A Modified Law of Gravitation taking Account of the Relative Speeds of Moving Masses A Preliminary Study

Scientific Paper
Title	A Modified Law of Gravitation taking Account of the Relative Speeds of Moving Masses. A Preliminary Study
Read in full	Link to paper
Author(s)	Bernard Guy
Keywords	gravitation, Theory of Relativity, dark matter, Dark Energy, galaxies, Pioneer anomaly, tri-dimensional time, Lorentz force, relative speeds
Published	2010
No. of pages	20

Read the full paper here

Abstract

A modified law of gravitation is proposed which takes account of the relative speeds of the moving masses. The law simulates a "supplement" of mass with respect to the standard Newtonian law. Its application to several gravitation problems provides a good order of magnitude for the apparent supplement of mass associated with the motion of the Pioneer satellites, that of stars in galaxies or galaxies in galaxy clusters, for the same value of one additional parameter. The law equally simulates a lack of attraction, for the later stages as compared to the early stages, for a system of expanding masses, imitating a repulsive force. The order of magnitude of the corresponding energy fits with what is found in the literature for the "acceleration" of universe expansion. The conceptual framework in which the law is proposed is sketched out: it is based on the assertion of the fundamental link between the space and time concepts, and on a better symmetry of the physical laws with respect to these parameters. The study is preliminary, it simply establishes orders of magnitude for the expected effects, by an approximate approach of the two-body problem. In the near future it seems interesting to perform quantitative computer simulations so as to check whether the proposed law resists to further confrontations with observational data. If it did, it would avoid in the same time the use of dark matter and dark energy. The present work also gives clues to help reconsider the theory of relativity, in continuity with the modified law of gravitation, and its links with gravitation and electromagnetism.

Overview

Bernard Guy, a geologist at the École nationale supérieure des mines de Saint-Étienne, proposes adding to Newton's inverse-square law a single velocity-dependent term, in exact analogy with the magnetic part of the Lorentz force. Where Newton's law depends only on the mutual distances of masses, the added term depends only on their relative speeds. It introduces one new constant, A, with the dimensions of an inverse velocity squared, provisionally fixed at 5×10⁻¹¹ m⁻²s².

The claim that motivates the paper is arithmetic economy. With that single value of A and no further adjustment, Guy obtains the right order of magnitude for four separate anomalies usually treated as unrelated: the anomalous sunward acceleration of Pioneer 10 and 11, the flat rotation curves of galaxies, the still larger velocity dispersions of galaxies in clusters, and the apparent acceleration of cosmic expansion. If the law survived proper testing, he argues, both dark matter and dark energy would become unnecessary. Guy is explicit and repeated about the paper's status: it is "preliminary and provisional," an order-of-magnitude exercise on the two-body problem, and computer simulation of the n-body case is the obvious next step.

The argument

A conceptual framework: time as movement

The added term is not introduced ad hoc but from Guy's longer programme on space and time, developed across a series of papers since 1997. Its premises are that time and space are "the same substance", and that the temporal parameter should itself be a three-component object t_x, t_y, t_z, since time is in practice marked by a movement — the sun's position in the sky, a photon's position in an atomic clock. Restoring symmetry between the three space and three time parameters, he holds, is what the Lorentz relations really express, and he credits J. A. Franco and Tsabary and Censor with the rigorous vectorial formulations.

Every physical quantity is then held to have two "faces": a spatial face g and a temporal face h. A law of physics equates the sum of derivatives of one face with respect to variables of one type to those of the other face with respect to the other type:

∑_i ∂g_i/∂t_i + ∑_j ∂h_j/∂x_j = 0, and the companion relation with x and t exchanged.

The general principle is that "laws of physics are globally invariant by exchanging space and time parameters." In electromagnetism this pairing is already realised: the electric field E is the spatial face, giving the force between static charges as a function of distances; the magnetic field B is the temporal face, giving the force between moving charges as a function of relative speeds.

The modified law

Gravitation, Guy observes, has only the spatial face — g = −GMr/r³ does not satisfy the paired relations. He therefore posits the missing temporal face

h = AGM(r×dr/dt)/r³,

and, by the same route that leads from Maxwell's equations to the motion of a charge, obtains the equation of motion

d²r/dt² = −GMr/r³ − A(dr/dt × (dr/dt × r)),

generalised for many masses by summing over pairs with relative separations r_ij. Crucially, Guy notes that the double vector product yields an attractive contribution whatever the relative orientation of r and v (his Fig. 1). Projecting it onto g gives the paper's working formula:

ΔM/M = Δg/g = Av²sin²θ,

where θ is the angle between radius and velocity vectors. The effect therefore vanishes when r and v are parallel — which, Guy points out, is exactly the geometry of a falling-body measurement in a terrestrial laboratory, and is why the term has gone unnoticed.

Why A is not 1/c²

In vacuum electromagnetism the analogous constant is ε₀μ₀ = 1/c². Guy's A is roughly 10⁶ times larger, and he offers a heuristic: following Assis's suggestion that gravitation may be an averaged macroscopic residue of electromagnetic forces, the internal particle speeds inside matter exceed bulk speeds by perhaps 10³, and since velocities enter squared, A(v_macro)² = (1/c²)(v_micro)² gives A ≈ 10⁶/c², close to the adopted value. He proposes keeping A = 1/c² for photons and the larger value for bulk matter, in loose parallel with the electromagnetic distinction between "in vacuum" and "in matter" formulations.

The four applications

Pioneer. At ~10¹³ m, g = GM_☉/R² ≈ 1.3×10⁻⁶ m/s², so the reported anomaly of 8.74×10⁻¹⁰ m/s² gives Δg/g = 6.7×10⁻⁴. With v ≈ 10⁴ m/s and θ ≈ 20°, the formula gives Av²sin²θ = 5.8×10⁻⁴. Guy notes that Pioneer 10's Δg/g is slightly the larger and its trajectory the more inclined, which fits the sin²θ dependence, and proposes A = (1/v²sin²θ)(Δg/g) as a constancy test across many spacecraft.
Galaxy rotation. Here r ⊥ v, so ΔM/M = Av². With v = 200–250 km/s, Av² ≈ 2 — a missing mass of order the visible mass, as reported.
Clusters. Relative velocities of 500–800 km/s raise v² by a factor of a few, so ΔM/M reaches tens; allowing sin²θ ≈ 10⁻¹ for non-perpendicular motions still leaves the observed order of ten times the visible mass. Same A throughout.
Cosmic acceleration. Guy first insists on the word: expansion is decelerating, but less than Newtonian gravity predicts, so distant objects appear farther away than expected. In his model, as bodies recede, θ between r and v tends to zero (Fig. 2), the velocity-dependent attraction switches off, and the comparison of late to early epochs reads as a repulsion. Taking early recession speeds of order 10⁸ m/s and sin²θ ≈ 10⁻³ gives Av²sin²θ ≈ 5×10², so an apparent dark-energy component of order 10² times the visible energy.

Self-criticism and comparison

Section 10 is largely a list of what has not been done. The two-body results must be checked against every standard success of Newtonian gravity and general relativity to confirm the new term hides within existing uncertainties. The n-body problem must be solved, and Guy asks openly whether Gauss's theorem — concentrating a spherically symmetric mass at its centre — even survives when relative velocities enter. He does perform one consistency check: for Mercury the factor Av² = 10⁻¹, and with sin²θ weighting of 10⁻¹ to 10⁻² the ratio of new to Newtonian coefficients is 10⁻²–10⁻³, against the observed 40/5600 = 7×10⁻³ for the residual perihelion advance — "close to the above numbers."

For photons he takes A = 1/c²; since the Newtonian calculation gives half the observed solar deflection, the added term contributes Δα/α = 1 and restores the right total. He acknowledges this "amounts to varying the speed of light and should be re-written in a relativistic manner."

He places his law among predecessors — Heaviside (1893), Assis, Jefimenko, Ragusa's modified Weber force, and especially Gruffat (2004), whose law he says "is the same as we proposed here" — and against MOND, which he credits with galaxy rotation curves but finds "less adapted" to clusters and Pioneer, silent on dark energy, and "ad hoc" for lacking a conceptual basis.

Assessment

The paper's most attractive feature is the one Guy himself emphasises: a single new constant, fixed once, is asked to carry four independent anomalies spanning eleven orders of magnitude in velocity and twenty in distance, and it delivers the right order of magnitude in each case. That is a real and unusual claim of economy. The sin²θ structure is also a genuine prediction with a built-in falsification route: the effect must vanish for radial motion, must scale as v², and must therefore differ measurably between spacecraft on differently inclined trajectories — a test Guy explicitly proposes and that could have been run on existing tracking data. The framework is not retrofitted to the anomalies either; the space–time duality argument was published a decade before, and the missing "temporal face" of the gravitational field is a coherent reason to expect a velocity term rather than an excuse for one. Guy's honesty about the paper's limits is exemplary and rarer than it should be: he lists the calculations he has not done, flags the Gauss-theorem question that could sink the galaxy application, and calls his own numbers "a rough guide."

The difficulties are correspondingly large, and most are ones Guy names. The central one is that order-of-magnitude agreement across four cases is weak evidence when each case involved a choice of poorly constrained inputs. In the cosmological application, sin²θ is estimated as "10⁻² to 10⁻⁴. Take 10⁻³" — a two-order-of-magnitude range collapsed to a single number chosen midway, in a calculation whose target is itself quoted only to an order of magnitude. The same latitude appears in the cluster case (sin²θ ≈ 10⁻¹) and in the heuristic for A itself, where v_micro/v_macro = 10³ is asserted "at heuristic title." With free angular factors available in every application, the claim that one value of A fits all four is considerably softer than it appears.

Empirically, the flagship case has since dissolved. The Pioneer anomaly was resolved by Turyshev and colleagues in 2012 — the year after this paper — as anisotropic thermal emission from the spacecraft's radioisotope generators and electronics, modelled and confirmed to account for essentially the whole 8.74×10⁻¹⁰ m/s². Guy could not have known this, but the calibration of A rests on that number, and with the anomaly gone the parameter loses its anchor and the "same value for four phenomena" claim its first term. The galaxy application faces a different and older problem: a term scaling as Av² with v roughly constant across a flat rotation curve does not obviously reproduce the radial shape of rotation curves, nor the tight Tully–Fisher relation, and Guy does not attempt either — he only matches the total mass discrepancy. The cluster and cosmology applications match a magnitude, not a curve; no v(z) is computed for comparison with the Type Ia supernova Hubble diagram that motivates dark energy in the first place.

There are also internal tensions. The velocity in the law is a relative velocity, but the applications quietly treat it as a velocity with respect to a centre of mass — the Sun for Pioneer, the galactic centre for stars — which is a frame choice, not a relative quantity, and the n-body summation Guy writes down does not obviously reduce to it. Taking one value of A for bulk matter and a different value of A = 1/c² for photons is a substantial extra assumption, and it is the photon value that rescues the solar light deflection — so the deflection success and the anomaly successes are not achieved by the same theory. Finally, the framework of tri-dimensional time is sketched in two pages and rests on prior unpublished or lightly published work; the step from equations (1) to the specific form of h is asserted to be "similar to the derivation from Maxwell equations" rather than carried out, which leaves the paper's claim to be non-ad-hoc resting on an analogy rather than a derivation.

Read as what it says it is — a preliminary exploration inviting simulation and test — the paper is a reasonable piece of work. Read as an alternative to dark matter and dark energy, it is a long way short, and the promised computer simulations would have to do most of the arguing.