Calculation of So-Called General Relativistic Phenomena by Advancing Newton's Theory of Gravitation, Maintaining Classical Conceptions of Space and Relativity

Scientific Paper
Title	Calculation of So-Called General Relativistic Phenomena by Advancing Newton's Theory of Gravitation, Maintaining Classical Conceptions of Space and Relativity
Read in full	Link to paper
Author(s)	Reiner Georg Ziefle
Keywords	perihelion, Mercury, relativity, GRT, pulsar, PSR 1913+16, gravitation, Paul Gerber, Newton, Einstein
Published	2003
Journal	Physics Essays
Volume	16
Number	3
No. of pages	10
Pages	375-384

Read the full paper here

Abstract

With the example of the motion of  Mercury around the sun it is shown, how Newton's Theory of Gravitation should be advanced by taking into consideration the finite velocity of gravitational expansion and the present concept of transference of forces by particles to be able to calculate so-called general relativistic phenomena, as the additional motion of Mercury's perihelion, the curvation of a light beam at the surface of the sun and the phenomena observed at the binary pulsar PSR 1913+16, maintaining classical conceptions of an Euklidean space and the Galileian principle of relativity.

Overview

Ziefle's paper, published in Physics Essays in 2003, proposes that the classic tests of general relativity can be recovered inside Newtonian gravitation, in a flat Euclidean space obeying Galilean relativity, by adding just two assumptions: that gravity propagates at the speed of light, and that it is carried by particles — gravitons. From these he derives a velocity-dependent correction factor to Newton's inverse-square law and applies it in turn to the perihelion advance of Mercury, to the deflection of starlight at the solar limb, and to the periastron advance and orbital-period decay of the binary pulsar PSR B1913+16.

The departure from the mainstream account is total in its foundations and deliberately modest in its results. Ziefle explicitly abandons curved spacetime and the relativistic principle, and instead asks the reader to accept that the closing speed between Mercury and a graviton from the Sun really can exceed c — "which is of course not possible in relativistic physics. But we have postulated that the Galilean principle of relativity should be valid, so that we want to assume, nevertheless, that this is possible." He also compares his approach against two other classical derivations in the same tradition, Paul Gerber's of 1898/1917 and Paul Marmet's of 1999, and argues that both have defects his does not, appealing finally to Ockham's razor.

The argument

The gravitational factor of motion

Gravitons stream from the Sun at speed c in all directions. If Mercury were at rest, the gravitons' speed relative to it would be c and the rate at which they arrive would have relative value 1. Because Mercury moves transversely with speed v, Ziefle composes the two velocities by the Pythagorean theorem to get the closing speed x = √(c² + v²), and in relative units

γ′ = √(1 + (v/c)²)

which he names the "gravitational factor of motion". The graviton encounter rate rises by γ′ for the Sun's gravitons meeting Mercury and, by the same reasoning, by γ′ again for Mercury's gravitons meeting the Sun. Multiplying, the gravitational interaction is enhanced by (γ′)² and Newton's law becomes

F = (γ′)²GMm/r²

He draws the immediate corollary that "G is not as constant as Newton thought", and that the Earth's 1 km/s annual variation in orbital speed should make measured G fluctuate slightly.

Mercury's perihelion

The chain from force to precession is short. If the acceleration rises by (γ′)², "the velocity of the planet must also increase by the same factor"; if the velocity rises by (γ′)², "in a certain time a larger angle is also traversed by the radius"; so every angular position maps as φ₂ = (γ′)²φ₁, and Δφ = φ₁(v/c)².

Taking the "median angular position of an elliptical planetary orbit" to be π, and then multiplying by 2π because "there results an alteration for each angular position along the whole route of Mercury's path from perihelion to perihelion", he obtains

Δφ = 2π²(v/c)²

and verifies it by the integral (1/2)(2π)²(v/c)². With Mercury's mean orbital speed v = 47.88 km/s, (v/c) = 1.5971 × 10⁻⁴, (γ′)² = 1.0000000255073, and Δφ = 5.03494 × 10⁻⁷ rad = 2.88481 × 10⁻⁵ degrees per revolution. At 4.1521 revolutions per year this is 0.011978° or 43.12″ per century, against Einstein's 43.03″ and the observed 43.11″ ± 0.45″.

He also gives the correct velocity profile of the orbit, v(φ) = v_min(1+e)/(1 − ecos φ), with e = 0.2056 and aphelion speed 38.86 km/s, and offers a second route through Newton's orbital energy E = −GMm/2a: if G rises by (γ′)², E becomes more negative, and "classical mechanics predicts that the orbiting velocity of a planet is larger if the energy E of an elliptical orbit is smaller."

Applying Kepler's second law, since area goes as the square of angle, he gets ΔA/A = [π(v/c)²]² = 6.42139 × 10⁻¹⁵, hence a period shorter than Newton's by 4.88 × 10⁻⁸ s per revolution, so that "the revolution of Mercury … must get slightly faster and faster with time".

Light deflection

For a photon, v = c, so (γ′)² = 1 + 1 = 2 exactly. The Newtonian deflection 2GM/c²r is therefore doubled to 4GM/c²r — "the correct value, as is predicted by Einstein's theory of general relativity."

PSR B1913+16

For the Hulse–Taylor binary Ziefle takes e = 0.617, period 7.75 h, masses 1.42 and 1.41 solar masses, and stellar speeds ranging 75–300 km/s with "median" 187.5 km/s. He then reduces this to 175 km/s by a factor cos i = 0.933, attributed to "an inclination (i) toward each other of about 21 angular degrees" between the two orbits. This gives v/c = 5.84 × 10⁻⁴ and (γ′)² = 1.000000341.

A further factor is then introduced. The ratio of the gravitational effect at periastron to that at apastron is (1+e)²/(1−e)² = 17.83, and Ziefle takes the arithmetic mean of this and 1, giving 9.415, as "the median relative gravitational effect caused in the center of mass by each star". Multiplying, Δφ = 2π²(v/c)² × 9.415 = 6.34 × 10⁻⁵ rad = 0.00363° per orbit; at 1131 orbits per year this is 4.1° per year, against an observed 4.0°–4.22°.

For the orbital decay he squares the mean angular shift and doubles it for the two stars: Δt/t = −2 × [π(v/c)²]² = −2.296 × 10⁻¹², against GR's −2.4 × 10⁻¹² and the observed (−2.30 ± 0.22) × 10⁻¹² — a shortening of 6.4 × 10⁻⁸ s per revolution, or 73 μs per year, "which is explained by present-day physicists by gravitational radiation."

Discussion of Gerber and Marmet

Ziefle criticises Paul Gerber on two counts: gravitational energy streaming outward from the attracting mass would require an observable secular loss of solar mass; and Gerber's decreasing gravitational interaction makes the orbital energy less negative, which by Newtonian mechanics slows the revolution rather than speeding it, contrary to what Gerber concluded. Marmet's 1999 derivation, based on mass–energy conservation and locally different units of mass, length and time on Mercury, is credited with getting the right answer but rejected as "not a pure classical physical theory, as it uses quantum mechanics to derive local values". Ziefle's own scheme is offered as the one requiring the fewest additional assumptions, while conceding that it makes the speed of gravitational propagation "noninvariant or nonconstant" between observers.

Assessment

The arithmetic, as arithmetic, is correct. Every number in the paper reproduces. (v/c) = 47.88/299792.458 = 1.59710 × 10⁻⁴; its square is 2.550744 × 10⁻⁸, matching the quoted 1.0000000255073 for (γ′)². 2π² × 2.550744 × 10⁻⁸ = 5.03497 × 10⁻⁷ rad, which is 2.884824 × 10⁻⁵°; 365.256/87.969 = 4.15210 revolutions per year; and 2.884824 × 10⁻⁵ × 415.210 × 3600 = 43.12″. The pulsar chain checks too: (175/c)² = 3.4074 × 10⁻⁷, 2π² × that = 6.731 × 10⁻⁶ rad, × 9.415 = 6.337 × 10⁻⁵ rad = 0.003631°, × 1131 = 4.11°/yr; and 2[π(v/c)²]² = 2.29 × 10⁻¹². The auxiliary data are right as well — Mercury's e = 0.2056, aphelion 38.86 km/s, mean 47.88 km/s, period 87.969 d; the velocity formula correctly returns Mercury's 58.98 km/s perihelion speed; (1.617/0.383)² = 17.82. There is no slip of a decimal point anywhere in the paper.

But the derivation drops a factor of π that the paper's own mapping forbids. Ziefle's premise is a single, explicit rule: every angular position is stretched, φ₂ = (γ′)²φ₁. Apply it to a complete revolution. At φ₁ = 2π the mapped position is 2π(γ′)², so the perihelion advance per orbit is 2π[(γ′)² − 1] = 2π(v/c)² = 1.6027 × 10⁻⁷ rad, which is 13.73″ per century — not 43.12″. The factor of π difference comes from the step where the "median angular position" π is multiplied in and then the whole is multiplied by 2π again. Equivalently, the integral ∫₀^2πφ dφ = 2π² that Ziefle offers as confirmation sums the displacement at every intermediate angle rather than reporting the displacement at the end of the circuit; the displacements are not independent contributions to be added, they are successive positions of the same point. The paper's own postulate, applied consistently, gives roughly a third of the observed precession.

The eccentricity dependence is missing, and that is what makes Mercury look like a success. Ziefle's result can be written in closed form. Because for a Kepler orbit the mean orbital speed 2πa/T equals √(GM/a) exactly, his Δφ = 2π²(v/c)² is Δφ = 2π²GM/ac². General relativity gives Δφ = 6πGM/[ac²(1−e²)]. The ratio is therefore

Δφ_Ziefle/Δφ_GR = (π/3)(1 − e²)

with no dependence on mass, distance or period. The two errors are of opposite sign: the geometric factor is π/3 = 4.72% too large, and the omission of 1/(1−e²) is 4.42% too small for Mercury. They cancel to 0.29% — which is the whole of the celebrated agreement. The formula is exact only for an orbit of eccentricity e = √(1 − 3/π) = 0.2123, and Mercury's eccentricity is 0.2056. Nothing in the derivation knows about eccentricity, so this is coincidence rather than physics, and it is testable on other bodies. For Venus (e = 0.0068) the formula predicts 9.03″ per century where GR gives 8.62″ and radar ranging measures the relativistic advance to a fraction of an arcsecond. For the Earth it predicts 4.02″ against 3.84″. Most decisively, for the highly eccentric asteroid 1566 Icarus (e = 0.827) it predicts 3.33″ per century against GR's ~10.1″ and Shapiro's radar measurement of 9.8″ ± 0.8″ — a factor of three, and far outside the error bar. Applied to the paper's own pulsar (e = 0.617) the same factor is 0.65, which is why an extra multiplier of 9.415 was needed there.

That 9.415 is a fitted number. It is introduced as "the median relative gravitational effect", but it is the arithmetic mean of the periastron and apastron values of a 1/r² quantity, which is not the average of anything over the orbit — the time-average of 1/r² over a Kepler orbit is 1/[a²√(1−e²)], giving a ratio to the apastron value of 3.3, not 9.4. Without the factor the prediction is 0.44°/yr instead of 4.1°/yr, a tenfold miss; with it, the answer lands on the observation. The projection factor that precedes it is worse: cos i = 0.933 is attributed to "an inclination toward each other of about 21 angular degrees" between the two stars' orbits, but the two orbits of a binary lie in the same plane by definition — the orbital inclination of PSR B1913+16 is a line-of-sight quantity of about 47°, and there is no 21° angle between the components' orbits to project by.

The orbital-decay match has no dimensional basis. Ziefle obtains a fractional period change by squaring an angle and doubling it. General relativity's prediction is the quadrupole formula, in which the decay depends on the component masses, the period to the −5/3 power, and an eccentricity enhancement factor (1 + 73e²/24 + 37e⁴/96)/(1−e²)^7/2 that is about 11.8 for this system. Ziefle's expression 2π²(v/c)⁴ contains none of these dependences; that it lands near 2.3 × 10⁻¹² is a numerical accident of this one system. He also states that the system "must get faster and faster with time, so that the system is losing energy" without naming any sink for that energy: in his framework nothing is radiated, and the orbit simply decays, which violates energy conservation in a theory whose selling point is classical conservatism. The same objection applies to his Mercury result, where the orbit is said to speed up secularly.

The steps from force to precession are asserted, not derived. "If the acceleration increases by the factor (γ′)², the velocity of the planet must also increase by the same factor" is not a mechanical statement: acceleration multiplied by a factor changes the rate of change of velocity, and for a circular orbit, where v² = Fr/m, a force multiplied by (1+ε) raises the speed by only (1+ε/2). The alternative energy route is no better: for a Kepler orbit the mean motion goes as |E|^3/2, not as |E|. The paper never integrates its modified force law over an orbit, which is the only way to obtain a precession honestly, and the two informal routes it offers give different powers of the same correction. A separate structural problem is that (γ′)² depends on speed alone and not on direction: a radially falling body and a transversely orbiting one at the same speed feel the same enhancement, which is not a force law that can be written down as a vector field, and cannot be tested against the many other cases where velocity-dependent gravity would show up.

Light bending, and what it costs elsewhere. Setting v = c gives exactly 2, and doubling the Newtonian 0.87″ to 1.75″ is the right answer — this is the paper's neatest result. But the same premise has consequences the paper does not follow up. If the gravitational force on a photon is twice the Newtonian value, the work done on a photon climbing out of a potential well is also doubled, so the gravitational redshift should be 2gh/c² rather than gh/c². The Pound–Rebka–Snider tower experiment measured the standard value to about 1%, and GPS satellite clocks confirm it continuously to parts in 10⁴. Ziefle's factor of two is therefore correct for one test and wrong by a factor of two for another that shares its premise. The related prediction that G should measurably fluctuate with the Earth's 1 km/s annual speed variation is at the level of (Δv·v)/c² ≈ 10⁻⁹ and is not obviously excluded, but it is offered without any comparison to the laboratory G record.

What is nevertheless of value. The paper is clear about its postulates, states them as postulates, does its arithmetic openly enough that a reader can check every step, and — unusually for the genre — engages seriously and critically with two rival classical derivations rather than ignoring them. The Gerber criticism is a real one: a theory in which gravitational energy streams outward from the source does owe an account of the source's mass loss, and Ziefle's observation that Gerber's decreasing interaction should slow rather than speed the orbit is a fair Newtonian point. And the light-bending result is a genuinely elegant coincidence of the framework. But the perihelion agreement, which is the paper's centrepiece, does not survive being tested against the paper's own premise (which gives 13.7″), against a second planet, or against Icarus; and the pulsar agreement rests on a factor that was chosen rather than derived. On its own terms the calculation is arithmetically clean and physically unfounded.

See also

• Reiner Georg Ziefle — the author
• Perihelion Precession of Mercury — the phenomenon at issue
• Paul Gerber, Paul Marmet — the rival classical derivations discussed in the paper
• Gravitational Lensing, Speed of Light, Equivalence Principle, Gravitational Waves
• Category:Gravity, Category:Relativity