Special Relativity Fails to Conserve Momentum

Scientific Paper
Title	Special Relativity Fails to Conserve Momentum
Read in full	Link to paper
Author(s)	Victor Nikolayevich Cochetkov
Keywords	Special Relativity, momentum, conservation laws, rotating system, Relativity
Published	2010
Journal	Proceedings of the NPA
Volume	7
No. of pages	5
Pages	80-84

Read the full paper here

Abstract

The article attempts to show that the use of the special theory of relativity, when considering the motion of a closed mechanical system in the inertial reference systems, can lead to non-compliance with the law of conservation of momentum.

Overview

This five-page conference paper argues that special relativity, applied to a specific closed mechanical system, predicts a total momentum that is not conserved from the point of view of a second inertial frame. The author, Victor Nikolayevich Cochetkov, an engineer at the Russian space-infrastructure agency TsENKI, treats this as evidence that the theory conflicts with the homogeneity of space from which momentum conservation is usually derived.

It is one of a series of closely related papers by Cochetkov built on the same construction — a rotating pair of masses examined from a boosted frame — and reaching the same conclusion. The others are discussed on this wiki under their own titles.

The argument

The system

The system is the simplest possible: two point bodies of equal rest mass M₀, joined by a massless string, rotating with angular speed ω about their common centre of mass O at radius R. In the frame O₂ in which O is at rest, the two bodies always move oppositely, so their momenta cancel and the total is zero at every instant.

The boost

Cochetkov then views the same system from a frame O₁, relative to which O₂ moves at constant speed V along the x-axis. He uses the Lorentz transformation and the relativistic velocity-composition formulas to find the two bodies' velocity components in O₁, and the relativistic expression P = M₀v/√(1−v²/c²) for each body's momentum.

The decisive step is that the two bodies are not sampled at the same phase of their rotation. Because simultaneity is relative, an instant of constant time t₁ in O₁ corresponds to different times t₂₁ and t₂₂ for the two bodies in O₂, so in O₁ one body is caught at a different point in its circular path than the other. Their transverse momenta therefore no longer cancel, and Cochetkov derives that conservation in O₁ would require his condition (66), which reduces to 1/c² = 0 — impossible for finite c. He concludes that momentum is not conserved.

The paper closes by acknowledging correspondence with a long list of physicists, and thanks David Bergman and Greg Volk in particular.

Assessment

The algebra is correct, and the effect it identifies is real: viewed from a boosted frame, the two bodies of a rotating pair are indeed caught at different phases, and the sum of the two point-mass momenta computed the way Cochetkov computes it does vary. This is not an arithmetic mistake, and the same feature appears, correctly, in his companion papers.

The conclusion does not follow, and the reason is a single omission that the paper's own idealisation builds in. The string is declared massless, and its tension is never entered into the momentum budget. But the tension is precisely what holds the rotating bodies on their circular path — it supplies the centripetal force — and in relativity a stressed body carries momentum by virtue of its stresses, not only through the rest mass of its parts. A taut, spinning string under tension is a stressed system, and when it is boosted it contributes a momentum term of exactly the order VωR/c² that Cochetkov finds "missing" from the two point masses. Restoring it makes the total conserved.

This is not an ad hoc rescue. It is the same accounting that resolves three classic problems in relativistic dynamics:

the Trouton–Noble experiment (1903), where a charged capacitor suspended in the moving Earth frame was expected to feel a turning couple and did not — resolved by the momentum stored in the stressed electromagnetic field;
the "4/3 problem" in the electromagnetic mass of the electron;
the general theorem, due to Max von Laue, that for any closed system the field and material stresses must be included for the four-momentum to transform as a four-vector.

In each case a naive sum over parts appears not to conserve momentum, and in each case the stresses supply the balance. Cochetkov's rotating dumbbell is a mechanical instance of the same thing, and the wiki's campaign summaries of his other papers reach this diagnosis independently: the missing term is the internal stress (tension) of the connecting body, the "hidden momentum" of a stressed system in motion.

Two further points are worth recording. First, the derived condition 1/c² = 0 is not a discovery but a restatement of the premise: it is what one gets by demanding that a stress-carrying system conserve momentum while refusing to count the stress, and its failure for finite c simply says that the stresses cannot be neglected. Second, the claim bears on a purely internal accounting question and makes no new observational prediction; the relevant experiment, Trouton–Noble, was performed and gave the null result relativity predicts, and later high-precision repetitions have confirmed it.

The paper is therefore best read as a clean, correctly worked illustration of why stresses must be included in relativistic momentum — arriving, by the sign of its own conclusion, at the opposite of what a full accounting shows.