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 10¹⁰ c. The so-called 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 cross-section, 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 ultra-dense 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 particle-gravity models has the potential to lead to a breakthrough in our post-Einsteinian 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.

            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 Poynting-Robertson 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 Poynting-Robertson 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 10⁷ 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 10¹⁰ c.

            General relativity (GR), of course, has an explanation for this experimental result that does not involve faster-than-light 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, 9-12 (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 energy-momentum density tensors [Rosen, N., "General Relativity and Flat Space.  I & II", Phys.Rev. 57, 147-153 (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. 66-76 (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 light-time, as electromagnetic forces do.  It is not even sufficient for each companion to accelerate via the full curvature that spacetime would have had one light-time ago.  Some information is being propagated between source and affected body faster than light unless we assume that the passive, non-intelligent 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.

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 well-tested 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 frame-independence postulate experimentally.  Only two historical experiments have made the attempt:  the Sagnac experiment in 1913, and the Michelson-Gale experiment in 1925.  Both utilized rotating reference frames, and both obtained non-zero fringe shifts in Michelson-Morley-type 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. 208-211 (1931).]  The modern formulation of this idea is referred to as the "Mansouri-Sexl" transformation.  [R. Mansouri and R.U. Sexl, "A test theory of special relativity: I. Simultaneity and clock synchronization", Gen.Rel.&Grav. 8, 497-513 (1977)  The respective equations for Einstein SR and the Lorentzian (Mansouri-Sexl) alternative are these:

 

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 dilation-contraction factor, , is always ³ 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 faster-than-light 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 pre-adjusted 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. 46-50.]  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 non-locality in quantum physics can be understood without invoking such radical hypotheses as "There is no deep reality!"

Particle Models of Gravity

            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, 9-13 (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 gravity-producing 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 resistance-induced acceleration.

            Absorption of the particle flux by matter ingredients would result in the heating of bodies through energy absorption.  This heat must be fully re-radiated 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 back-scattering into the shadow of particles colliding with other particles.

            This back-scattering diminishes the long-range 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 back-scattering 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 light-years.

            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 finite-range-gravity 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 Tully-Fisher 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 T-F 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/r² 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.

            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, 334-338 (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 light-years 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 non-Newtonian 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, n² a³ = GM_S (where n = mean motion,  a = semi-major axis, M_S = mass of Sun), radar observations of Mercury's mean motion n₁ and semi-major axis a₁ are used to determine GM_S.  This value is then used for the Earth's orbit, for which a₃ (semi-major axis of third planet) is much better determined by ranging data than n₃ because n₃ becomes indeterminate from radar data for a circular orbit.  So n₃ is effectively measured with respect to n₁ rather than independently determined.  When the radar-determined orbits are compared with optical data over the past century or more, the optical data being very sensitive to the true value of  n₃ for Earth, the error in n₃ determined from radar through Kepler's law and n₃ 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 = n₃ (a₃ - a₁) / (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: n₁ a₁ / [2 r_G Ö(1 - e₁²)] = 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 5-10 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 +41.9±0.5"/cy.  [L.V. Morrison and C.G. Ward, "An analysis of the transits of Mercury: 1677-1973", Mon.Not.Roy.Astr.Soc. 173, 183-206 (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 one-sigma 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 Eotvos-type experiments.  [See T. Van Flandern, "Are gravitational and inertial masses equal?", MRB 4, 1-10 (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 cross-sectional area over mass.

            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 400-kg satellites are massive enough, to be very little affected by most non-gravitational forces such as atmospheric drag or solar radiation pressure.  And both satellites are covered all over their outer surfaces with retro-reflectors 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 110°, which causes its orbit plane to precess forward.  Lageos 2 is in a direct orbit with an inclination of 53°, 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 non-gravitational 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 in-track 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 in-track acceleration (negative because it operates just as a drag force would) in units of 10^-12 m/s² is plotted against time, shown as a 2-digit 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, 4881-4886 (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 rapid-spin 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 cm²/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 non-gravitational 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.