A DIRECT DERIVATION OF THE MASS-VELOCITY RELATION

Xu Shaozhi, Xu Xiangquan, Bejing Control Device Research Institute with Curt Renshaw, Tele-Consultants, Inc.

The well-verified relation of apparent mass increase with velocity is derived from a direct approach utilizing concepts of finiteness of the transmitting velocity of force (TVF), effective action, and the coupled effect of the of the TVF for two EM fields. A more appropriate interpretation of the Lorentz factor is developed, whereby when a particle is moving at a speed v under an EM field, it is the effective action on the particle which is effected, and not the mass of the particle itself. Since the only way to measure the mass of such particles is by measuring the impact of the action of EM fields upon their trajectories, the concept of mass increase with velocity is inherently protected against refutation in all such experiments to date, while the experiments themselves provide no test or verification of the theory of mass increase with velocity.

Introduction

A question is often put to the physicists and others who dissent from the special theory of relativity (SRT): why are there so many experiments which successfully conform to the predictions of SRT? Here we point out the following:

(a) A distinction must be made between "purely observational" effects and genuine electrodynamic problems; that is, the problems of so-called "electrodynamics of moving bodies" by Einstein [1] should be separated into: (1) the "purely observational" effects, such as the Michelson-Morley experiment, radial and transverse Doppler effects, stellar aberration, and the constancy of the velocity of light; and (2) those which involve the concepts of mutual action, such as particle acceleration under an EM field.

 

(b) For the first sort of phenomena, there is sufficient evidence for us to conclude that they cannot be interpreted by SRT, that all "proofs" of SRT are miss-proofs because they were obtained from experiments that violate one or both of the two postulates of SRT ([2]-[8], for example), and that SRT at best provides an equivalence: a mathematical form which produces correct results but misinterprets the underlying physics. Each of these types of phenomena can be explained by a Galilean transformation, provided the initial assumptions are not more restrictive than our actual knowledge of the physics entails (i.e. Einstein's second postulate).

 

(c) The second sort of experiments should be interpreted via the dynamic processes of mutual action, thus providing an appropriate interpretation of the Lorentz factor as it relates to apparent mass increase with velocity in the realm of particles accelerated under an EM field, the only realm under which such an apparent velocity dependent mass increase has been observed.

 

Transmitting Velocity of Force and Effective Action

Classical mechanics has an implicit assumption that the result caused by a mutual action between two bodies is independent of the relative velocity between them. Thus all such results are derived based on center of mass of the system, etc., and are further based on the concept of "instantaneous action-at-a-distance." But, the fact that an EM field propagates with a finite velocity provides the insight that instantaneous action-at-a-distance is impossible. It is therefore necessary to introduce into classical mechanics the concept of the transmitting velocity of a force (TVF). In the case of EM fields, this velocity is apparently equal to the propagation velocity of the fields, whether one considers such a velocity to be frame invariant, measured with respect to a presumed aether, or simply measured with respect to the relative velocity between source and sink.

When you ride a bicycle at full speed, you may not be able, despite your best efforts, to further accelerate your bike due to "pedal-idle." This phenomenon makes you unable to apply your effort to accelerating the bike, because your pedaling cannot equal or exceed the revolution of the wheel. This concept illustrates that the effective action exerted on a body will become zero when the motion of the particle being acted upon causes the effective value of the TVF to become zero.

We can use the standard equations to determine the force on a charged particle in a magnetic field of some fixed value B. However, the geometry of this equation as viewed from the laboratory frame is equivalent to that as viewed in the particle's frame only for small velocities v_o, where the direction of B can be considered normal to the direction of v_o. It is well known that the photon is the carrier of the electromagnetic force, and also that such forces must interact at a velocity of c with respect to the observer. We can therefore assign to the vector B a velocity component equal to c.

As the magnitude of v_o increases, the lines of the B field no longer appear normal to the direction of v_o, in the reference frame of the particle, but instead are directed back along the hypotenuse of the triangle of figure 1. In the figure, the value of the cross product in the force equation is given by:

Figure 1

In a particle accelerator, we wish to provide a constant acceleration to a particle of mass m, to keep it, say, confined to motion in a circle of defined radius. Referring to Newton:

Substituting (1) into (2) yields:

Thus we see that, for a given B field, the effective acceleration of a fast moving particle, a', is given by the acceleration for low velocities, a, divided by g.

The Apparent Mass-Velocity Relationship

For the electrodynamics of a moving body, we see that:

If we let m₀ denote the rest mass of the particle, and m denote the mass of the moving particle, then we could state mathematically that:

This interpretation, which ignores the TVF, is the one used in SRT to deduce that the mass of the particle increases with velocity by the Lorentz factor, g. However, realizing the impossibility of instantaneous action-at-a-distance and the results of equation (4) we see that:

Thus while we could attribute our inability to continually accelerate the particle to a mass increase as in (5), it is clear from this presentation that the preferred interpretation is that of a decrease in the effective force on the particle (4), due to the effective reduction in the TVF in the frame of the moving particle. In fact, if there were an increase in mass with velocity, this effect would be in addition to the effects of the effective reduction in TVF in the frame of the particle, and the effects of the applied field would be even less than we observe.

The reason that we cannot distinguish between the mass increase of (5) and the unchanging mass of (6) is that both equations are mathematically equivalent, and there is no independent means to verify the mass of these particles. The mass is determined by measuring the effect of a given EM field at various velocities, thus we are using the effect we are observing to measure the effect itself. If you step on a scale that says you way only twenty pounds, you cannot verify that reading by stepping back on the same scale--an independent means of verification is required. That this independent verification is missing makes the statement that this experiment "successfully conforms to the predictions of SRT" a circular argument at best.

Summary

Although the Lorentz group, as has been shown in previous papers [2][3][5][8] to be generally invalid, it does have application to the effective force on a moving charged particle in an EM field. The reduction in effective action as a result of change in effective TVF can be mathematically represented as an effective increase in mass, even though no such increase in mass actually occurs. It is clearly a mistake to ascribe a physical interpretation of the genuine electrodynamics of moving bodies based on any observation without careful attention to the underlying assumptions, in this case the assumption that the TVF will be always c regardless of the relative velocity of source and observer.

It is important to note that the mass-energy relation (E = mc²) can be derived independently of the apparent mass-velocity relation, and completely without regards to SRT. Einstein provided a clear derivation of this relation from first principles alone [10].

The fact that a charged particle with an initial velocity v < c cannot gain a relative speed greater than c under an EM field does not imply that the speed of a particle with respect to an observer is unable to exceed c. It simply states that other means of acceleration are required.

The new concepts presented in this paper of TVF and effective action can be expected to be of potential significance and to apply, in principle, to any form of dynamical field.

Einstein himself stated, "Unthinking respect for authority is the greatest enemy of the truth." The question today is not whether SRT is valid, but rather how physics is to be awakened from its unthinking respect for the authority of SRT, which is retarding the advance of science. Many premised postulates in theoretical science, especially basic physics, must be re-examined thoroughly, ensuring that presumed first principles are indeed first principles, unencumbered by purely observational results, in order that man-kind may begin to scale new heights in science as fully as is possible.

Acknowledgments

We should like to thank our colleagues Cheng Jianmin, Li Benyuan, William Kallfelz, Wang Yufang, and Bo Yongan for their kind help. We are also grateful to Dr. T. E. Phipps, J., R. J. Hannon, Dr. G. P. Rodrigue and Senior-advisor of the Chinese J. of Systems Engineering and Electronics, Prof. Wu Shouping, for their useful discussions or valuable help.

References

[1] Einstein, A., et al., The Principle of Relativity, Methuen, London 1923; pp. 42-46

[2] Xu Shaozhi and Xu Xiangquan, "A Reexamination of the Lorentz Transformation," Galilean Electrodynamics, Vol. 3, no. 1, pp 5-8; no. 3, pg 60 (1992)

[3] Xu Shaozhi and Xu Xiangquan, "Systematical Scrutiny into Special Relativity," Chinese J. of Systems Engineering and Electronics, Vol. 4, no. 2, pp 75-85, 1993

[4] Xu Shaozhi and Xu Xiangquan, "On the Relativity of Simultaneity," Apeiron, Vol. 16, pp 8-11 (1993); Vol. 19, pp 34-37 (1994)

[5] Xu Shaozhi and Xu Xiangquan, "Investigation of Einstein-Lorentz Group," preprints, 1992.

[6] Xu Shaozhi and Xu Xiangquan, "Space-Time, Motion and Light Velocity Problems," presented in the 3rd International Conference on Space, Time and Gravitation, St. Petersburg, May 23-27, 1994.

[7] Wu Shouping and Xu Shaozhi, "Investigation of Invariance Problem in Special Relativity," Chinese J. of Systems Engineering and Electronics (Chinese Edition), Vol. 16, no. 5 (1994)

[8] Fox, J. G., "Experimental Evidence for the Second Postulate of Special Relativity," American Journal of Physics, vol. 30, pp 297-300 (1962)

[9] Renshaw, Curt, "The Radiation Continuum Model of Light and the Galilean Invariance of Maxwell's Equations," submitted Galilean Electrodynamics (1994), presented at the 1995 annual meeting of the AAAS in Norman, OK, 21-24 May, 1995.

[10] Born, Max, Einstein's Theory of Relativity, Dover Publications, New York, 1965 pp. 283-286

Questions? Comments?crenshaw@teleinc.com