Download g= -(1-~)dt2+(1- ~r! g= -(1-~)dt2+(1

1996MNRAS.281..659S
Mon. Not. R. Astron. Soc. 281, 659-665 (1996)
On the notions of gravitational and centrifugal force in static
spherically symmetric space-times
Sebastiano Sonego 1 * and Marc Massar2t
lIntemational School for Advanced Studies, Via Beirut 2-4, 34014 Trieste, Italy
2RGGR, Universite Libre de Bruxelles, Campus Plaine CP 231, 1050 Brussels, Belgium
Accepted 1996 February 20. Received 1995 December 18
ABSTRACT
We present an assessment of some recent suggestions for performing a split between
gravitational and inertial forces in general relativity. We prove, by considering a
family of simple static space-times, that any such split leads necessarily to a
centrifugal force that points in the direction of the local, rather than global, outward
direction. We also show that Abramowicz's expression for the centrifugal force
emerges spontaneously when separating the variables in the Hamilton-Jacobi and
Klein-Gordon equations. Furthermore, we explain in some detail why the whole
idea is not necessarily in contradiction with the equivalence principle.
Key words: black hole physics - gravitation - relativity - waves.
1 INTRODUCTION
In 1974, Abramowicz & Lasota pointed out the existence of
a regime of anomalous dynamical behaviour for test particles in the Schwarzschild space-time (Abramowicz &
Lasota 1974, 1986). For the metric!
g=
-(1- ~)dt2+(1- ~r!
dr2+r2(d82+sin 28d(l),
(1.1)
one can write the four-velocity of a particle moving on a
circular orbit as
(1.2)
where 0 is a parameter, and r = (1 - 2M/r - 02r2 sin28) -112
because of the normalization u~u~= -1. If the motion is
uniform, i.e., 0 = constant, and we assume for simplicity
that the orbit lies in the equatorial plane corresponding to
8=n/2, the components of the acceleration a~:=u'V,u~
are
(1.3)
For a particle with mass m, the thrust P (i.e., the force of
non-gravitational origin) necessary in order to maintain it in
the orbit identified by rand 0 is given by ma~; we have then
the following situation.
(i) r> 3M. The thrust on a static particle (0 = 0) points in
the global outward direction. 2 As 101 increases, the intensity of the thrust decreases, until it vanishes when 0 2=M/r\
when the particle is geodesic and the motion Keplerian. For
greater values of I0 I, the thrust points in the global inward
direction, and its intensity increases with I0 I.
(ii) r = 3M. The thrust has a fixed value, independent of
I0 I, and points in the global outward direction.
(iii) 3M> r > 2M. The thrust points in the global outward
direction, and its intensity increases with I0 I.
In the first case, this behaviour is qualitatively in agreement
with the Newtonian-based idea that, in the rest frame of the
particle, the thrust should balance the sum of a velocityindependent attractive gravitational force and of a repulsive
centrifugal force that increases with speed. However, for
3M?:.r > 2M, it is obvious that this picture cannot be
maintained.
Of course, in general relativity one is not obliged to split
the particle acceleration into a gravitational and a centrifugal part, because Einstein's equivalence principle guaran-
* E-mail: [email protected]
tE-mail: [email protected]
'We work in units in which c =G = 1, and use the metric signature
+ 2. Greek indices 11, v, ... denote tensor components in some
chart.
2For reasons that will become clear later, we adopt the following
terminology. The global outward direction is the one along which
the area of the surface r=constant tends to infinity; the global
inward direction is the opposite one.
© 1996 RAS
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.281..659S
660 S. Sonego and M Massar
tees that any such distinction is locally unobservable, hence
fundamentally devoid of physical meaning. The equivalence
principle, however, holds also in pre-relativistic physics.
Hence, strictly speaking, splitting the force that acts on a
particle into a gravitational and an inertial part should be
meaningless even in Newtonian mechanics. As a matter of
fact, one could avoid introducing the notion of inertial
forces at all, and end up with Newton-Cartan's theory - a
reformulation of Newtonian mechanics where the somewhat metaphysical notion of 'absolute space' is not needed
(Misner, Thorne & Wheeler 1973, pp. 289-303). However,
when discussing specific problems, it is in most cases convenient to treat gravitational and inertial forces separately,
even though such a distinction has, admittedly, no operational foundation at a local level. In practice, the splitting is
performed by selecting a class of privileged observers; this
defines a reference frame in which no inertial forces are
present, and in which therefore only gravity acts. A further
hypothesis, that the gravitational force should not depend
on the observer's velocity, allows one to remove completely
any ambiguity and to arrive at a unique form of the splitting.
Of course, if one does not want to introduce absolute space,
there is no a priori notion of privileged observers; it is
therefore a rather surprising experimental fact that a class
of observers exists (the so-called inertial observers) which
leads to a definition of gravitational and inertial forces that
turns out to be very simple and practically useful. The physical reasons for this circumstance are ultimately unclear
(although they are often referred to as 'Mach's principle'),
but for our purposes it is sufficient to recognize that the
existence of such observers - hence of convenient separate
notions of gravitational and inertial forces - reflects the
existence of some particular symmetry in our physical
universe.
We have made this long digression in order to justify the
introduction of a split between gravitational and inertial
forces in general relativity. The situation, in fact, is not very
different from the one in Newtonian theory, when the latter
is freed from the concept of absolute space. If space-time
has enough symmetry to justify the introduction of a class of
privileged observers (as happens, e.g., in static space-times
for static observers, or in cosmological models for observers
orthogonal to the hypersurfaces of homogeneity), we may
expect that a split of this type could still be useful, either for
computational purposes or simply because it helps one's
intuition in understanding the dynamical behaviour even in
extreme relativistic regimes. It is nevertheless clear from the
example discussed at the beginning that at least one of the
following two Newtonian prejudices should be abandoned.
(1) The centrifugal force points always in the global outward direction, and increases with the particle speed.
(2) The gravitational force in the rest frame of the particle does not depend on the particle speed.
The choice of one or the other of these two possibilities has
led, in the past five years, to some controversy. In this paper
we shall show that hypothesis (1) must definitely be abandoned, because rejecting (2) alone is not sufficient in order
to account for the behaviour of particles, when space-times
sufficiently different from Schwarzschild are considered.
Although, in the literature on the subject, considerable
emphasis is placed on the behaviour of orbiting gyroscopes,
our arguments do not require a discussion of this topic, and
we shall thus focus exclusively on the mechanics of point
particles. More generally, we want to concentrate only on
the basic ideas necessary in order to make a comparison
between the two approaches, and therefore we shall not
discuss technical details or generalizations of the analysis.
The structure of this article is as follows. In the next
section we present a short review of two extreme interpretations of the phenomena described above. In Section 3 we
introduce static models of space-time where no gravitational field is present, and in which nevertheless particle
dynamics presents anomalies of the same type. In Section 4
we show that, when separating variables in the HamiltonJacobi and Klein-Gordon equations, a centrifugal potential
that violates property (1) appears spontaneously; this provides further evidence for the idea that the notion of centrifugal force should be modified when dealing with general
relativistic situations. Our conclusions are presented in
Section 5.
2 SPLITTING THE FORCE
If we think of the particle in its rest frame as being in
eqUilibrium under the action of the thrust F 1', a gravitational
force GI'=m/t, and a centrifugal force CI'=mcl', we are led
to write
/t+cl'= -al'.
(2.1)
In order to identify separately the terms/t aad cl', however,
it is necessary to propose additional hypotheses.
For a static particle, the thrust must balance only the
gravitational force. We have then, from (1.3) with 0=0 and
(2.1),
M
/t= -c5~2'
r
(2.2)
If we want to preserve the same form of the gravitational
part for an observer ul' with arbitrary a, we find that the
components of the centrifugal part cl' are
1'_
C
~I'
-Ur
r-3M
n2
(2.3)
:L3ur.
r-2M-Or
Whereas the gravitational acceleration points always in the
global inward direction (i.e., gravity is always attractive), the
centrifugal acceleration given by (2.3) changes sign as r
crosses the value 3M. For r> 3M, cl' points in the global
outward direction, analogously to what happens in the
Newtonian case. However, for r = 3M we have cl' = 0, and for
3M> r > 2M we see that cl' points in the global inward
direction.
This splitting of the acceleration, aI', into an ordinary
gravitational part and an unusual centrifugal one accounts
easily for the paradoxical behaviour described in the previous section (Abramowicz & Lasota 1986; Abramowicz &
Prasanna 1990), and can be straightfOlwardly generalized to
the case of an axially symmetric static space-time (Abramowicz 1990). Strictly speaking, however, we have only shifted
the difficulty, because the expression (2.3) for the centrifugal acceleration is ad hoc and has no kinematical justifica© 1996 RAS, MNRAS 281, 659-665
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.281..659S
Gravitational and centrifugal forces
tion yet. This problem can be solved by introducing the
concept of 'optical reference geometry' (Abramowicz, Carter & Lasota 1988).
For the Schwarzschild space-time, the optical reference
geometry is defined by a metric h on the hypersurfaces
t = constant that differs from the usual directly projected
one h by a conformal factor (1 - 2M/r):
h=(l-
~ -2 dr2+(1- ~ ->(d82+ sin28 d(l).
(2.4)
A possible justification for adopting h rather than h as
metric on the sub manifold t = constant is the following. In
general relativity, only the space-time metric g is determined unambiguously; hence, if a hypersurface !/ is
assigned in a space-time vii, there is only one metric on !/
which is intrinsically defined - the directly projected metric
h, induced by g through the embedding of !/ in vii. However, if the space-time possesses additional structure, as for
example a Killing vector field or, more generally, a privileged vector field, we have the freedom to make use of these
extra elements in defining metrics on !/ that may differ
from h. Of course, different metrics give !/ different geometrical properties, correspond to different operational
procedures for measuring distances and serve different purposes. The optical reference geometry, of which (2.4) is an
example, is defined for a static space-time by the relation
h = ( - 'v't) -Ih,
(2.5)
where rt = <5:' is the time-like Killing vector field, and is
particularly suitable for the analysis of optics and particle
dynamics (Abramowicz et al. 1988; Abramowicz 1992).
Without entering into umiecessary details, we can get a
feeling of the way in which the use of the optical reference
geometry explains the strange behaviour of the centrifugal
force (2.3), simply by looking at Fig. 1, which represents an
embedding diagram of the surface t = constant, 8 = n/2 of
Schwarzschild space-time, with the metric (2.4) (Abramowicz et al. 1988). If we consider a particle constrained on a
similar surface in ordinary three-dimensional Euclidean
space, and focus only on the dynamically relevant tangential
component of the centrifugal acceleration, we recover the
same qualitative behaviour of eJl (Abramowicz & Prasanna
661
1990; Abramowicz 1992; Abramowicz & Szuszkiewicz
1993). In particular, we understand the phenomenon of
centrifugal force reversal as related to the reversal of the
local concepts of inward and outward when r = 3M is
crossed. The centrifugal force points always in the direction
of the local outward direction, and its reversal when
3M> r > 2M simply means that in this region the local and
global notions of outward do not coincide.
An alternative splitting of the acceleration aJl into a gravitational and a centrifugal part can be given which preserves
property (1). Then, of course, the expression of the gravitational force in the particle rest frame will contain its speed
with respect to a static observer. One way to fix the precise
form of this dependence is to require that the gravitational
force measured by some observer could be written as m d 2p/
dr2, where p and r are, respectively, the observer's proper
length in the radial direction and proper time (de Felice
1991). Using the fact that, for a static observer, the gravitational acceleration points in the global inward direction, and
has absolute valueg(O)=(l-2M/r)-I!2 M/r2, it is then easy
to see that the observer uJl should feel a gravitational part
~ -1/2 _l---:--:-M
g(Q)= ( 1- -r-}
r-2M _Q2r3 - ; '
(2.6)
also directed to the global inward direction. By subtraction,
the centrifugal part has the absolute value
_ ( _ 2M)112
Q2r2
23'
r
r-2M-Qr
e(Q)- 1
(2.7)
and its direction coincides always with the direction of
global outward. On writing g(Q) =hJlvgIlgV, where
hJlv: =gJlV + uJlu" and a similar expression for e (Q), we have
the form of the components gil and eJl in this approach.
As we said, the centrifugal force (2.7) is always repulsive.
The anomalous dynamical behaviour described in the introduction now finds its root in the property of gil of being
Q-dependent, and, more precisely, of increasing its strength
when Q increases. It is possible to see that, for r=3M, (2.6)
and (2.7) vary with Q exactly in the same way, and that for
3M > r > 2M the gravitational force increases more than the
centrifugal one when Q is increased (de Felice 1991; Bar-
Figure 1. Embedding diagram for part of the surface t = constant, () = n/2 of Schwarzschild space-time, with the optical reference geometry
(2.4) (Abramowicz et al. 1988).
© 1996 RAS, MNRAS 281, 659-665
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.281..659S
662 S. Sonego and M. Massar
rabes, Boisseau & Israel 1995). This explanation can be
contrasted with the one based on (2.2) and (2.3) in that it
suggests a dynamical, rather than geometrical, origin of the
anomaly.
(3.3)
As a first simple case exhibiting anomalous behaviour, let
us choose Einstein's static universe, which corresponds to
[(X) =sinx, with X E (0, n). Equation (3.3) becomes
3 A SIMPLER EXAMPLE
As far as the analysis in the Schwarzschild space-time is
concerned, no compelling evidence can be provided in
favour of either of the two views outlined in the previous
section. As a matter of fact, only aesthetic (hence inconclusive) or practical arguments have been formulated until
now by the supporters of the first approach. On the other
hand, the counter-intuitive idea of a centrifugal force reversal and the need to resort to the uncommon notion of
optical reference geometry have been often criticized. In
this section we shall discuss particle dynamics in a simpler
static space-time for which I'/~I'/P= -1, hence h =h and no
objection of the second kind can be raised. This choice has,
however, a further and even more important advantage. As
we shall see in a moment, the gravitational part of the
acceleration for circular orbits in such a space-time turns
out to vanish according to both the approaches described in
Section 2. Any anomalous behaviour of the thrust FP must
therefore be taken as incontestable proof that the centrifugal force in general relativity, under suitable circumstances, does present the surprising qualitative features of
(2.3).
Let us consider a family of static, spherically symmetric
space-times with the metric
g= -dt2+RZ[dXz+[(X)Z(d02+sinZOdql)],
(3.1)
where R is a constant parameter, 0 and qJ are usual angular
coordinates, t E IR, and X E (0, Xmax), with Xmax possibly
infinite. We leave for the moment the function [(X)
unspecified, apart from reasonable requirements of regularity.3 The fact that, if the Einstein field equation is
imposed, (3.1) might turn out to imply an unphysical stressenergy-momentum tensor (except perhaps if a non-vanishing cosmological constant is introduced) is completely
irrelevant for our discussion.
For a particle with four-velocity (1.2) in the metric
(3.1),
the
normalization
condition
gives
r=
[1 - QZRZ[ (X)Z sin z 0] -liZ. The calculation of the acceleration is straightforward, and leads to
where a prime denotes the derivative with respect to X. For
a static observer (0.=0) we have a"=O; this means that the
gravitational force vanishes identically in this space-time,
regardless of which splitting procedure (as discussed in the
previous section) is used. As a consequence, the centrifugal
force is unambiguously identified as c~ = - aP• In order to
simplify the rest of the discussion, we restrict ourselves to
considering orbits on the equatorial plane, 0 = rrJ2; the
centrifugal force then reads
3For example, one must have f' (0) = 1 in order for the geometry to
be locally Minkowskian when X= O.
(3.4)
which changes sign as X crosses the value n/2. The embedding diagram of the surface t = constant, 0 = n/2, is a sphere
with parallels that correspond to the circular orbits of particles; this shows very clearly the link between the reversal of
the centrifugal force (3.4) and the reversal of the notion of
the local outward direction.
One might still argue that in the Einstein universe there is
no global outward direction, and that the unavoidable
reversal of the centrifugal force in this example is not sufficient to justify the idea that the same should happen when a
global outward does exist, as for example in Schwarzschild
space-time. The weakness of this argument is evident if we
consider the metric (3.1) with X E (0, + CD) and [(X) such
that the embedding diagram of the surface t = constant,
0= n/2 is qualitatively similar to the one of Fig. 2 [a possible
choice would be, for example, [(X) = X - (1.X z exp( -/h),
with (1., f3 > suitably arranged]. We have now a space-time
in which no gravitational force is acting and the global
notion of outward is well-defined, and in which nevertheless
the thrust (i.e. the centrifugal force) vanishes for X= X± [the
zeros of f' (X )], and points in the global inward direction
when X E (X _, X+). Furthermore, we see again from the
embedding diagram that the direction of the centrifugal
force coincides with the one identified by the local outward
direction. It is easy to check that this property holds indeed
for all the metrics in the class (3.1).
°
4 CENTRIFUGAL POTENTIAL FOR FREE
PARTICLES AND WAVES
The previous discussion makes the idea of a centrifugal
force pointing in the local, rather than global, outward
direction less difficult to accept. In this section we show that
attractive centrifugal potentials emerge spontaneously
when one tries to solve the Hamilton-Jacobi equation with
the method of separation of variables, and that the same
happens during the resolution of the wave equation for a
scalar field. We shall work in a general static spherically
symmetric space-time, the metric of which can always be
written as
g= -A(r)Zdtz+B(r)Zdrz+rZ(d02+sinzOdqJZ),
(4.1)
with A and B suitable functions.
The Hamilton-Jacobi equation for a free particle with
massm,
g~'V"SVvS +mz=O,
(4.2)
can be completely separated, introducing the three constants of motion L z : = oS/oqJ, e: = (OS/OO)2 + L;/sinzO, and
E: = aS/ot (representing, respectively, the azimuthal component of the angular momentum, the square of its absolute
© 1996 RAS, MNRAS 281, 659-665
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.281..659S
Gravitational and centrifugal forces
663
Figure 2. Embedding diagram for part of the surface t=constant, (J=n/2 of the static space-time (3.1) withf(X) = X- CXX2 exp( - PX).
value, and the particle energy, all with respect to a static
observer at infinity). Using the relation
dr
1
as
m---dt -B 2 or'
where
t
(4.3)
fugal potential Ve: =A2L 2/2m 2r2 gives rise to an 'acceleration' - d Ve/dr that coincides with (2.3).
Let us now consider a free scalar field ¢ obeying the
Klein-Gordon equation
(4.6)
is the particle proper time, one finally arrives at
where p, is a constant. Separating the variables as
(4.7)
(4.4)
where Y 1m (e, <p) is a spherical harmonic, and introducing the
coordinate
where the effective potential V (r) is given by4
1 2
2
L2
V(r)=-A +A -----z:z.
2
2mr
(4.5)
It is natural to identify the two terms on the right-hand side
of (4.5) with a gravitational and a centrifugal potential,
respectively. For the Schwarzschild space-time A =
B- 1 = (1- 2Mlr)1!2, and one can easily check that the centri-
r*:=f dr B(r)
A(r) ,
which generalizes the Regge-Wheeler 'tortoise coordinate'
of Schwarzschild space-time, we arrive at the one-dimensional Schrodinger equation (see also Futterman, Handler
& Matzner 1988)
{
4V (r) differs from the effective potential Y (r) defined elsewhere
(see, e.g., Misner et al. 1973, pp. 656, 659-661), in that2V= 0. We
prefer to call V, rather than Y, 'effective potential', because it
reduces directly to the Newtonian effective potential (apart from
an additive constant) in the non-relativistic limit.
(4.8)
d2
+ [2
w dr*2
1 d2r
r dr*2
2
2 21(I + 1)]} (rR/) =0.
- - -A p, -A - -
r2
(4.9)
The potential term in (4.9) contains the same centrifugal
part that we have already discussed in connection with the
Hamilton-Jacobi equation.
© 1996 RAS, MNRAS 281,659-665
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.281..659S
664 S. Sonego and M Massar
5 CONCLUSION
In this article we have discussed the possibility of splitting
the force that acts on a particle in a static space-time into a
gravitational and an inertial part, and we have compared
two specific suggestions for doing so. At first sight, the very
concepts of 'gravitational' and 'inertial' force sound wrong
in general relativity, because they seem to contradict the
equivalence principle. However, we have already argued in
the introduction that the situation is fundamentally the
same in Newtonian theory, where nevertheless no objections are raised to the use of inertial forces although no
local experiment can be performed that allows one to distinguish them from forces of gravitational origin. If spacetime possesses some symmetry defining a class of observers
that can reasonably be thought of as non-accelerating, then
a distinction between gravity and inertia can be made which
might turn out to be practically useful even in the context of
general relativity. As this turns out indeed to be the case
(Abramowicz & Prasanna 1990; Abramowicz & Miller
1990), it seems that there is no reason - apart from prejudice - to prevent oneself from taking advantage of this
new computational and intuitive paradigm.
Having clarified the motivation and meaning of the splitting, we can summarize the results of this paper by saying
that any notion of centrifugal force in general relativity must
necessarily violate property (1) (see Introduction), and that
the idea of a local reversal of the notions of inward and
outward should be taken seriously. The alternative extreme
possibility, to attribute the anomalous dynamical behaviours entirely to the gravitational strength, can be ruled out
by considering the examples treated in Section 3, in which
no such argument can be formulated for the simple reason
that there is no gravitational field at all. Of course, our
discussion leaves open the possibility that both properties
(1) and (2) should be dropped, but at the moment there
seems to be no reason for doing that, especially if we consider that Abramowicz's expression (2.3) for the centrifugal
force emerges independently within the context of the
Hamilton-Jacobi and wave equations. Incidentally, the
latter result suggests that waves also 'feel' the reversal of the
centrifugal force, and it would be very interesting to see
whether there are properties of scattering from black holes
that can easily be explained with the help of this idea.
The notion of inertial forces is ultimately a geometric
one; it should not therefore be surprising that, in a nonEuclidean space, the centrifugal force has properties that
differ from those in a Euclidean space. The only difficulty is
that the centrifugal force turns out to be linked not to the
directly projected geometry h of space, but rather to the
optical reference geometry it given by (2.5); this fact is
sometimes used in order to argue that the entire subject is
artificial. However, we have already noted in Section 2 that
the choice of a three-geometry is not dictated by any fundamental principle in the theory of general relativity, and can
thus be made on purely utilitarian grounds. 5 This remark,
together with the previous arguments in favour of AbramoSOne might say that the arbitrariness of the choice of a threegeometry reflects just the arbitrariness of making a split between a
gravitational and a centrifugal force. When a three-geometry is
chosen, the centrifugal force is also uniquely determined.
wicz's extension of the notion of centrifugal force, makes
the use of it perfectly legitimate even for those cases in
which 1J~1JP oF -l.
In this paper we have deliberately restricted ourselves to
considering circular orbits in static, spherically symmetric
space-times, for which we believe that both the notion of a
centrifugal force and its interpretation in terms of the optical reference geometry are well justified. The extension to
more complicated cases requires some caution, because of
the possible occurrence of effects that cannot be accounted
for by forces with simple Newtonian analogues. A very
important example is provided by Kerr space-time, in
which the anomalous dynamical behaviour of particles in
circular orbits is a-dependent (de Felice & UsseglioTomasset 1991; de Felice 1995), and cannot thus be attributed only to the effect of a three-geometry, but requires
the additional introduction of a Lense-Thirring force
(Abramowicz, Nurowski & Wex 1995).6 Furthermore, any
extension of the method requires that a family of privileged
observers n~ be singled out, in order to define uniquely the
notion of gravitational force. The generalization to arbitrary
space-times considered until now (Abramowicz 1993;
Abramowicz et al. 1993), however, allows for an infinite
number of such observers, and therefore leaves the split
between gravity and inertia completely arbitrary from the
outset (Sonego & Massar 1995). It seems plausible that
further progress along this line will require additional conditions to be imposed on n P, which will lead automatically to
the intuitive choice in simple cases (non-accelerating observers in Minkowski space-time; static observers in static
space-times; observers orthogonal to the hypersurfaces of
homogeneity in cosmological models).
ACKNOWLEDGMENTS
It is a pleasure to thank Dr P. Nardone and R. Rosin for
their help in the preparation of the figures. SS would like
also to thank Professor D. W. Sciama for hospitality at the
Astrophysics Sector of SISSA. This work was partially supported by the Directorate-General for Science, Research
and Development of the Commission of the European
Communities (DG XII-B) under contract No. CIl */CT94/
0004.
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© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
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Gravitational and centrifugal forces
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