HERE.\message{more}
Maxwell's equations simplify since the presence of the spatial metric
determinant in the $e_0$ Lie derivative term is eliminated.
??MORE HERE\message{more}
Since the lapse $M$ is a stationary function,
the proper time parameters along each threading curve are affinely
related to the coordinate time parametrization.
Again due to stationarity,
the dual lapse alone serves as the potential for the dual
gravitoelectric field
\beq g(m)^\flat =-d(m)(\ln M) =-d(\ln M)
\eeq
The threading gravitoelectric field
measures the spatial variation in the affine relationship between the
coordinate parametrization (equivalently Killing parametrization)
of the threading congruence and the proper time
parametrization, vanishing only when the two are globally affinely related,
the case of a constant lapse function.
\Section{Synchronization gap and Sagnac effect}
One also has a nice interpretation of the threading shift $\overeq M$
in the stationary case.
Its integral along a curve in a slice represents the amount
of simultaneous translation along the threading congruence
necessary to maintain the orthogonality to the threading congruence
of the tangent of the resulting
curve. This represents a local synchronization
of clocks along the new curve \Cite{Landau and Lifshitz 1975}.
The integral of the dual shift around a closed curve in the slice yields
the ``synchronization defect" in the terminology of
Baza\'nski \Cite{1988}
which is in turn closely related to the Sagnac
effect \Cite{Post 1967, Ashtekar and Magnon 1975,
Van Bladel 1984, Henrickson and Nelson 1985}
involving null rather than spacelike curves.
This phenomenon in the stationary case is best described by interpreting
the spacetime region $\four M$ together with the parametrized
nonlinear reference frame
as a (local) line bundle with group $\flow(e_0)$ and base space equal to the
quotient space $\four M/\flow(e_0)$, for which $e_0$ is a fundamental
vector field. This line bundle has a natural connection whose horizontal
subspaces are just the local rest spaces of $m$. The associated connection
1-form is just $\theta^0=\omega^0-\overeq M$ and the condition
that it transform under the (trivial) adjoint representation of the group
is just $\Lie_{e_0}\theta^0=-(e_0 M_a)\omega^a=0$
which is satisfied by the
condition of stationarity.
A compatible slicing is just a particular local trivialization.
The spacetime metric is just a fiber invariant Kaluza-Klein metric on this
bundle, with the dual lapse describing the metric along the fiber direction
or vertical subspace
and the spatial metric the metric on the horizontal subspace. If $s :
\four M/\flow(e_0) \to \four M$ is a section,
i.e., a valid compatible slice hypersurface,
then pulling down the
connection 1-form $ s^\ast \theta^0 $% =-s^\ast \overeq M$
%%%%%%%%%%%%%%%%%%%%%%%%%%%% CHECK:
to the quotient space
yields a 1-form which is the projection of the
sign reversed dual shift 1-form associated with this hypersurface.
For the section $s_t$ which maps up to the slice $\Sigma_t$, this is
just the projection of $-\overeq M$.
Given any curve in a slice starting at a particular point,
one can project the curve down to the quotient space
and then lift it back up to the spacetime horizontally to obtain a new
curve emanating from the same point as the original curve. Such a curve
describes a synchronization of %%%%%%%%% proper time
clocks carried by the observers
whose worldlines intersect the original (and final) curve. The integrability
of this synchronization operation depends on the vanishing of the
curvature of the connection which is just the projected exterior derivative
of the connection 1-form
\beq D\theta^0 = P(m)d\theta^0 = -d(m) \overeq M = -d \overeq M
\eeq
which is proportional to the rotation tensor of $m$, whose spatial dual leads
to the gravitomagnetic vector field in the threading point of view.
(The proportionality factor of the dual lapse function leads to a field which
is invariant under reparametrization of the threading congruence.)
A flat connection leads to a family of integral submanifolds of the
horizontal distribution
which correspond to the elements of the new slicing orthogonal to the threading
congruence. This is the static subcase.
The parallel transport on the quotient space associated with this bundle
connection transports coordinate clock time around in ``space" in a way
which represents local synchronization. Parallel transport around a closed
curve leads to a translation along the fiber by an amount which represents
the synchronization defect of the curve. One can always introduce another
uniquely defined translation along the fiber above the given curve in the
quotient space by instead following the future-directed null curve, leading
to a ``future-null lift" of the curve. This represents a null synchronization
rather than a spacelike synchronization of time parameters along the fibers.
Suppose one has a parametrized curve segment $\{C(\lambda) \relv \lambda\in
[0,\lambda_0]\}$ in the quotient space. This represents a sheaf of threading
curves in the spacetime or a ``Sagnac tube" \Cite{Ashtekar and Magnon 1975}.
From any point $c(0)$ lying above the initial
point $C(0)$ in the quotient space (i.e., lying on the threading curve
$C(0)$), one can construct three parametrized curves through $c(0)$
in this sheaf which project down to $C$ at corresponding
parameter values. The first curve $c$ is just the representative
of $C$ lying in the slice containing $c(0)$. The second
$c_S$ is the horizontal lift of $C$ through $c(0)$
consisting of those points on the sheaf which are spatially
synchronized relative to the threading. The third curve $c_N$ is
is the future-directed null representative of $C$, consisting of
those points of the sheaf related by a null synchronization of the
threading time parameters. (See Figure \ref{fig:lifts}.)
\fig{Lifts from the quotient space}{TO DO}{lifts}
Define $t^{SD}(\lambda)$ to be the amount of translation of $c_S(\lambda)$
along the fiber from $c(\lambda)$, i.e., the difference in value of the
time function from $c(\lambda)$ to $c_S(\lambda)$. Similarly let
$t^{SE}(\lambda)$ be the corresponding quantity for $C_N(\lambda)$. When
$C$ is a simple closed curve with $C(\lambda_0)=C(0)$,
then $c$ is also a simple closed curve.
For notational convenience, let a minus sign subscript denote
denote the same curves with the reverse orientation and a plus sign subscript
the original curves. This leads to new parametrized
curves $c_{S-}$ and
$c_{N-}$ defined in the same way relative to the curve $C_-$,
and corresponding values $t^{SD}_{\pm}$ and
$t^{SE}_{\pm}$ for the translation intervals for these curves
and the originals after one revolution in the quotient space.
The synchronization defect for the simple closed curve $C$,
or equivalently for $c$, is just the quantity
\beq
\Delta t^{SD} =t^{SD}_+ = -t^{SD}_-
\eeq
and the Sagnac effect is
\beq
\Delta t^{SE} =t^{SE}_+ - t^{SE}_- =2\Delta t^{SD}\ .
\eeq
The Sagnac effect is just a slightly different geometric realization of the
synchronization effect for a stationary spacetime in a stationary nonlinear
reference frame. Figure \ref{fig:sagnac} illustrates this situation.
\fig{Sagnac tube geometry}{TO DO}{sagnac}
To derive this result, let $q$ be the unit spatial vector field relative
to the threading at each point of the above sheaf which is obtained by
horizontally lifting the tangent $C^\prime(\lambda)$ and normalizing it.
Then $k_\pm=m\pm q$ specify the two null directions tangent to the sheaf at
each point.
The spatial synchronization curve $c_S$ is the integral curve
of $q$ through $c(0)$, while the future-null synchronization curve
$c_N$ is the integral curve of $k_+$ through $c(0)$, each suitably
reparametrized.
The first curve $c_S$ is an integral curve of the distribution determined by
$\theta^{\top }$ or equivalently by $\theta^0=-M^{-1}m^\flat =dt-\overeq M$;
the second curve $c_N$
is an integral curve of the distribution determined by
$-M^{-1}k^\flat = dt-(\overeq M + M^{-1}q^\flat)$.
(Here integral curve is used in the sense of a partial integral
submanifold of the distribution determined by a 1-form.)
Apart from Lie translation along $e_0$, the three tangents $c^\prime$,
$c_S^\prime$, and $c_N^\prime$ all have the same spatial projection
so the integral of $\overeq M$ along the three curves is the same
\beq \eqalign{
\int_c \overeq M =& \int_{c_S} \overeq M = \int_{c_N} \overeq M \cr
=& \int_{c_S} dt = t^{SD}(\lambda_0)\cr
}\eeq
and equals the change in the time coordinate along the spatial synchronization
curve $c_S$ since $ (dt -\overeq M)(c_S^\prime) =0$.
When $c$ is a simple closed curve, this is just the synchronization defect
\beq
\Delta t^{SD} = t^{SD}_+ = \oint_c \overeq M\ ,
\eeq
while the integral on $c_{S-}$ gives the opposite result due to the
reverse orientation.
On the other hand the null circuit times are
\beq
t^{SE}_{\pm} = \int_{c_{N\pm}} (\overeq M \pm M^{-1}q^\flat)\ .
\eeq
Again, since apart from Lie translation along $e_0$ the tangents
$c_{N+}^\prime$, $-c_{N-}^\prime$, $c_{S+}^\prime$, and $c^\prime$
all have the same spatial projection by construction, the integrals
\beq
\int_{c_{N+}} M^{-1} q^\flat = - \int_{c_{N-}} M^{-1} q^\flat =
\int_{c_{S+}} M^{-1} q^\flat = \int_{c_{S+}} M^{-1} ds
\eeq
are all equal
and give the integral of the reciprocal of the
dual lapse with respect to the arclength of the
spacelike spatial synchronization curve $c_{S+}$.
For the same reason the integrals
\beq \eqalign{
\int_{c_{N+}} \overeq M = - \int_{c_{N-}} \overeq M =&
\int_{c_{S+}} \overeq M \cr
=& \oint_c \overeq M = \Delta t^{SD}
}\eeq
are all equal to the Sagnac defect,
so the average null circuit time is
\beq
t^{SE}_{avg} = \half (t^{SE}_+ + t^{SE}_-) = \int_{c_{S+}} M^{-1}ds\ ,
\eeq
while the difference is the Sagnac effect
\beq
\Delta t^{SE} = t^{SE}_+ - t^{SE}_- = 2 \Delta t^{SD}\ .
\eeq
Using Stoke's theorem the Sagnac effect can be re-expressed
as a surface integral over any simple %spacelike
2-surface $\Xi$ %in the slice
whose boundary is the closed curve $c$
\beq \eqalign{
\half \Delta t^{SE} &= \oint _{c} \overeq M
=\int_\Xi d \overeq M
=\int_\Xi d(m) \overeq M
=\int_\Xi M^{-1}[\dual{}^{_{(m)}} H(m)^\flat]\ ,\cr
&=\int_\Xi M^{-1}\vec H(m)\leftcontract \eta(m)
=\int_\Xi M^{-1} \vec H(m) \cdot_m d\vec S_m
\ .\cr}
\eeq
This formula is the threading point of view expression for the
four-dimensional integral of Ashtekar and Magnon \Cite{1975}.
The final form of the integral uses the classical notation $d\vec S_m$ for the
projection orthogonal to $m$ of the
vector differential of surface area on the 2-surface $\Xi$. Note that
the spacetime differential of $\overeq M$ equals its spatial differential
since it is a stationary field.
Thus one can interpret the Sagnac
effect as the quantity of projected
flux contained in the closed curve
of the quotient of the dual gravitomagnetic vector field
by the dual lapse function.
The term ``projected flux" or equivalently
``projected surface integral" is necessary to distinguish the
classical notation $d\vec S_m$ from the differential of surface area
$d\vec S$ of the surface $\Xi$.
For example, in the case of a spacelike slicing, if $\Xi$ is an interior
surface of $c$ lying in the slice, then
\beq \eqalign{
\vec H(m)\cdot_m d\vec S_m |_\Xi &=
H(m)\leftcontract \eta(m)|_\Xi
=\half H(m)^a \eta(m)_{abc} \,\omega^b\wedge\omega^c |_\Xi\cr
& =\half \gamma_L H(m)^a \eta_{abc} \,\theta^b\wedge\theta^c |_\Xi
= [\Pscr H(m)]\leftcontract \eta |_\Xi
= \vec H(m)\cdot_n d\vec S_n |_\Xi \cr
}\eeq
shows that an additional gamma factor becomes explicit when rewritten
in terms of the differential of surface area $d\vec S_n$
on the surface $\Xi$. The notation $|_\Xi$ indicates the restriction of
a differential form to the surface $\Xi$.
This same integral may be expressed by projecting everything down to the
quotient space, where the projection of the threading spatial metric
$P(m)^\flat$ provides a Riemannian metric.
One obtains
an ordinary surface integral of the projected dual
gravitomagnetic vector field divided by the projected
dual lapse function over the
projection of the surface $\Xi$ into the quotient space, or equivalently
as a line integral of the projected dual shift over the closed
curve $C$ which bounds $\Xi$, in each case using the projected
threading spatial
metric on the quotient space
\beq
\half \Delta t^{SE} = \oint _{C} \overeq M
=\int_\pi(\Xi) M^{-1} \vec H(m) \cdot d\vec S
\ .
\eeq
In the case that the slicing is spacelike and $\Xi$ lies entirely in
the slice of the closed curve $c$,
this may also be
transformed into an integral in terms of the slicing spatial gravitational
fields
\beq \eqalign{
\Delta t^{SE}
&=\int_\Xi N^{-1} \gamma_L{}^2 \left[\Pscr\vec H(m) \right]
\leftcontract \eta\cr
&=\int_\Xi N^{-1} \gamma_L{}^4
\left[ \vec H +(\vec H \cdot_n \vec v)\vec v
+ 2\vec v \times_n (\vec g + (\SYM \bivec H)\cdot_n \vec v) \right]
\leftcontract \eta\ ,\cr }
\eeq
since the restriction of $\eta(m)$ to $\Xi$ is the same as that
of $\gamma_L \eta$.
It is also useful to introduce the relative Sagnac effect in order to
put the magnitude of the Sagnac effect into perspective
\beq
{\half \Delta t^{SE} \over \Delta t^{null}}
= {\int_\Xi M^{-1} \vec H(m) \cdot_m d\vec S_m \over \int_c M^{-1} ds}\ .
\eeq
This quantity
is just the ``corrected" projected
surface integral of the gravitomagnetic field
divided by the ``corrected" arclength of the bounding curve, where
``corrected" refers to the additional factor of $M^{-1}$ which converts
the proper time along the congruence to coordinate time.
Each of the time intervals discussed above is an interval of coordinate
time. However, as noted above,
along a given threading curve, the dual lapse $M$ is constant
so the proper time interval along a given threading curve $C$
which corresponds to the Sagnac effect
or the null transit times
can be obtained just by multiplication of the latter interval by the
value of the dual lapse on that threading curve.
%This is locally independent of the curve
%only when the dual shift 1-form is the differential of a function, in which
%case the same is true of $\theta^0$, defining a new slicing of spacetime
%orthogonal to the threading. This is the static subcase. %%%%%%%%%%%%%%???
%???CLARIFY???
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Rotating spatial Cartesian coordinates in flat spacetime}
No discussion of spatial gravitational forces would be complete without
relating them to the familiar notions of centrifugal and Coriolis forces
which arise in everyday experience. It is these forces which have been
generalized by the splitting discussion of
the acceleration equation, so it
is important to understand exactly how they fit into the general picture
as a special case.
The point of view in which they are discussed in classical mechanics is
neither the slicing or threading point of view but rather the reference
point of view.
%They appear not directly in
%the guise of either the
%slicing or threading spatial gravitational forces, but rather those which
%appear in the reference point of view decomposition of the geodesic equation.
Consider adapted coordinates $\{t,x^a\}$ for a nonlinear reference frame
in an arbitrary spacetime. The reference decomposition of the geodesic
equation considered as a (contravariant) vector field equation
is just its coordinate decomposition.
The calculation of the spatial reference projection of that equation
may be found in Forward's discussion \Cite{1961} of linearized
general relativity based on M\o ller's first edition discussion \Cite{1952} of
spatial gravitational forces using the coordinate time as the parameter for
the geodesic, but modified to use the spatial coordinate decomposition of the
geodesic equation rather than the threading decomposition. Linearization allows
one to sidestep the some of the complications which arise in the general case,
but the logarithmic time derivative of the
coordinate gamma factor $\Gamma$ defined by \Eq(\ref{eq:coordgamma})
leads to an additional acceleration-dependent term proportional to
the velocity. (Compare with \Eq(6.25) of Misner, Thorne and Wheeler
\Cite{1973}.)
However, in flat spacetime with a coordinate system $\{t,x^a\}$ consisting of a
rigidly rotating system of orthonormal Cartesian spatial coordinates and
the usual time coordinate $t$ of the related nonrotating inertial coordinates,
the complications wash out and lead to the usual formulas
\beq \eqalign{
\ddot t & = 0\ ,\cr
\ddot {\vec x} &= \vec g_{(ref)} + \dot{\vec x} \times \vec H_{(ref)} \ .\cr
}\eeq
Here the dot is the usual time coordinate derivative and the usual
vector notation is employed. The spatial
forces are
\beq \eqalign{
& \vec g_{(ref)} = -\vec\Omega \times (\vec\Omega \times \vec x)\ ,\cr
& \vec H_{(ref)} = 2\vec\Omega\ ,\cr
}\eeq
where $\vec\Omega$ is the constant vector
describing the angular velocity of the rotating system,
identified with a constant vector field.
The reference gravitoelectric field is exactly the centrifugal force
(per unit mass) and the reference gravitomagnetic field vector is
just twice the global rotation of the coordinate system, leading to
the Coriolis force (per unit mass).
To see how these relate to the slicing and threading fields, one must
evaluate them for the nonlinear reference frame for which these coordinates are
adapted. The slicing is a flat slicing of Minkowski spacetime but
the threading is an inhomogeneous tilted threading relative to that slicing,
and it is timelike only within a certain cylinder in space called the light
cylinder
inside of which the
velocity of rotation is less than the speed of light. Thus the slicing
point of view holds everywhere,
%although the physical interpretation of
%``Space" breaks down outside the light cylinder,
while the threading
point of view is limited to the interior of the light cylinder.
Since the spatial metric is Euclidean in the slicing point of view, one can
use the customary vector notation unambiguously in expressing the line element
(instead of metric to allow the dot product notation)
\beq \eqalign{
& ds^2 = - dt^2 + ( d\vec x + \vec\Omega \times \vec x dt) \cdot
( d\vec x + \vec\Omega \times \vec x dt)\ ,\cr
& N=1\ ,\qquad \vec N = \vec\Omega \times \vec x \ ,\qquad
g_{ab}=\delta_{ab}\ .\cr
}\eeq
The differentials $ d\vec x + \vec\Omega \times \vec x dt$ are the rotated
differentials of the nonrotating Cartesian coordinates.
The slicing spatial gravitational fields in this notation are
\beq
\vec g(n)=0 = \SYM \bivec H(n)\ ,\qquad
\vec H(n) = 2 \vec\Omega = \vec H_{(ref)}
\eeq
and the expansion tensor vanishes $\theta(n)=0$.
The Lorentz boost parameters are
\beq
\vec v= \vec N\ , \qquad \gamma_L =(1-v^2)^{-1/2}\ ,
\eeq
leading to the threading quantities
\beq \eqalign{
& M=\gamma_L{}^{-1}\ , \qquad
\overeq M = \gamma_L{}^2 \delta_{ab} v^a dx^b\ , \qquad
\gamma_{ab} = \delta_{ab} +\gamma_L{}^{-2} M_a M_b\ ,\cr
& g(m)_a = \gamma_L{}^2 \delta _{ab}g(m)_{(ref)}^b\ ,\qquad
H(m)^a = \gamma_L{}^2 H_{(ref)}^a\ .\cr
}\eeq
Thus it is the gravitoelectric force in the threading point of view, apart from
a time reparametrization and a projection,
which corresponds to the centrifugal force.
The slicing gravitomagnetic vector field directly equals the reference
gravitomagnetic vector field, which in turn is twice the global angular velocity
of the coordinate system, and leads to the Coriolis force.
The threading gravitomagnetic vector field
differs from twice the constant global rotation vector
by the time reparametrization and a projection
in such a way that it equals the inhomogeneous
local rotation of the threading congruence.
Since the slicing spatial velocity differs from the reference spatial
velocity by the velocity of the threading curves
\beq
\nu(n)^a = d x^a / d t + N^a \ ,
\eeq
the slicing second-order acceleration equation takes the form
\beq \eqalign{
D(U,\Pscr,n)^2 x^a / d t^2
&= d^2 x^a / d t^2 \cr
&= [ \vec\nu(n) \times_n \vec H(n) ]^a
+ N^a{}_{|b} N^b \cr
&= [ d\vec x /dt \,\times_n \vec H(n) ]^a
+ 2 N_{[b|}{}^a{}_] N^b+ N^a{}_{|b} N^b \cr
&= [ d\vec x /dt \,\times_n \vec H(n) ]^a
+ N_{b|}{}^a N^b \ .\cr
}\eeq
The resultant shift derivative term corresponds to the reference
gravitoelectric field, i.e., the centrifugal force term.
For the threading point of view where there is no relative velocity term
\beq
\nu(m)^a = M^{-1} d x^a / d t \ ,
\eeq
the second-order acceleration equation is instead
\beq\eqalign{
D(U,m)^2 x^a / d t^2
&= d^2 x^a / d t^2 + \Gamma(m)^a{}_{bc} d x^b / d t \,\, d x^c / d t \cr
&= M^2 [ -2\vec g(m)\cdot_m \vec\nu(m) \, \vec\nu(m)
+ \vec g(m) + \vec \nu(m) \times_m \vec H(m) ]^a \ .\cr
}\eeq
Here the quadratic term in the spatial velocity involving the gravitoelectric
field cancels the connection term, leaving a direct correspondence between
the threading and reference spatial gravitational forces.
The spatial curvature in the threading point of view is easily evaluated from
the splitting of the vanishing spacetime curvature tensor.
Since the expansion tensor is zero, one has in the projected
computational frame
\beq\eqalign{
& \four R^{ab}{}_{cd} = 0 \rightarrow \cr
& \qquad R\ncov(m)^{ab}{}_{cd} = R\sym(m)^{ab}{}_{cd} =
- 2 \omega(m)^a{}_{[c} \omega(m)^b{}_{d]}
- 2 \omega(m)^{ab} \omega(m)_{cd} \ .\cr
}\eeq
The double-sided spatial dual gives the equivalent Einstein tensor
\beq
G\ncov(m)^{ab} =
G\sym(m)^{ab} = 3 \omega(m)^a \omega(m)^b
= 3 \gamma_L{}^4 \Omega^a \Omega^b \ .
\eeq\message{sign??}%
%pig
\message{!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!}
\Section{Stationary axially-symmetric case:
rotating Minkowski, G\"odel and Kerr spacetimes}
A very interesting class of stationary spacetimes are the
stationary axisymmetric spacetimes,
which are characterized by the existence of
a pair of commuting Killing vector fields
$\xi_{(t)}$ and $\xi_{(\phi)}$,
respectively timelike and spacelike in some region of the spacetime,
associated with the isometric action in this region
of the abelian group $R\times SO(2,R)$ of time translations
and rotations around an axis of symmetry.
The axial Killing vector field
$\xi_{(\phi)}$ is uniquely defined by the condition that its one-parameter
subgroup have closed spacelike
orbits diffeomorphic to $S^1$ with periodicity of $2\pi$.
When spatial infinity exists, at least in the direction orthogonal to the
axis of symmetry,
one may define $\xi_{(t)}$
as the unique linearly independent Killing vector field which at spatial
infinity is timelike, has unit length and is orthogonal to $\xi_{(\phi)}$.
In this same region of spacetime
there exists a privileged class of observers, the so called
``stationary observers", which are
characterized by timelike worldlines contained in a single orbit of the group
and invariant under the action of the group.
A family of stationary observers which covers the stationary region of the
spacetime obviously is a preferred candidate for a threading congruence.
Such a family has a timelike unit four-velocity $u$ which is
tangent to the orbits of the symmetry group and commutes with its
generators.
Such a four-velocity $u$ is characterized by
a single angular velocity function $\Omega$ which is invariant under the
group (a stationary function)
\beq \eqalign{
& u = |\xi_{(\Omega)}^\alpha \xi_{(\Omega)}{}_\alpha |^{-1/2}
\xi_{(\Omega)}\ ,\cr
& \xi_{(\Omega)} = \xi_{(t)} +\Omega\xi_{(\phi)}\ ,
\qquad \xi_{(t)} \Omega =\xi_{(\phi)} \Omega =0\ .\cr}
\eeq
The condition that $u$ be timelike restricts the angular velocity to lie
between two values $\Omega_{min}$ and $\Omega_{max}$.
The one-parameter family of families of stationary observers for which
$\Omega$ is a constant have worldlines which are orbits of a one-parameter
subgroup of the symmetry group generated by the Killing vector field
$\xi_{(\Omega)}$.
Such observers are called Killing observers and are associated with a
stationary parametrized nonlinear reference frame with threading tangent
$e_0=\xi_{(\Omega)}$.
The components of the metric will be stationary functions in the associated
computational frame, leading to stationary spatial fields in
the slicing and threading splittings.
However, just as in flat spacetime in rotating cylindrical coordinates,
there will always be an orbit of some
maximum circumference beyond which $u$ will be spacelike
and the threading point of view breaks down, unless $\Omega=0$. These special
observers have zero rotation at spatial infinity and are
commonly referred to as the static observers. This is the usual threading
associated with the Boyer-Lindquist \Cite{1967} coordinates for a black hole,
and the associated threading point of view is valid only outside the
ergosphere, where the value $\Omega=0$ lies in the allowed range
$[\Omega_{min},\Omega_{max}]$.
As one moves inward into the ergosphere, the threading switches
from being timelike to spacelike through the outer boundary (the so-called
static limit) where it is null.
The non-Killing stationary observers have differential rotation relative
to the Killing observers. If one
takes a threading point of view with $e_0= \xi_{(\Omega)}$ in this case,
the metric becomes explicitly time-dependent, i.e., one obtains
nonstationary spatial fields from the associated splittings of the
spacetime metric.
Of all families of stationary observers, a rotation-free congruence,
if it exists, provides a preferred slicing of the stationary region
by the family of orthogonal hypersurfaces it defines. Such a slicing
is stationary, i.e., taken into itself under the isometry group.
The most convenient parametrized nonlinear reference frame is therefore the
stationary one
with a Killing threading associated with the choice $e_0=\xi_{(t)}$ and such a
preferred slicing, leading to stationary spatial fields in both points of
view.
For the orthogonally transitive case, where the group orbits admit a family
of orthogonal 2-surfaces, Carter \Cite{1969}
has shown that there exists a unique rotation-free stationary
family of observers with $u$ orthogonal to the axial Killing vector field
$\xi_{(\phi)}$. These are called the ``locally nonrotating
observers" \Cite{Bardeen 1970, \MTW\ 1973}
or the ``zero angular momentum observers" (ZAMO's)
\Cite{Thorne et al 1986, Thorne and Macdonald 1982}.
This provides such a spacetime with a preferred
slicing, while the static observers (the Killing observers
which are nonrotating at spatial infinity) provide a preferred threading.
Black hole spacetimes have an additional preferred threading which is
adapted to the geometric structure of the spacetime in a more subtle way.
This is the Carter threading associated with the canonical orthonormal
frame defined by the separability properties of various geometric equations
on the spacetime \Cite{Carter 1968, 1973}.
This orthonormal frame is the one in which the Boyer-Lindquist
representation of the metric is usually given
\Cite{\MTW\ 1973}, although it is not usually explicitly referred to.
For the charged black hole, this family of observers (the ``Carter observers"
with unit velocity $u\carter$)
measures parallel electric and magnetic fields, for example, as was used
by Damour and Ruffini \Cite{1975} in quantum electrodynamical considerations
for black holes.
In any case,
the electric and magnetic parts of the Weyl curvature associated
with this splitting are also aligned, i.e., proportional, a simple statement
that is difficult to find in the literature in spite of all the
articles which have used an orthonormal frame adapted to the Carter
observers. This is easily derived using the more transparent formulas
of Marck \Cite{1983} which also corrects some sign errors in the Carter
expressions.
\begincomment
This is easily confirmed using
STENSOR \Cite{H\"ornfeldt 1988}
and SHEEP \Cite{Fricke 1977}.\endcomment
All of these threadings and the single slicing are well behaved and have
the appropriate causal behavior for both the slicing and threading
points of view outside of the ergosphere.
The Boyer-Lindquist \Cite{1967}
coordinates $\{t,r,\theta,
\phi\}$ are adapted coordinates for the nonlinear reference frame consisting of
this preferred slicing and the static threading associated with $\xi_{(t)}$.
The field $m$
associated with the static observers is boosted with respect to $n$
in the direction of the axial orbits opposing the corotating direction
(negative angular momentum), while the field
$u\carter$ is boosted in the corotating direction
(positive angular momentum). This latter direction becomes null leaving the
ergosphere on its inner boundary, the event horizon, where the radial
direction also becomes null, switching the causal nature of the slicing.
Thus the slicing point of view for this nonlinear reference frame is valid
only outside the horizon, while the threading point of view is valid
only outside the ergosphere.
The two standard choices of metric variables for an orthogonally transitive
stationary axisymmetric spacetime, as given by Demiansky \Cite{1985}
for example,
correspond to the slicing and threading decompositions respectively
\beq \eqalign{
\four \gm &=- \tilde f dt\otimes dt + \tilde f{}^{-1}\rho^2(d\phi -
\tilde\omega dt)\otimes (d\phi -\tilde\omega dt) +
e^{2\mu} (d\rho\otimes d\rho +dz\otimes dz)\ ,\cr
&=- f (dt-\omega d\phi)\otimes (dt-\omega d\phi) +
f{}^{-1}[\rho^2(d\phi \otimes d\phi) + e^{2\gamma} (d\rho\otimes d\rho
+dz\otimes dz)\ ,\cr
& N^2=\tilde f\ , \quad (g_{ab}) = \diag(\tilde f{}^{-1}\rho^2,
e^{2\mu},e^{2\mu})\ ,\quad N^\phi =-\tilde\omega\ ,\cr
& M^2= f\ ,\quad (\gamma_{ab})= f^{-1}\diag(\rho^2, e^{2\gamma},
e^{2\gamma})\ , \quad M_\phi = \omega\ .\cr}
\eeq
The threading variables prove more useful in obtaining exact
solutions for these models.
The simplest example of this class is flat spacetime with a nonlinear
reference frame associated with rotating cylindrical coordinates
$(t,\rho,\phi,z) = (\bar t, \bar \rho, \bar\phi -\Omega \bar t, \bar z)$,
where the barred coordinates are standard cylindrical coordinates, and the
unbarred angular coordinate plane rotates about the $z$ axis with constant
angular velocity $\Omega\ge0$ in the counterclockwise direction.
The Killing vector fields described above are $\xi_{(t)}=\partial/\partial
\bar t$, $\xi_{(\phi)}=\partial/\partial \bar{\phi}$ and
$\xi_{(\Omega)}=\partial/\partial t$.
In addition, the coordinates are adapted to the Killing vector field
$\partial/\partial z$ which extends the axial symmetry of the
coordinate computational frame to cylindrical symmetry.
In the slicing point of view one has the decomposition of the metric
\beq
N=1\ , \quad \vec N = \Omega\partial_\phi\ , \quad
P(n)^\flat = d\rho\otimes d\rho +\rho^2 d\bar\phi\otimes d\bar\phi
+ dz\otimes dz \ ,
\eeq
where
\beq
v=\rho\Omega\ , \quad \gamma_L= (1-v^2)^{-1/2}\ , \quad
d\bar\phi =d\phi+\Omega dt \ .
\eeq
The spatial metric on the time slices is flat, the gravitoelectric field
is zero and the gravitomagnetic field is spacetime-homogeneous
\beq
\vec g=0\ , \quad \half \vec H = \Omega\partial_z\ , \quad
\SYM \bivec H =0\ .
\eeq
In the threading point of view valid up to the the radial coordinate
value $\rho=\Omega^{-1}$,
one has instead a spatially inhomogeneous splitting
\beq \eqalign{
&M=\gamma_L{}^{-1}\ , \quad \overeq M = (\gamma_L \rho)^2 \Omega d\phi\ ,\quad
P(m)^\flat = d\rho\otimes d\rho
+(\gamma_L\rho)^2 d\phi\otimes d\phi + dz\otimes dz
\ ,\cr}
\eeq
and the spatially inhomogeneous spatial gravitational fields
\beq
\vec g(m){}^\flat=(\gamma_L\Omega)^2 \rho d\rho \ , \quad
\half \vec H(m) = \gamma_L{}^2 \Omega\partial_z\ .
\eeq
The Sagnac effect for a $\phi$-coordinate circle
traced out in the co-rotating direction of increasing $\phi$ is
\beq
\half \Delta t^{SE} =
\oint_{S^1} \overeq M = 2(\pi \rho^2) \gamma_L{}^2 \Omega \ ,
\eeq
where the integral over $S^1$ is shorthand for the integral over $\phi$
in the positive direction.
This relativistic result was given implicitly by Landau and Lifshitz
\Cite{1975}.
The average time for a null circuit of the Sagnac tube is
\beq
\Delta t^{avg} = \oint_{S^1} M^{-1} ds = 2\pi \rho \gamma_L{}^2 \Omega
\eeq
so the relative Sagnac effect is
\beq
{\half \Delta t^{SE} \over \Delta t^{avg} } = \rho \Omega =v\ ,
\eeq
which is the relative speed at a given radius.
In this example, the slicing gravitomagnetic vector field is just the constant
vector field of the global angular velocity of the
associated rotating cartesian coordinate system,
while the threading gravitomagnetic vector field is the inhomogeneous
local rotation of the inhomogeneous congruence of
time coordinate lines which becomes singular at
the radius at which the rotational velocity equals the speed of light,
beyond which the time lines are spacelike and the threading point of
view breaks down.
On the other hand the spacetime homogeneous cosmological
models \Cite{Ryan and Shepley 1975}
expressed in a stationary reference frame have spacetime homogeneous
threading congruences and set up spacetime homogeneous spatial
gravitational fields in the threading point of view.
The G\"odel solution \Cite{G\"odel 1949}
is the most famous such cosmological model,
a solution of the Einstein equations with dust and
nonzero cosmological constant, but
the stationary reference frame in which it is usually presented
has a timelike homogeneous slicing, so only the threading point of view
can be considered. The comoving fluid nonlinear reference frame is geodesic
leading to zero gravitoelectric field, leaving only a spacetime
homogeneous gravitomagnetic field associated with the local rotation
of the fluid flow lines.
Jacobs and Wilkins \Cite{1988}
discuss the G\"odel solution in this light using a linear analogy with
electromagnetism.
When the stationary nonlinear reference frame of a spacetime homogeneous
cosmological model has a spatially homogeneous spacelike
slicing, the same is true of the slicing spatial gravitational fields,
namely they are also spacetime homogeneous.
Ozsv\'ath and Sch\"ucking \Cite{1969}
describe a class of such dust
models with $S^3$ spatial slices, using not only a spatially homogeneous
stationary reference frame but introducing also the normal and comoving
perfect fluid reference frames for the same slicing. The latter two
reference frames are spatially homogeneous but nonstationary
and set up time-dependent spatially
homogeneous spatial gravitational fields, just like the nonstationary
spatially homogeneous cosmological models. This is analogous to the
nonstationary reference frames associated with non-Killing stationary
observers in the stationary axially symmetric case.
\endinput
\Section{Gyroscope precession}
An obvious question to ask is, what does all of this formalism have to do
with the classic gyro precession formula? This formula,
describing the precession of a gyroscope in the field of a rotating body,
seems to have been first obtained in its present form by Schiff \Cite{1960}
within linearized GR and was later extended to the PPN theory, as reviewed
for example by Misner, Thorne and Wheeler \Cite{1973},
and more recently discussed within linearized GR in
a very elegant way by Thorne \Cite{1989}.
Physically gyros define operationally what it means to be
locally nonrotating, so
if there is rotation of the spin of a gyro, it has to be a relative rotation
with respect to something which is locally rotating.
This is exactly what the splitting formalism is set up to measure and which
leads to the introduction of the concept of the gravitomagnetic field,
the measurement of which is the goal
of the long awaited Stanford gyroscopic precession experiment
\Cite{Everitt 1979}
It is this problem which has provided much of the motivation for talking
about ``gravitomagnetism."
However, the famous precession formula is not just a splitting version
of the equation for Fermi-Walker transport along a worldline of a vector
which is spatial with respect to the worldline's 4-velocity $u$,
namely eq. (\label{eq:fwtsplit}).
This formula describes how the spin vector $S$ changes with respect
to the threading point of view nonlinear reference frame. It does not
directly
describe a rotation, since it must take into account the nonorthonormality
of the spatial projected computational frame and changes in the magnitude
of the spatial spin vector.
What then is the classic precession formula?
For a black hole spacetime, it is possible to complete the 4-velocity of
the gyro worldline to an orthonormal frame which is linked to the
Boyer-Lindquist nonlinear reference
frame and then measure the precession of the gyro itself with respect to this
special orthonormal frame.
Since the Boyer-Linquist nonlinear reference frame itself
%establishes a rigid link between the local computational frame and the
defines what it means to be nonrotating with respect to the
asymptotically flat region of the spacetime, this has a physical interpretation
in terms of rotation with respect to that asymptotic region, i.e., rotation
relative to the ``distant stars".
The Boyer-Lindquist spatial coordinates $\{r,\theta,\phi\}$ are orthogonal
so both the coordinate derivatives $\{e_a\}$ and coordinate differentials
$\{\omega^a\}$ are orthogonal and can be normalized and then completed
uniquely to an (axially symmetric stationary)
orthonormal spacetime frame or dual frame. Normalizing the spatial
coordinate derivatives leads
to the slicing orthonormal frame $\{n,e_{\hat a}\}$
with dual frame $\{\omega^\bot,\theta^{\hat a}\}$
while normalizing the spatial coordinate differentials leads to the
threading orthonormal frame $\{m,\epsilon_{\hat a}\}$
with dual frame $\{\omega^\top, \omega^{\hat a}\}$.
One can boost each
of these two orthonormal frames uniquely to align them with the 4-velocity
of an arbitrary gyro worldline
\beq \eqalign{
& B(u,n) \{ n,e_{\hat a} \} = \{ u,E_{(sl)a} \}\ ,\cr
& B(u,m) \{ m,\epsilon_{\hat a} \} =
B(u,m) B(m,n) \{ n,e_{\hat a} \} = \{ u,E_{(th)a} \}\ .\cr
}\eeq
The two orthonormal frames so obtained are related to each other
by the time-independent Thomas rotation determined by the composition
of the two boosts $B(u,m)$ and $B(m,n)$. Thus either one may be
used to calculate the angular velocity of the gyro spin vector with
the same result. This result describes the precession of the gyro
with respect to the orthogonal
spatial computational frame vector fields seen by the gyro, since
modulo normalization the boost is equivalent to spatial projection.
Once aberration of starlight is taken into account, this may be used
to measure the precession of the gyro in its own rest frame relative
to the ``distant stars", while the splitting discussion describes
the measurement of the spin of the moving gyro with respect to the
changing observers along its worldline.
A similar situation exists in the PPN theory, where the PPN spatial
coordinates are orthogonal to the lowest nontrivial order and
hence the previous discussion holds. The derivation of the gyro precession
within the PPN theory is described in detail in Misner, Thorne
and Wheeler \Cite{1973} using the threading point of view choice of orthonormal
frame tied to the PPN coordinate grid. (Schiff \Cite{1960}
omits this discussion for a rotating body, quoting only the result.)
Thorne \Cite{1989} has given a very physical explanation for each of the
terms in the expression for the precession angular velocity
within linearized GR, showing how each is associated with either
the gyro acceleration, the ``spatial curvature", an ``induced gravitomagnetic
field" effect, and the actual gravitomagnetic field.