Suppose "pushing gravity" says that gravitational attraction is caused by an influx of particles (or really any momentum carrying radiation) from infinity. This radiation would be analogous in nature to the cosmic microwave background. The idea then is that all bodies absorb some of this radiation, and when they absorb the radiation they feel a force. The more massive the body, the more efficient it is at absorbing the radiation. Let us supposed specifically, that the rate of absorption of the particles by a unit of volume is directly proportional to the mass density.

If a body is sitting in space, the particle flux is isotropic, so it doesn't feel a net force. If you have two bodies near one another, the body on the right blocks some of the leftward moving particles, so that the body on the left has less particles hitting it from the right; thus, it would feel a net rightward force. By the same logic, the body on the right would feel a net leftward force, so there would appear there was an attractive force at a distance at work if you didn't know about these particles. The magnitude of the attractive force would depend on the mass of the body blocking particles (because mass determines how strongly it blocks particles) and on the mass of the body that we're considering the force for (because the magnitude of the force imbalance will depends on how strongly *it* absorbs particles. It might look like gravity.

Newton's Law of Gravitation can be written:

**F** = -G m ∫ ρ(**x**) **e**_{r} / r^{2} dV

Where ρ is mass density, **r** is the vector that points from the point **x** to the point at which the force is being considered, and dV is an element of volume. **r** has magnitude r and points in the direction of the unit vector **e**_{r}. For simplicity, suppose the mass m we're finding the force on is sitting at the origin, then we can rewrite this integral in spherical polar coordinates as

**F** = -G m ∫ ρ(r,θ,φ) **e**_{r} dr dΩ

where dΩ is an element of solid angle ( dΩ = sin(φ) dθ dφ ).

In that form, it's starting to look pretty similar to what you might get from this pushing gravity thing. Imagine mass m is sitting at the origin being bombarded by particles from every direction. Now you place this other body with mass density ρ out there in space. Along each radial line that it intersects, if it blocks particles in proportion to mass density, then it looks like you might get the total blocked by adding up the effect for the element at each distance r. Then you might add up the effect for each solid angle blocked. Sounds like it should agree with the equation above quite nicely.

...well, not quite. Suppose that the incoming flux in an element of solid angle is I. Now, when it hits the first element of matter, some amount αρI of those particles is absorbed, according to our theory. That means that the flux of particles that hit the next closer bit of matter is less (because some of them have already been absorbed). If we make I(r) the flux as a function of distance and the initial flux I(∞), then it will obey a differential equation like

dI/dr = α ρ(r) I(r)

So, I(r) = I(∞) Exp(-∫ α ρ(r') dr') where the integral is from r to ∞.

The equation for gravity from that body becomes

**F** = -G m ∫ ρ(r,θ,φ) Exp(-∫ α ρ(r',θ,φ) dr') **e**_{r} dr dΩ

Meaning that gravity is "screened", so that a bigger body will gravitate less than one would expect under Newtonian gravity. This also means that the gravitational forces holding a large body together will be weaker. It seems that the more massive an object is, the more pronounced these effects should be. One wonders if compact objects like neutron stars could exist in such a theory. This would also cause the Roche limit of bodies in orbit to be smaller.

The other effect of the screening is that unlike in Newtonian gravity, this theory doesn't obey the principle of superposition. Namely, you can't get the gravitational force due to masses A and B on mass C just by adding them anymore, because if B is between A and C, then A blocks some of the incoming particles and B has less to block, so B exerts a weaker force on C than if A were not there. So, for example, during a lunar eclipse the force of the earth on the moon would be weaker than normal according to this theory.

A further implication is that the tides should be larger than they would be otherwise. The tides happen primarily because the gravitational pull of the moon (and other celestial bodies) is stronger on the water near the moon and weaker on water further away. This difference in force, called tidal force, causes high tide on the near and far points of the earth and low tide in between. Since the earth and moon are in a line, then this theory predicts that the force on water on the far side of the earth should be weaker. Larger tidal forces, hence larger tides.

If we have these particles coming in from every direction, then, like the CMBR it should have a average rest frame where the particle flux is isotropic, on average. However, if one moves relative to this rest frame it will no longer be isotropic. A body moving relative to the particle rest frame will experience a head wind and so it will experience drag. How fast you have to go for this to be appreciable depends on how fast the particles are moving. If they are moving quite fast, the drag might be small enough to not be noticeable in most cases. If the particles are massless (as they must be if they are not self-interacting), then they should all move at the speed of light in all reference frames according to special relativity; however, the ones that make up the head wind will still be more energetic, due to blue shift, still resulting in drag. All the same things can be said of the CMBR, but the force it exerts on almost any body is minuscule. It should be noted that this drag will *not* account for inertia, because it depends on velocity, not acceleration. In fact, when the effect is noticeable it will cause an apparent violation of both Newton's first and second laws.

#### A Step in the Wrong Direction

In recent years many people have sought to modify gravity in order to account for things like the galactic rotation curve. It seems that as you move away from the center of our galaxy, things are orbiting more rapidly than would be expected given the distribution of matter we can see and Newton's law of gravitation. Currently, the most popular explanation for this is that there's more matter we can't see, called dark matter. In this pushing gravity theory, all matter in the galaxy would feel less of a gravitational force (since some of the gravity is “screened”, or absorbed); thus the orbits of all objects would be slower than according to Newton. What's more, as you move out toward the edge of the galactic disk, the extra exponential decay term in the force law (given above) would mean that the inward force (due to the center of the galaxy and the far side of the disk) would die off more rapidly than in the Newtonian theory while the outward force (due to matter further out in the disk) would increase more quickly. Thus, the net inward force would fall off more rapidly, so pushing gravity would predict that orbital velocities would decrease toward the outside of the disk even more rapidly than in Newtonian gravity. The new theory would do exactly the opposite of what would be the biggest goal of a modified theory of gravity.

#### We Already Know Newtonian Gravity is Wrong

As the title says, Newtonian gravity has already been discarded because it did not make sense with special relativity. Instead, we have adopted general relativity (GR), which has so far met every experimental test. So it seems weird to try and knock off Newtonian gravity when that's already been done. Now we can ask if this theory could reproduce the phenomena of GR that have already been observed, directly or indirectly. Some of these effects would include the following: gravitational time dilation, gravitational redshift, gravitational lensing, gravitational radiation, and cosmological redshift.

You might hope to address gravitational redshift and gravitational lensing by supposing that the particles couple not only to mass but to energy-momentum (like GR). That requires, of course, a more complex theory, since then you're talking about it depending on a tensor, not a scalar. Even allowing you find a good way to do that, it seems like light would get redshifted traveling in any direction and not preferentially when traveling away from a massive object.

Cosmological redshift you might actually try to explain in terms of the drag effect I mentioned earlier, supposing that because light travels really fast, then when it travels for a long time this effect would become appreciable. You'd have to pick the way the force depends on energy-momentum in order that you'd still get Hubble's law. No obvious problem here, but not an obvious success either. Then if you want to try to explain the more recent amendments to Hubble's law (that the universe's expansion is accelerating)...well, you might be able to do it, but good luck figuring out how.

Gravitational radiation has never been directly observed, but the effects of it have been observed indirectly in the decay of a binary star system [1]. Since the pushing gravity theory doesn't obey the superposition principle, it seems like it might be hard to come up with any sort of linear wave theory from it. On the other hand, GR isn't linear either, only in the weak field approximation, so I'm not sure how problematic this is. One might also try to claim that the decay is due to this drag, not gravitational radiation forces, but in that case you'd have to show it predicts the correct form of the decay (which GR does).

The main objection I can't get past is that of gravitational time dilation. Clocks on the surface of the earth run more slowly than those in orbit. This has been experimentally measured and actually must be compensated for by the GPS. I can't really fathom how a rain of particles can slow the passage of time without some other major additions to the theory.

#### Experimental Tests

One could test the validity of this theory in various ways. The gravity due to any extended body should be slightly different in this theory, but that is generally hard to test to high accuracy. Usually the best technique is to setup an experiment where the force would be zero according to Newtonian gravitation but not in pushing gravity; one can then test to see if it is zero. This is the sort of test that has been used to see if gravity behaves differently on small length scales. A simple setup would be two source masses of the same density that are each a portion of a sphere. The two source masses should be setup with a test mass, so that the test mass lies at the center of two concentric, imaginary spheres. Each source mass should lie on one of the imaginary spheres and they should be opposite one another. According to Newton's law of gravitation, attraction should depend only on the thickness times the solid angle subtended, while pushing gravity says a thicker shell should have less of a pull. Unfortunately, gravity is weak enough that testing its strength in a laboratory, as in the Cavendish experiment, is quite difficult and likely would not be possible at the required level of precision. One should see an effect in any system of the 3 or more bodies whenever some portion of each body lies on the same line, for example if during a solar or lunar eclipse. In many systems these alignments are not very common (since usually the orbits of the three bodies are not coplanar), so this is a possible avenue of investigation but might also be too difficult. Probably the easiest experimental test would be to compare to some of the predictions of GR. There is experimental evidence for some of the effects predicted by GR, as I mentioned above, and if pushing gravity does not predict them as well then we already know it is wrong without the need for further experiments.

#### Preliminary Conclusions

So, it looks to me like the theory is pretty clever in how closely it mimics Newtonian gravity, but it looks like is has some weird features and it seems unlikely it could explain all the observations of GR effects. With no more Big Bang, I wonder how you'd account for things like the CMBR and cosmic abundances. There is also the odd fact that gravity is absorbed in this theory but never emitted. Generally, most of physics is time reversal invariant at the microscopic level, so if radiation can be absorbed it can normally be emitted (the time reverse of absorption), and this seems like it would be necessary for the theory to be sane from a thermodynamic viewpoint. If objects emitted these gravity causing particles, however, that would lead to anti-gravity, which has certainly never been observed. That might be part of the motivation for people exploring this theory, though it may also stem from a desire to explain Newtonian type gravity without action at a distance. Generally, scientists explore a new physical theory either because they have an observation they can't explain with any previously known theory (e.g. the blackbody spectrum) or because they have two different theories that explain two things separately but don't make sense together (e.g. quantum physics and general relativity). It doesn't seem like there's any compelling reason to explore this theory as it completely fails to solve the one problem that actually might cause us to want a different (classical) theory of gravity and doesn't seem to stem from other questions (like explorations of short range behavior of gravity motivated by the possibility of extra dimensions). As far as I can tell this theory (or variants of it) have not been seriously considered within mainstream physics.

This whole discussion has been addressing the form of “pushing gravity” I mentioned at the beginning, my interpretation of what the theory is claiming; however, this may be a bit of a straw man. It looks like a more complete discussion of this and related theories is given in the eprint gr-qc/0310081 [2], so if one were inclined to make a more in depth investigation that might be a place to start. It looks like what I've discussed is probably akin to the original theory of Le Sage, which he proposed in the eighteenth century as an alternative to action at a distance. Several modern variants of this idea have apparently been put forward and it may be that some are more sophisticated and have some of the relativistic features that this theory lacks, perhaps being modifications of the formulation of general relativity as a Yang-Mills field theory. I am not an expert in gravitation and have not studied this issue in depth, so the only further thing I can say is that the experts in that field do not seem to be terribly interested in this class of theories, perhaps for some of the reasons I have illustrated.

Sources:

- Taylor, J. H., 1994, “Binary pulsars and relativistic gravity,” Rev. Mod. Phys. 66, 711–719
- gr-qc/0310081 (http://arxiv.org/pdf/gr-qc/0310081)