International Journal of Modern Physics D, Vol. 6, No. 1 (1997) 1–38 [reformatted 2001]
❢
c World Scientific Publishing Company
THE INTRINSIC DERIVATIVE AND CENTRIFUGAL FORCES
IN GENERAL RELATIVITY: I. THEORETICAL FOUNDATIONS
DONATO BINI
Istituto per Applicazioni della Matematica C.N.R., I–80131 Napoli, Italy and
International Center for Relativistic Astrophysics, University of Rome, I–00185 Roma, Italy
PAOLO CARINI
∗
GP-B, Hansen Labs, Stanford University, Stanford, CA 94305, USA and
International Center for Relativistic Astrophysics, University of Rome, I–00185 Roma, Italy
ROBERT T. JANTZEN
Department of Mathematical Sciences, Villanova University, Villanova, PA 19085, USA and
International Center for Relativistic Astrophysics, University of Rome, I–00185 Roma, Italy
Received 17 December 1996
Everyday experience with centrifugal forces has always guided thinking on the close rela-
tionship between gravitational forces and accelerated systems of reference. Once spatial
gravitational forces and accelerations are introduced into general relativity through a
splitting of spacetime into space-plus-time associated with a family of test observers,
one may further split the local rest space of those observers with respect to the direction
of relative motion of a test particle world line in order to define longitudinal and trans-
verse accelerations as well. The intrinsic covariant derivative (induced connection) along
such a world line is the appropriate mathematical tool to analyze this problem, and by
modifying this operator to correspond to the observer measurements, one understands
more clearly the work of Abramowicz et al who define an “optical centrifugal force” in
static axisymmetric spacetimes and attempt to generalize it and other inertial forces to
arbitrary spacetimes. In a companion article the application of this framework to some
familiar stationary axisymmetric spacetimes helps give a more intuitive picture of their
rotational features including spin precession effects, and puts related work of de Felice
and others on circular orbits in black hole spacetimes into a more general context.
Keywords
: gravitoelectromagnetism–inertial forces
1.
Introduction
Ever since Einstein unified space and time into spacetime, people have been
trying to break them apart again. Spacetime splittings play an important role in
many aspects of gravitational theory, not only in helping interpret 4-dimensional
geometry in terms of our more familiar space-plus-time perspective, but also in
mathematical analysis of various problems of the theory. Although there are many
variations on the idea of reintroducing space and time, all share a common founda-
tion of introducing a family of test observers in spacetime who measure spacetime
∗
Present Address: Physics Department, Amherst College, Amherst, MA 01002, USA.
1
2
D. Bini, P. Carini and R.T. Jantzen
quantities mathematically by the orthogonal decomposition of the tangent spaces
to the spacetime manifold into their local rest spaces and local time directions.
1, 2
Such a construction leads to a “reference frame” or “reference system” or other
variations of this terminology whose effect in general is to contribute inertial forces
to the spatialforce equation due to the motion of the famil
y of test observers.
Although one can agree on these concepts in simple nonrelativistic situations, the
richness of general relativity and of the geometry of spacetime allows many varia-
tions of their possible generalizations to the latter theory. There is not necessarily
a single “correct” generalization of any given concept, but simply different ways of
measuring different quantities, some of which may be more usefulthan others.
Because of the equivalence principle, inertial forces have always entrigued people
in connection with gravitationaltheory. Indeed the concept of centrifugalforce is
usefulin nonrelativistic mechanics, and in recent years Abramowicz et al
3–9
have
shown that a certain generalization of this concept can give a nice physical interpre-
tation to certain properties of strong static gravitational fields in general relativity,
although attempts to extend it first to stationary and then to arbitrary spacetimes
have been somewhat problematic.
10–17
Here we place that work and related studies
of de Felice and others
18–24
in the more generalcontext of gravitoelectromagnetism,
the framework which encompasses all the various splitting approaches to general rel-
ativity and provides a clean description of the possible choices of curved spacetime
generalizations of centripetal acceleration and centrifugal and Coriolis forces. This
is done not to exaggerate the importance of splitting spacetime but to help clarify
the link between our three-dimensional world view and nonrelativistic common ex-
perience on the one hand and the interpretation of concepts related to rotation and
acceleration in general relativity on the other.
2.
The nonrelativistic background
Before exploring the spacetime picture, it is worth recalling the classic example
from nonrelativistic mechanics of a rigidly rotating Cartesian coordinate system and
its relation to the ideas of centrifugalforce and centripetalacceleration. It is here
that all our intuition for these concepts has its roots.
Let
x
i
=
R
i
j
(
t
)¯
x
j
be the coordinate transformation between “space-fixed” or-
thonormalcoordinates
{
¯
x
i
}
and rotating “body-fixed” such coordinates
{
x
i
}
with
a common origin in Euclidean space, borrowing from the usual terminology of the
rigid body problem in classical mechanics. The body-fixed components of the an-
gular velocity of the rotating system are defined by
˙
R
i
k
R
−
1
k
j
=
δ
ik
kjm
Ω
m
.
(2.1)
Both these and the space-fixed components ¯
Ω
i
=
R
−
1
i
j
Ω
j
are constant in the case
of a rotation about a fixed axis with constant angular velocity about that axis.
If one evaluates the first and second time derivatives of the coordinates of a
trajectory, one easily finds
˙¯
x
i
=
R
−
1
i
j
[
V
j
+ ˙
x
j
]
,
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
3
¨
¯
x
i
=
R
−
1
i
j
[
A
j
+ 2(Ω
×
˙
x
)
j
+ ¨
x
j
]
,
(2.2)
where
V
i
= (Ω
×
x
)
i
=
δ
ij
jkm
Ω
k
x
m
is the relative velocity field of the body-fixed
points relative to the space-fixed points, and
A
i
= (Ω
×
(Ω
×
x
))
i
+ ( ˙
Ω
×
x
)
i
is the
relative acceleration field, both evaluated along the trajectory.
If ¨
¯
x
i
= ¯
F
i
is the space-fixed force per unit mass equation of motion for a point
particle of mass
m
moving under the influence of a force
m
¯
F
i
, then the acceleration
relation above may be rewritten in the form
¨
x
i
=
F
i
+
g
i
+ ( ˙
x
×
H
)
i
,
(2.3)
by moving the additionalterms due to the time dependence of the transformation
to the other side of the equation where they appear as “inertialforces” due to the
motion of the body-fixed points to which the coordinate system is attached. One
has a “gravitoelectric-like” force
g
i
=
−
A
i
due to the acceleration of those points,
and the remaining “gravitomagnetic-like” force, the Coriolis force, which is due
to the changing orientation of the body-fixed axes, which in turn is a manifesta-
tion of the relative motion of the body-fixed points. The latter force involves a
“gravitomagnetic-like field”
H
i
= 2Ω
i
. The “gravitoelectric-like” force consists of
the centrifugalforce directed away from the axis of rotation and an additionalforce
due to the changing angular velocity.
In the more familiar case of a time-independent angular velocity, these two
noninertial force fields admit a scalar and a vector potential respectively.
The
scalar potential is just half the square of the magnitude of the relative velocity field
Φ
=
1
2
δ
ij
V
i
V
j
,
g
i
=
−
δ
ij
grad
j
Φ
,
(2.4)
while this field itself serves as the vector potential
H
i
= (curl
V
)
i
.
(2.5)
Since each body-fixed point is undergoing circular motion at constant velocity, the
relative acceleration field is exactly the familiar centripetal acceleration associated
with this simple motion, so the centrifugal force is just the sign-reversal of this
centripetal acceleration. The Coriolis force can be interpreted as due to the local
vorticity of the flow of the body-fixed points in space (half the curl of the velocity
field), which is equal to the global constant angular velocity vector Ω
i
.
3.
The Spacetime Setting
The great simplication of this discussion compared to a corresponding one in
terms of a geometric splitting of spacetime is the common Newtonian time used
by both systems of spatialcoordinates. In a spacetime discussion, one must also
take into account the change in the local time direction, which complicates matters,
4
D. Bini, P. Carini and R.T. Jantzen
especially the relationship between a sign-reversed centripetal acceleration and a
corresponding centrifugalforce. One also must re-interpret the time derivative in a
way which makes geometric sense, and there are a number of distinct ways of doing
this, depending on how changes in fields are measured along general world lines.
Consider only the case of time-independent angular velocity. By adding the
additionaltime coordinate transformation
t
= ¯
t
(3.1)
to the originalspatialcoordinate transformation, one obtains a transformation from
inertialcoordinates
{
¯
t,
¯
x
i
}
in Minkowski spacetime to noninertialcoordinates
{
t, x
i
}
which may be interpreted using the spacetime geometry. The time lines of each such
coordinate system sharing the same time coordinate hypersurfaces in Minkowski
spacetime may be interpreted as the world lines of a family of test observers (when
timelike), representing the trajectories of the space-fixed and body-fixed points.
The first set are inertial (zero acceleration) observers with zero relative velocity,
whose world lines are the orthogonal trajectories to the family of time hypersurfaces,
while the second are noninertial (accelerated) observers in relative motion and not
admitting any orthogonal family of hypersurfaces. Both families of world lines are
the flow lines of Killing vector fields of Minkowski spacetime (therefore having zero
expansion tensor), the first with zero vorticity and the second with nonzero vorticity.
In this description, it is only the families of time lines and time hypersurfaces
which play a central role, not the specific choice of spatial coordinates which para-
metrize the families of time lines. It is in fact useful to introduce new spatial
coordinates adapted to the orbits of the body-fixed points, and their corresponding
rotating system, namely nonrotating and rotating cylindrical coordinates adapted
to the axis of the rotation. For example, if one chooses the axis of rotation so that
¯
Ω
i
= Ω
i
= Ω
δ
i
3
, one can choose the usualcylindricalcoordinates
{
¯
ρ,
¯
φ,
¯
z
}
in place of
{
¯
x
i
}
, and rotating cylindrical coordinates
{
ρ, φ, z
}
which differ only by
φ
= ¯
φ
−
Ω¯
t
.
However, a description in terms of quantities measured by each family of test ob-
servers involves their local proper times, which in the second case are not associated
with any global time function. Global time functions not directly measuring ob-
server proper time occur in the context of “observer-adapted” coordinate systems.
A system of coordinates can be adapted to the observer congruence in one of two
ways.
1
For a generalcongruence with nonzero vorticity, the appropriate coordinates
are comoving, leading to an approach called the threading point of view, while for
a vorticity-free congruence, one can use more generalcoordinates adapted to the
family of orthogonal hypersurfaces admitted by the congruence, leading to an ap-
proach called the slicing point of view. In each case, the adapted local coordinates
{
t, x
i
}
lead to the introduction of explicit potentials for the various gravitational
forces.
A slicing together with a transversal threading (congruence) describes exactly
the structure on which each of these two points of view is built, here called a
“nonlinear reference frame.” In the flat Minkowski space example with inertial or
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
5
rotating observers, one has the geodesically parallel slicing orthogonal to the paths
of the originalinertialobservers, together with the rotating observers which provide
a new threading of this slicing (when those observers are defined). Rotating Carte-
sian or cylindrical coordinates on Minkowski spacetime are examples of coordinates
adapted to the nonlinear reference frame associated with this slicing and threading.
In the threading point of view the threading congruence serves both as the observer
congruence as well as the curves along which evolution is measured, while in the
slicing point of view these two roles are separated: the orthogonal trajectories to
the slicing form the observer congruence, while the measured quantities are evolved
along the distinct threading congruence.
This is a nonlinear reference frame adapted to the stationary axisymmetry of the
flat Minkowski spacetime. A similar geometrically privileged nonlinear reference
frame exists in any stationary axisymmetric spacetime, in particular in the Kerr
black hole spacetimes (with the Schwarzschild black hole as their static limit) and
in the G¨
odelspacetime. For such spacetimes with some of the same symmetries
which characterize the uniformly rotating observers in Minkowski spacetime, one
might hope to define generalized centrifugal and Coriolis forces, but without these
symmetries one must rethink the point of departure.
This is not the whole story about centrifugal force in classical mechanics, since it
makes its appearance in at least two other familiar contexts. As discussed in detail
by Abramowicz,
5
one is the train, plane, car context in which one has an acceler-
ated platform to which a local reference frame is attached, essentially the previous
problem with additional motion of the origin of coordinates. Any point fixed in
this local platform will then experience accelerations tangential and transverse to
its direction of motion, and the transverse acceleration can be interpreted in terms
of a centrifugalforce in the localreference frame due to its instantaneous rotation
about the center of the osculating circle associated with the curvature of its path. If
the point is in motion with respect to the local platform, Coriolis effects are felt as
well, but the details are more complicated than the simpler rigid body discussion.
A third context in which the centrifugal force is usually introduced is in the
discussion of motion in a centralpotential.
5
Here one introduces a polar coordinate
system in the plane of the motion and then expresses the equation of motion in
that coordinate system. The radialcomponent of this equation for generalmotion
then contains what is interpreted as a centrifugalforce term due to the curvature
of the circular angular coordinate lines. This term is just the sign-reversal of the
centripetal acceleration for motion confined to these coordinate lines and enters the
equation of motion through a Christoffelsymbolterm associated with this curva-
ture, quadratic in the angular speed. Although there is no rotating frame in this
discussion, one may be introduced by letting the new system rotate about the cen-
ter of force so that the particle in motion has a fixed new angular coordinate. In
this rotating frame, the same centrifugalforce term is then realized as in the rigid
body discussion as a rotating frame effect, but rotating about the center of force,
not the instantaneous radialdirection associated with the curvature of the particle
6
D. Bini, P. Carini and R.T. Jantzen
path. For circular motion, this centrifugal force is just the sign-reversal of the cen-
tripetal acceleration of the particle path, but for general noncircular motion, the
two quantities are not simply linked.
In other words, the “fictitious” centrifugalforce is a convenience that only has
meaning with respect to some implied reference frame, and in the same problem can
play different roles depending on which frame is chosen. For circular trajectories,
allof these various aspects of centrifugalforce come together, presenting the most
usefulapplication of the concept.
In general relativity the best hope of having a useful generalization of centrifugal
force lies in the static axisymmetric case with circular trajectories. Consideration
of noncircular trajectories in that case or relaxing the symmetry even to stationar-
ity already introduces difficulties which make the whole discussion rather unclear.
However, the idea of a relative centripetal acceleration viewed by an observer, being
well defined in a single reference frame, sidesteps the questions involving two dif-
ferent reference frames that seem to be tied up with various aspects of centrifugal
force and it is therefore reasonable to generalize it to an arbitrary spacetime.
The key difference between the nonrelativistic description of inertial forces and
the framework of general relativity is that in the latter context, the effects of a
gravitationalfield due to the presence of matter are intertwined with those of the
accelerated motion of the field of observers used to establish a given reference frame
in order to “measure” the gravitationalforces. Only under specialsymmetry con-
ditions does it seem to make sense to try to separate the two. Moreover, focusing
attention on a single test particle world line without referring it to an independent
family of test observers (which are not adapted to the particular worldline) reduces
the usefulness of a space-plus-time interpretation of the motion, except possibly in
reference to the spacetime Frenet-Serret frame which is completely determined by
the worl d l ine al one.
25–28
4.
A family of test observers
A splitting of a general spacetime equipped with a Lorentzian metric
g
αβ
(signa-
ture
-+++
) and the covariant derivative
∇
α
associated with its symmetric connection
is accomplished locally by specifying a future-pointing unit timelike vector field
u
α
(
u
α
u
α
=
−
1). This field may be interpreted as the four-velocity of a family of
test observers whose proper time parametrized world lines (let
τ
u
denote such a
parameter on each world line) are integral curves of
u
α
.
The orthogonaldecomposition of each tangent space into the localrest space
and local time direction of the observer extends to all the tensor spaces above
it and to the algebra of spacetime tensor fields, and may be referred to as the
measurement process associated with the family of test observers. Tensors or tensor
fields which have no component along
u
α
are called spatial (with respect to
u
α
). The
fully covariant and contravariant forms of the spatial projection tensor
P
(
u
)
α
β
=
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
7
δ
α
β
+
u
α
u
β
are referred to as the corresponding forms of the spatialmetric
P
(
u
)
αβ
=
g
αβ
+
u
α
u
β
.
(4.1)
Similarly let
η
(
u
)
αβδ
=
u
δ
η
δαβδ
be the spatialunit antisymmetric tensor associated
with the spatialmetric, where
η
0123
= 1 =
η
(
u
)
123
in a time-oriented, oriented
orthonormalframe having
u
α
as its first element. This tensor may be used to define
a spatial duality operation for antisymmetric spatial tensor fields in an obvious way.
As described in,
1
one may also spatially project various derivative operators
so that the result of the derivative of any tensor field is always spatial; such
derivatives naturally occur in expressing tensor equations in space-plus-time form.
Two usefulspatialderivatives are the spatialLie derivative
£
(
u
)
X
=
P
(
u
)
£
X
,
for any spatialvector fiel
d
X
α
, and the spatialcovariant derivative
∇
(
u
)
α
=
P
(
u
)
P
(
u
)
β
α
∇
β
, where the projection on all free indices is implied after the ap-
plication of the derivative.
Similarly three useful temporal derivatives are the
temporalLie derivative
∇
(lie)
(
u
) =
P
(
u
)
£
u
, the Fermi-Walker temporal deriv-
ative
∇
(fw)
(
u
) =
P
(
u
)
u
α
∇
α
, and the corotating Fermi-Walker temporal derivative
∇
(cfw)
(
u
) related to the first two by the kinematical fields (acceleration
a
(
u
)
α
, vor-
ticity
ω
(
u
)
α
β
, expansion
θ
(
u
)
α
β
) of the observer congruence,
a
(
u
)
α
=
∇
(fw)
(
u
)
u
α
,
ω
(
u
)
αβ
=
P
(
u
)
γ
α
P
(
u
)
δ
β
∇
(
u
)
[
γ
u
δ
]
,
θ
(
u
)
αβ
=
P
(
u
)
γ
α
P
(
u
)
δ
β
∇
(
u
)
(
γ
u
δ
)
=
1
2
∇
(lie)
(
u
)
g
αβ
=
1
2
∇
(lie)
(
u
)
P
(
u
)
αβ
,
(4.2)
through the relations
∇
(cfw)
(
u
)
X
α
=
∇
(fw)
(
u
)
X
α
+
ω
(
u
)
α
β
X
β
=
∇
(lie)
(
u
)
X
α
+
θ
(
u
)
α
β
X
β
(4.3)
valid only for a spatial vector field
X
α
, but easily extended to any spatial tensor field
in the usualway. It is also convenient to use extend the notation
£
(
u
)
X
=
P
(
u
)
£
X
to any vector field
X
α
, spatial or not. All indices are spatially projected when these
spatialdifferentialoperators are applied to a tensor field. The geometricalmeaning
of these derivatives is discussed in.
1
For spatial fields, the Fermi-Walker temporal
derivative coincides with the spacetime Fermi-Walker derivative along the observer
congruence, explaining the terminology.
Note that the spatialcovariant derivative and the ordinary and corotating Fermi-
Walker temporal derivatives of the spatial metric are all zero, so index-shifting of
spatialfields commutes with these derivatives. The Lie temporalderivative of the
spatialmetric instead equals twice the expansion tensor of the observer congruence,
so index shifting of a spatialtensor being differentiated by this operator leads to
additionalterms involving this tensor.
It is usefulto introduce a vorticity or rotation vector fiel
d using the spatial
duality operation
ω
(
u
)
α
=
1
2
η
(
u
)
αβγ
ω
(
u
)
βγ
.
(4.4)
8
D. Bini, P. Carini and R.T. Jantzen
This in turn may be used to rewrite the contraction of the vorticity tensor with a
spatialvector field as a spatialcross-product
ω
(
u
)
α
β
X
β
=
−
η
(
u
)
α
βγ
ω
(
u
)
β
X
γ
=
−
[
ω
(
u
)
×
u
X
]
α
.
(4.5)
Finally the shear tensor is defined as the spatial tracefree part of the expansion
tensor
σ
(
u
)
αβ
=
θ
(
u
)
αβ
−
1
3
θ
(
u
)
γ
γ
P
(
u
)
αβ
.
(4.6)
5.
Measuring the intrinsic derivative along a parametrized curve
Given any parametrized curve in spacetime, with parameter
λ
and tangent
V
(
λ
)
α
, the spacetime connection induces a connection on the curve whose derivative
is called either the “intrinsic” or “absolute” derivative along it.
29
This derivative
D/dλ
is uniquely defined by the condition that if one extends a tensor smoothly
off the curve to a tensor field on spacetime, the action of this intrinsic derivative
on the tensor at a point on the curve equals the action of the covariant directional
derivative
V
(
λ
)
α
∇
α
on the extended tensor field at that point
DT
α...
β...
/dλ
=
V
(
λ
)
γ
∇
γ
T
α...
β...
.
(5.1)
In this equation it is important to note that its right hand side is understood to be
the value on the worldline of the derivative of the extended tensor field. The usual
sloppy notation of this equation which does not distinguish between the original
and the extended tensor field nor indicate its validity only on the curve itself must
always be understood in this context.
For a vector defined only along the curve, this leads to the usual formula in
terms of the ordinary parameter derivative of the components along the world line
plus the connection coefficient correctional terms
DX
α
/dλ
=
dX
α
/dλ
+ Γ
α
βγ
V
(
λ
)
β
X
γ
.
(5.2)
In a previous article
1
this derivative has been referred to as the totalcovariant
derivative along
V
(
λ
)
α
or along the parametrized curve.
If one performs an orthogonalprojection of the intrinsic derivative along such a
parametrized curve in the measurement process associated with the family of test
observers, one is led to a new derivative operator along that curve which is natural to
call either the Fermi-Walker spatial intrinsic derivative or the Fermi-Walker total
spatialcovariant derivative. For example, if
X
α
is any vector defined along the
curve, this derivative is defined by
D
(fw)
(
V
(
λ
)
, u
)
X
α
/dλ
=
P
(
u
)
α
β
DX
β
/dλ .
(5.3)
If one extends
X
α
to a smooth vector field off the curve, then for this extended
vector field, expressing this derivative in terms of the measurement of the tangent
vector itself
V
(
λ
)
α
=
V
(
λ
)
(
||
u
)
u
α
+ [
P
(
u
)
V
(
λ
)]
α
(5.4)
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
9
leads to the natural pairing of its temporal part
V
(
λ
)
(
||
u
)
=
−
u
γ
V
(
λ
)
γ
and spa-
tialpart [
P
(
u
)
V
(
λ
)]
α
=
P
(
u
)
α
γ
V
(
λ
)
γ
with the corresponding projections of the
covariant derivative acting on the smooth extension
D
(fw)
(
V
(
λ
)
, u
)
X
α
/dλ
=
{
V
(
λ
)
(
||
u
)
∇
(fw)
(
u
) + [
P
(
u
)
V
(
λ
)]
β
∇
(
u
)
β
}
X
α
,
(5.5)
where the individualterms on the right hand side depend on how the extension is
made. These individualterms have no meaning without such an extension, a fact
which has caused some confusion in attempts to generalize centrifugal and other
noninertialforces to curved spacetimes.
12–15
By replacing the Fermi-Walker temporal derivative in the expression for the
Fermi-Walker intrinsic derivative acting on the extended tensor field with the Lie
temporalderivative or the corotating Fermi-Walker temporalderivative respectively,
one defines the corresponding totalspatialcovariant derivatives of a spatialvector
field
D
(tem)
(
V
(
λ
)
, u
)
X
α
/dλ
=
{
V
(
λ
)
(
||
u
)
∇
(tem)
(
u
) + [
P
(
u
)
V
(
λ
)]
β
∇
(
u
)
β
}
X
α
tem=fw
,
cfw
,
lie
,
(5.6)
where again the righthand side expression only makes sense for an extended vector
field, but leads to well-defined derivatives for a spatial vector defined only on the
parametrized curve.
These three derivatives of spatial vectors along the world line differ among them-
selves only by a linear transformation of the local rest space
D
(cfw)
(
V
(
λ
)
, u
)
X
α
/dλ
=
D
(fw)
(
V
(
λ
)
, u
)
X
α
/dλ
+
V
(
λ
)
(
||
u
)
ω
(
u
)
α
β
X
α
=
D
(lie)
(
V
(
λ
)
, u
)
X
α
/dλ
+
V
(
λ
)
(
||
u
)
θ
(
u
)
α
β
X
α
,
(5.7)
expressions which may be used together with Eq. (5.3) to define the corotating and
Lie such derivatives in terms of the Fermi-Walker one when acting on spatial fields.
The ordinary and corotating Fermi-Walker totalspatialcovariant derivatives of the
spatialmetric vanish so they commute with index shifting on spatialfields, and one
may introduce spatialorthonormaltriads along the curve for which one of these
derivatives vanishes. These are natural to call “relative Fermi-Walker” or “relative
corotating Fermi-Walker” propagated spatial frames along the parametrized curve.
The Lie such derivative of the spatialmetric vanishes only if the expansion tensor
vanishes, in which case it coincides with the corotating Fermi-Walker total spatial
covariant derivative. Thus in generalthe “relative Lie” propagated spatialframes
along the parametrized curve will not remain orthonormal if initially so when the
expansion tensor is nonzero.
The three different choices of derivative correspond to the three possible ways
of evolving spatial frames into the future, the first two of which preserve inner
products. They each measure differences with respect to the associated relative
propagated spatial frames along the given curve. The relationship between the
relative Lie transport and the relative Fermi-Walker transport along the observer
10
D. Bini, P. Carini and R.T. Jantzen
congruence itself leads to the physical intepretation of the rotation, expansion, and
shear of that congruence.
6.
Reparametrization of a parametrized curve
Two new parametrizations may be introduced for any parametrized curve by
solving the following ordinary differential equations
dτ
(
V
(
λ
)
,u
)
/dλ
=
V
(
λ
)
(
||
u
)
,
d
(
V
(
λ
)
,u
)
/dλ
=
||
P
(
u
)
V
(
λ
)
||
,
(6.1)
(using the notation
||
Y
||
=
|
Y
α
Y
α
|
1
/
2
) corresponding to the limiting sequence of
temporaland spatialarclength differentials seen by the test observers whose paths
are crossed by the curve. The solutions lead to valid reparametrizations as long as
the right hand side of the differentialequation does not vanish, so that an invertible
relationship exists between the old and new parametrizations. When one of the
two right hand sides vanishes identically, the parameter becomes a proper interval
parameter (proper distance orthogonal to the observer family or proper time along
it respectively).
The derivatives of these new parameters are in turn related to the derivatives of
the spacetime intervalby the usualrelation
[
ds/dλ
]
2
=
−
[
dτ
(
V
(
λ
)
,u
)
/dλ
]
2
+ [
d
(
V
(
λ
)
,u
)
/dλ
]
2
,
(6.2)
while their quotient up to sign defines the relative speed of the curve as seen by the
observer family
±
ν
(
V
(
λ
)
, u
) =
d
(
V
(
λ
)
,u
)
/dτ
(
V
(
λ
)
,u
)
=
||
P
(
u
)
V
(
λ
)
||
/V
(
λ
)
(
||
u
)
.
(6.3)
The relative velocity itself and (when nonzero) the unit vector defining its direction
are themselves defined by
ν
(
V
(
λ
)
, u
)
α
= [
P
(
u
)
V
(
λ
)]
α
/V
(
λ
)
(
||
u
)
,
(6.4)
and
ˆ
ν
(
V
(
λ
)
, u
)
α
= [
P
(
u
)
V
(
λ
)]
α
/
||
P
(
u
)
V
(
λ
)
||
.
(6.5)
The totalspatialcovariant derivatives along the parametrized curve may be re-
expressed in terms of the new parametrizations by the chain rule
D/dλ
= [
dλ/dλ
]
D/dλ
= [
dλ
/dλ
]
−
1
D/dλ ,
(6.6)
where
D
here stands for any of the intrinsic derivative operators and
λ
for either
new parametrization.
Since the Fermi-Walker and corotating Fermi-Walker such derivatives respect
spatial orthogonality, one may use them to introduce a relative spatial Frenet-Serret
frame of each type along the parametrized curve, using the relative spatial arclength
parametrization to generalize the usual objects on a Riemannian 3-manifold. The
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
11
spatialunit vector ˆ
ν
(
V
(
λ
)
, u
)
α
plays the role of the “relative unit tangent,” and
the direction and length of its derivative with respect to the relative spatial ar-
clength yields the “relative unit normal” and “relative spatial curvature” of each
type, while the spatial cross-product of the relative unit tangent and normal defines
the “relative unit bi-normal,” whose derivative in turn leads to the “relative tor-
sion.” For a stationary spacetime with test observers following the trajectories of
timelike Killing vector field, the relative Frenet-Serret structure for the corotating
case corresponds to the usualsuch structure on the observer quotient space with the
projected Riemannian metric, suggesting that it is a generalization worth consider-
ing. Such a relative spatial Frenet-Serret structure should be clearly distinguished
from the spacetime Frenet-Serret structure for the curve.
25–28
7.
Measuring the intrinsic derivative along a test particle world line
Suppose one considers the world line of a nonzero rest mass test particle in
spacetime parametrized by the particle’s proper time
τ
U
, letting
U
α
denote its unit
timelike tangent vector, the four-velocity of the test particle. This vector may be
split using the orthogonal decomposition associated with the family of test observers
with four-velocity
u
α
U
α
=
γ
(
U, u
)[
u
α
+
ν
(
U, u
)
α
] =
E
(
U, u
)
u
α
+
p
(
U, u
)
α
.
(7.1)
Here the spatialvector
ν
(
U, u
)
α
is the relative velocity of
U
α
with respect to
u
α
,
and
γ
(
U, u
) = [1
−
ν
(
U, u
)
2
]
−
1
/
2
is its associated gamma factor, while
ν
(
U, u
) =
[
ν
(
U, u
)
α
ν
(
U, u
)
α
]
1
/
2
is the relative speed. Similarly
p
(
U, u
)
α
=
γ
(
U, u
)
ν
(
U, u
)
α
is
the three-momentum (per unit mass) observed by the test observers, with magnitude
p
(
U, u
), while
E
(
U, u
) =
γ
(
U, u
) is the energy (per unit mass) as seen by the test
observers. (The tilde notation of
1
used for per unit mass quantities will be dropped
for simplicity.) Either set of quantities satisfies the identity
γ
(
U, u
)
2
=
γ
(
U, u
)
2
ν
(
U, u
)
2
+ 1
,
E
(
U, u
)
2
=
p
(
U, u
)
2
+ 1
,
(7.2)
imposed by the unit nature of
u
α
. Finally the two new relative parametrizations of
the world line are here defined by
dτ
(
U,u
)
/dτ
U
=
γ
(
U, u
)
,
d
(
U,u
)
/dτ
U
=
γ
(
U, u
)
ν
(
U, u
)
,
(7.3)
with
d
(
U,u
)
/dτ
(
U,u
)
=
ν
(
U, u
)
.
(7.4)
Adopting the observer proper time parametrization, the spatialprojection of
the total covariant derivative along the world line
D/dτ
(
U,u
)
=
γ
(
U, u
)
−
1
D/dτ
U
defines the Fermi-Walker totalspatialcovariant derivative, which together with its
two spatial generalizations can be expressed in the following way for a spatial vector
field
X
α
defined along the world line and which has been extended off the world
line for the right hand side to make sense
D
(tem)
(
U, u
)
X
α
/dτ
(
U,u
)
= [
∇
(tem)
(
u
) +
ν
(
U, u
)
β
∇
(
u
)
β
]
X
α
,
tem=fw
,
cfw
,
lie
.
(7.5)
12
D. Bini, P. Carini and R.T. Jantzen
These three derivatives of spatial vectors along the world line differ among them-
selves only by a linear transformation of the local rest space
D
(cfw)
(
U, u
)
X
α
/dτ
(
U,u
)
=
D
(fw)
(
U, u
)
X
α
/dτ
(
U,u
)
+
ω
(
u
)
α
β
X
α
=
D
(lie)
(
U, u
)
X
α
/dτ
(
U,u
)
+
θ
(
u
)
α
β
X
α
,
(7.6)
expressions which may be used to define the Lie and corotating Fermi-Walker such
derivatives in terms of the Fermi-Walker one when acting on spatial fields.
8.
Relative acceleration: longitudinal and transverse parts
Applying these derivatives to the relative velocity itself leads to a relative accel-
eration vector for each one
a
(tem)
(
U, u
)
α
=
D
(tem)
ν
(
U, u
)
α
/dτ
(
U,u
)
,
(8.1)
differing from each other in the same way as the above three derivatives
a
(cfw)
(
U, u
)
α
=
a
(fw)
(
U, u
)
α
+ [
ω
(
u
)
×
u
ν
(
U, u
)]
α
=
a
(lie)
(
U, u
)
α
+
θ
(
u
)
α
β
ν
(
U, u
)
β
.
(8.2)
The first of these equations just reflects the relative rotation of the relative Fermi-
Walker and corotating Fermi-Walker transported axes along the world line. Apart
from a gamma factor, the rate of change of the spatialmomentum is related to the
relative acceleration by an additional term along the direction of relative motion
involving the rate of change of the energy (per unit mass)
E
(
U, u
) =
γ
(
U, u
) of the
test particle
D
(tem)
(
U, u
)
p
(
U, u
)
α
/dτ
(
U,u
)
=
ν
(
U, u
)
α
d
ln
γ
(
U, u
)
/dτ
(
U,u
)
+
γ
(
U, u
)
a
(tem)
(
U, u
)
α
.
(8.3)
One may further decompose both the relative acceleration and the observed
rate of change of spatialmomentum into longitudinaland transverse components
with respect to the observed motion of the test particle using the relative motion
projectors
1, 30
P
u
(
U, u
)
(
||
)
α
β
= ˆ
ν
(
U, u
)
α
ˆ
ν
(
U, u
)
β
,
P
u
(
U, u
)
(
⊥
)
α
β
=
P
(
u
)
α
β
−
P
u
(
U, u
)
(
||
)
α
β
.
(8.4)
For the ordinary and corotating Fermi-Walker cases, this decomposition is equiva-
lent to the terms arising from the product rule when these quantities are represented
as the scalar product of their magnitude and direction
ν
(
U, u
)
α
=
ν
(
U, u
)ˆ
ν
(
U, u
)
α
,
p
(
U, u
)
α
=
p
(
U, u
)ˆ
ν
(
U, u
)
α
,
(8.5)
where
p
(
U, u
) =
γ
(
U, u
)
ν
(
U, u
). Consider first the relative accelerations, which
decompose into two terms
a
(tem)
(
U, u
)
α
=
ˆ
ν
(
U, u
)
α
dν
(
U, u
)
/dτ
(
U,u
)
+
ν
(
U, u
)
D
(tem)
(
U, u
)ˆ
ν
(
U, u
)
α
/dτ
(
U,u
)
=
a
(
||
)
(tem)
(
U, u
)
α
+
a
(
⊥
)
(tem)
(
U, u
)
α
,
tem=fw
,
cfw
,
(8.6)
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
13
which define respectively their components parallel (“tangential” to the observed
orbit, or longitudinal) and perpendicular (“normal” or transverse) to the relative
direction of motion, most naturally called the longitudinal and transverse relative
accelerations, conforming to traditional names for these quantities.
For the Lie case the derivative of the unit relative velocity is not orthogonal to
the velocity vector
ˆ
ν
(
U, u
)
α
D
(lie)
(
U, u
)ˆ
ν
(
U, u
)
α
/d
(
U,u
)
=
−
2
θ
(
u
)
αβ
ˆ
ν
(
U, u
)
α
ˆ
ν
(
U, u
)
β
(8.7)
unless the expansion tensor of the observer congruence vanishes (in which case the
Lie and corotating Fermi-Walker derivatives of the various types agree) or unless the
relative motion is along a direction in which the observer expansion is zero. Thus
one must actually project this derivative in order to accomplish the “direction-of-
relative-motion” orthogonal decomposition.
The transverse relative acceleration for the ordinary and corotating Fermi-Wal-
ker cases
a
(
⊥
)
(tem)
(
U, u
)
α
=
ν
(
U, u
)
D
(tem)
(
U, u
)ˆ
ν
(
U, u
)
α
/dτ
(
U,u
)
=
ν
(
U, u
)
2
D
(tem)
(
U, u
)ˆ
ν
(
U, u
)
α
/d
(
U,u
)
,
(8.8)
where Eq. (7.4) has been used to re-parametrize the derivative of the unit velocity
vector, is exactly what one calls the centripetal acceleration in the case of the usual
inertialobservers in Minkowski spacetime, so it is naturalto callit the “relative
centripetalacceleration.”
One may next decompose the rate of change of spatialmomentum in the same
way
D
(tem)
(
U, u
)
p
(
U, u
)
α
/dτ
(
U,u
)
=
ˆ
ν
α
dp
(
U, u
)
/dτ
(
U,u
)
+
γ
(
U, u
)
ν
(
U, u
)
D
(tem)
(
U, u
)ˆ
ν
(
U, u
)
α
/dτ
(
U,u
)
=
ˆ
ν
α
dp
(
U, u
)
/dτ
(
U,u
)
+
γ
(
U, u
)
a
(
⊥
)
(tem)
(
U, u
)
α
,
(8.9)
where the second equality only holds for the ordinary and corotating Fermi-Walker
cases. The second term is proportional to the relative acceleration for those cases.
The first term is parallel to the direction of motion and itself contains both the
longitudinal relative acceleration scalar
dν
(
U, u
)
/dτ
(
U,u
)
as well as the effect of the
changing three-energy (per unit mass) of the test particle when the derivative is
expanded by the product rule.
Consider the derivatives of the unit velocity vector for the two Fermi-Walker
cases, which are related to each other by
D
(cfw)
(
U, u
)ˆ
ν
(
U, u
)
α
/d
(
U,u
)
=
D
(fw)
(
U, u
)ˆ
ν
(
U, u
)
α
/d
(
U,u
)
−
ν
(
U, u
)
−
1
[
ω
(
u
)
×
u
ˆ
ν
(
U, u
)]
α
.
(8.10)
If each were the uniquely defined intrinsic derivative with respect to the arclength
of the unit tangent to a curve in a three-dimensionalRiemannian manifold, the unit
14
D. Bini, P. Carini and R.T. Jantzen
vector
η
(tem)
(
U, u
)
α
specifying its direction (the relative unit normal) would be the
first normalto the curve and its magnitude
κ
(tem)
(
U, u
)
≥
0 (the relative curvature)
would be the curvature of that curve, the reciprocal of which would define a radius
of curvature (the relative radius of curvature)
ρ
(tem)
(
U, u
) = 1
/κ
(tem)
(
U, u
) when
the curvature is nonzero. This leads to the representation
D
(tem)
(
U, u
)ˆ
ν
(
U, u
)
α
/d
(
U,u
)
= 1
/ρ
(tem)
(
U, u
)
η
(tem)
(
U, u
)
α
(8.11)
of the unit velocity derivative and
a
(
⊥
)
(tem)
(
U, u
) =
ν
(
U, u
)
2
/ρ
(tem)
(
U, u
)
(8.12)
for the magnitude of the relative centripetal acceleration, which takes its familiar
form in terms of the relative radius of curvature. Eq. (8.2) shows that the two
accelerations differ by a term orthogonal to the relative direction of motion. As
noted above, in the stationary case these concepts reduce to the analogous quantities
in the Riemannian 3-manifold of the observer quotient space. For an arbitrary
spacetime these generalizations can be studied to understand the sense in which
they generalize the more familiar concepts, but for now they will be taken merely
as formaldefinitions.
For any test particle trajectory, the relative centripetal acceleration is zero when
the rel ative curvature
κ
(tem)
(
U, u
) vanishes, corresponding to the limit of infinite
radius of curvature. This enables one to define trajectories for which the curvature
vanishes identically as “relatively straight” with respect to the ordinary or corotat-
ing Fermi-Walker totalspatialcovariant derivative. Relative motion for which this
is true may be called the case of purely linear relative acceleration (for each type),
examined recently in the static case by Rindler and Mishra
31, 32
and the present
authors.
30
On the other hand, if the longitudinal relative acceleration vanishes,
as it does for the case of constant relative speed in the ordinary and corotating
cases, one has the case of purely transverse relative acceleration, the case stud-
ied extensively for circular orbits by Abramowicz et al in the static case
3–9
and by
Abramowicz and coworkers in the stationary Kerr spacetime and stationary axisym-
metric spacetimes.
10–14
De Felice has studied this same case for Schwarzschild and
Kerr without decomposing the 4-force and using the angular velocity relative to the
static (or distantly nonrotating) observers as his key variable.
18–22
Barrab`
es, Bois-
seau, and Israel
23
have done the same, but using the locally nonrotating observer
relative velocity as the key variable.
It is worth noting that in the case of orthogonality of the vorticity vector and
the relative velocity as occurs for the circular orbits the vanishing of the corotat-
ing Fermi-Walker relative curvature implies that the angular velocity-like quantity
ν
(
U, u
)
/ρ
(fw)
(
u
) of the center of relative curvature in the local rest space equals the
magnitude of the vorticity.
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
15
9.
Spatial gravitational forces
The spatialprojection of the four-acceleration
a
(
U
)
α
=
DU
α
/dτ
U
,
(9.1)
when rescaled to take into account the differences in proper times, is the apparent
three-acceleration as seen by the test observers
A
(
U, u
)
α
=
γ
(
U, u
)
−
1
P
(
u
)
α
β
DU
β
/dτ
U
=
D
(fw)
(
U, u
)[
γ
(
U, u
)
u
α
+
p
(
U, u
)
α
]
/dτ
(
U,u
)
=
D
(fw)
(
U, u
)
p
(
U, u
)
α
/dτ
(
U,u
)
−
F
(G)
(fw)
(
U, u
)
α
,
(9.2)
and it can be rewritten in terms of the other two totalspatialcovariant derivatives
in a single form
A
(
U, u
)
α
=
D
(tem)
(
U, u
)
p
(
U, u
)
α
/dτ
(
U,u
)
−
F
(G)
(tem)
(
U, u
)
α
(9.3)
where tem = fw
,
cfw
,
lie.
The spatialgravitationalforce (per unit mass)
F
(G)
(tem)
(
U, u
)
α
=
γ
(
U, u
)
D
(tem)
(
U, u
)
u
α
/dτ
(
U,u
)
=
γ
(
U, u
)[
g
(
u
)
α
+
H
(tem)
(
u
)
α
β
ν
(
U, u
)
β
]
,
(9.4)
is a Lorentz-like force determined by the gravitoelectric
g
(
u
)
α
and gravitomagnetic
H
(tem)
(
u
)
α
β
fields which in turn are simply related to the kinematical fields of the
observer congruence
g
(
u
)
α
=
−
a
(
u
)
α
,
H
(fw)
(
u
)
α
β
=
ω
(
u
)
α
β
−
θ
(
u
)
α
β
,
H
(cfw)
(
u
)
α
β
=
2
ω
(
u
)
α
β
−
θ
(
u
)
α
β
,
H
(lie)
(
u
)
α
β
=
2
ω
(
u
)
α
β
−
2
θ
(
u
)
α
β
.
(9.5)
It is useful to introduce a single gravitomagnetic vector field which determines the
antisymmetric part of all of the various gravitomagnetic tensor fields
H
(
u
)
α
= 2
ω
(
u
)
α
,
(9.6)
namely so that
2
ω
(
u
)
α
β
ν
(
U, u
)
β
= [
ν
(
U, u
)
×
u
H
(
u
)]
α
.
(9.7)
The symmetric part of the gravitomagnetic tensor field is just a multiple of the
expansion tensor for each case.
If the acceleration of the test particle equals a spacetime force
a
(
U
)
α
=
f
(
U
)
α
and one introduces the rescaled spatial projection
F
(
U, u
)
α
=
γ
(
U, u
)
−
1
P
(
u
)
α
β
f
(
U
)
β
,
(9.8)
16
D. Bini, P. Carini and R.T. Jantzen
then the spatialprojection of the force equation (“spatialequation of motion”) may
be written in the form
D
(tem)
(
U, u
)
p
(
U, u
)
α
/dτ
(
U,u
)
=
F
(G)
(tem)
(
U, u
)
α
+
F
(
U, u
)
α
.
(9.9)
The spatialgravitationalforce represents the combined inertialforces due to the
motion of the family of test observers. It arises in the same way as the noninertial
forces in nonrelativistic mechanics, namely as a part of the total acceleration which is
moved to the opposite side of the “acceleration equals force per unit mass” equation
with a sign change.
Space curvature effects are encoded in the totalspatialcovariant derivative it-
self. Suppose one considers a world line segment which starts and ends on a single
observer world line, and one transports a spatial vector along both paths (general
world line and observer world line) from the initial to the final point using the
transport associated with one of the three kinds of totalspatialcovariant deriva-
tives. In each case the two finalvectors willhave distinct directions, and in the
Lie case, different magnitudes in general, due to curvature effects associated with
the spatial metric, similar to the case of two such paths in the simpler case of a
fixed Riemannian three-manifold. For a timelike Killing vector field test observer
congruence in a stationary spacetime, where the Lie and corotating totalspatial
covariant derivatives coincide, the corotating Fermi-Walker space curvature effect
is exactly that due to the curvature of the natural projected Riemannian metric
on the quotient space of observer world lines. For circular orbits in a stationary
axisymmetric spacetime, this effect may be calculated explicitly using the tangent
cone to the embedding of the plane of the orbit, as discussed in appendix 1.A of
Arnold.
33
Since index shifting does not commute with the Lie totalspatialcovariant deriv-
ative, letting this derivative act instead on
p
(
U, u
)
α
leads to an additional expansion
term in the Lie spatialgravitationalforce. Conveniently introducing a “flattened”
Lie totalspatialcovariant derivative by
D
(lie
)
(
U, u
)
X
α
/dλ
=
P
(
u
)
αβ
D
(lie
)
(
U, u
)
X
β
/dλ ,
(9.10)
and a “flattened” Lie spatialgravitationalforce with a corresponding gravitomag-
netic field
H
(lie
)
(
u
)
α
β
= 2
ω
(
u
)
α
β
,
(9.11)
one has the analogous form of the force equation
D
(lie
)
(
U, u
)
p
(
U, u
)
α
/dτ
(
U,u
)
=
F
(G)
(lie
)
(
U, u
)
α
+
F
(
U, u
)
α
.
(9.12)
This notation facilitates the comparison of the different choices without requiring
index shifting.
10.
Massless test particles
Consider a massless test particle following a null path with affine parameter
λ
P
and tangent
P
α
locally expressible as
dx
α
/dλ
P
in terms of local coordinates.
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
17
Interpreting
P
α
as the 4-momentum directly, rather than the 4-momentum per
unit mass of the previous discussion for a massive test particle, one may essentially
make the substitution (
U
α
, τ
U
, γ
(
U, u
)
, f
α
, F
α
)
→
(
P
α
, λ
P
, E
(
P, u
)
, f
α
, F
α
) in that
discussion to reinterpret the results and formulas in the new context. Since the
speed now satisfies
ν
(
U, u
) =
d
(
U,u
)
/dτ
(
U,u
)
= 1
,
(10.1)
the relative velocity is a unit vector ˆ
ν
(
P, u
)
α
=
p
(
P, u
)
α
/E
(
P, u
), while the en-
ergy
E
(
P, u
) and magnitude of the spatialmomentum
p
(
P, u
)
α
are now related by
E
(
P, u
) =
p
(
P, u
). The spatialequation of motion then becomes simply
D
(tem)
(
U, u
)
p
(
U, u
)
α
/dτ
(
U,u
)
=
F
(G)
(tem)
(
U, u
)
α
+
F
(
P, u
)
α
,
(10.2)
where the spatialgravitationalforce is
F
(G)
(tem)
(
U, u
)
α
=
E
(
P, u
)[
g
(
u
)
α
+
H
(tem)
(
u
)
α
β
ν
(
U, u
)
β
]
.
(10.3)
For null geodesics the (affine-parameter-dependent) forces
f
(
P
)
α
and
F
(
P, u
)
α
are
zero, but accelerated photon motion is important in the discussion of certain rel-
ativistic phenomena like the Sagnac effect, where photons traveling in opposite
directions around a loop via mirrors or fiber optics are indeed accelerated.
For a massless test particle, the ordinary and corotating Fermi-Walker tangential
relative accelerations are automatically zero since the relative velocity is a unit
vector, and the relative centripetal acceleration has exactly the same form as for
the nonzero mass case
a
(
⊥
)
(tem)
(
P, u
) =
ν
(
P, u
)
2
/ρ
(tem)
(
P, u
) = 1
/ρ
(tem)
(
P, u
)
,
tem=fw
,
cfw
.
(10.4)
This acceleration vanishes for null trajectories which undergo “relatively straight”
relative motion.
11.
Observer-adapted spatial frames
The projection formalism is greatly simplified if expressed in terms of a spacetime
frame adapted to the splitting of each tangent space defined by the test observer
family. Let
{
E
a
α
}
be a spatialframe, i.e., such that it is a basis of each localrest
space
LRS
u
. It is convenient to express the above results in terms of the observer-
adapted spacetime frame
{
u
α
, E
a
α
}
, with dualframe
{−
u
α
, W
a
α
}
. Spatialfields
then only have spatially-indexed frame components nonzero, like the spatial metric
h
ab
=
P
(
u
)
ab
. Note that observer-adapted spatialframe components are distinct
from the Latin-indexed coordinate components; unless otherwise indicated Latin
indices in formulas will refer to the frame components.
Let the frame derivatives of functions be denoted by the comma notation
u
(
f
) =
f
,
0
,
E
a
α
∂
α
f
=
f
,a
.
(11.1)
18
D. Bini, P. Carini and R.T. Jantzen
To express derivatives of tensor fields, one needs the temporaland spatialderivatives
of the spatial frame vectors themselves, as well as their Lie brackets
∇
(tem)
(
u
)
E
a
α
=
C
(tem)
(
u
)
b
a
E
b
α
,
∇
(
u
)
E
a
E
b
α
= Γ(
u
)
c
ab
E
c
α
,
(
P
(
u
)[
E
a
, E
b
])
α
=
C
(
u
)
c
ab
E
c
α
,
(11.2)
where
C
(cfw)
(
u
)
a
b
=
C
(fw)
(
u
)
a
b
+
ω
(
u
)
a
b
=
C
(lie)
(
u
)
a
b
+
θ
(
u
)
a
b
.
(11.3)
For example, if
X
α
=
X
a
E
a
α
is a spatialvector field, one has
∇
(tem)
(
u
)
X
a
=
X
a
,
0
+
C
(tem)
(
u
)
a
b
X
b
,
∇
(
u
)
b
X
a
=
X
a
,b
+ Γ(
u
)
a
bc
X
c
,
(11.4)
while if it is only defined along the world line, one has
D
(tem)
(
U, u
)
X
a
/dτ
(
U,u
)
=
dX
a
/dτ
(
U,u
)
+
C
(tem)
(
u
)
a
b
X
b
+ Γ(
u
)
a
bc
ν
(
U, u
)
b
X
c
.
(11.5)
The frame components of the spatialconnection are easily expressed in terms
of the spatialmetric derivatives and the spatialstructure functions of the spatial
frame
Γ(
u
)
abc
=
1
2
(
h
{
ab,c
}
−
+
C
(
u
)
{
abc
}
−
)
,
(11.6)
where
A
{
abc
}
−
=
A
abc
−
A
bca
+
A
cab
.
For an orthonormal frame, the Fermi-Walker and corotating Fermi-Walker frame
coefficients
C
(tem)
(
u
)
a
b
are antisymmetric. For the specialSerret-Frenet orthonor-
malspatialframe associated with the observer congruence, only two independent
such coefficients exist; in the Fermi-Walker case, since they arise from the spatial
projection of the Fermi-Walker derivatives of the frame vectors along the observer
world lines, they are just the first and second torsions of those world lines.
28
(Note
that according to the definition,
28
a single trajectory of a nontrivial quasi-Killing
vector field is a Killing vector field trajectory, but the family of such trajectories
for a single quasi-Killing vector field consists of trajectories not of one single Killing
vector field.)
12.
Spatial gravito-potentials
The discussion of the measurement by a congruence of test observers of tensor
fields and of tensor differential equations using no other spacetime structure may
be referred to as the congruence point of view. (Originalreferences for spacetime
splittings are given elsewhere.
1
) While in a generalized sense the 4-velocity
u
α
serves as a 4-vector potentialfor the gravitoel
ectric and gravitomagnetic vector
force fields in this point of view (a partial splitting of spacetime), scalar and spatial
vector potentials analogous to those in electromagnetism may be defined only by
introducing certain equivalence classes of local coordinates which are adapted to
the congruence of observer world lines in some way (a full splitting of spacetime).
For a generalcongruence with nonzero vorticity, the appropriate coordinates are
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
19
comoving, leading to an approach called the threading point of view, which merely
represents the generaldiscussion of observer measured quantities in an adapted
coordinate system.
For a specialcase of a vorticity-free congruence where the
gravitomagnetic vector field vanishes, it is more natural to refer to the partial
splitting as the hypersurface point of view, and for the full splitting one can use more
generalcoordinates adapted to the family of orthogonalhypersurfaces admitted by
the congruence (the time coordinate hypersurfaces), allowing the time coordinate
lines to be determined by a second independent congruence. One then has the choice
of representing all the hypersurface-forming observer-measured quantities directly
(the hypersurface point of view) or of working in a hybrid two-congruence approach
called the slicing point of view, in which the evolution is described in terms of the
second congruence. This latter approach is the one well known from the work of
Arnowit, Deser, and Misner.
34, 35
The gravitomagnetic vector field reappears in
this latter approach as a relative velocity effect introduced through the use of a
new temporal derivative along the time lines rather than along the observer world
lines.
36
Thus for a given spacelike slicing and timelike threading (together forming
a “nonlinear reference frame”), one obtains two distinct families of observers and
three distinct points of view which agree only when the slicing and threading are
orthogonal.
Let
{
x
α
}
=
{
t, x
a
}
with
x
0
=
t
be a set of local coordinates (said to be “adapted
to the nonlinear reference frame”) for which the time coordinate hypersurfaces (con-
stant
t
) belong to the given slicing of spacetime and the time coordinate lines (con-
stant
x
a
) belong to the given threading congruence. Introduce also the vector field
tangent to the time coordinate lines
e
0
α
=
δ
α
0
.
12.1.
The threading point of view
First consider the threading point of view, which is especially useful in a station-
ary spacetime where it enables one to interpret the spacetime geometry in terms
of the quotient space geometry on the space of Killing observers. The adapted
coordinates are comoving coordinates for the 4-velocity of the observer congruence
u
α
=
m
α
=
M
−
1
δ
α
0
, and the observer world lines coincide with the time coordinate
lines.
The spacetime line element (covariant metric) takes the form
ds
2
=
−
M
2
(
dt
−
M
a
dx
a
)
2
+
γ
ab
dx
a
dx
b
,
=
M
2
[
−
(
dt
−
M
a
dx
a
)
2
+ ˜
γ
ab
dx
a
dx
b
]
,
(12.1)
where
γ
ab
parametrizes the spatialmetric and is the matrix inverse of
γ
ab
=
P
(
m
)
ab
, whil e ˜
γ
ab
=
M
−
2
γ
ab
parametrizes the “opticalspatialmetric” ˜
P
(
m
)
αβ
=
M
−
2
P
(
m
)
αβ
obtained by a conformalrescaling of the spatialmetric, using the ter-
minology of Abramowicz et al.
3
The “spatialderivatives”
a
=
∂/∂x
a
+
M
a
∂/∂t
=
a
α
∂/∂x
α
define the basis vector fields
{
a
α
}
of the local rest space of the test ob-
servers (which together with
m
α
form an observer-adapted frame for which
20
D. Bini, P. Carini and R.T. Jantzen
C
(lie)
(
m
)
a
b
= 0 and
C
(
m
)
a
bc
= 0), and the matrix
γ
ab
is the matrix of their inner
products. The associated components of the spatialconnection
a
α
∇
α
(
m
)
b
β
= Γ(
m
)
c
ab
c
β
,
(12.2)
expressable as
Γ(
m
)
c
ab
=
1
2
γ
cd
(
γ
da,b
−
γ
ab,d
+
γ
bd,a
)
,
(12.3)
where
f
,a
=
a
α
∂f /∂x
α
, may be used to evaluate the spatial covariant derivative of
a spatialvector field
X
α
=
X
a
a
α
(parametrized by its spatially-indexed contravari-
ant coordinate components
X
a
) entirely in terms of the components in that frame
in the usualway.
1
Analogous tilde expressions hold for the components ˜
Γ(
m
)
c
ab
of
the opticalconnection ˜
∇
(
m
)
α
.
The shift 1-form
M
α
=
M
a
δ
a
α
is a spatial1-form which determines the shift of
the orientation of the local rest spaces of the test observers away from the coordinate
time hypersurfaces, while the lapse function
M
relates coordinate time along the
time lines to the test observer proper time. The combination
ν
(
n, m
)
α
=
M M
a
δ
α
a
is the relative velocity field of the normal trajectories to the coordinate time hy-
persurfaces. The lapse and shift serve as scalar and vector potentials for the grav-
itoelectric and gravitomagnetic vector fields of the test observer congruence, while
the spatialmetric generates the symmetric part of the gravitomagnetic tensor field,
which is proportionalto the expansion tensor
g
(
m
)
α
=
−
a
(
m
)
α
=
−∇
(
m
)
α
ln
M
−
£
(
m
)
e
0
M
α
,
H
(
m
)
α
=
2
ω
(
m
)
α
=
M η
(
m
)
αβγ
∇
(
m
)
β
M
γ
=
M
[
∇
(
m
)
×
m
M
]
α
,
(12.4)
θ
(
m
)
αβ
=
1
2
£
(
m
)
e
0
P
(
m
)
αβ
.
In the observer-adapted frame only the spatially indexed components (distinct from
the coordinate components of this type) of these fields are nonzero, and the Lie
derivatives reduce to the partialderivatives of these components with respect to
the
t
coordinate, for example
θ
(
m
)
ab
=
1
2
∂
t
γ
ab
.
(12.5)
The totalspatialcovariant derivative of the spatialmomentum becomes explicitly
D
(tem)
(
U, m
)
p
(
U, m
)
a
/dτ
(
U,m
)
(12.6)
=
dp
(
U, m
)
a
/dτ
(
U,m
)
+
γ
(
U, m
)
C
(tem)
(
m
)
a
b
ν
(
U, m
)
b
−
F
(SC)
(
U, m
)
a
,
where
F
(SC)
(
U, m
)
a
=
−
γ
(
U, m
)Γ(
m
)
a
bc
ν
(
U, m
)
b
ν
(
U, m
)
c
(12.7)
defines the “space curvature” force in the threading point of view.
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
21
12.2.
The hypersurface and slicing points of view
In the hypersurface and slicing points of view, given a family of spacelike hyper-
surfaces with unit normal
n
α
, the spacetime line element in adapted local coordi-
nates takes the form
ds
2
=
−
N
2
dt
2
+
g
ab
(
dx
a
+
N
a
dt
)(
dx
b
+
N
b
dt
)
=
N
2
[
−
dt
2
+ ˜
g
ab
(
dx
a
+
N
a
dt
)(
dx
b
+
N
b
dt
)]
.
(12.8)
Here
g
ab
=
P
(
n
)
ab
parametrizes the spatialmetric and is the matrix inverse of
g
ab
, whil e ˜
g
ab
=
N
−
2
g
ab
parametrizes the “opticalspatialmetric” ˜
P
(
n
)
αβ
=
N
−
2
P
(
n
)
αβ
. The spatialcoordinate vectors themselves
e
a
α
=
δ
α
a
form a spatial
frame (which together with
n
α
form an observer-adapted frame with
C
(
n
)
a
bc
= 0
and
C
(lie)
(
n
)
a
b
=
N
−
1
∂N
a
/∂x
b
) which may be used to express the spatialcovariant
derivative of a spatialvector field
X
α
=
X
b
g
ba
e
a
α
(parametrized by its spatially-
indexed covariant coordinate components
X
a
) in terms of the associated connection
components, namely
e
a
α
∇
α
(
n
)
e
b
β
= Γ(
n
)
c
ab
e
c
γ
,
(12.9)
expressable as
Γ(
n
)
c
ab
=
1
2
g
cd
(
g
da,b
−
g
ab,d
+
g
bd,a
)
,
(12.10)
where here
f
,a
=
∂f /∂x
a
. Analogous tilde expressions hold for the components
˜
Γ(
n
)
c
ab
of the opticalconnection ˜
∇
(
n
)
α
.
The 4-velocity of the associated test observers is the unit normal
n
α
=
N
−
1
[
δ
α
0
−
N
a
δ
α
a
]
,
(12.11)
and their world lines are the orthogonal trajectories to the time hypersurfaces. The
shift vector field
N
α
=
N
a
δ
α
a
is a spatialvector field which determines the shift
of the time lines away from these orthogonal trajectories, while the lapse function
N
relates the coordinate time along the observer world lines to the observer proper
time. The combination
ν
(
e
0
, n
)
α
=
N
−
1
N
a
δ
α
a
is the relative velocity field of the
threading curves.
The distinguishing feature of the slicing point of view compared to the hyper-
surface point of view or the threading point of view for the same congruence of
observers (the latter requiring a comoving coordinate system) is that it uses a new
Lie temporal derivative along the threading curves rather than along the observer
world lines
∇
(lie)
(
n, e
0
) =
N
−
1
£
(
n
)
e
0
=
∇
(lie)
(
n
) +
N
−
1
£
(
n
)
N
.
(12.12)
This in turn leads to a new Lie spatial total covariant derivative along a test particle
worldline
D
(lie)
(
U, n, e
0
)
X
α
/dτ
(
U,n
)
=
[
∇
(lie)
(
n, e
0
) +
ν
(
U, n
)
β
∇
(
n
)
β
]
X
α
(12.13)
=
D
(lie)
(
U, n
)
X
α
/dτ
(
U,n
)
−
[
N
−
1
∇
(
n
)
β
N
α
]
X
β
.
22
D. Bini, P. Carini and R.T. Jantzen
Expressing the two Lie totalcovariant derivatives in the observer-adapted frame
leads to
D
(lie)
(
U, n
)
X
a
/dτ
(
U,n
)
=
dX
a
/dτ
(
U,n
)
+ Γ(
n
)
a
bc
ν
(
U, n
)
b
X
c
+
X
b
N
−
1
∂N
a
/∂x
b
=
dX
a
/dτ
(
U,n
)
+ Γ(
n
)
a
bc
[
ν
(
U, n
)
b
−
ν
(
e
0
, n
)
b
]
X
c
+
X
b
N
−
1
∇
(
n
)
b
N
a
=
D
(lie)
(
U, n, e
0
)
X
a
/dτ
(
U,n
)
+
X
b
N
−
1
∇
(
n
)
b
N
a
.
(12.14)
When
X
a
=
ν
(
U, n
)
a
, these equations define the slicing Lie relative acceleration
a
(lie)
(
U, n, e
0
)
α
and its relation to the hypersurface quantity, namely
a
(lie)
(
U, n, e
0
)
a
−
a
(lie)
(
U, n
)
a
=
−
ν
(
U, n
)
b
N
−
1
∇
(
n
)
b
N
a
.
(12.15)
The spatialequation of motion of a test particle then takes the form
D
(lie)
(
U, n
)
p
(
U, n
)
α
/dτ
(
U,n
)
=
γ
(
U, n
)[
g
(
n
)
α
+
H
(lie)
(
n
)
α
β
ν
(
U, n
)
β
] +
F
(
U, n
)
α
(12.16)
in the hypersurface point of view and
D
(lie)
(
U, n, e
0
)
p
(
U, n
)
α
/dτ
(
U,n
)
=
γ
(
U, n
)[
g
(
n
)
α
+
H
(lie)
(
n, e
0
)
α
β
ν
(
U, n
)
β
]
+
F
(
U, n
)
α
,
D
(lie)
(
U, n, e
0
)
p
(
U, n
)
α
/dτ
(
U,n
)
=
γ
(
U, n
)[
g
(
n
)
α
+
H
(lie
)
(
n, e
0
)
αβ
ν
(
U, n
)
β
]
+
F
(
U, n
)
α
(12.17)
in the slicing point of view, where the hypersurface Lie gravitomagnetic tensor field
is just minus twice the expansion tensor when the equations of motion are expressed
in contravariant form, but zero in the covariant form
H
(lie)
(
n
)
αβ
=
N
−
1
[2
∇
(
n
)
(
α
N
β
)
−
£
(
n
)
e
0
P
(
n
)
αβ
] =
−
2
θ
(
n
)
αβ
,
H
(lie
)
(
n
)
αβ
=
0
(12.18)
and the slicing gravitomagnetic tensor fields are
H
(lie)
(
n, e
0
)
αβ
=
N
−
1
[
∇
(
n
)
α
N
β
−
£
(
n
)
e
0
P
(
n
)
αβ
]
,
H
(lie
)
(
n, e
0
)
αβ
=
N
−
1
∇
(
n
)
α
N
β
.
(12.19)
The contravariant and covariant forms of the equation of motion differ by a term
arising from the Lie temporalderivative of the spatialmetric. For the contravariant
form of the equation of motion the symmetric part of the gravitomagnetic tensor
differs from
−
2
θ
(
n
)
αβ
by the missing reversed-index shift derivative term which
would symmetrize the term which is present. This missing term now contributes
by its absense to the slicing gravitomagnetic vector field but is restored by an extra
shift derivative term in the corresponding second order acceleration equation due
to the relative velocity of the observers and evolution curves, thus restoring the
analogy with the threading point of view at that level.
1
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
23
The lapse and shift serve as scalar and vector potentials for the gravitoelectric
and gravitomagnetic vector fields in the slicing point of view. The gravitoelectric
field is still the sign-reversed acceleration of the test observers in both points of
view, but the slicing gravitomagnetic vector field is now a result of the relative
motion of the time lines with respect to the observer world lines
g
(
n
)
α
=
−
a
(
n
)
α
=
−∇
(
n
)
α
ln
N ,
H
(lie)
(
n
)
α
= 0
,
H
(lie)
(
n, e
0
)
α
=
N
−
1
η
(
n
)
αβγ
∇
(
n
)
β
N
γ
=
N
−
1
[
∇
(
n
)
×
n
N
]
α
,
(12.20)
In terms of the gravitomagnetic vector field, the slicing spatial equation of motion
takes the form
D
(lie)
(
U, n, e
0
)
p
(
U, n
)
α
/dτ
n
=
γ
(
U, n
)
{
g
(
n
)
α
+
1
2
[
ν
(
U, n
)
×
n
H
(lie)
(
n, e
0
)]
α
+
H
(SYM)
(lie)
(
n, e
0
)
α
β
ν
(
U, n
)
β
}
.
(12.21)
All of these expressions have simple analogous forms when expressed in the observer-
adapted frame. The totalspatialcovariant derivative of the spatialmomentum
becomes explicitly
D
(lie)
(
U, n
)
p
(
U, n
)
a
/dτ
(
U,n
)
=
dp
(
U, n
)
a
/dτ
(
U,n
)
−
F
(SC)
(
U, n
)
a
+
p
(
U, n
)
b
∂N
a
/∂x
b
,
(12.22)
D
(lie)
(
U, n, e
0
)
p
(
U, n
)
a
/dτ
(
U,n
)
=
dp
(
U, n
)
a
/dτ
(
U,n
)
−
F
(SC)
(
U, n, e
0
)
a
where
F
(SC)
(
U, n
)
a
=
−
γ
(
U, n
)Γ(
n
)
a
bc
ν
(
U, n
)
b
ν
(
U, n
)
c
,
F
(SC)
(
U, n, e
0
)
a
=
−
γ
(
U, n
)Γ(
n
)
a
bc
[
ν
(
U, n
)
b
−
N
−
1
N
b
]
ν
(
U, n
)
c
(12.23)
defines the “space curvature” force in the two points of view.
12.3.
Stationary spacetimes
Suppose
e
0
α
(and therefore
m
) is a timelike Killing vector field in some open
submanifold of a stationary spacetime, implying that
θ
(
m
)
αβ
= 0, and suppose
that
n
α
is also timelike in some other open submanifold overlapping with the first.
One then has a nonlinear reference frame adapted to the stationarity in which
one may consider threading, slicing, and hypersurface points of view. The Kerr
black hole spacetimes in Boyer-Lindquist coordinates which are adapted to the
distantly nonrotating (static) observers determining the threading congruence and
to the locally nonrotating observers determining the slicing congruence are a good
example to keep in mind.
All Lie derivatives of stationary fields along
e
0
α
vanish. This eliminates the
complication of the mixing of time coordinate derivatives with spatial coordinate
derivatives in the spatialderivatives of component fields in threading point of view
when differentiating stationary tensor fields, since their various components are
24
D. Bini, P. Carini and R.T. Jantzen
independent of the time coordinate. In allcases the spatialmetric projects to a
Riemannian metric on the threading quotient space where the relative motion takes
place and all calculations become much more straightforward. The scalar and vec-
tor potentials then become potentials in the usual sense on this space since spatial
projection is unnecessary and the curland gradient are just the ones on this Rie-
mannian observer quotient space. Furthermore the threading observer expansion
tensor vanishes although the slicing observer one does not, leading to the coinci-
dence of the Lie and corotating Fermi-Walker derivatives in the threading point of
view. The temporalderivative observer-adapted frame structure functions in the
threading point of view therefore satisfy
C
(cfw)
(
m
)
a
b
=
C
(lie)
(
m
)
a
b
= 0, and the
two associated total spatial covariant derivatives coincide as well. In the case of the
slicing observer-adapted frame, one instead has
C
(
n
)
(cfw)
a
b
−
θ
(
n
)
a
b
=
C
(
n
)
(lie)
a
b
=
N
−
1
∂N
a
/∂x
b
,
(12.24)
but for relative motion along a spatial Killing direction
ν
(
U, n
)
b
∂N
a
/∂x
b
= 0, so
the shift derivative term does not contribute to the relative curvature and the cen-
tripetal acceleration in that case, making the Lie relative curvature the simplest
such curvature in the hypersurface point of view. However, the expansion term
does lead to an additional term in the corotating Fermi-Walker centripetal accel-
eration which is linear in the relative velocity; a similar linear term arises also in
the the slicing point of view expression due to the relative velocity of the observers
themselves relative to the threading.
13.
Spatial coordinate line curvature in stationary spacetimes with ad-
ditional symmetry
Suppose
e
0
α
is a timelike Killing vector field in a stationary spacetime with
an additional spacelike Killing vector field for which
t
and
x
3
respectively are co-
moving coordinates, so that the metric only depends on the remaining two coordi-
nates. Assume also that the spatial coordinates are orthogonal, a situation which
describes many interesting stationary spacetimes, including rotating black holes
(Kerr spacetimes) in Boyer-Lindquist coordinates and the G¨
odelspacetime in the
usual Cartesian-like or cylindrical-like coordinates.
Consider a test particle worldline following a Killing trajectory in the 2-surface
of the coordinates
t
and
x
3
with the remaining two coordinates fixed, implying that
in either the associated threading or slicing point of view, the unit velocity vector
has a single nonvanishing constant spatial coordinate component. Constant speed
circular orbits in the above-mentioned spacetimes are of this type, for example.
Consider the threading point of view, for example, where ˆ
ν
(
U, m
)
a
=
γ
33
−
1
/
2
δ
a
3
and
D
(lie)
(
U, m
)ˆ
ν
(
U, m
)
a
/d
(
U,m
)
= Γ(
m
)
a
33
(ˆ
ν
(
U, m
)
3
)
2
=
−
γ
aa
−
1
∂
(ln
γ
33
1
/
2
)
/∂x
a
,
(13.1)
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
25
or, since the spatialcoordinates are orthogonal, in terms of the associated ortho-
normal(“physical”) components indicated with the “hat” notation
D
(lie)
(
U, m
)ˆ
ν
(
U, m
)
ˆ
a
/d
(
U,m
)
=
−
γ
aa
−
1
/
2
∂
(ln
γ
33
1
/
2
)
/∂x
a
=
κ
(3
, m
)
ˆ
a
,
(13.2)
which can only be nonzero for
a
= 1
,
2. The Lie relative curvature
κ
(lie)
(
U, m
)
will just be the square root of the sum of the squares of the these two physical
components; in fact for circular orbits only one of these will be nonzero. (For the
case of additional symmetry like stationary cylindrically symmetric spacetimes of
which the G¨
odel spacetime is an example, one can evaluate this spatial curvature
for each spatial coordinate line along a Killing vector field, with the result for a
general timelike Killing trajectory being obtained in a simple way from the indi-
vidual results, i.e., as for a helical Killing trajectory in G¨
odel.) The corresponding
optical relative curvature has the same formula with the tilde metric and derivative,
the latter of which is described in the appendix. A similar formula holds for the
physicalobserver-adapted frame components in the hypersurface point of view
D
(lie)
(
U, n
)ˆ
ν
(
U, n
)
ˆ
a
/d
(
U,n
)
=
−
g
aa
−
1
/
2
∂
(ln
g
33
1
/
2
)
/∂x
a
=
κ
(3
, n
)
ˆ
a
,
(13.3)
and for its optical generalization. All formulas continue to hold for massless test
particle trajectories as well.
These coordinate curvature quantities
κ
(3
, u
)
ˆ
a
will be referred to as the signed
Lie relative curvatures (“signed” since they may take all real values). In each of
these four formulas, the relative curvature of these special trajectories corresponds
exactly to the coordinate line curvature in the quotient space Riemannian geometry
or its optical version for the relevant observer point of view (threading, slicing, or
hypersurface). In the static case, all of these points of view coincide and only two
distinct relative Lie curvatures exist.
14.
Circular orbits in stationary axisymmetric spacetimes
Now specialize to the case of a stationary axisymmetric spacetime with coor-
dinates
{
t, z, ρ, φ
}
, where
t
and
φ
are comoving with respect to the Killing vector
fields associated with the stationary and axial symmetries respectively, and the
latter three coordinates are orthogonal, as occurs for the stationary cylindrically
symmetric G¨
odel and rotating Minkowski spacetimes. Only the single threading
observer-adapted spatialframe vector
ε
φ
α
=
δ
α
φ
+
M
φ
δ
α
t
differs from the orthog-
onalspatialcoordinate frame vectors.
For circular orbits, the observer-adapted physical components of the relative
velocity along the angular direction are
ν
(
U, m
)
ˆ
φ
=
M
−
1
γ
φφ
1
/
2
˙
φ/
(1
−
M
φ
˙
φ
)
,
ν
(
U, n
)
ˆ
φ
=
N
−
1
g
φφ
1
/
2
( ˙
φ
+
N
φ
)
,
(14.1)
while
ν
(
U, n
)
ˆ
φ
−
ν
(
e
0
, n
)
ˆ
φ
=
N
−
1
g
φφ
1
/
2
˙
φ ,
(14.2)
26
D. Bini, P. Carini and R.T. Jantzen
where the coordinate angular velocity ˙
φ
=
dφ/dt
along the test particle world line
is constant for the stationary circular orbits of constant speed to be considered
here. Thus the test particle world line is a Killing trajectory and the ordinary time
derivative terms vanish in the totalspatialcovariant derivatives along the test world
line.
Attention will be confined to those circular orbits for which the
z
-derivative
relative curvature expression vanishes
κ
(
φ, m
)
ˆ
z
=
κ
(
φ, n
)
ˆ
z
= 0
,
κ
(
φ, m
)
ˆ
ρ
=
−
γ
ρρ
−
1
/
2
∂
(ln
γ
1
/
2
φφ
)
/∂ρ ,
κ
(
φ, n
)
ˆ
ρ
=
−
g
ρρ
−
1
/
2
∂
(ln
g
1
/
2
φφ
)
/∂ρ .
(14.3)
This limits one to the equatorial circular orbits in the Kerr spacetime. The Lie rel-
ative centripetal acceleration then has the single nonzero physical observer-adapted
frame component
a
(lie)
(
U, u
)
(
⊥
) ˆ
ρ
=
κ
(
φ, u
)
ˆ
ρ
|
ν
(
U, u
)
ˆ
φ
|
2
,
u
=
m, n .
(14.4)
Since
θ
(
n
)
φφ
= 0, the hypersurface point of view Lie relative acceleration is for-
tuitously orthogonal to the relative velocity and so directly represents the relative
centripetalacceleration.
The nonzero observer-adapted physicalcomponents of the various fields are
g
(
m
)
ˆ
ρ
=
−
(
γ
ρρ
)
−
1
/
2
(ln
M
)
,ρ
,
H
(
m
)
ˆ
z
=
M
(
γ
ρρ
γ
φφ
)
−
1
/
2
M
φ,ρ
,
g
(
n
)
ˆ
ρ
=
−
(
g
ρρ
)
−
1
/
2
(ln
N
)
,ρ
, H
(
n, e
0
)
ˆ
z
=
N
−
1
(
g
ρρ
g
φφ
)
−
1
/
2
N
φ,ρ
,
(14.5)
and
H
(
n, e
0
)
(ˆ
ρ
ˆ
φ
)
=
1
2
H
(
n
)
ˆ
ρ
ˆ
φ
=
−
θ
(
n
)
ˆ
ρ
ˆ
φ
=
1
2
N
−
1
(
g
φφ
/g
ρρ
)
1
/
2
N
φ
,ρ
=
1
2
C
(lie)
(
n
)
ˆ
φ
ˆ
ρ
=
1
2
C
(cfw)
(
n
)
ˆ
φ
ˆ
ρ
,
(14.6)
where the comma indicates the coordinate partialderivative here.
The physical 4-force responsible for the motion of the test particle on the circular
orbit is related to the relative force by equation (9.8). Since the projection
P
(
u
)
acts as the identity in the radialdirection orthogonalto the relative motion, one
has the simpler relationship between the 4-force, the 4-acceleration, and the relative
forces
f
(
U
)
ˆ
ρ
=
γ
(
U, u
)
F
(
U, u
)
ˆ
ρ
=
γ
(
U, u
)[
−
F
(SC)
(
U, u
)
ˆ
ρ
−
F
(G)
(
U, u
)
ˆ
ρ
] =
a
(
U
)
ˆ
ρ
.
(14.7)
The middle equality is the equation of motion for the circular orbit, while the last
equality is the value of the acceleration of the orbit. The same considerations apply
to circular orbits in the equatorial plane of the Kerr spacetime and its Schwarzschild
limit in terms of the usual “spherical” radial coordinate in that plane.
Iyer and Vishveshwara have given the complete Serret-Frenet frame formulas
for the stationary axisymmetric Killing trajectories (arbitrary constant speed cir-
cular orbits) in Kerr, G¨
odel, Minkowski and several other spacetimes.
28
These
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
27
depend only on the single ratio of the two Killing vector components of the tan-
gent vector
δ
α
t
+
ωδ
α
φ
, or equival entl y of
ω
−
1
δ
α
t
+
δ
α
φ
because of its normal-
ization. The limit
ω
−
1
= 0 of their formula for the curvature
κ
of the associ-
ated Killing trajectory gives exactly
|
κ
(
φ, n
)
ˆ
ρ
|
, with their frame vectors reducing to
(
e
α
(0)
, e
α
(1)
, e
α
(2)
) = (
−
e
α
ˆ
φ
, e
α
ˆ
ρ
,
−
n
α
).
15.
The Abramowicz et al Approach Demystified
By introducing the opticalmetric in the threading or slicing points of view,
the full spacetime metric is conformal to one with unit lapse and the same shift
1-form or vector field respectively, the overall conformal factor being the square of
the lapse function in each point of view. Since null geodesics are invariant under
spacetime conformaltransformations, one may re-express the spatialequation of
motion for a massless test particle in terms of the new line element, thus absorbing
the scalar potential part of the gravitoelectric force term into the spatial geometry
itself (since the rescaled metric has a unit lapse). In the case of a static spacetime
with a nonlinear reference frame adapted to the threading congruence of a timelike
Killing vector field, so that the shift field is zero and the adapted coordinates are
Gaussian normalwith respect to the new geometry, the spacetime geodesics of the
rescaled spacetime metric project down to geodesics of the observer quotient space
with the opticalmetric. Thus the geodesics of the opticalspatialmetric on this
space are the paths of light rays, and the optical geometry measures deviations
from these paths. Of course timelike geodesics are not invariant under conformal
transformations, so the equations of motion of a massive test particle still contain
an explicit gravitoelectric term when re-expressed in this way.
This idea, closely related to the general relativistic Fermat principle
37
and older
discussions of Møller,
38
is the origin of a long series of papers by Abramowicz and
coworkers. The key difficulty in understanding their calculations associated with
this idea lies in the obscure representation both of the intrinsic derivative along the
test particle world line and of the kinematical quantities of the observer congru-
ence (which are not clearly identified) in terms of nongeometrical derivatives of a
vector field on spacetime possessing the given world line as an integral curve. The
very special circumstances of circular orbits in their applications also give false im-
pressions of the general case; in particular the various centripetal accelerations and
gravitoelectric and gravitomagnetic forces there are all transverse to the direction
of relative motion, and so reside in the intersection of the local rest spaces of the
observers and the test world line. The most recent formulation of this work
14
is
simply the hypersurface point of view description of the decomposition of the spatial
projection of the 4-acceleration of the test world line.
The details of the spacetime conformal transformation are straightforward and
are discussed in the appendix. The new spatialequation of motion in the threading
point of view for a massless test particle following a null geodesic turns out to be
˜
D
(tem)
(
P, m
)˜
p
(
P, m
)
α
/dτ
(
P,m
)
28
D. Bini, P. Carini and R.T. Jantzen
=
M
−
1
˜
E
(
P, m
)[
−
£
(
m
)
e
0
M
α
+
H
(tem
)
(
m
)
αβ
ν
(
P, m
)
β
]
,
(15.1)
where the flat notation
5
in the gravitomagnetic field is only needed for the Lie case
tem=lie
, while in the slicing point of view it is
˜
D
(lie)
(
P, n, e
0
)˜
p
(
P, n
)
α
/dτ
(
P,n
)
=
N
−
1
˜
E
(
P, n
)
H
(lie
)
(
n, e
0
)
αβ
ν
(
P, n
)
β
.
(15.2)
The covariant rather than contravariant form of this equation is given for comparison
with the form often found in the literature.
14
Here the explicit factors of the lapse
correct for the proper time and the other untransformed spatialquantities stil
l
present.
Thus in the case of a static spacetime with an orthogonalslicing and threading
adapted to the timelike vorticity-free Killing vector field, the threading and slicing
points of view coincide (
u
=
m
=
n
), the shift and expansion tensor both vanish,
allthe various totalspatialcovariant derivatives agree, and the equation of motion
reduces to the geodesic equation in the time-independent geometry of the observer
quotient space
˜
D
(tem)
(
P, u
)˜
p
(
P, u
)
α
/dτ
(
P,u
)
= 0
.
(15.3)
In this specialcase only, one can interpret the opticaltotalspatialcovariant deriv-
ative as measuring the deviation of particle motion from “optically straight line
paths” in the quotient space, as advocated by Abramowicz et al.
3–5
In the station-
ary case additionalgravitomagnetic tensor effects deflect the nullgeodesics from
the opticalspatialgeodesics in the observer quotient spaces.
The spatial equation of motion for massive test particles may also be expressed
in terms of the conformally rescaled quantities. Using the results of the appendix
one finds for the threading point of view
˜
D
(tem)
(
U, m
)˜
p
(
U, m
)
α
/dτ
(
U,m
)
=
−
γ
(
U, m
)
−
1
∇
(
m
)
α
ln
M
−
γ
(
U, m
)
£
(
m
)
e
0
M
α
+
γ
(
U, m
)
H
(tem
)
(
m
)
αβ
ν
(
U, m
)
β
+
F
(
U, m
)
α
,
(15.4)
where the flat subscript is only needed in the Lie case. Correspondingly the expres-
sion for the spatial projection of the test particle 4-acceleration (just the gamma
factor times the apparent three-acceleration) can be written
γ
(
U, m
)
A
(
U, m
)
α
=
γ
(
U, m
)ˆ
ν
(
U, m
)
α
dp
(
U, m
)
/dτ
(
U,m
)
+
γ
(
U, m
)
2
[
ν
(
U, m
)
2
D
(tem)
(
U, m
)ˆ
ν
(
U, m
)
α
/d
(
U,m
)
+
∇
(
m
)
α
ln
M
+
£
(
m
)
e
0
M
α
−
H
(tem
)
(
m
)
αβ
ν
(
U, m
)
β
]
.
(15.5)
This in turn can be rewritten in terms of the opticalmetric and the naturalconfor-
mally rescaled quantities introduced in the appendix as
γ
(
U, m
)
A
(
U, m
)
α
=
γ
(
U, m
)˜
ˆ
ν
(
U, m
)
α
d
˜
p
(
U, m
)
/dτ
(
U,m
)
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
29
+
γ
(
U, m
)
2
{
ν
(
U, m
)
2
˜
D
(tem)
(
U, m
)˜
ˆ
ν
(
U, m
)
α
/d
˜
(
U,m
)
+
γ
(
U, m
)
−
2
∇
(
m
)
α
ln
M
+
£
(
m
)
e
0
M
α
−
H
(tem
)
(
m
)
αβ
ν
(
U, m
)
β
}
.
(15.6)
Similarly, expressing the spatial projection of the 4-acceleration in the hypersurface
point of view leads to
γ
(
U, n
)
A
(
U, n
)
α
=
γ
(
U, n
)˜
ˆ
ν
(
U, n
)
α
d
˜
p
(
U, n
)
/dτ
(
U,n
)
+
γ
(
U, n
)
2
[
ν
(
U, n
)
2
˜
D
(lie)
(
U, n
)˜
ˆ
ν
(
U, n
)
α
/d
˜
(
U,n
)
+
γ
(
U, n
)
−
2
∇
(
n
)
α
ln
N
−
H
(lie)
(
n
)
α
β
ν
(
U, n
)
β
]
=
γ
(
U, n
)˜
ˆ
ν
(
U, n
)
α
d
˜
p
(
U, n
)
/dτ
(
U,n
)
+
γ
(
U, n
)
2
[
ν
(
U, n
)
2
˜
D
(lie
)
(
U, n
)˜
ˆ
ν
(
U, n
)
α
/d
˜
(
U,n
)
+
γ
(
U, n
)
−
2
∇
(
n
)
α
ln
N
]
(15.7)
and in the slicing point of view
γ
(
U, n
)
A
(
U, n
)
α
=
γ
(
U, n
)˜
ˆ
ν
(
U, n
)
α
d
˜
p
(
U, n
)
/dτ
(
U,n
)
+
γ
(
U, n
)
2
[
ν
(
U, n
)
2
˜
D
(lie)
(
U, n, e
0
)˜
ˆ
ν
(
U, n
)
α
/d
˜
(
U,n
)
+
γ
(
U, n
)
−
2
∇
(
n
)
α
ln
N
−
H
(lie
)
(
n, e
0
)
α
β
ν
(
U, n
)
β
]
.
(15.8)
In each point of view the key difference between the originaland the confor-
mally rescaled versions of these equations is the removal of the gamma squared
factor which multiplies the lapse derivative term, which comes about from the dif-
ference term between the two covariant derivatives and the identity (7.2). For purely
transverse relative accelerated motion in which the spatial acceleration lies in the
common rest subspace
LRS
U
∩
LRS
u
(as in circular motion), then gamma times
the spatial acceleration is the 4-acceleration itself, which can therefore be expressed
as the sum of the logarithmic gradient of the lapse plus gamma squared times an
“opticalcentripetalacceleration”
˜
a
(
⊥
)
(tem)
(
U, u
)
α
= ˜
ν
(
U, u
)
2
˜
D
(tem)
(
U, u
)˜
ˆ
ν
(
U, u
)
α
/d
˜
(
U,u
)
(15.9)
plus additional terms due to the gravitomagnetic vector force and possible temporal
derivatives. This is in some sense the decomposition of Abramowicz, which focuses
on the opticalcentripetalacceleration, used to define an “opticalcentrifugalforce”
as gamma squared times the sign reversalof the opticalcentripetalacceleration
5
˜
f
(
⊥
)
(tem)
(
U, u
)
α
=
−
γ
(
U, u
)
2
˜
a
(
⊥
)
(tem)
(
U, u
)
α
.
(15.10)
The square of the gamma factor enters from the change of proper time between the
observer and test particle in the second derivative defining the acceleration.
30
D. Bini, P. Carini and R.T. Jantzen
One can also introduce the optical relative curvatures ˜
κ
(tem)
(
U, u
) and
˜
κ
(lie)
(
U, n, e
0
) as the magnitude of the opticalderivatives ˜
D
(tem)
(
U, u
)˜
ˆ
ν
(
U, u
)
α
/d
˜
(
U,u
)
and ˜
D
(lie)
(
U, n, e
0
)˜
ˆ
ν
(
U, n
)
α
/d
˜
(
U,n
)
respectively, but all of these will only be relevant
in the static case. This leads to the “optically straight” world lines which play a
key role in the “reversal of the centrifugal force” discussion of Abramowicz et al.
Note finally that presence of the inverse square of the gamma factor multiplying
the logarithmic gradient of the lapse in the conformally rescaled force equations
above shows clearly that the ultrarelativistic limit for geodesic motion approaches
the free photon behavior in stationary spacetimes for which the Lie derivative term is
zero. As the relative speed approaches 1 and the gamma factor becomes increasingly
large, this term drops out and the observer arclength parametrization approaches
the proper time parametrization. One is then left with a balance of the optical
centripetalaccel
eration and the gravitomagnetic force as occurs for photons, as
noted by Abramowicz et al
14
for the case of circular orbits in stationary axially
symmetric spacetimes.
In a l ong review,
12
both the threading and hypersurface points of view were
mentioned in the context of stationary axisymmetric spacetimes, and both points
of view were applied to the Kerr spacetime in separate articles,
10, 11
although the
ambiguity of the individualforce terms in this approach was not addressed untilthe
gauge-fixing discussion of a later version of the force decomposition.
13
Although
it is not easy to decipher (complicated by sign inconsistencies and minor errors),
the subsequent presentation of the Abramowicz et aldefinition of noninertialforces
for generalspacetimes
14
is just the hypersurface point of view decomposition (15.7)
of the sign-reversed spatial projection of the test particle 4-acceleration expressed
in terms of the Lie totalspatialcovariant derivative and the contravariant relative
velocity, with the gravitoelectric force and longitudinal and transverse relative accel-
erations shuffled among themselves by the conformal transformation. They refer to
the latter two as the Euler and centrifugal forces when sign-reversed and conformally
shuffled. The gravitomagnetic tensor force is entirely due to the expansion tensor,
which they refer to as the Coriolis force with its extra gamma factor, while their
gravitational force is just the gravitoelectric vector field itself without the gamma
factor which occurs in the gravitoelectric force. The “ACL gauge” they have cho-
sen for the extension of
U
α
off its worldline in the hypersurface point of view in
order to make their derivative expressions meaningfulis just
∇
(lie)
(
n
)ˆ
ν
(
U, n
)
α
= 0
(inconsistent with their covariant derivative formula for this derivative). One could
also repeat their discussion with the equally valid but distinct corresponding con-
travariant gauge condition
∇
(lie)
(
n
)ˆ
ν
(
U, n
)
α
= 0, as well as switch to the threading
point of view as alluded to in an earlier article.
12
None of their forces coincide with the relative forces measured by the observers,
and in the case of more generalmotion in which the various forces (apart from the
longitudinal relative acceleration) are no longer all transverse, they do not lie in the
local rest space of the test world line, so their interpretation in terms of forces that
would be seen with respect to axes which comove with the test world line (apart
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
31
from proper time adjustments) are no longer valid. This inconsistency appears in
the claim that the various forces they introduce lie in the “comoving frame of the
particle”
13
when in fact they lie in the local rest space of the observer (the exception
being the intersection of the two subspaces orthogonal to the relative 2-plane of the
motion).
Perhaps one can best characterize the limitations of the approach and its de-
velopment by saying that it was born in a very special context and then attempts
were made to generalize it, rather than realizing it as a specialization of an already
generalapproach for arbitrary spacetimes, the tools of which have been around for
a long time, but simply lacked a unifying umbrella. In particular, the key accelera-
tion potentialequation
u
β
∇
β
u
α
=
±∇
β
Φ underlying this approach is inappropriate
for nonstationary spacetimes where an additionalspatialprojection is needed for
the gradient term and an additionalLie derivative of a vector potentialis needed
for a threading point of view. However, this said, the originalapplication to static
spacetimes where the optical centripetal acceleration does reverse when the relative
signed opticalcurvature expression ˜
κ
(
φ, u
)
ˆ
ρ
associated with the
φ
coordinate circles
changes sign is a very beautifulgeometrization of the relative motion of massive
and massless test particles, for which credit is clearly due for its recognition and
description.
16.
Concluding Remarks
By implementing relatively straightforward ideas about special relativistic space-
plus-time splitting in the context of general relativity, using a notation which allows
one to examine all the possibilities for generalizing concepts which do not have
unambiguous extensions into the more generalarena, a foundation has been built
which enables one to analyse any particular problem that involves the description of
idealized observations in a given spacetime. Not only are such questions fascinating,
but their answers are often sufficiently subtle that much confusion has arisen even
in the case of relatively simple spacetimes. Indeed the case of “rigid rotation” in
flat spacetime itself still leads people astray in their attempts to come to terms with
such questions.
Armed with the present tools, one can examine the traditional test cases for
investigating these ideas,
39
namely Minkowski spacetime in rotating coordinates,
the black hole spacetimes of Kerr and Schwarzschild, and the spacetime which was
the first to dramatically challenge our intuition about rotation in general relativity
nearly half a century ago, the G¨
odelspacetime. This willbe done in a companion
article, leading to a much clearer understanding not only of these spacetimes but
of the tools themselves for studying other spacetimes.
Acknowledgements
We thank Remo Ruffini of the InternationalCenter for Relativistic Astrophysics
at the University of Rome and Francis Everitt of the Gravity Probe B Relativity
32
D. Bini, P. Carini and R.T. Jantzen
Mission group at Stanford University for their support and encouragement of this
work, as well as the many relativists before us who have laid the groundwork for
our analysis.
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Inertial Frame Dragging and Mach’s Principle in General Relativity
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A.
Conformal transformations
A.1.
General considerations
Following the abstract index notation of Wald,
40
suppose one has a pair of
conformally related metrics
˜
g
ab
=
σ
2
g
ab
,
(A.1)
each with its associated symmetric connection
∇
and ˜
∇
. Then under the additional
conformaltransformation of a vector
˜
v
a
=
σ
−
1
v
a
,
˜
v
a
=
σv
a
,
˜
v
a
˜
v
a
=
v
a
v
a
(A.2)
which preserves its magnitude, Eq. (D.5) of Wald
v
a
˜
∇
a
v
b
=
v
a
∇
a
v
b
+ (2
v
b
v
c
−
g
bc
v
d
v
d
)
∇
c
(ln
σ
)
,
(A.3)
is easily transformed to
σ
2
˜
v
a
˜
∇
a
˜
v
b
=
v
a
∇
a
v
b
+ (
v
b
v
c
−
g
bc
v
d
v
d
)
∇
c
(ln
σ
)
=
v
a
∇
a
v
b
−
v
d
v
d
P
(
v
)
bc
∇
c
(ln
σ
)
,
(A.4)
where the last equality is only valid in the case
v
d
v
d
= 0.
P
(
v
)
a
b
is just the tensor
which projects orthogonalto
v
a
in that case, necessary since the either covariant
derivative of a unit vector must be orthogonalto its direction, given that the metric
is covariant constant. Using the latter fact, one obtains the covariant form of this
equation
˜
v
a
˜
∇
a
˜
v
b
=
v
a
∇
a
v
b
+ (
v
b
v
a
−
δ
a
b
v
c
v
c
)
∇
a
(ln
σ
)
=
v
a
∇
a
v
b
−
v
c
v
c
P
(
v
)
a
b
∇
a
(ln
σ
)
.
(A.5)
A.2.
Conformal transformations of spatial quantities
For a given family of test observers with 4-velocity
u
α
and a given timelike world
line of a test particle with 4-velocity
U
α
, consider a conformaltransformation of
the spatialmetric
˜
P
(
u
)
αβ
=
σ
2
P
(
u
)
αβ
.
(A.6)
One can introduce a new spatialcovariant derivative associated with the new spatial
metric and use it to re-express the various totalcovariant derivatives of the relative
velocity of a test particle. For our present purposes this is really only useful in
34
D. Bini, P. Carini and R.T. Jantzen
the context of a nonlinear reference frame where the spacetime metric may be
conformally rescaled by the inverse square of the lapse function, corresponding to
a conformalfactor
σ
equalto the reciprocalof the lapse function,
σ
=
M
−
1
or
σ
=
N
−
1
respectively (the optical gauge), leading to the optical spatial metric
whose components in an observer-adapted frame have been denoted by
γ
ab
and
g
ab
respectively. However, the “antioptical” gauge
σ
=
M
or
σ
=
N
respectively is
important in certain analyses of the gravitational field equations themselves in the
case of stationary spacetimes and in post-Newtonian approximations.
From their definitions, the spatial arclength parameter, the relative speed, and
the relative velocity unit vector must transform in the following way
d
˜
(
U,u
)
/d
(
U,u
)
=
σ ,
˜
ν
(
U, u
) =
d
˜
(
U, u
)
/dτ
(
U,u
)
=
σd
(
U, u
)
/dτ
(
U,u
)
=
σν
(
U, u
)
,
˜
ˆ
ν
(
U, u
)
α
=
σ
−
1
ˆ
ν
(
U, u
)
α
,
(A.7)
implying that the relative velocity vector
˜
ν
(
U, u
)
α
=
ν
(
U, u
)
α
=
P
(
u
)
α
β
dx
β
/dτ
(
U,u
)
(A.8)
is invariant.
One may also introduce the conformally rescaled energy and momentum (per
unit mass)
˜
E
(
U, u
)
=
σ
−
1
E
(
U, u
) =
σ
−
1
γ
(
U, u
)
,
˜
p
(
U, u
)
=
σ
−
1
p
(
U, u
) =
σ
−
2
γ
(
U, u
)˜
ν
(
U, u
)
,
(A.9)
˜
p
(
U, u
)
α
=
σ
−
2
p
(
U, u
)
α
˜
p
(
U, u
)
α
=
p
(
U, u
)
α
satisfying
˜
E
(
U, u
)
2
−
˜
p
(
U, u
)
2
=
σ
−
2
.
(A.10)
This choice for the conformal scaling of these last two quantities is made so that in
the special case of a Killing observer where
u
α
=
M
−
1
ξ
α
, with
σ
−
1
taken to be the
magnitude
M
of the Killing vector
ξ
α
, then the conformally rescaled spatial metric
is the optical spatial metric and the conformally rescaled energy is the conserved
quantity ˜
E
(
U, u
) =
−
p
(
U
)
α
ξ
α
which is constant along the test particle world line,
while a Killing component of the covariant spatial momentum remains a conserved
quantity if it is not rescaled.
Finally, for a null geodesic where the equations of motion are conformally invari-
ant, the new affine parameter can be defined by
d
˜
λ
P
/dλ
P
=
σ
2
as in Eq. (D.6) of
Wald, leading to this same choice of conformal transformation for the momentum
4-vector
P
α
=
dx
α
/dλ
P
and its corresponding 1-form as for the spatialmomentum
and its 1-form in the case of a massive particle. This is easily seen from the action
which gives the equations for affinely parametrized null geodesics as its Lagrangian
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
35
equations
g
αβ
(
dx
α
/dλ
P
)(
dx
β
/dλ
P
)
dλ
P
.
(A.11)
This choice of transformation for
λ
P
leaves the action invariant.
One may introduce a new spatialcovariant derivative ˜
∇
(
u
)
α
associated with the
conformally rescaled spatial metric by using the appropriate difference connection
as in Eqs. (D.3) and the sign reversalof (3.1.7) of Wald
40
˜
∇
(
u
)
α
X
β
=
∇
(
u
)
α
X
β
−
X
γ
C
γ
αβ
,
C
γ
αβ
=
[2
δ
γ
(
α
∇
(
u
)
β
)
−
g
αβ
∇
(
u
)
γ
](ln
σ
)
.
(A.12)
This in turn may be used to introduce a conformally rescaled total spatial covariant
derivative of each type, by replacing the spatial covariant derivative in the definition
valid for vector fields by the conformally rescaled derivative. For example, for a
spatialcovector one has
˜
D
(tem)
(
U, u
)
X
β
/dτ
(
U,u
)
=
D
(tem)
(
U, u
)
X
β
/dτ
(
U,u
)
−
X
γ
C
γ
αβ
ν
(
U, u
)
α
.
(A.13)
One can easily re-express the three-acceleration or rate of change of spatial
momentum in terms of this new derivative, using an immediate consequence of Eq.
(A.12)
X
α
˜
∇
(
u
)
α
X
β
=
X
α
∇
(
u
)
α
X
β
−
X
α
X
α
∇
(
u
)
β
(ln
σ
)
.
(A.14)
For example, for a congruence of test particle world lines one would have
D
(tem)
(
U, u
)
p
(
U, u
)
β
/dτ
U
=
E
(
U, u
)
∇
(tem)
(
u
)
p
(
U, u
)
β
+
p
(
U, u
)
α
∇
(
u
)
α
p
(
U, u
)
β
=
˜
D
(tem)
(
U, u
)˜
p
(
U, u
)
β
/dτ
U
+
p
(
U, u
)
2
∇
(
u
)
β
(ln
σ
)
,
(A.15)
where
˜
D
(tem)
(
U, u
)˜
p
(
U, u
)
β
/dτ
U
=
E
(
U, u
)
∇
(tem)
(
u
)
p
(
U, u
)
β
+
p
(
U, u
)
α
˜
∇
(
u
)
α
p
(
U, u
)
β
=
D
(tem)
(
U, u
)
p
(
U, u
)
β
/dτ
U
−
p
(
U, u
)
2
∇
(
u
)
β
(ln
σ
)
(A.16)
defines the rescaled total spatial covariant derivatives for either a congruence of test
particle world lines or a single such world line respectively. Raising the index on
these equations introduces an extra term from the temporalderivative of the con-
formal factor. These formulas yield the results for the threading and hypersurface
points of view in the opticalgauge for
u
=
m, n
, while replacing
n
by
n, e
0
in the
appropriate places in the hypersurface Lie form of these equations yields the slicing
version.
One can similarly transform the derivative of the unit velocity vector needed to
evaluate the relative centripetal acceleration, leading to the conformally rescaled
quantities corresponding to the relative curvature and radius of curvature and the
36
D. Bini, P. Carini and R.T. Jantzen
relative centripetal acceleration. If ˆ
ν
(
U, u
)
α
were actually a vector field on spacetime
rather than being defined only along a single world line, one could decompose its
spatially projected intrinsic derivative in the following way
D
(tem)
(
U, u
)ˆ
ν
(
U, u
)
β
/d
(
U,u
)
= [1
/ν
(
U, u
)]
D
(tem)
(
U, u
)ˆ
ν
(
U, u
)
β
/dτ
(
U,u
)
= [1
/ν
(
U, u
)][
∇
(tem)
(
u
) +
ν
(
U, u
)ˆ
ν
(
U, u
)
α
∇
(
u
)
α
]ˆ
ν
(
U, u
)
β
.
(A.17)
Now re-express this in terms of the conformally rescaled quantities using the results
of the first section of the appendix for the unit velocity vector to re-express the
spatial covariant derivative in it in terms of a conformally rescaled derivative ˜
∇
(
u
)
α
.
One finds
D
(tem)
(
U, u
)ˆ
ν
(
U, u
)
β
/d
(
U,u
)
= ˜
D
(tem)
(
U, u
)˜
ˆ
ν
(
U, u
)
β
/d
˜
(
U,u
)
+
P
u
(
U, u
)
(
⊥
)
β
α
∇
(
u
)
α
ln
σ
−
ˆ
ν
(
U, u
)
β
/ν
(
U, u
)
∇
(fw)
(
u
) l n
σ ,
(A.18)
where the first of the following equalities
˜
D
(tem)
(
U, u
)˜
ˆ
ν
(
U, u
)
β
/d
˜
(
U,u
)
= [1
/
˜
ν
(
U, u
)][
∇
(tem)
(
u
) + ˜
ν
(
U, u
)˜
ˆ
ν
(
U, u
)
α
˜
∇
(
u
)
α
]˜
ˆ
ν
(
U, u
)
β
=
D
(tem)
(
U, u
)ˆ
ν
(
U, u
)
β
/d
(
U,u
)
−
P
u
(
U, u
)
(
⊥
)
α
β
∇
(
u
)
α
ln
σ
+ ˆ
ν
(
U, u
)
β
/ν
(
U, u
)
∇
(fw)
(
u
) l n
σ
(A.19)
defines the equivalent action of the new derivative on a congruence of test particle
world lines, while the second defines it for a single such world line.
The magnitude of this conformalderivative of the conformalunit velocity de-
fines the conformally rescaled relative curvature and its reciprocal the conformally
rescaled radius of curvature, for the ordinary and corotating Fermi-Walker cases.
Multiplying the last relationship by the conformal square of the velocity gives the
conformally rescaled relative centripetal acceleration and its relationship to the orig-
inalone as long as the totalspatialcovariant derivative is orthogonalto the relative
direction of motion
˜
a
(
⊥
)
(tem)
(
U, u
)
β
=
σ
2
[
a
(
⊥
)
(tem)
(
U, u
)
β
−
ν
(
U, u
)
2
P
u
(
U, u
)
(
⊥
)
α
β
∇
(
u
)
α
ln
σ
+
ν
(
U, u
)
β
∇
(fw)
(
u
) l n
σ
]
,
tem=fw
,
cfw
.
(A.20)
Note that unless the observer temporal derivative of the conformal factor is zero,
the “conformal derivative” of the unit velocity is not orthogonal to the unit velocity
itself, even if it is before the conformal rescaling. The same is true of the conformally
rescaled relative centripetal acceleration.
In the stationary case with
σ
chosen for the opticalgauge, this temporalderiva-
tive vanishes while the spatial derivative produces the gravitoelectric field, and this
Intrinsic Derivatives and Centrifugal Forces in General Relativity
. . .
37
relationship can be rewritten in the form (using first Eq. (7.2) and then Eqs. (9.1)–
(9.4))
γ
(
U, u
)
2
σ
−
2
˜
a
(
⊥
)
(tem)
(
U, u
)
β
=
P
u
(
U, u
)
(
⊥
)
α
β
{
γ
(
U, u
)
2
[
a
(
⊥
)
(tem)
(
U, u
)
α
−
g
(
u
)
α
] +
g
(
u
)
α
}
(A.21)
=
P
u
(
U, u
)
(
⊥
)
α
β
[
a
(
U
)
α
+
g
(
u
)
α
+
γ
(
U, u
)
2
H
(tem)
(
u
)
αδ
ν
(
U, u
)
δ
]
.
Thus if in addition the transverse gravitomagnetic force is zero, one sees that the
opticalrelative centripetalacceleration changes sign when the transverse spatial
projection of the test particle acceleration just balances the transverse gravitoelec-
tric field, i.e., exactly opposes the transverse observer acceleration
γ
(
U, u
)
2
σ
−
2
˜
a
(
⊥
)
(tem)
(
U, u
)
β
=
P
u
(
U, u
)
(
⊥
)
α
β
[
a
(
U
)
α
−
a
(
u
)
α
]
,
(A.22)
γ
(
U, u
)
2
a
(
⊥
)
(tem)
(
U, u
)
β
=
P
u
(
U, u
)
(
⊥
)
α
β
[
a
(
U
)
α
−
γ
(
U, u
)
2
a
(
u
)
α
]
,
where for comparison the analogous relation for the relative centripetal accelera-
tion itself is given (just the transverse analog of the static case longitudinal ac-
celeration relation (3.13) of
30
). For static circular orbits, the optical relative cen-
tripetalacceleration then becomes outward pointing when the transverse test par-
ticle 4-acceleration becomes larger in magnitude than the transverse test observer
4-acceleration. This is the famous effect of the “reversal of the centrifugal force”
motivating the work of Abramowicz et al. The additional squared gamma factor
in the relative centripetal acceleration compared to the optical relative centripetal
acceleration prevents the reversal in the former case.
Corrections
This reformatted version contains one typo correction: line 4 after 15.10 (top of
p. 31 in the originalarticle) where “the opticalderivatives ˜
D . . .
” should have the
two uppercase D derivative symbols with an over tilde.