Abstract. Gravity has no aberration, and propagation delays cannot be used
without destroying angular momentum conservation at an unacceptable rate. Even the curved spacetime explanation ("gravity
is just geometry") breaks down when masses and speeds are large, as in binary
pulsars. But if gravity or spacetime
curvature information is carried by classical propagating particles or waves, a
modern Laplace experiment places a lower limit on their speed of 10^{10}
c. The socalled Lorentzian
modification of special relativity permits such speeds without need of
tachyons. But there are other
consequences. If ordinary gravity is
carried by particles with finite collision crosssection, such collisions would
progressively diminish its inverse square character. Gravity would gradually convert to inverse linear behavior on the
largest scales. Curiously, at all
scales greater than about 2 kiloparsecs, gravity can be modeled without need
for dark matter by an inverse linear law. The orbital motions of Mercury and Earth may also show traces of this
effect. Moreover, if gravity were
carried by particles, a collapsed ultradense mass between two bodies could
shield each of them from the gravity of the other. Anomalies are seen in the motions of certain artificial Earth
satellites during eclipse seasons that behave like shielding of the Sun's
gravity. Certain types of radiation
pressure might cause a similar behavior, but require far more free parameters
to model. Each of these effects of
particlegravity models has the potential to lead to a breakthrough in our
postEinsteinian understanding of gravitation. This would also change our views of the nature of time in relativity
theory.
Properties of Gravity
Gravity has some
curious properties. One of them is that its effect on a body is apparently
completely independent of the mass of the affected body. As a result, heavy and
light bodies fall in a gravitational field with equal acceleration. Another is
the seemingly infinite range of gravitational force. Truly infinite range is
not possible for forces conveyed by carriers of finite size and speed  a point
we will elaborate in Part II.
Another curious
property of gravity is its apparently instantaneous action. By way of contrast,
light from the Sun requires about 500 seconds to travel to the Earth. So when
it arrives, we see the Sun in the sky in the position it actually occupied 500
seconds ago rather than in its present position. (Figure 1.) This difference
amounts to about 20 seconds of arc, a large and noticeable amount to
astronomers.
From our perspective,
the Earth is standing still and the Sun is moving. So it seems natural that we
see the Sun where it was 500 seconds ago, when it emitted the light now
arriving. From the Sun's perspective, the Earth is moving. It's orbital speed
is about 10^{4} c, where c is the speed of light. So light from the
Sun strikes the Earth from a slightly forward angle because the Earth tends to
"run into" the light. The forward angle is 10^{4} radians (the ratio
of Earth's speed to light speed), which is 20 arc seconds, the same displacement
angle as in the first perspective. This displacement angle is called
aberration, and it is due entirely to the finite speed of light. Note that
aberration is a classical effect, not a relativistic one. Frame contraction and
time dilation effects are four orders of magnitude smaller, since they are
proportional to the square of the ratio of speeds.
Figure 1 
Now we naturally
expect that gravity should behave similarly to light. Viewing gravity as a
force that propagates from Sun to Earth, the Sun's gravity should appear to
emanate from the position the Sun occupied when the gravity now arriving left
the Sun. From the Sun's perspective, the Earth should "run into" the
gravitational force, making it appear to come from a slightly forward angle
equal to the ratio of the Earth's orbital speed to the speed of gravity
propagation.
This slightly forward
angle will tend to accelerate the Earth, since it is an attractive force that
does not depend on the mass of the affected body. Such an effect is observed in
the case of the pressure of sunlight, which of course does depend on the mass of
the affected body. The slightly forward angle for the arrival of light produces
a deceleration of the bodies it impacts, since light pressure is a repulsive
force. Bodies small enough to notice, such as dust particles, tend to spiral
into the Sun as a consequence of this deceleration, which in turn is caused by
the finite speed of light. This whole process is called the PoyntingRobertson
effect.
But observations
indicate that none of this happens in the case of gravity! There is no
detectable delay for the propagation of gravity from Sun to Earth. The
direction of the Sun's gravitational force is toward its true, instantaneous
position, not toward a retarded position, to the full accuracy of observations.
And no perceptible change in the Earth's mean orbital speed has yet been
detected, even though the effect of a finite speed of gravity is cumulative over
time. Gravity has no perceptible aberration, and no PoyntingRobertson effect 
the primary indicators of its propagation speed. Indeed, Newtonian gravity
explicitly assumes that gravity propagates with infinite speed.
The Speed of Gravity
The absence of
detectable aberration implies that, to the extent that gravity is a propagating
force, its speed of propagation must be very high compared to that of light. In
the early 19^{th} century, Laplace
(Mechanique Celeste; English translation
reprinted by Chelsea Publ., New York, 1966) used the possible error in the
determination of the absence of an acceleration of the Earth's orbital speed to
set a lower limit to the speed of gravity of about 10^{7} c. Using the
same technique with modern observations, Van Flandern (Dark Matter, Missing
Planets and New Comets, North Atlantic Books, 1993) improved that lower
limit to 10^{10} c.
General relativity (GR),
of course, has an explanation for this experimental result that does not involve
fasterthanlight propagation. GR suggests that gravity is not a force that
propagates. Instead, the Sun curves spacetime around it; and the Earth simply
follows the nearest equivalent of a straight line available to it through this
curved spacetime. But it has been known since the time of Sir Arthur Eddington
that the curved spacetime explanation is not required by general relativity [see
Van Flandern, T., "Relativity with Flat Spacetime", MRB 3, 912
(1994)] or certain other variants that preserve agreement with the classical
observational tests of the theory. Other authors have proposed minor
modifications of the field equations to replace spacetime curvature tensors with
gravitational energymomentum density tensors [Rosen, N., "General Relativity
and Flat Space. I & II", Phys.Rev. 57, 147153 (1940)]. Indeed,
there is even some direct experimental evidence against the curved spacetime
explanation that is provided by neutron interferometers.
["The Role of Gravity in Quantum Mechanics", D.M. Greenberger and A.W.
Overhauser, Sci.Amer. 242, May, pp. 6676 (1980).] The results
are incompatible with the geometric weak equivalence principle because the
interference depends on mass.
A general problem with purely
geometric explanations of gravity is that they ignore causality. How does
spacetime far from a large mass get its curvature updated without detectable
delay so that orbiting bodies accelerate through space toward the true,
instantaneous position of the source of gravity?
In particular, computer
experiments show that binary pulsars are especially sensitive to this test. To
satisfy observations, it is not sufficient that each massive companion of a
binary pulsar acts from its retarded position; nor from its linearly
extrapolated position over one lighttime, as electromagnetic forces do. It is
not even sufficient for each companion to accelerate via the full curvature that
spacetime would have had one lighttime ago. Some information is being
propagated between source and affected body faster than light unless we assume
that the passive, nonintelligent processes of nature are capable of orbit
determination and extrapolation!
We can further illustrate this
dilemma for GR with two examples involving black holes. A black hole emits no
light because escape velocity is greater than c. Yet it still has gravity.
This is explained as due to the presence of a "fossilized" field, a curvature of
spacetime outside the black hole's event horizon that remains after the star
that created the hole collapsed. But the black hole may well be an orbiting
companion of a normal star. How does the "fossilized" field know about
accelerations of the center of mass behind the event horizon caused by the
normal star, so that it can accurately keep pace?
There are two problems here. 1)
The curvature of spacetime created by the normal star is sufficiently different
at points inside the event horizon of the black hole from what it is for points
outside that nothing outside the event horizon could remain in proximity to
something inside for very long without some sort of linkage across the horizon.
2) The curved spacetime generated by the normal star should require an infinite
time to reach the center of mass of the black hole, leaving the singularity in
the black hole unaware of the current state of curvature of spacetime that it
must respond to without detectable delay.
Figure 2 
The second example consists of two
identical black holes that make a close approach, and then recede again to
infinity. (Figure 2.) Despite the complex interactions between black holes
when they draw close, an observer riding the balance point between the two could
remain there indefinitely, and recede again to infinity, without experiencing
strong gravitational forces or being drawn toward either hole, because of the
balance and symmetry of the example. This would be true even if the event
horizons of the two holes came to overlap, allowing the observer to peer into
the spacetime formerly hidden behind both event horizons!
Such paradoxes could not be
constructed if GR were not trying to insist that gravitational information must
not propagate faster than light. But abandoning the light speed limit does not
mean abandoning GR. The main properties of the theory, including its
satisfaction of the four classical observational tests, can be retained in flat
spacetime versions of the theory as in the papers already cited.
So the first new property of
gravity proposed here is a propagation speed far greater than the speed of
light. This flatly contradicts a corollary of special relativity (SR), wherein
it is proved that no communication faster than light speed in forward time is
possible. SR is also a welltested and confirmed theory. But the emphasis has
been on all the experimental tests that SR has already passed, demonstrating the
reality of time dilation, space contraction (indirectly), and the increase in
inertial mass with speed, as well as the independence of measured light speed on
the motion of its source, etc.
However, it is often forgotten
that there are two postulates underlying SR, not just one. And the first
postulate, called the "covariance" postulate, requires that no inertial frame be
"special" since all are equivalent for formulating the laws of physics. Since
almost all SR experiments of the past have been done in the "laboratory", it has
not been possible to confirm this frameindependence postulate experimentally.
Only two historical experiments have made the attempt: the Sagnac experiment in
1913, and the MichelsonGale experiment in 1925. Both utilized rotating
reference frames, and both obtained nonzero fringe shifts in
MichelsonMorleytype experiments performed on rotating platforms. Both
published results claiming to be experimental contradictions of SR. However, SR
has long since developed an "explanation" for these results, and incorporated
the Sagnac effect for rotating frames as a standard part of the model.
Both Sagnac and Michelson favored
an alternate formulation of SR that allows a "universal time", as originally
advanced by Lorentz. [H.A. Lorentz, Lectures on theoretical physics,
vol. 3, Macmillan & Co., London, pp. 208211 (1931).] The modern formulation of
this idea is referred to as the "MansouriSexl" transformation. [R. Mansouri
and R.U. Sexl, "A test theory of special relativity: I. Simultaneity and clock
synchronization", Gen.Rel.&Grav. 8, 497513 (1977) The respective
equations for Einstein SR and the Lorentzian (MansouriSexl) alternative are
these:
Lorentzian SR equations:
_{} 
Einstein SR equations:
_{} 
Both of these transformations
relate coordinate X and time T in one inertial frame ("the
laboratory") to x and t in a frame moving relative to the
laboratory in the X direction with speed v. The
dilationcontraction factor,
_{}, is
always Â³
1.
These two sets of
equations differ only by the term
_{}in the
Einstein SR time transformation when it is expressed in the second form shown.
The reality of this term has never been tested by past experiments because the
term is always zero (or at least constant) when only a single clock represents
time in the "moving" frame. This is because the single clock in that frame is
usually placed at x = 0 by definition of the origin in that frame. Yet
this term plays a crucial role in the formula for the addition of velocities in
SR, which in turn plays the central role in the proof that nothing can propagate
faster than light speed in forward time.
In short, this
fundamental tenet of modern physics, the impossibility of fasterthanlight
propagation in forward time, rests on an experimentally unverified aspect of the
theory of special relativity. This fact is a frequent topic of discussion in
the journal Galilean Electrodynamics, one of whose aims is a fuller
exposition and understanding of the role of special relativity and its
alternatives from both theoretical and experimental perspectives.
Can we now perform the
crucial test required to choose between these two forms of SR? The Global
Positioning System (GPS) is a network of 24 satellites carrying atomic clocks on
board, now in various orbits around the Earth. When two or more GPS satellite
clocks are compared with ground clocks, Einstein SR requires that the satellites
cannot simultaneously be synchronized with one another and with the ground
clocks because the term
_{}cannot
simultaneously be zero between all pairs of relatively moving clocks. Since the
dimensions of the satellite orbits are about 80 light milliseconds in radius,
and since v/c for the satellites is about 10^{5}, the predicted
discrepancy can be on the order of 800 nanoseconds, and easily detectable.
For the actual GPS
satellite network, the rate of each orbiting clock has been preadjusted while
still on the ground so that the average length of the second will be the same
for orbiting clocks as for ground clocks. This is equivalent to setting
g
= 1 in the preceding transformation equations. That should not affect the
distinguishing term in question here, since no
g
factor appears in that term. Nonetheless, simultaneous and continuing
synchronization between all satellites and all ground clocks to a precision of a
few nanoseconds has already been achieved.
On the face of it,
this extraordinary fact tells us that the term
_{}is not
present in the transformations relating real clocks, and seems to comprise the
first experimental contradiction of Einstein SR in favor of its Lorentzian
cousin. In Einstein SR, there should be no distant simultaneity between
relatively moving clocks. And even if epoch and rate offsets are introduced to
synchronize relatively moving clocks at one instant, that synchronization could
not be maintained as each satellite clock continually changes its inertial frame
through its orbital motion.
Whether or not GPS
clocks can still be related to each other with the SR time transformation named
after Lorentz is still under debate. But Lorentz transformations are just one
of a family of transformations in which the speed of light is constant. [H.P.
Robertson and T.W. Noonan, Relativity and Cosmology, W.B. Saunders Co.,
Philadelphia (1968); cf. pp. 4650.] The dependence of the speed of light on
the speed of the observer depends on the method of synchronization of clocks,
since speed measurement involves more than one point in space and instant of
time.
What the GPS system
has shown is that, in the classical "Twins Paradox" problem, the traveling twin
could have carried along a second clock preset in epoch and rate such that it
always reads the correct time back on Earth throughout the traveler's journey.
For that is what GPS satellites clocks are  clocks in relatively moving frames
that maintain their synchronization with ground clocks even as they travel at
high relative speeds and change frames relative to the ground clocks.
So if there is no
_{}term in
the transformations, then the proof that nothing can propagate faster than light
fails, and there is no longer a need for elaborate rationale to explain the
simple fact that gravity has no aberration. The explanation can be simply that
gravity propagates much faster than light. In like manner, elaborate arguments
for nonlocality in quantum physics can be understood without invoking such
radical hypotheses as "There is no deep reality!"
Particle Models of Gravity
What
is a "Particle Model" of Gravity?
The
basic idea behind particle models of gravity is that space is filled
with a flux of rapid, randomly moving particles, so tiny that ordinary
matter is almost transparent to them. (Neutrinos, for example, pass
through the Earth virtually without noticing.)
Then
objects on Earth feel a downward force because more such particles
strike them from above than from below because the Earth absorbs some of
the particles coming from below.
In
general, any two bodies in space would shadow one another from some
particle impacts (Figure 3), resulting in an acceleration of each body
toward the other that depends directly on the number of "matter
ingredients" (mass) within the other body, and inversely on the square
of the distance between the two bodies.
Figure 3

The 18^{th}
century physicist LeSage is usually credited with the first particle model of
gravity, although LeSage himself says he was inspired by even earlier writers.
[G.L. LeSage, Berlin Mem. 404
(1784).] A flux of tiny, rapidly moving particles in
space is an elegant way to explain the gravitational force, including
relativistic effects. [see inset; also, T. Van Flandern, "Relativity with flat
spacetime", MRB 3, 913 (1994).]
However, particle
gravity models imply the existence of certain properties that are not as yet
discovered. For example, it implies that bodies will experience resistance as
they move through the particle flux. Of course, if the particles are very tiny
and transfer momentum primarily by means of a very high speed rather than by
being relatively massive, the resistance they pose to bodies moving through the
particle medium will be minimal. Indeed, the ratio of the mass of one particle
to the mass of a single "matter ingredient" within a body is constrained to be
quite small by the absence of an observable resistance for larger bodies moving
through the particle medium. (A matter ingredient is defined as the largest
element of mass within a body that will totally absorb any gravityproducing
particles it encounters.) A future detection of such resistance would weigh
heavily in favor of particle models; but it would be difficult to distinguish it
from the effects of tidal friction or other causes of orbital acceleration. The
acceleration seen in binary pulsars would set an upper limit to the possible
size of such a resistanceinduced acceleration.
Absorption of the
particle flux by matter ingredients would result in the heating of bodies
through energy absorption. This heat must be fully reradiated into space, and
the body must be in thermodynamic equilibrium, or the particle flux would cause
it to melt. In that connection, it is perhaps noteworthy that the six largest
planets all appear to radiate more heat back into space than they take in from
the Sun. It has been traditional to attribute this excess to radioactivity in
the planetary cores, although no observational evidence supports this
conjecture. We now see another possibility for the heat excess of large
planets, and a contributor to the radiative energy of stars.
The Range of Gravity
Another property of
particle models of gravity is a finite range for the force. Generally, one body
seems to attract another at any distance. In particle models this is because
the bodies shadow one another from some particle impacts, resulting in a net
push toward one another. But these particles must have finite dimensions,
however small they may be; and finite speeds, however fast they may be. So
there must exist some characteristic distance,
r_{G},
that a particle can travel before it will likely run into another similar
particle and change course. If two large bodies are separated by much more than
the distance r_{G},
the shadowing effect they have on one another will be diluted or canceled by the
backscattering into the shadow of particles colliding with other particles.
Objections to Particle Models of Gravity
Â·
If
particle collisions with matter are elastic, momentum is conserved and
no (gravitational) net force will result. [Ans: Particle collisions
must be inelastic. Particles lose velocity and raise the temperature of
the impacted mass.]
Â·
The
temperature of matter would be continually raised by particle
collisions. [Ans: Matter must radiate energy isotropically to maintain
an equilibrium. This is analogous to radiation pressure from light.]
Â·
Particles must travel very rapidly to convey the necessary momentum to
matter, yet produce no detectable frictional resistance to motion. [Ans:
The minimum particle speed consistent with experimental data is 10^{10}
c. This is also consistent with the lack of detectable friction.]
Â·
Matter must be mostly empty space to make shielding effects very small.
[Ans: It is now accepted that matter is mostly empty space.]
Â·
The
range of the force between bodies cannot be infinite because of
backscatter of particles colliding with other particles. [Ans: The
range of gravitational force may in fact be limited to about 2 kpc.]

This backscattering
diminishes the longrange force of gravity. At sufficiently great distances,
gravity essentially disappears. For example, the following formula for
gravitational acceleration represents one possible modification of the Newtonian
law of gravity to account for the range limitation effect in particle gravity
models. It assumes that backscattering into the shadow between bodies occurs
uniformly with distance, and at a rate that is proportional to the size of the
shadow's particle deficit:
_{}
In this formula, G is the
gravitational constant, M the mass of the attracting body, r
is the distance of some point from the attracting body, r_{G}
is the characteristic range of gravity (the mean distance a particle travels
before collision), e is the base for natural logarithms, arrows
over variables indicate vectors, and dots over variables indicate time
derivatives. This reduces to the Newtonian gravity formula as r_{G}
approaches infinity.
It might appear at first glance,
since the force of gravity diminishes rapidly toward zero over distances much
greater than r_{G}, that large structures held together
only by gravity could not be much larger than r_{G} in any
direction. But in fact much larger structures are possible. For example, if
two globular clusters of stars, each of radius r_{G}, are
so close that their outer stars intermingle, they can orbit each other because
many of their individual stars are closer than r_{G} and
therefore attract each other strongly. A chain of attraction operates, wherein
stars attract only other stars within about r_{G}, but the
most widely separated stars are forced to recognize each other's existence
because of all the intermediate attractions of stars between them.
If additional globular clusters
joined the original two, they could fill a plane out to many times r_{G}
with globular clusters, and each would strongly attract its immediate neighbors
so that the entire ensemble would remain bound together. But if another
globular cluster were added out of the plane, the force of its neighbors would
cause it to move on an orbit that passes through the plane, thereby causing a
merger with some of the clusters already in the plane through dynamical
friction. So structures can enlarge well beyond the characteristic distance
r_{G} in two spatial dimensions, but generally not in three
dimensions because of forced mergers.
Moreover, if the cluster mergers are
suitably placed, the structure may easily resemble a bar instead of a disk. But
as the bar's length gets well beyond the characteristic distance r_{G},
outer stars along the bar will find themselves unable to travel fast enough to
keep up with the bar's rotation, and will tend to lag behind  leading in a
natural way to spiral structure. (Spiral structure is still poorly understood
in the standard model, and needs to be explained as "density waves" to avoid the
problem of spiral arms winding up.) These simple consequences of a finite range
of gravity lead to descriptions that closely resemble real galaxies, giving us
some hint why galaxies form into disks with bars and spiral arms. Meanwhile
stars, star clusters, planets, and structures smaller than r_{G}
are generally spherical in shape, not flat like galaxies. Even the solar system
as a whole, which might be thought of as "flat" if the planets are included,
actually has 99.9% of its total mass in one central spherical star.
Thus, one can see that large, planar
structures such as galaxies are possible even though their largest dimensions
greatly exceed r_{G}. But spherical structures such as
galactic haloes are limited to sizes that do not exceed r_{G}
in radius by more than a small factor.
But other than for galaxy forms,
does the universe actually behave in the manner described by the modified
Newtonian law? Apparently, it does. Consider a typical disk galaxy such as our
own Milky Way. The characteristic size of galaxy haloes and typical disk
thickness imply that the characteristic scale distance for gravity, r_{G},
may be about 2 kiloparsecs (kpc), which is about 6000 lightyears.
Our Sun is located in the middle of
the galactic disk at about 10 kpc from the galaxy center. Under these
conditions, our Sun would feel practically no force at all from the galactic
center despite its great mass. Instead, the Sun would feel mainly the
attraction of the stars within about 2 kpc in all directions around it. But
there are more stars toward the galactic center than away from it because the
density of stars increases toward the galactic center. So the Sun feels a net
attraction toward the galactic center arising entirely from stars in its own
vicinity. And this attraction is ultimately what makes the Sun orbit the
galaxy. We are bound to stars up to about 2 kpc closer to the center than we
are, and those stars are in turn bound to other stars 2 kpc closer yet to the
center, and so on.
Now notice what can happen to our
understanding of galaxies if we work deductively from this finiterangegravity
premise. Stars and clusters in the dense central regions of the galaxy will
orbit in the halo as long as they don't stray too far from the center. The
transverse orbital velocity of those stars must be a function of the total mass
interior to them. But stars that do stray too far will start to spiral away,
retaining the transverse velocity appropriate for the halo of the galaxy they
reside in. So there will be a relation between the velocity of stars in the
disk and the mass of the halo. Note that the TullyFisher relation for galaxies
is an empirical formula that relates the rotational velocity of galaxies to
their intrinsic luminosity. But luminosity should be a very good indicator of
the halo mass, since there is no need for dark matter in this model. Therefore,
our picture so far already gives a theoretical basis for the TF relation 
something the standard model does not do.
If stars are continually being fed
into spiral arms from the galaxy halo, then gradually spiraling away as they
orbit the halo, the mean density of stars in any given ring of width dr
will decrease with 1/r, where r is the radius of the
ring. This is because roughly the same number of stars are entering and leaving
each ring at any given time through the spiraling process, and each ring has a
circumference proportional to r. So the total mass within each
ring of width dr may be taken as constant, while its volume
increases as r, so its density must vary with 1/r.
This accomplishes two things for our model: The decreasing density of stars in
each ring is just what we need to ensure that all stars will feel a net
attraction toward the galactic center from their immediate neighbors; and the
magnitude of that apparent net attraction will drop off with 1/r.
So our model predicts that stars in galaxies will act as if there were a
1/r attraction from the center instead of a 1/r^{2}
attraction. And indeed, that is just what is observed! Even the mean radius of
globular clusters increases linearly with distance from the galactic center. So
galactic disks would not have an outer edge, but simply fade into invisibility
as the number density of stars gets less and their average age gets older.
Mainstream astronomers assume that
the Newtonian law of gravity still holds, so there must exist invisible "dark
matter" in amounts that, for unknown reasons, increase radially in galaxies with
r, thereby canceling one power of r in the inverse
square attraction of the center. These astronomers speak of the M/L
ratio of galaxies, where M is mass and L is
luminosity or light. This would be unity if most mass were luminous, but is
generally much larger because of inferred dark matter.
Figure 4. M/L (masstolight ratio) versus r
(linear scalesize in lightyears).

Other astronomers take note that the
universe simply seems to better obey an inverse linear law at large scales than
an inverse square law, even if they don't understand why. See, for example,
figure 4. [Data taken from A.E. Wright, M.J. Disney, and R.C. Thompson,
"Universal gravity: was Newton right?", Proc.Astr.Soc.Australia 8,
334338 (1990).] This illustrates the inferred M/L ratios over a
variety of scales. Note that the general trend is linear, even over three
orders of magnitude in scale. The same authors discuss their computer
experiments showing that an inverse linear law of gravity is also more effective
in predicting the observed shapes of interacting galaxies than is an inverse
linear law.
Note also that the mean trend line
would intercept the horizontal axis, corresponding to M/L = 1, at
about 3000 lightyears or 1 kpc. This suggests that the estimate of 2 kpc based
on galactic disk thicknesses and halo sizes may correspond to 2 r_{G}
because the force of inverse square gravity probably remains fairly effective in
binding stars out to a distance of roughly 2 r_{G}. All
uncertainties considered, r_{G}, the range of gravity,
probably lies somewhere between 1 and 2 kpc, but may be closer to 1 kpc.
For small values of r,
the nonNewtonian exponential factor in the gravitational acceleration formula
simplifies to (1  r/r_{G}). For the Earth,
this factor differs from unity by 4.85 x 10^{9} kpc / r_{G}
. For Mercury, this difference would be 1.9 x 10^{9} kpc / r_{G},
since it varies linear with orbit size. For any given distance from the Sun,
the factor is constant, and therefore behaves as if the gravitational constant
G were slightly modified and slightly variable with distance.
Observationally, orbit
determinations using radar ranging data are dominated by Mercury observations
for determining the effective value of G because of Mercury's
large eccentricity. In Kepler's third law, n^{2} a^{3}
= GM_{S} (where n = mean motion, a
= semimajor axis, M_{S} = mass of Sun), radar
observations of Mercury's mean motion n_{1} and semimajor
axis a_{1} are used to determine GM_{S}.
This value is then used for the Earth's orbit, for which a_{3}
(semimajor axis of third planet) is much better determined by ranging data than
n_{3} because n_{3} becomes
indeterminate from radar data for a circular orbit. So n_{3}
is effectively measured with respect to n_{1} rather than
independently determined. When the radardetermined orbits are compared with
optical data over the past century or more, the optical data being very
sensitive to the true value of n_{3} for Earth, the error
in n_{3} determined from radar through Kepler's law and
n_{3} determined optically would be a function of the
difference between the effective value of G for Mercury and that
for Earth: (n_{radar}  n_{optical})_{3}
= n_{3} (a_{3}  a_{1})
/ (2 r_{G}) = 0.19 / r_{G} arc
seconds per century ("/cy). In the latter form, r_{G}
must be measured in kpc.
At the same time, the difference in
effective gravitational constant between a planet's perihelion and its aphelion
causes the longitude of perihelion to rotate by a comparable amount. For
Mercury, this rotation rate is: n_{1} a_{1}
/ [2 r_{G}
Ã–(1
 e_{1}^{2})] = 0.52 / r_{G}
"/cy. Since Mercury's perihelion direction dominates the determination of a
fixed direction in inertial space for the radar data, this motion will cause a
retrograde rotation of the radar inertial frame at the rate just specified,
which is not negligible.
The combination of the two effects
just described, one for the Earth's mean motion and the other for the direction
of the origin, will cause the radar mean motion of the Earth to exceed the
optical mean motion by 0.71 / r_{G} "/cy. Such a
discrepancy is actually observed, has a magnitude of just about this size, and
has remained an unexplained puzzle over the past 510 years. This a priori
derivation of the effect not only lends support to the basic idea of particle
models for gravity, but also suggests that r_{G} probably
is close to 1 kpc.
The observed excess perihelion rate
is estimated to be +41.9Â±0.5"/cy.
[L.V. Morrison and C.G. Ward, "An analysis of the transits of Mercury:
16771973", Mon.Not.Roy.Astr.Soc. 173, 183206 (1975).] This
should presumably be increased by the correction for equinox motion for the
reference system used, +1.2 "/cy. The result lies within a onesigma error of
both theories, and therefore favors neither.
The model predictions are
sufficiently well satisfied by the observational data that the entire model
should now be tested against observations of other planets to determine if it is
consistent with all existing solar system data.
Gravitational Shielding
Yet another way in
which particle gravity models differ from Newtonian gravity is in the ability of
matter to shield other matter from the effects of gravity. Ordinary matter must
be extremely porous to the flux of particles responsible for gravity, consistent
with our knowledge that ordinary matter is indeed made up mostly of empty
space. But since matter ingredients (MIs), by definition, totally absorb all
flux particles that strike them, there must exist some density of matter so
great that it lacks space between MIs, and through which no flux particles can
penetrate. Other matter behind such a solid wall of MIs could absorb no flux
particles, and therefore could not contribute to the gravitational field of the
body it resides in.
In the 19^{th}
century, J.C. Maxwell used the analogy of a swarm of bees blocking sunlight. If
two equal swarms of bees are superimposed, twice as much light will be blocked 
unless the swarms are so dense that some bees overlap bees in the other swarm,
in which case less than twice as much light is blocked. If one swarm is so
dense that it blocks all the light, then the second swarm adds nothing to the
light loss.
For particle gravity,
this means that dense matter might have more than one matter ingredient (MI)
along the same path of a flux particle, but only the first MI encountered
absorbs the flux particle. If matter were sufficiently dense, no flux particles
could penetrate beyond a certain depth, and only the outer layers of a body
would contribute to its external gravitational field. The body's gravitational
mass and its matter content would be different. The ratio of gravitational to
inertial mass would depart from unity  a condition not at all in conflict with
the results of Eotvostype
experiments. [See T. Van Flandern, "Are gravitational and inertial masses
equal?", MRB 4, 110 (1995).]
This theoretical
effect is usually referred to as "gravitational shielding", since a portion of
the gravitational field that would exist in Newtonian gravity is shielded. At a
point in space, the gravitational acceleration induced by a body of mass M
at a distance r when another body intervenes is:
_{}
where
r
is the density of the intervening body over the short distance dr,
the integral must be taken through the intervening body along the vector joining
the point in space and body M, and K_{G} is
the shielding efficiency factor in units of crosssectional area over mass.
Figure 5 
To test for such an
effect in nature, one needs to examine a test body orbiting near a relatively
dense intermediate mass, where the intermediate mass occasionally intervenes in
front of a more distant large mass. (Figure 5.) We then seek evidence that the
distant mass exerts less than its full effect on the test body at times when the
intermediate mass is aligned between the other two. But there is no a priori
way to be certain how big this effect (the size of
K_{G})
might be, because it depends on the amount of empty space between MIs.
What is probably the
most suitable test case for this effect in the solar system arises from the two
Lageos artificial satellites. The Earth's core provides the dense intermediate
mass, and the Sun is then the large distant mass. Both satellites are in orbits
high enough, and the 400kg satellites are massive enough, to be very little
affected by most nongravitational forces such as atmospheric drag or solar
radiation pressure. And both satellites are covered all over their outer
surfaces with retroreflectors that bounce back light along the incoming
direction. This enables these satellites to have their positions measured by
laser ranging from ground stations. In that way, the orbits can be determined
with a precision on the order of a centimeter or better.
Lageos 1 has been in
orbit for 20 years, and Lageos 2 for about 4 years. Both are in nearly circular
orbits roughly an Earth radius high, and circle the globe roughly once every
four hours. Lageos 1 revolves retrograde with an inclination of 110Â°, which
causes its orbit plane to precess forward. Lageos 2 is in a direct orbit with
an inclination of 53Â°, precessing backward. As a consequence, Lageos 2 has
"eclipse seasons"  periods of time when the satellite enters the Earth's shadow
on every orbit for up to 40 minutes  that are more frequent and more variable
in length than for Lageos 1. Then as precession changes orbit orientation, each
satellite may go many months continuously in sunlight, without eclipses. For
Lageos 2, it is possible for two consecutive eclipse seasons to merge into one
long season, as happened in late 1994 through early 1995.
Figure 6. Shading denotes eclipse seasons.
The significance of eclipses for this
discussion is that these are periods when any gravitational shielding effect
that may exist would be operative. Of course, several other types of
nongravitational forces also operate only during eclipses. Solar radiation
pressure shuts off only during eclipses, as does much of the thermal radiation
from the Earth. Light, temperature, and charged particles are all affected, and
at the one centimeter level, these must all be considered.
Both Lageos satellites
exhibit anomalous intrack accelerations that were unexpected. See figures 6
and 7 showing this effect for each satellite. [Thanks to Erricos Pavlis at NASA
Goddard Space Flight Center for supplying this data.] The anomalous intrack
acceleration (negative because it operates just as a drag force would) in units
of 10^{12} m/s^{2} is plotted against time, shown as a 2digit
year. The onset and end of eclipse seasons are indicated with vertical lines.
An average negative acceleration throughout the data can be explained as a
combination of radiation, thermal, and charge drag forces. But the data shows
substantial deviations from this average drag, especially during eclipse
seasons, and these are not so easily explained. [D.P. Rubincam, "Drag of the
Lageos satellite", JGR 95, 48814886 (1990).]
Figure 7. Shading denotes eclipse seasons.
Theoretical gravitational shielding effect appears above observed anomalous
acceleration for comparison.
Other authors, most recently V.J.
Slabinski ["A numerical solution for Lageos thermal thrust: the rapidspin
case", preprint], have succeeded in modeling the bulk of the anomalous
acceleration, including the eclipse season variations, for Lageos 1. But this
was accomplished by using about a dozen empirical corrections, and the
assumption that albedo variations over the satellite surface combined with spin
orientation and precession to produce these variations. The surface of Lageos 1
was supposed to be very uniform and highly reflective. For these models to be
viable, it must be assumed that some factor, perhaps rocket exhaust at the time
of injection into orbit, dirtied the surface and produced these albedo
variations. Lageos 2 was launched with care to avoid any repetition of such
problems. Yet the preliminary data available so far suggest that the anomalous
acceleration during eclipse seasons is still present.
The top portion of Figure 7, placed
above for easy comparison, shows the theoretical gravitational shielding effect,
calculated with the single parameter K_{G}. = 2 x 10^{18}
cm^{2}/g. The amplitude of the effect would be essentially the same for
Lageos 1 and Lageos 2. Lageos 1 is affected by radiation forces and/or other
effects that perhaps sometimes reinforces and sometimes goes contrary to the
hypothetical shielding effect. But the data clearly allows (though it does not
require) a gravitational shielding effect.
What an exciting discovery such a
finding would be! It has been proposed to launch a satellite inside a large,
hollow spherical shell. The shell would protect the inside satellite from all
nongravitational forces. The shell would have sensors and rockets that would
allow it to adjust its own orbit to keep the interior satellite always near its
center, no matter what forces act on the shell. But the interior satellite
would move under the influence of gravitational forces alone, protected from all
external radiative, thermal, and charge influences. Such a configuration would
allow the unambiguous detection of a gravitational shielding effect, if one does
exist.