The Special Theory of Relativity: Linear Example of Infringement of Laws of Preservation of an Impulse: Difference between revisions
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In article attempt to show becomes that use of the special theory of relativity can lead to infringement of the law of preservation of an impulse of the closed mechanical system consisting of bodies, located on one line and which interaction has constant character, in inertial systems of readout. | In article attempt to show becomes that use of the special theory of relativity can lead to infringement of the law of preservation of an impulse of the closed mechanical system consisting of bodies, located on one line and which interaction has constant character, in inertial systems of readout. | ||
==Overview== | |||
This short paper by Victor Nikolayevich Cochetkov — the byline on the PDF reads "Kochetkov Victor Nikolayevich," chief specialist at the Russian FSUE "TSENKI" centre for space ground-based infrastructure — is a single, tightly constructed thought experiment aimed at [[Special Relativity]]. Its claim is that a completely ordinary closed mechanical system, treated by the standard relativistic rules, comes out with a momentum ("impulse") that varies in time in an inertial frame, which would contradict conservation of momentum. The paper is the linear counterpart of an earlier planar version published by the author in the ''Journal of Vectorial Relativity'' (vol. 6, 2011, 65–76), cited as reference [1]; the standard formulae are taken from the Yavorsky and Detlaf physics handbook (Nauka, Moscow, 1980). | |||
The English is a machine translation from Russian, and its vocabulary needs decoding: "impulse" is momentum, "system of readout" is reference frame, "preservation" is conservation, "weight" is mass, "unclenched" means relaxed or extended. Once translated back, the argument is entirely conventional in its methods — Lorentz coordinate and velocity transformations, the relativistic momentum ''p'' = ''M''<sub>0</sub>''V''/√(1−''V''<sup>2</sup>/''c''<sup>2</sup>) — and its force rests on the relativity of simultaneity. What departs from the mainstream is the conclusion drawn: rather than reading the result as an incomplete accounting, Cochetkov reads it as an internal contradiction in the theory. | |||
==The argument== | |||
===The system=== | |||
Two point bodies, 1 and 2, each of rest mass ''M''<sub>0</sub>, are joined by a perfectly elastic spring 3 whose mass is negligible compared with theirs. Under the spring's action they oscillate symmetrically back and forth along a single straight line about their common centre of mass, point ''S''. The system is placed in an inertial frame ''O''<sub>2</sub>''x''<sub>2</sub>''y''<sub>2</sub>''z''<sub>2</sub> with ''S'' at rest at the origin and the bodies on the ''x''<sub>2</sub> axis. | |||
The cycle is described in detail. At ''t''<sub>2</sub> = 0 the spring is fully compressed, holding maximum potential energy, and both bodies are at rest with ''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S'' and ''O''<sub>2</sub> all coincident — a configuration the author says is assumed to be achieved structurally. The spring then relaxes and drives the bodies apart; at ''t''<sub>2''m''</sub> it is fully relaxed, its potential energy zero, and the bodies have maximum speed and kinetic energy. The spring then stretches and decelerates them until at ''t''<sub>2''t''</sub> they stop with the spring fully extended at maximum potential energy. The motion then reverses through the mirror-image half-cycle to ''t''<sub>2''n''</sub>, the period. Throughout, symmetry gives | |||
: ''x''<sub>21</sub> = −''x''<sub>22</sub> and ''V''<sub>21</sub> = −''V''<sub>22</sub> | |||
so the momentum in the rest frame of ''S'' is identically zero at every instant. | |||
===Transforming to a moving frame=== | |||
A second inertial frame ''O''<sub>1</sub>''x''<sub>1</sub>''y''<sub>1</sub>''z''<sub>1</sub> is introduced, with parallel axes, with ''O''<sub>2</sub>''x''<sub>2</sub>''y''<sub>2</sub>''z''<sub>2</sub> moving along ''O''<sub>1</sub>''x''<sub>1</sub> at constant speed ''V'', and with the clocks zeroed when the origins coincide. Cochetkov writes down the [[Lorentz Transformation|Lorentz transformations]] for the coordinates of each body separately, and the velocity-addition relations connecting ''V''<sub>21</sub> to ''V''<sub>11</sub> and ''V''<sub>22</sub> to ''V''<sub>12</sub>. | |||
The decisive step is his equations (11)–(16). Transforming the times, and imposing the condition that the two bodies be considered at the ''same'' instant ''t''<sub>11</sub> = ''t''<sub>12</sub> in the moving frame, he obtains | |||
: ''t''<sub>21</sub> + ''Vx''<sub>21</sub>/''c''<sup>2</sup> = ''t''<sub>22</sub> + ''Vx''<sub>22</sub>/''c''<sup>2</sup> | |||
Since by construction ''x''<sub>21</sub> ≥ 0 and ''x''<sub>22</sub> ≤ 0, it follows that ''t''<sub>21</sub> < ''t''<sub>22</sub> whenever the bodies are separated. In other words, one instant in frame 1 corresponds to two ''different'' instants in frame 2 for the two bodies — the ordinary relativity of [[Simultaneity]], stated correctly. | |||
===The momentum does not stay constant=== | |||
The momenta of the two bodies in frame 1 are written in the standard relativistic form, ''P''<sub>11</sub> = ''M''<sub>0</sub>''V''<sub>11</sub>/√(1−''V''<sub>11</sub><sup>2</sup>/''c''<sup>2</sup>) and likewise for body 2, and the system's momentum is their sum ''P''<sub>1</sub> = ''P''<sub>11</sub> + ''P''<sub>12</sub>. Because the two bodies are sampled at different phases of the oscillation — that is what ''t''<sub>21</sub> ≠ ''t''<sub>22</sub> means — the sum does not stay fixed. Cochetkov's figure 6 shows ''P''<sub>1</sub> oscillating about the value 2''M''<sub>0</sub>''V''/√(1−''V''<sup>2</sup>/''c''<sup>2</sup>). His conclusion: for a closed system in an inertial frame the momentum "should be necessarily constant (not to depend on size of the moment of time)," and here it is not. | |||
===The consistency condition=== | |||
Section 3 turns the result around and asks what would be required for conservation to hold. Two instants are compared. At ''t''<sub>1''o''</sub> = 0 both bodies sit at the common origin with the spring fully compressed, so ''V''<sub>11''o''</sub> = ''V''<sub>12''o''</sub> = ''V'' and the total momentum is ''P''<sub>1''o''</sub> = 2''M''<sub>0</sub>''V''/√(1−''V''<sup>2</sup>/''c''<sup>2</sup>). At the later instant ''t''<sub>1''t''</sub>, chosen so that body 1 is momentarily at rest in frame 2 (''V''<sub>21''t''</sub> = 0, hence ''V''<sub>11''t''</sub> = ''V''), body 2 cannot also be at rest in frame 2, because its corresponding time ''t''<sub>22''t''</sub> is necessarily later than ''t''<sub>21''t''</sub> by the simultaneity result (16). So ''V''<sub>22''t''</sub> ≠ 0 and therefore ''V''<sub>12''t''</sub> ≠ ''V''. | |||
Setting ''P''<sub>1''o''</sub> = ''P''<sub>1''t''</sub> and solving gives the requirement ''V''<sub>12''t''</sub> = ''V'', equivalently ''V''<sub>22''t''</sub> = 0, which in turn demands ''t''<sub>21''t''</sub> = ''t''<sub>22''t''</sub>. The paper ends on the resulting standoff, stated plainly: conservation of momentum requires ''t''<sub>21''t''</sub> = ''t''<sub>22''t''</sub>, while special relativity requires ''t''<sub>21''t''</sub> < ''t''<sub>22''t''</sub> "owing to not of a simultaneity" of events that are simultaneous in the other frame. Hence, Cochetkov concludes, "use of the special theory of relativity by consideration of separate examples can lead to default of the law of preservation of an impulse of the closed mechanical system in inertial systems of readout." | |||
==Assessment== | |||
What is genuinely good about this paper is its discipline. It invents no new mechanics, introduces no aether and postulates no new force; it takes the textbook transformations at face value, applies them to the simplest possible bound system, and follows the arithmetic where it leads. The relativity-of-simultaneity result (16) is derived correctly, and the observation that the two ends of an oscillating bound system are necessarily sampled at different internal phases in a moving frame is exactly right and is the physically interesting content of the exercise. The paper is also honest in structure: rather than resting on the first result, section 3 works backwards to identify precisely which condition would have to fail, and states the contradiction in its sharpest form instead of hiding it in prose. | |||
The difficulty is that one term has been left out of the bookkeeping, and it is the term that carries the whole effect. The system is described as consisting of "bodies 1 and 2 (and springs 3)," but the spring is stipulated to be massless and never appears in any momentum expression: ''P''<sub>1</sub> is defined throughout as ''P''<sub>11</sub> + ''P''<sub>12</sub> alone. In special relativity a spring holding potential energy ''U'' holds an equivalent [[Mass]] ''U''/''c''<sup>2</sup>, and when that spring is in motion it carries momentum accordingly; more than that, a stressed elastic body in motion carries an additional momentum contribution from its internal stress, which for a longitudinal spring under tension or compression is of order ''σV''/''c''<sup>2</sup> per unit volume. These are not negligible corrections at the order being tested — they are precisely of order ''V''/''c''<sup>2</sup> times the energy exchanged, which is the same order as the variation Cochetkov finds in ''P''<sub>11</sub> + ''P''<sub>12</sub>. The relativistic statement of conservation applies to the total energy-momentum of the closed system, obtained by integrating the full stress-energy tensor over a spacelike surface, not to a sum over the point particles only. Once the spring's energy and stress are included, the oscillation in the particle sum is compensated exactly, and the total is constant. | |||
This is not an ad hoc rescue invented for the occasion; it is the same accounting that resolves the classic Trouton-Noble experiment and the related "hidden momentum" of a current loop in an electric field, where a purely mechanical tally of the moving parts likewise appears to fail until the stresses in the supporting structure are counted. That the resolution is well known does not make the paper's construction worthless — it is a clean and unusually explicit demonstration of why the naive tally fails — but it does mean the conclusion drawn is not licensed by the calculation performed. | |||
Two smaller points weaken the presentation. The stipulation that at ''t'' = 0 both bodies occupy the same point, with a fully compressed spring between them, is physically impossible for extended bodies and is waved through as something "achieved structurally"; since this configuration supplies the reference momentum ''P''<sub>1''o''</sub> in section 3, the idealization is load-bearing rather than cosmetic. And the paper's own framing of the standoff is telling: it presents "conservation requires ''t''<sub>21''t''</sub> = ''t''<sub>22''t''</sub>" as an independent physical demand, when in fact that requirement follows only from the incomplete momentum expression it was derived from. The relativity of simultaneity is not in conflict with conservation of momentum; it is what makes the compensating stress terms necessary. | |||
==See also== | |||
* [[Victor Nikolayevich Cochetkov]] | |||
* [[Special Relativity]] | |||
* [[Simultaneity]] | |||
* [[Lorentz Transformation]] | |||
* [[Time Dilation]] | |||
* [[Length Contraction]] | |||
* [[Mass]] | |||
[[Category:Scientific Paper|special theory relativity linear example infringement laws preservation impulse]] | [[Category:Scientific Paper|special theory relativity linear example infringement laws preservation impulse]] | ||
Latest revision as of 12:26, 21 July 2026
| Scientific Paper | |
|---|---|
| Title | The Special Theory of Relativity: Linear Example of Infringement of Laws of Preservation of an Impulse |
| Read in full | Link to paper |
| Author(s) | Victor Nikolayevich Cochetkov |
| Keywords | special relativity, conservation of momentum, closed mechanical systems, weight, speed, symmetry, space, time, relativity |
| Published | 2011 |
| No. of pages | 15 |
Read the full paper here
Abstract
In article attempt to show becomes that use of the special theory of relativity can lead to infringement of the law of preservation of an impulse of the closed mechanical system consisting of bodies, located on one line and which interaction has constant character, in inertial systems of readout.
Overview
This short paper by Victor Nikolayevich Cochetkov — the byline on the PDF reads "Kochetkov Victor Nikolayevich," chief specialist at the Russian FSUE "TSENKI" centre for space ground-based infrastructure — is a single, tightly constructed thought experiment aimed at Special Relativity. Its claim is that a completely ordinary closed mechanical system, treated by the standard relativistic rules, comes out with a momentum ("impulse") that varies in time in an inertial frame, which would contradict conservation of momentum. The paper is the linear counterpart of an earlier planar version published by the author in the Journal of Vectorial Relativity (vol. 6, 2011, 65–76), cited as reference [1]; the standard formulae are taken from the Yavorsky and Detlaf physics handbook (Nauka, Moscow, 1980).
The English is a machine translation from Russian, and its vocabulary needs decoding: "impulse" is momentum, "system of readout" is reference frame, "preservation" is conservation, "weight" is mass, "unclenched" means relaxed or extended. Once translated back, the argument is entirely conventional in its methods — Lorentz coordinate and velocity transformations, the relativistic momentum p = M0V/√(1−V2/c2) — and its force rests on the relativity of simultaneity. What departs from the mainstream is the conclusion drawn: rather than reading the result as an incomplete accounting, Cochetkov reads it as an internal contradiction in the theory.
The argument
The system
Two point bodies, 1 and 2, each of rest mass M0, are joined by a perfectly elastic spring 3 whose mass is negligible compared with theirs. Under the spring's action they oscillate symmetrically back and forth along a single straight line about their common centre of mass, point S. The system is placed in an inertial frame O2x2y2z2 with S at rest at the origin and the bodies on the x2 axis.
The cycle is described in detail. At t2 = 0 the spring is fully compressed, holding maximum potential energy, and both bodies are at rest with S1, S2, S and O2 all coincident — a configuration the author says is assumed to be achieved structurally. The spring then relaxes and drives the bodies apart; at t2m it is fully relaxed, its potential energy zero, and the bodies have maximum speed and kinetic energy. The spring then stretches and decelerates them until at t2t they stop with the spring fully extended at maximum potential energy. The motion then reverses through the mirror-image half-cycle to t2n, the period. Throughout, symmetry gives
- x21 = −x22 and V21 = −V22
so the momentum in the rest frame of S is identically zero at every instant.
Transforming to a moving frame
A second inertial frame O1x1y1z1 is introduced, with parallel axes, with O2x2y2z2 moving along O1x1 at constant speed V, and with the clocks zeroed when the origins coincide. Cochetkov writes down the Lorentz transformations for the coordinates of each body separately, and the velocity-addition relations connecting V21 to V11 and V22 to V12.
The decisive step is his equations (11)–(16). Transforming the times, and imposing the condition that the two bodies be considered at the same instant t11 = t12 in the moving frame, he obtains
- t21 + Vx21/c2 = t22 + Vx22/c2
Since by construction x21 ≥ 0 and x22 ≤ 0, it follows that t21 < t22 whenever the bodies are separated. In other words, one instant in frame 1 corresponds to two different instants in frame 2 for the two bodies — the ordinary relativity of Simultaneity, stated correctly.
The momentum does not stay constant
The momenta of the two bodies in frame 1 are written in the standard relativistic form, P11 = M0V11/√(1−V112/c2) and likewise for body 2, and the system's momentum is their sum P1 = P11 + P12. Because the two bodies are sampled at different phases of the oscillation — that is what t21 ≠ t22 means — the sum does not stay fixed. Cochetkov's figure 6 shows P1 oscillating about the value 2M0V/√(1−V2/c2). His conclusion: for a closed system in an inertial frame the momentum "should be necessarily constant (not to depend on size of the moment of time)," and here it is not.
The consistency condition
Section 3 turns the result around and asks what would be required for conservation to hold. Two instants are compared. At t1o = 0 both bodies sit at the common origin with the spring fully compressed, so V11o = V12o = V and the total momentum is P1o = 2M0V/√(1−V2/c2). At the later instant t1t, chosen so that body 1 is momentarily at rest in frame 2 (V21t = 0, hence V11t = V), body 2 cannot also be at rest in frame 2, because its corresponding time t22t is necessarily later than t21t by the simultaneity result (16). So V22t ≠ 0 and therefore V12t ≠ V.
Setting P1o = P1t and solving gives the requirement V12t = V, equivalently V22t = 0, which in turn demands t21t = t22t. The paper ends on the resulting standoff, stated plainly: conservation of momentum requires t21t = t22t, while special relativity requires t21t < t22t "owing to not of a simultaneity" of events that are simultaneous in the other frame. Hence, Cochetkov concludes, "use of the special theory of relativity by consideration of separate examples can lead to default of the law of preservation of an impulse of the closed mechanical system in inertial systems of readout."
Assessment
What is genuinely good about this paper is its discipline. It invents no new mechanics, introduces no aether and postulates no new force; it takes the textbook transformations at face value, applies them to the simplest possible bound system, and follows the arithmetic where it leads. The relativity-of-simultaneity result (16) is derived correctly, and the observation that the two ends of an oscillating bound system are necessarily sampled at different internal phases in a moving frame is exactly right and is the physically interesting content of the exercise. The paper is also honest in structure: rather than resting on the first result, section 3 works backwards to identify precisely which condition would have to fail, and states the contradiction in its sharpest form instead of hiding it in prose.
The difficulty is that one term has been left out of the bookkeeping, and it is the term that carries the whole effect. The system is described as consisting of "bodies 1 and 2 (and springs 3)," but the spring is stipulated to be massless and never appears in any momentum expression: P1 is defined throughout as P11 + P12 alone. In special relativity a spring holding potential energy U holds an equivalent Mass U/c2, and when that spring is in motion it carries momentum accordingly; more than that, a stressed elastic body in motion carries an additional momentum contribution from its internal stress, which for a longitudinal spring under tension or compression is of order σV/c2 per unit volume. These are not negligible corrections at the order being tested — they are precisely of order V/c2 times the energy exchanged, which is the same order as the variation Cochetkov finds in P11 + P12. The relativistic statement of conservation applies to the total energy-momentum of the closed system, obtained by integrating the full stress-energy tensor over a spacelike surface, not to a sum over the point particles only. Once the spring's energy and stress are included, the oscillation in the particle sum is compensated exactly, and the total is constant.
This is not an ad hoc rescue invented for the occasion; it is the same accounting that resolves the classic Trouton-Noble experiment and the related "hidden momentum" of a current loop in an electric field, where a purely mechanical tally of the moving parts likewise appears to fail until the stresses in the supporting structure are counted. That the resolution is well known does not make the paper's construction worthless — it is a clean and unusually explicit demonstration of why the naive tally fails — but it does mean the conclusion drawn is not licensed by the calculation performed.
Two smaller points weaken the presentation. The stipulation that at t = 0 both bodies occupy the same point, with a fully compressed spring between them, is physically impossible for extended bodies and is waved through as something "achieved structurally"; since this configuration supplies the reference momentum P1o in section 3, the idealization is load-bearing rather than cosmetic. And the paper's own framing of the standoff is telling: it presents "conservation requires t21t = t22t" as an independent physical demand, when in fact that requirement follows only from the incomplete momentum expression it was derived from. The relativity of simultaneity is not in conflict with conservation of momentum; it is what makes the compensating stress terms necessary.