Special Relativity: Depending on the Definition of the Momentum of a Closed System of Bodies from Time

Scientific Paper
Title	Special Relativity: Depending on the Definition of the Momentum of a Closed System of Bodies from Time
Read in full	Link to paper
Author(s)	Victor Nikolayevich Cochetkov
Keywords	The special theory of a relativity, the law of conservation of momentum of the closed mechanical system, dependence of weight of a body on speed of its movement, symmetry of space and time, relativity
Published	2011
No. of pages	12

Read the full paper here

Abstract

The article attempts to show a concrete example, that the application of the special theory of relativity, when considering the motion of a closed mechanical system of bodies in inertial reference systems, can lead to the fact, that the momentum of a closed system will be a function of time.

Overview

V N Cochetkov, an engineer at the Russian federal space-infrastructure enterprise TSENKI in Moscow, published this note in the Journal of Vectorial Relativity in 2011. It is not a philosophical objection to special relativity but a single worked calculation, carried through to numerical graphs, of one mechanical arrangement viewed from two inertial frames. The claim is narrow and testable: apply the Lorentz Transformation, the relativistic velocity-addition rule and the velocity dependence of mass to every part of a rotating system, sum the momenta at a fixed instant of the moving frame, and the total does not stay constant. If so, the law of conservation of momentum fails for a closed mechanical system in some inertial frame.

Cochetkov states his motivation in the introduction. The velocity dependence of mass is, in the standard treatment, derived from the requirement that momentum and energy be conserved in collisions — that is, for systems "whose interaction is instantaneous in nature". He proposes to test the same relation on a system "whose interaction is ongoing", where the parts are in permanent contact rather than meeting briefly. The rotating dumbbell is his chosen example precisely because the interaction never stops.

The argument

The mechanical system

The system consists of two point bodies, 1 and 2, each of rest mass M_o, joined by a string 3 of rest mass m_o uniformly distributed along its length. The bodies rotate at angular speed ω about the common centre of mass O_c, each at distance R from it. In the frame Oxyz the centre O_c is held fixed at the origin, the rotation is counter-clockwise in the Oxy plane, and at t = 0 body 1 lies on the positive x axis with body 2 diametrically opposite. In this frame the arrangement is manifestly symmetric: at every instant the two bodies have equal speeds and opposite velocities, and the total momentum vanishes.

A second inertial frame O 'x y z ' moves at constant speed V along Ox, with axes parallel and origins coinciding at t = t ' = 0. To treat the string, Cochetkov divides it at rest into 2n equal segments and places a point body of rest mass m_o/2n at the centre of each, indexing the segments between O_c and body 1 as i-points and those between O_c and body 2 as j-points, each at its own radius R_i or R_j.

The transformation and the simultaneity condition

For every one of these 2n + 2 masses the paper writes out, in full, the position and velocity components in Oxyz as functions of the phase ωt; then the transformed coordinates, the transformed time, and the transformed velocity components in the primed frame using the standard Lorentz transformation and velocity-composition formulae. Momentum components in the primed frame follow from the relativistic mass–velocity relation applied to each point mass individually.

The pivot of the whole argument is the step Cochetkov makes explicit as his equation (56). To speak of the momentum of the system at a primed instant t ', all the individual primed times t '₁, t '₂, t '_1i, t '_2j must be set equal to that common t '. Because the masses sit at different x coordinates, this single primed instant corresponds to different unprimed times for each of them — different phases of the rotation. Equation (56) is the resulting simultaneity condition, an implicit relation of the form t − (V/c²)R cos(ωt) = t '√(1 − V²/c²), one for each mass, and it is solved numerically for each t '.

The numerical example and the result

The worked case takes V/c = 0.9, ωR/c = 0.8, a string-to-body rest-mass ratio of 0.1 and n = 10 segments per half-string. The procedure is: choose t '; invert (56) to find the corresponding t₁, t₂, t_1i, t_2j; evaluate each mass's momentum components; and sum.

Five figures present the outcome, each plotted both for the two bodies alone and for the bodies plus string. The longitudinal component P '_x oscillates in a band running from about −5 to −11 in units of M_oc; the transverse component P '_y swings between roughly −1 and +1; the magnitude |P '| ranges over about 5 to 11 M_oc; and the angle α' between the momentum vector and the O 'x ' axis oscillates between about −5° and +6°. A final figure gives the string's own contribution separately, of order ±0.4 M_oc. The conclusion drawn is that in the primed inertial frame the closed system "has variable in magnitude and direction of the vector of the momentum P ' in the time t ', ... which contradicts the law of conservation of momentum". The paper thanks Michael H. Brill of Physics Essays for help and support, and cites a single reference, the Yavorsky and Detlaf physics handbook.

Assessment

The paper deserves credit for being concrete where most objections of this kind are rhetorical. It defines one system completely, applies the textbook transformations to every constituent without exception, discretises the string rather than waving at it, states its numerical inputs, and publishes the curves. It also identifies, correctly and without fuss, the feature that does the work: relativity of simultaneity means that a single instant in the boosted frame slices the rotating system at different phases, so the cancellation that holds in the rest frame cannot be assumed to survive the boost. That the effect should be large here is easy to confirm from the stated inputs. The phase offset between the two bodies induced by the boost is Δ(ωt) = 2γ(V/c)(ωR/c), and with γ = 1/√(1 − 0.81) = 2.294 this is 2 × 2.294 × 0.9 × 0.8 = 3.30 radians, or about 189°. The two bodies, already half a turn apart, are being sampled almost another half turn apart — so in the primed frame they are caught at nearly the same phase, their transverse momenta add instead of cancelling, and the sum swings by an amount of order the individual momenta. That is exactly what the figures show, and the magnitudes are right: a body at ωR = 0.8c boosted by 0.9c reaches 0.988c at best alignment, γ ≈ 6.5, giving p ≈ 6.5 M_oc each and a maximum around 11–13 M_oc for the pair. The arithmetic is sound.

What does not follow is the conclusion. The quantity Cochetkov computes — the sum of the mechanical momenta of the constituent particles, evaluated on a hyperplane of constant t ' — is not the total momentum of the system, and there is no theorem of relativity that says it should be conserved. The total momentum is the integral of the energy–momentum tensor over that hyperplane, and it includes the contribution of the internal stresses. The paper's own construction makes clear that those stresses are present and are not optional: the two bodies travel in circles, so something must supply the centripetal force, and that something is the tension in string 3. Yet the string is modelled as 2n free point masses carrying only γmv, with no tension term anywhere. A stressed body carries momentum density in a frame in which it moves — the T⁰ⁱ components pick up the stress T^xx, T^yy under a boost — and when that contribution is included the total is constant, as the divergence-free condition on the stress tensor and the boundedness of the system require.

This is not an obscure repair. It is von Laue's 1911 resolution of the Trouton–Noble paradox, and the same omission generates the celebrated "4/3 problem" in the electromagnetic mass of the electron and the apparent torque on a charged capacitor in motion. In every case the naive sum of particle momenta over a boosted slice appears to vary or to point the wrong way, and in every case the discrepancy is exactly the stress term that was left out. Cochetkov's example is a mechanical version of the same construction, and it is a clean and instructive one — but it demonstrates that mechanical momentum alone is not a four-vector for a stressed system, which has been known for over a century, rather than that momentum is not conserved.

Two further difficulties are worth naming. First, the model is internally strained in a way that also matters: an ideal string transmitting tension between two bodies at ωR = 0.8c is a rigid-body idealisation, and rigid bodies are precisely what relativity forbids — the Herglotz–Noether theorem shows that no Born-rigid motion with rotation of this kind exists. Cochetkov's introduction sets out to replace the "instantaneous" interactions of collision theory with an "ongoing" one, but an inextensible string is an instantaneous interaction in disguise, propagating force along its length at infinite speed. Second, the paper offers no independent check on its own numerics. A calculation whose entire content is that a sum fails to vanish ought to be validated on a case where the answer is known — the same code with ω = 0, or with V = 0, or in the limit m_o → 0 and ωR ≪ c where the Newtonian answer must be recovered. No such test is reported, and the graphs are read off figures rather than tabulated.

Set against measurement, the conclusion also has to contend with the fact that relativistic momentum conservation is what accelerator physics is built on. Every invariant-mass reconstruction from decay products, every beam-energy calculation, every missing-momentum measurement used to infer a neutrino, assumes that Σγmv is conserved in the laboratory frame for systems whose parts were prepared in a different frame; these succeed to fractions of a per cent daily. Any genuine failure of the kind claimed here would show up first there. The paper is careful, honest and clearly presented, and its calculation is worth doing; but what it has found is the stress term, not a broken conservation law.

See also

• Victor Nikolayevich Cochetkov
• Special Relativity
• Lorentz Transformation
• Simultaneity
• Mass
• Inertia
• Michael H Brill