A New Equation for Gravity

Scientific Paper
Title	A New Equation for Gravity
Read in full	Link to paper
Author(s)	Bob de Hilster
Keywords	gravity, moon
Published	2009
No. of pages	12

Read the full paper here

Abstract

This paper describes the development of a new equation for gravity. The theory behind the equation is based on the work of Georges L. Le Sage. He proposed that there are "ultramundane corpuscles" (particles) coming at us from space. Most of these particles pass through objects, but a small number of them push the object. This causes a reduction in the number of partcles that leave the object. If the object is in space, then it is being pushed on all sides equally and there is no net force. In the case of the earth and the moon, the number of particles that pass through the earth to the moon are reduced. So there are fewer particles pushing the moon away from the earth then there are particles pushing the moon toward the earth. This net force is gravity. This new equation for gravity is based on this theory and is developed using four postulates.

Overview

This is Robert de Hilster's working statement of a particle-shadowing gravity equation built directly on Le Sage's eighteenth-century "ultramundane corpuscles". The premise is stated plainly: Newton and Einstein produced mathematics with no mechanism, Le Sage produced a mechanism with no working mathematics, and other researchers such as Majorana "start with Newton's equation and then add terms to fit their theory". De Hilster's aim is a formula that is not a modification of Newton at all but a computation of the actual particle flux imbalance on an orbiting body, from which orbital velocities and periods can then be predicted.

The document is candidly a work in progress — it is a printout of a wiki work-area page dated 15 January 2009, complete with a section headed "Junk", a placeholder "30??? days", an unfilled "xxxx m/s" and a reference to "Figure X". The author flags his own uncertainties throughout, including a note that one of his key substitutions "is not clear... is valid". It should be read as a research notebook rather than a finished result, and its most interesting content is arguably the list of things that did not work.

The argument

Four postulates

1. A particle — an entity with mass and velocity, imposing itself on an object from all directions; "the source of the particle is not known".
2. Reduction rate — as particles pass through an object, a small number do not pass through. The reduction rate is the percentage removed per unit distance travelled through the object, initially assumed linear.
3. Pushing force — having mass and velocity, each interacting particle imparts a push in its own direction of travel.
4. Reduction factor — the reduction rate is proportional to the density of the material traversed, equal to a reduction factor Rf times density.

Postulate 4 is new to this paper. De Hilster explains that the earlier version made the reduction rate proportional to mass, which failed an elliptical-orbit test: the computed velocity came out different "if the orbiting body were a lead ball instead of the moon", because density involves volume and the two volumes differ enormously. Recasting the reduction in terms of density removed that dependence.

Building the equation

The construction is geometric rather than field-theoretic. A plane is chosen through the centre of gravity of the Moon and through the Earth; within that plane, pairs of opposed particle paths are considered. Along the path that traverses the Earth the flux is attenuated to (1 − Z_e·RR_e), where Z_e is the chord length through the Earth and RR_e the Earth's reduction rate; the two opposed particles are combined into a number of particles imposed, N_pi, and multiplied by the sine of the plane angle to extract the vertical component.

A double summation over 180 planes and 180 angle pairs — 64,800 paths — replaces integration. De Hilster defends this deliberately: the first postulate makes gravity a discrete entity, "How can one integrate when the gravity field is discontinuous?"; discontinuous configurations such as Majorana's lead cube can be summed exactly but only approximated by an integral; and the chord lengths Z require supporting equations that an integral cannot easily accommodate. His summary is "It is more important to have the math fit the physics, rather than forcing the physics to fit the math." Multiplying through by the Moon's own absorption term and by F_g, the mean force one particle imposes, gives the force equation.

The five terms and how they are set

N_p is the number of paths; N_g the number of particles per path in a short interval; Z_x the chord through object x; Rf the reduction factor, with units m²/kg so that (1 − Z_x·Rf·D_x) is dimensionless; and F_g the mean force per particle in newtons. Only Z is fixed by the configuration. The procedure is explicit: assume values, compute, "adjust the value of Rf and F_g such that the new equation will come closer to the measured value", and repeat until the calculated force equals the measured one. De Hilster does not disguise this — "The answer is forced so that the parameters can be determined."

Results claimed

Fitting the Moon's sidereal period of 27.321 days for an elliptical orbit yields N_p = 1.62×10⁸, N_g = 1, Rf = 2.7216×10⁻¹³ m²/kg and F_g = 4.068×10⁵ N per particle. The velocity used Kepler's g = V²/R with a cosine correction for the small angle between the Kepler perpendicular and the ellipse tangent — about which the author adds: "It is not clear that adding the cosine of the angle to Kepler's equation is valid."

The second result invokes a companion paper, "An Equation for G" (NPA 2008), which produced a curve of G against altitude. At zero altitude the curve gives G = 6.46×10⁻¹¹ (forced to the measured value); at the Moon's distance it gives 5.64×10⁻¹¹; and the shuttle's ~17,321 mph at a few hundred miles requires 6.48×10⁻¹¹. That the Moon's and shuttle's velocities "fell right on top of the curve" is the paper's emotional climax — "It Was Amazing!" — and the basis of its two conclusions: G is not a constant, and the equation for G is valid.

De Hilster nonetheless lists his own caveats: the four fitted terms "are not the absolute values", Rf varies between experiments, N_g = 1 gives fractional particles in individual calculations, and doubling N_p simply halves F_g — only the product N_p·N_g·F_g ≈ 6.58×10¹³ is determined. He also says outright that a per-particle force of that order "seems ridiculously high".

Assessment

What is attractive here is the honesty and the concreteness. De Hilster states a mechanism, reduces it to a computable object, runs it, and reports where it wobbles — including a postulate he had to discard because a lead ball and the Moon gave different answers. The double-summation choice is defensible on its own terms for shadowing problems with sharp-edged bodies, and the recognition that only the product N_p·N_g·F_g is constrained is exactly the right observation about a degenerate parameter set. Few papers in this area are so explicit that "the answer is forced".

That candour, however, exposes the method's central weakness. With three adjustable quantities tuned until the computed force equals the measured force, reproducing the Moon's period is not a test of the theory — it is the definition of the fit. The paper's own criterion, that "one failure can refute the equation", cannot be met by a procedure that has no residual left to fail. The shuttle comparison is offered as an independent check, but a standard Newtonian orbit at the quoted altitude of 159.1 miles gives about 17,350 mph against the curve's 17,391 mph, so the agreement distinguishes nothing: any relation of the form v ∝ r^−1/2 passes through both the shuttle and lunar points.

The load-bearing claim that G varies with altitude does not survive a check against the paper's own numbers. For a circular orbit G = v²R/M; inserting de Hilster's own quoted values — v = 1023 m/s, R = 3.844×10⁸ m, M_e = 5.98×10²⁴ kg — gives 6.73×10⁻¹¹, essentially the textbook constant, not the 5.64×10⁻¹¹ the paper says is required at the Moon's distance. The "Junk" section quotes yet a third figure, 6.3×10⁻¹¹, for the same quantity. Three mutually inconsistent values for G at the Moon appear in a twelve-page document, and the one used to argue that G is not constant is the one that does not follow from the stated velocity and radius.

The comparison with Newton is likewise misstated. The paper leaves "Velocity would be 1018 m/s, with an orbital time of 30??? days" unresolved, implying Newton fails. The velocity is right, but the period that follows from it is 2πR/v ≈ 27.4 days, and the full two-body Kepler result using G(M_e + M_m) gives about 27.3 days — matching the observed 27.321 days to roughly a tenth of a per cent, with no free parameters at all. There is thus no Newtonian discrepancy for the new equation to repair. A related internal slip: F_g is tabulated as 4.068×10⁵ N per particle, but the closing section objects that it "can't be 4E+13 Newtons" — apparently confusing F_g with the product N_p·N_g·F_g.

The claim that the equation "does not use the inverse square law" also needs qualifying. It does not write 1/R², but the double summation over paths is a solid-angle calculation, and the solid angle the Earth subtends at the Moon falls as 1/R²; in the weak-shadowing limit the geometry supplies the inverse square automatically. That is a strength of Le Sage models, not something the paper has escaped. Finally, the two classical objections to corpuscular gravity go unaddressed. Particles carrying momentum and being absorbed must also deposit energy and exert a velocity-dependent drag; lunar laser ranging measures the Moon's semi-major axis increasing at 3.8 cm per year, a rate fully accounted for by tidal friction and leaving no room for the secular orbital decay that absorption drag would produce. De Hilster's own summary — that much more work is needed and that "the detailed facts about the particle should be determined" — is the fairest description of where the paper stands.

See also

• Bob de Hilster
• Georges-Louis Le Sage
• Quirino Majorana
• Isaac Newton
• Gravity
• Newton's Third Law
• General Relativity