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Abstract. The following is a slightly updated excerpt from my reply to Victor Slabinski's "Notes on Gravitation in the Meta Model" as originally published in the Meta Research Bulletin. Some of this material has already been incorporated into our respective chapters in the book Pushing Gravity; but some has not. This article should be considered a supplement to those who have read these two chapters in the book.

Correspondent Wayne Hayes in 1995 was the first to persuade me that C-gravitons (CGs) could not produce a net force by reflections, but would have to be absorbed by matter if the model were to survive. Subsequently, I learned a good deal more about the history of the subject, and why that same conclusion had been reached over a century ago. [Thompson, William (Lord Kelvin), "On the ultramundane corpuscles of Le Sage", The London Edinburgh and Dublin Philosophical Magazine and Journal of Science, London: Taylor and Francis, May 1873, Vol. XLV, Fourth series), p. 326f. Lord Kelvin also concluded that the "corpuscles" (= C-gravitons) must have a finite mean free path, so the range of gravity must be finite.] So that part of your manuscript came as no surprise. The Meta Research Bulletin carried two articles on this subject, including references to the Lageos data and your own work. [MRB 6, 23-29 & 38-50 (1996).]

The theoretical framework you developed in your manuscript is certainly very helpful and of considerable interest. Indeed, I am excited about the possibilities your line of thinking triggers. But first, I have a number of comments and points for discussion on the rest of what you wrote.

I first note the a priori expectation that  K_abs << K_scat  because of the almost total transparency of ordinary matter. Since all the processes under discussion should occur at all scales, according to the Meta Model picture of the universe, I like to use macroscopic analogies. I think of CG interactions with a matter ingredient (MI) as dynamically similar to comets interacting with a star. Most comet approach orbits result in path deflection ("scattering"). Only the rare collision orbit results in absorption.

I follow your development through eq. (18). But you begin section 2 with an assumption that I do not agree with - that absorption implies the flux of CGs is decreasing with time. This point was also part of the history of consideration of flux models, and was debated by Maxwell, Kelvin, and others long before us. I suppose that every colliding CG is eventually bounced or ejected back into space, but at a lower velocity than it initially had. This preserves the total number of CGs, yet transfers energy to the MI as it must to produce a net force, and likewise heats the MI. That heat increases until the MI emits a photon, releasing all the excess energy it has acquired from CG impacts and preserving thermodynamic equilibrium with its surroundings. Emitted photons are radiated isotropically into space. But (according to the Meta Model) all photons are losing energy by friction back to the CG medium as they propagate through it, and this photon energy loss is what gives rise to cosmological redshift. This energy transfer to the CG medium should then balance the books, ensuring that the CG medium as a whole is not running down.

At Eqn. (19) of your manuscript, you derive a formula for the heat flux of masses due to CG heating. The heat flow from the Earth’s interior is measured to be 10²⁸ ergs/year, or about 1/5000 of the solar contribution that arises from a solar constant of 1.36 x 10⁶ ergs cm^-2 s^-1. Therefore, the constraint you propose becomes Nm_gv_g³ K_abs ≤ 8.4 x 10^-9 cm² s^-3 None of the excess heat flows from the outer planets set any tighter constraint than this.

In TEST 3A, you suggest that CG heating might be a solution to the solar neutrino flux deficit problem. However, the Sun’s heat flux is of order 10⁴¹ ergs/year. The preceding constraint for Earth tells us that less than 10^-7 of all emitted solar heat could be due to CGs because the CG effect is proportional to mass. If the CG model has anything to say about the solar neutrino problem, it is more likely to be via the mechanism of gravitational shielding, which implies that higher densities of matter exist in the solar interior than are inferred from its external gravity field. This increased density might well slow the exodus of neutrinos out of the Sun, leading to a lower observed neutrino flux.

Your discussion in section 3 attempts to evaluate the "decrease in graviton flux with time". I would interpret this instead as the total CG energy loss through collisions with matter, which must equal the CG energy gain through cosmological redshift of photons. So the formulas you derive are useful, but the interpretation perhaps needs to be altered. In particular, my interpretation suggests that the characteristic time constant for CG decay in eqn. (26) may be the same as the characteristic time constant for photon decay, which is the inverse of the Hubble constant. If that is true, then we have the relation K_absv_g = 10¹³ cm³ g^-1 s^-1 , which assumes a baryon density of about 1/40 of the critical density and a Hubble constant of 60 km/s/Mpc. This constraint is extra useful, even though very approximate, because it is an equality rather than an upper or lower limit. It replaces your eqn. (38) in section 5.

Next, we have drag in section 4, leading to eqn. (29). The first thing I did with that equation was to change force into acceleration, because acceleration is what we observe and force is what we infer from observation. And that immediately shows the remarkable fact that the drag acceleration of bodies caused by the CG medium is independent of mass! This makes sense because the drag really acts on MIs rather than on the body as a whole, and it doesn’t much matter whether an MI is part of a large or small body. It also eliminates a potential fatal flaw in the theory, since mass-dependent drag would surely have astronomically observable consequences.

Likewise, since all bodies in a gravitationally bound unit such as the solar system share a common velocity with respect to the CG medium, there are no observable consequences except for the drag caused by each body’s peculiar velocity relative to the barycenter of the system. The size of the effect would be proportional to v P (v is orbital velocity, P is period) regardless of the mass of the primary. For circular orbits, this is proportional to the semi-major axis, a. So the overall effect on the Earth’s orbit would be 10⁴ times greater than for Lageos, and the greater accuracy of Lageos observations cancels only about 10³ of that. So, on balance, the Earth’s orbit sets a more stringent limit on possible drag than does Lageos by about one order of magnitude.

Substituting some numbers for Earth’s orbit, F_drag / m_⊕ < 0.75 x 10^-13 m s^-2 and v_orbit = 30 km s^-1 (note units), we arrive at the constraint K_abs v_g > 2 x 10¹¹ cm³ g^-1 s^-1, which is consistent with the estimate made earlier based on the Hubble constant. Planetary radar ranging should eventually allow this drag effect to be detected, if it exists. But it is far too small to be seen in Lageos. We can also take the constraint from lack of gravitational aberration that v_g > 6 x 10²⁰ cm s^-1, which implies K_abs < 1.7 x 10^-8 cm² g^-1. [MRB 6, 49-62 (1997).]

Next, your eqn. (40) tells us that v_g / [K_abs + K_scat (1 - <cos θ>)] < 0.13 g cm^-1 s^-1. Then since K_abs << K_scat, as I noted at the outset, L >> 1. From K_abs v_g = 10¹³ cm³ g^-1 s^-1 and v_g > 6 x 10²⁰ cm s^-1, we can replace eqn. (43) with L > 3 x 10²⁹. This represents approximately the square of the ratio of the mean distance between MIs to the diameter of a single MI. For comparison, the corresponding number for stars in the solar neighborhood is about 3 x 10¹⁵. Clearly, the large ratio of surrounding space to collisional cross-section area for MIs explains why matter is generally transparent to CGs.

Finally, we reach your eqn. (45), which now reads K_abs + K_scat (1 - <cos θ>) > 5 x 10²⁰ cm² g^-1, and which you think is fatal to the model. I do not follow your logic in this step. We already know that the model requires L to be a huge number. But that refers to MIs, not to macroscopic bodies. Suppose there were, say, 10⁴⁰ MIs in a one-gram body. Then each would have a mass of 10^-40 g, and a CG-interaction cross section < 1.7 x 10^-48 cm². Why would that preclude short-range interactions on a macroscopic scale?

If I am correct that the model survives this scrutiny, then it is surely worth further exploration. Perhaps you (or readers) will think of how to link in the "finite range of gravity" and "gravitational shielding" constraints mentioned in the earlier MRB articles. These might allow some even more interesting constraints to be placed on parameters in this model, and predictions that can be made and tested.

[The following is added in response to Dr. Slabinski's second note.]

The diagram below (courtesy of Boris Starosta) illustrates what I call "the meta cycle", which you described well. C-gravitons can be absorbed by matter ingredients as they create gravitational force. The energy deposited heats each MI, which continues to gain energy (driving all atomic motions and vibrations within atoms) until it explodes. The explosion (analogous to a supernova explosion) releases all the C-gravitons (with lower velocities) that were absorbed, one or more larger particles (alpha or beta decay; analogous to a pulsar shot from a supernova explosion), and a wave into the light-carrying medium (spontaneous emission of a "photon"; analogous to a supernova shock wave). This is the mechanism of radioactivity, which is generally credited as the source of the excess heat radiated into space by at least the five largest planets in our solar system. The heat radiation from planets and light from stars travels great distances through the C-graviton medium, losing energy to it by friction. It that way, the CG medium recovers the energy lost during absorption events, and the mean speed of CGs remains constant over time.

A bonus from the cycle is a possible energy mechanism for exploding planets. If something blocks the excess heat flow from masses -- for example, formation of an insulating layer between core and mantle in the interior of a planet -- then heat and energy within that planet will continue to build up until released explosively.

This cycle merely involves the conversion of conserved energy from form to form. Because no new physics is needed, no modification of Maxwell's equations is called for.