A Modified Law of Gravitation taking Account of the Relative Speeds of Moving Masses. A Preliminary Study: Difference between revisions

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
Published	2010
No. of pages	20

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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, of the Ecole nationale supérieure des mines de Saint-Etienne, offers a velocity-dependent modification of Newton's law of gravitation and argues that a single new constant, fixed once and never re-tuned, reproduces the right order of magnitude for four separate anomalies that mainstream astrophysics currently treats as evidence for two unseen substances. The additional term is built by analogy with the Lorentz Force: just as a static electric field accounts for forces between charges at rest and a magnetic field accounts for forces between charges in relative motion, Guy proposes that the ordinary gravity field g has a partner field h generated by the relative motion of masses. The paper is explicitly labelled preliminary: it works only the two-body problem, and only to orders of magnitude.

The departure from the standard account is twofold. Empirically, if the extra term is real then the "missing mass" inferred in galaxies and clusters and the repulsion attributed to Dark Energy are both artefacts of using an incomplete force law — no new matter and no new energy component is required. Conceptually, Guy grounds the proposal in his longer-running programme on space and time, in which time is not a scalar fourth dimension but a three-component quantity t_x, t_y, t_z with the meaning of a movement (the position of the sun in the sky, of a photon in an atomic clock). On that reading the framework is "twice three-dimensional and not four-dimensional", and General Relativity itself is something to be reconsidered rather than extended.

The argument

Space, time and the two faces of a physical quantity

Guy's starting premise is that space and time are "the same substance" and that the laws of physics should be globally invariant under exchange of space and time parameters. Every physical quantity is then given two faces, a "spatial" face g and a "temporal" face h. A law of physics equates the sum of the derivatives of one face with respect to variables of one type to the derivatives of the other face with respect to the other type, giving the pair of relations

Σ_i ∂g_i/∂t_i + Σ_j ∂h_j/∂x_j = 0 and Σ_i ∂g_i/∂x_i + Σ_j ∂h_j/∂t_j = 0

which Guy reads as very general conservation laws, citing Tsabary and Censor (2005) and Franco (2006) for similar structures. In Electromagnetism the identification is immediate: the electric field is the spatial face, used for forces between static charges as a function of mutual distances; the magnetic field is the temporal face, used for forces between moving charges as a function of relative speeds. Passage between three-dimensional time and a scalar time uses t² = t_x² + t_y² + t_z² together with a choice of the direction along which the time-defining movement is taken.

The modified law

Gravity, Guy observes, does not by itself satisfy these relations: the g field has no partner to balance it. He therefore posits the pair

g = −GM r / r³ and h = AGM (dr/dt × r) / r³

and derives, in the same manner in which the force on a moving charge follows from Maxwell's equations, the equation of motion for the two-body problem:

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

The first term is Newton. The second, involving a double vector product of the relative velocity with the radius vector, is the gravitational analogue of the magnetic force. A is a new constant with dimensions of an inverse squared speed; Guy provisionally sets A = 5 × 10⁻¹¹ SI (m⁻²s²). A summed version is written for a distribution of masses m_j.

Crucially, the added term is shown to be attractive whatever the relative orientation of r and v, so it always mimics extra mass. Projecting it on the ordinary gravity direction gives the paper's working formula

ΔM/M = Δg/g = Av² sin²θ

where θ is the angle between radius vector and relative velocity. When r and v are parallel the effect vanishes entirely — which, Guy notes, is exactly the configuration of a laboratory free-fall measurement on Earth, so the new term escapes terrestrial detection. When they are perpendicular the supplement is simply Av².

Why A is not 1/c²

In vacuum electromagnetism the corresponding constant is ε₀μ₀ = 1/c². Guy argues that gravitation applied to macroscopic matter is more like electromagnetism "in matter", where εμ can exceed its vacuum value by many orders of magnitude. Following the suggestion (attributed to Assis, 1992) that gravitation is a macroscopic average of forces of electromagnetic origin, he offers the heuristic A·(macroscopic velocity)² = (1/c²)(microscopic velocity)². With an internal-to-bulk speed ratio of about 10³ this gives A ≈ 10⁶/c², close to the adopted 5 × 10⁻¹¹. For massless photons he instead takes A = 1/c².

Four applications

For the Pioneer spacecraft, an anomalous Δg of 8.74 × 10⁻¹⁰ m/s² at distances of order 10¹³ m gives Δg/g = 6.7 × 10⁻⁴. With v = 10⁴ m/s and a trajectory inclination of some 20°, the formula returns Av²sin²θ = 5.8 × 10⁻⁴ — the same order. Guy proposes the testable constancy (1/v²sin²θ)(Δg/g) = A across many spacecraft.

For flat galaxy rotation curves, peripheral stars move at 200–250 km/s essentially perpendicular to the radius, so Av² = 5 × 10⁻¹¹ × 4 × 10¹⁰ ≈ 2 — an apparent missing mass of order the visible mass, as reported. For galaxies in clusters, relative velocities of 500–800 km/s raise v² by a factor of a few, and ΔM/M rises to tens, matching the larger cluster discrepancy at the same A.

For cosmic expansion, Guy first insists on what "acceleration" means in the Big Bang picture: the recession speed still decreases, only less than Newtonian gravity predicts. As a system of mutually receding bodies evolves, θ tends toward zero, so the always-attractive extra term progressively switches off; writing g_II = g_I + g_repulsive, the shortfall behaves as an effective repulsion following the same 1/r² law with apparent repulsive mass AMv². Taking early speeds of ~10⁸ m/s and small angles gives a ratio of order 10² between apparent dark and visible energy.

Consistency checks

Guy checks that the new term does not spoil the solar system. For Mercury (v = 48 km/s) Av² ≈ 10⁻¹, and after angular weighting the ratio of the new coefficients β_ij to the Newtonian α_ij is 10⁻²–10⁻³, against the observed perihelion ratio 40/5600 ≈ 7 × 10⁻³. For light bending, the Newtonian calculation gives half the observed deflection; with A = 1/c² the extra term contributes Δα/α = 1, restoring the full value.

Assessment

The attractive feature of this paper is its economy and its falsifiability. One constant, fixed at 5 × 10⁻¹¹ SI, is carried unchanged through the Pioneer anomaly, galactic rotation, cluster dynamics and cosmic acceleration, and lands within an order of magnitude in each case. That is a genuinely different discipline from fitting each anomaly separately, and Guy states plainly the test that would break it: the quantity (1/v²sin²θ)(Δg/g) must come out the same for every spacecraft. His criticism of MOND — that it is an ad hoc profile that works for rotation curves but not for clusters, spacecraft or expansion — is fair, and his placement of the proposal in the older tradition of velocity-dependent gravity (Oliver Heaviside 1893, Assis, Jefimenko, Ragusa, Gruffat) is honest about priority, including the acknowledgement that Gruffat (2004) proposed the same law.

The difficulties are equally plain, and several are conceded. The derivation of equation (3) from the field pair (2) is asserted rather than shown: the reader is referred to the analogy with the Lorentz force and to third-party papers on scalar-to-vector time, and no explicit calculation appears. The parameter A is not predicted, only fitted and then rationalised after the fact by a heuristic ratio of microscopic to macroscopic speeds that is chosen (10³) precisely to land near the required value — so the single-constant economy is weaker than it looks. The cosmological application is the loosest: the angles between radius and velocity vectors are simply "difficult to estimate, put a few degrees", and the factor sin²θ = 10⁻³ that produces the 10² energy ratio is inserted by hand. The choice A = 1/c² for photons versus A = 5 × 10⁻¹¹ for matter is a second free choice presented as one.

Against measurement, the sharpest problem is that the treatment is Newtonian and applies to the Pioneer anomaly a formula that vanishes for radial motion — yet the anomaly is reported as a nearly constant sunward acceleration on trajectories that become increasingly radial, whereas Av²sin²θ should fall as sin²θ shrinks and as the craft decelerate. (The anomaly has since been widely attributed to anisotropic thermal recoil, which the paper predates.) In clusters, the Gravitational Lensing mass — a static measurement in which the source galaxies' relative velocities play no role — agrees with the dynamical missing mass; a velocity-dependent term should decouple the two. The supernova evidence for acceleration is a luminosity-distance relation over redshift, not a directly measured deceleration of a two-body system, so the identification of ΔE/E with the observed departure is at best schematic. And the promise that the framework will "reconsider" relativity is left as a programme: the metric, the invariance of ds², and the observed Time Dilation of moving clocks are named as things to be redesigned, not redesigned here.

Guy is candid about all of this — the paper calls itself exploratory, asks repeatedly for the n-body simulations that would decide the matter, and does not claim the anomalies are explained. Judged as what it says it is, a preliminary order-of-magnitude exploration of one idea, it is coherent and worth the computation it asks for; judged as a replacement for Dark Matter and dark energy, it is not yet a calculation.

@@ Line 13: / Line 13: @@
 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, of the Ecole nationale supérieure des mines de Saint-Etienne, offers a velocity-dependent modification of Newton's law of gravitation and argues that a ''single'' new constant, fixed once and never re-tuned, reproduces the right order of magnitude for four separate anomalies that mainstream astrophysics currently treats as evidence for two unseen substances. The additional term is built by analogy with the [[Lorentz Force]]: just as a static electric field accounts for forces between charges at rest and a magnetic field accounts for forces between charges in relative motion, Guy proposes that the ordinary gravity field ''g'' has a partner field ''h'' generated by the relative motion of masses. The paper is explicitly labelled preliminary: it works only the two-body problem, and only to orders of magnitude.
+The departure from the standard account is twofold. Empirically, if the extra term is real then the "missing mass" inferred in galaxies and clusters and the repulsion attributed to [[Dark Energy]] are both artefacts of using an incomplete force law — no new matter and no new energy component is required. Conceptually, Guy grounds the proposal in his longer-running programme on space and time, in which time is not a scalar fourth dimension but a three-component quantity ''t''<sub>''x''</sub>, ''t''<sub>''y''</sub>, ''t''<sub>''z''</sub> with the meaning of a movement (the position of the sun in the sky, of a photon in an atomic clock). On that reading the framework is "twice three-dimensional and not four-dimensional", and [[General Relativity]] itself is something to be reconsidered rather than extended.
+==The argument==
+===Space, time and the two faces of a physical quantity===
+Guy's starting premise is that space and time are "the same substance" and that the laws of physics should be globally invariant under exchange of space and time parameters. Every physical quantity is then given two faces, a "spatial" face '''''g''''' and a "temporal" face '''''h'''''. A law of physics equates the sum of the derivatives of one face with respect to variables of one type to the derivatives of the other face with respect to the other type, giving the pair of relations
+: Σ<sub>''i''</sub> ∂''g''<sub>''i''</sub>/∂''t''<sub>''i''</sub> + Σ<sub>''j''</sub> ∂''h''<sub>''j''</sub>/∂''x''<sub>''j''</sub> = 0 and Σ<sub>''i''</sub> ∂''g''<sub>''i''</sub>/∂''x''<sub>''i''</sub> + Σ<sub>''j''</sub> ∂''h''<sub>''j''</sub>/∂''t''<sub>''j''</sub> = 0
+which Guy reads as very general conservation laws, citing Tsabary and Censor (2005) and Franco (2006) for similar structures. In [[Electromagnetism]] the identification is immediate: the electric field is the spatial face, used for forces between static charges as a function of mutual distances; the magnetic field is the temporal face, used for forces between moving charges as a function of relative speeds. Passage between three-dimensional time and a scalar time uses ''t''<sup>2</sup> = ''t''<sub>''x''</sub><sup>2</sup> + ''t''<sub>''y''</sub><sup>2</sup> + ''t''<sub>''z''</sub><sup>2</sup> together with a choice of the direction along which the time-defining movement is taken.
+===The modified law===
+Gravity, Guy observes, does not by itself satisfy these relations: the ''g'' field has no partner to balance it. He therefore posits the pair
+: '''''g''''' = −''GM'' '''''r''''' / ''r''<sup>3</sup> and '''''h''''' = ''AGM'' (d'''''r'''''/d''t'' × '''''r''''') / ''r''<sup>3</sup>
+and derives, in the same manner in which the force on a moving charge follows from Maxwell's equations, the equation of motion for the two-body problem:
+: d<sup>2</sup>'''''r'''''/d''t''<sup>2</sup> = −''GM'' '''''r'''''/''r''<sup>3</sup> − ''A'' (d'''''r'''''/d''t'' × (d'''''r'''''/d''t'' × '''''r''''')) ''GM''/''r''<sup>3</sup>
+The first term is Newton. The second, involving a double vector product of the relative velocity with the radius vector, is the gravitational analogue of the magnetic force. ''A'' is a new constant with dimensions of an inverse squared speed; Guy provisionally sets ''A'' = 5 × 10<sup>−11</sup> SI (m<sup>−2</sup>s<sup>2</sup>). A summed version is written for a distribution of masses ''m''<sub>''j''</sub>.
+Crucially, the added term is shown to be ''attractive'' whatever the relative orientation of '''''r''''' and '''''v''''', so it always mimics extra mass. Projecting it on the ordinary gravity direction gives the paper's working formula
+: Δ''M''/''M'' = Δ''g''/''g'' = ''Av''<sup>2</sup> sin<sup>2</sup>θ
+where θ is the angle between radius vector and relative velocity. When '''''r''''' and '''''v''''' are parallel the effect vanishes entirely — which, Guy notes, is exactly the configuration of a laboratory free-fall measurement on Earth, so the new term escapes terrestrial detection. When they are perpendicular the supplement is simply ''Av''<sup>2</sup>.
+===Why A is not 1/c<sup>2</sup>===
+In vacuum electromagnetism the corresponding constant is ε<sub>0</sub>μ<sub>0</sub> = 1/''c''<sup>2</sup>. Guy argues that gravitation applied to ''macroscopic'' matter is more like electromagnetism "in matter", where εμ can exceed its vacuum value by many orders of magnitude. Following the suggestion (attributed to Assis, 1992) that gravitation is a macroscopic average of forces of electromagnetic origin, he offers the heuristic ''A''·(macroscopic velocity)<sup>2</sup> = (1/''c''<sup>2</sup>)(microscopic velocity)<sup>2</sup>. With an internal-to-bulk speed ratio of about 10<sup>3</sup> this gives ''A'' ≈ 10<sup>6</sup>/''c''<sup>2</sup>, close to the adopted 5 × 10<sup>−11</sup>. For massless photons he instead takes ''A'' = 1/''c''<sup>2</sup>.
+===Four applications===
+For the Pioneer spacecraft, an anomalous Δ''g'' of 8.74 × 10<sup>−10</sup> m/s<sup>2</sup> at distances of order 10<sup>13</sup> m gives Δ''g''/''g'' = 6.7 × 10<sup>−4</sup>. With ''v'' = 10<sup>4</sup> m/s and a trajectory inclination of some 20°, the formula returns ''Av''<sup>2</sup>sin<sup>2</sup>θ = 5.8 × 10<sup>−4</sup> — the same order. Guy proposes the testable constancy (1/''v''<sup>2</sup>sin<sup>2</sup>θ)(Δ''g''/''g'') = ''A'' across many spacecraft.
+For flat galaxy rotation curves, peripheral stars move at 200–250 km/s essentially perpendicular to the radius, so ''Av''<sup>2</sup> = 5 × 10<sup>−11</sup> × 4 × 10<sup>10</sup> ≈ 2 — an apparent missing mass of order the visible mass, as reported. For galaxies in clusters, relative velocities of 500–800 km/s raise ''v''<sup>2</sup> by a factor of a few, and Δ''M''/''M'' rises to tens, matching the larger cluster discrepancy at the same ''A''.
+For cosmic expansion, Guy first insists on what "acceleration" means in the [[Big Bang]] picture: the recession speed still decreases, only less than Newtonian gravity predicts. As a system of mutually receding bodies evolves, θ tends toward zero, so the always-attractive extra term progressively switches off; writing ''g''<sub>II</sub> = ''g''<sub>I</sub> + ''g''<sub>repulsive</sub>, the shortfall behaves as an effective repulsion following the same 1/''r''<sup>2</sup> law with apparent repulsive mass ''AMv''<sup>2</sup>. Taking early speeds of ~10<sup>8</sup> m/s and small angles gives a ratio of order 10<sup>2</sup> between apparent dark and visible energy.
+===Consistency checks===
+Guy checks that the new term does not spoil the solar system. For Mercury (''v'' = 48 km/s) ''Av''<sup>2</sup> ≈ 10<sup>−1</sup>, and after angular weighting the ratio of the new coefficients β<sub>''ij''</sub> to the Newtonian α<sub>''ij''</sub> is 10<sup>−2</sup>–10<sup>−3</sup>, against the observed perihelion ratio 40/5600 ≈ 7 × 10<sup>−3</sup>. For light bending, the Newtonian calculation gives half the observed deflection; with ''A'' = 1/''c''<sup>2</sup> the extra term contributes Δα/α = 1, restoring the full value.
+==Assessment==
+The attractive feature of this paper is its economy and its falsifiability. One constant, fixed at 5 × 10<sup>−11</sup> SI, is carried unchanged through the Pioneer anomaly, galactic rotation, cluster dynamics and cosmic acceleration, and lands within an order of magnitude in each case. That is a genuinely different discipline from fitting each anomaly separately, and Guy states plainly the test that would break it: the quantity (1/''v''<sup>2</sup>sin<sup>2</sup>θ)(Δ''g''/''g'') must come out the same for every spacecraft. His criticism of MOND — that it is an ad hoc profile that works for rotation curves but not for clusters, spacecraft or expansion — is fair, and his placement of the proposal in the older tradition of velocity-dependent gravity ([[Oliver Heaviside]] 1893, Assis, Jefimenko, Ragusa, Gruffat) is honest about priority, including the acknowledgement that Gruffat (2004) proposed the same law.
+The difficulties are equally plain, and several are conceded. The derivation of equation (3) from the field pair (2) is asserted rather than shown: the reader is referred to the analogy with the Lorentz force and to third-party papers on scalar-to-vector time, and no explicit calculation appears. The parameter ''A'' is not predicted, only fitted and then rationalised after the fact by a heuristic ratio of microscopic to macroscopic speeds that is chosen (10<sup>3</sup>) precisely to land near the required value — so the single-constant economy is weaker than it looks. The cosmological application is the loosest: the angles between radius and velocity vectors are simply "difficult to estimate, put a few degrees", and the factor sin<sup>2</sup>θ = 10<sup>−3</sup> that produces the 10<sup>2</sup> energy ratio is inserted by hand. The choice ''A'' = 1/''c''<sup>2</sup> for photons versus ''A'' = 5 × 10<sup>−11</sup> for matter is a second free choice presented as one.
+Against measurement, the sharpest problem is that the treatment is Newtonian and applies to the Pioneer anomaly a formula that vanishes for radial motion — yet the anomaly is reported as a nearly constant sunward acceleration on trajectories that become increasingly radial, whereas ''Av''<sup>2</sup>sin<sup>2</sup>θ should fall as sin<sup>2</sup>θ shrinks and as the craft decelerate. (The anomaly has since been widely attributed to anisotropic thermal recoil, which the paper predates.) In clusters, the [[Gravitational Lensing]] mass — a static measurement in which the source galaxies' relative velocities play no role — agrees with the dynamical missing mass; a velocity-dependent term should decouple the two. The supernova evidence for acceleration is a luminosity-distance relation over redshift, not a directly measured deceleration of a two-body system, so the identification of Δ''E''/''E'' with the observed departure is at best schematic. And the promise that the framework will "reconsider" relativity is left as a programme: the metric, the invariance of d''s''<sup>2</sup>, and the observed [[Time Dilation]] of moving clocks are named as things to be redesigned, not redesigned here.
+Guy is candid about all of this — the paper calls itself exploratory, asks repeatedly for the ''n''-body simulations that would decide the matter, and does not claim the anomalies are explained. Judged as what it says it is, a preliminary order-of-magnitude exploration of one idea, it is coherent and worth the computation it asks for; judged as a replacement for [[Dark Matter]] and dark energy, it is not yet a calculation.
+==See also==
+* [[Bernard Guy]]
+* [[Dark Matter]]
+* [[Dark Energy]]
+* [[Gravity]]
+* [[General Relativity]]
+* [[Lorentz Force]]
+* [[Expanding Universe]]
+* [[Time]]
 [[Category:Scientific Paper|modified law gravitation taking account relative speeds moving masses preliminary study]]
 [[Category:Relativity|modified law gravitation taking account relative speeds moving masses preliminary study]]
+[[Category:Gravity|modified law gravitation taking account relative speeds moving masses preliminary study]]
+[[Category:Cosmology|modified law gravitation taking account relative speeds moving masses preliminary study]]