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To colleagues interested in trying to reach a common understanding of relativistic mass - would you consider contributing a paper to

• a volume on relativistic mass (if interested see Call for papers above for contact info)
• a special session on relativistic mass will be included in the program of the Third Hermann Minkowski Meeting on the Foundations of Spacetime Physics (see Call for Papers).

• Max Born's explicit warning about the danger of improper understanding of mass in relativity:

  "In ordinary language the word mass denotes something like amount of substance or quantity of matter, these concepts themselves being defined no further... In physics, however, as we must very strongly emphasize, the word mass has no meaning other than... the measure of the resistance of a body to changes of velocity." [Max Born, Einstein's Theory of Relativity (Dover Publications, New York 1965), p. 33]

• Richard Feynman's brilliant explanation of its role in spacetime physics and its experimental confirmation:

  "What happens if a constant force acts on a body for a long time? In Newtonian mechanics the body keeps picking up speed until it goes faster than light. But this is impossible in relativistic mechanics. In relativity, the body keeps picking up, not speed, but momentum, which can continually increase because the mass is increasing. After a while there is practically no acceleration in the sense of a change of velocity, but the momentum continues to increase. Of course, whenever a force produces very little change in the velocity of a body, we say that the body has a great deal of inertia, and that is exactly what our formula for relativistic mass says (see Eq. 15.10 [p=mv=m₀v(1-v²/c²)^-1/2]) - it says that the inertia is very great when v is nearly as great as c. As an example of this effect, to deflect the high-speed electrons in the synchrotron that is used here at Caltech, we need a magnetic field that is 2000 times stronger than would be expected on the basis of Newton's laws. In other words, the mass of the electrons in the synchrotron is 2000 times as great as their normal mass, and is as great as that of a proton! That m should be 2000 times m₀ means that 1 - v²/c² must be 1/4,000,000, and that means that v differs from c by one part in 8,000,000, so the electrons are getting pretty close to the speed of light." [The Feynman Lectures on Physics. The New Millennium Edition. Vol. 1 (Basic Books, New York 2010), p. 15.9.]

• to read (and eventually reply to) the note "Relativistic Mass is an Experimental Fact" posted online on 17 January 2023.

During the last four decades physicists have witnessed (or rather endured), as Max Jammer put it, "what has probably been the most vigorous campaign ever waged against the concept of relativistic mass" [M. Jammer, Concepts of Mass in Contemporary Physics and Philosophy (Princeton University Press, Princeton 2000) p. 51]

One may reasonably wonder whether that campaign would have been even launched, if Feynman had been alive at that time: "Newton's Second Law, which we have expressed by the equation F = d(mv)/dt, was stated with the tacit assumption that m is a constant, but we now know that this is not true, and that the mass of a body increases with velocity... Actually, the correctness of the [relativistic mass] formula has been amply confirmed by the observation of many kinds of particles, moving at speeds ranging up to practically the speed of light." [The Feynman Lectures on Physics. The New Millennium Edition. Vol. 1 (Basic Books, New York 2010), p. 15.1. See also Sec. 16-4 Relativistic mass]

During the last four decades the concept of relativistic mass has been continuously questioned by a number of physicists¹ and even attacked (!?) by some over-confident colleagues, mostly particle physicists, rarely experts in spacetime physics, e.g. by using in scientific papers even expressions like "the virus of relativistic mass"; see the three-page summary below or the Appendix On Relativistic Mass to a new publication of five works by Einstein). It seems two facts might have given rise to that fashion:

• unlike proper (rest) mass, relativistic mass is not an invariant; but by exactly the same argument one should talk only about the invariant proper time and deny the use of coordinate time, which, like relativistic mass, is the time component of a four-vector (the position four-vector), and which, like the relativistic mass, is found experimentally to change relativistically, i.e., to "dilate"
• there are some difficulties (rather open questions) involved in the relativistic mass (e.g., the relativistic mass cannot be merely substitute the mass in the Newtonian equations and mass appears to behave as a tensor) and some colleagues might have decided that it is easier to reject the notion of relativistic mass altogether instead of facing and dealing with the open questions which it revealed (if that is indeed the case, it would hardly be physics at its best).

It should be reminded that one of the major open questions in physics is why a particle resists its acceleration and now the experimentally confirmed prediction of relativity that that resistance increases as the particle's velocity increases makes that open question even more urgent. That is why, facing, recognizing and dealing with the experimental fact of the relativistic increase of mass is of utmost importance since it turns out to be one of the deepest open questions in spacetime physics.² Research on it is one of the examples of physics at its best.

It should be stressed that the experimental evidence proving the relativistic increase of mass, provided by particle accelerators, is overwhelming - very powerful electric fields are used to accelerate electrons, protons and other charged particles. These particles need greater forces for their acceleration as their velocities approach the velocity of light c. Those greater and greater forces are needed to overcome the increasing resistance with which the particles oppose their acceleration as their velocities approach c. In other words, as (Newtonian) mass of particles is the measure of the resistance they offer to their acceleration, the greater forces are needed to accelerate particles whose masses (being the measure of their increasing resistance to their acceleration) relativistically increase.

But even before the accelerators the early experiments provided the experimental verification of the prediction of the 1905 Einstein's special relativity that the mass of a particle depends on its velocity (the velocity dependence of mass was first predicted by the electron theory - that the electromagnetic mass of charged particles increases as their velocities increase - and generalized by Einstein for all particles: "these results as to the mass are also valid for ponderable material points"). For example, the 1908 Bucherer experiment and the 1916 Guye and Lavanchy experiment.

The Bucherer experiment allowed two interpretations (either the electron charge or its mass are velocity dependent) and independent experiments ruled out the interpretation that the electron charge decreases as its velocity increases. So, it is exceedingly clear that this experiment would be impossible if the mass of electrons did not increase as their velocities increase. See details of why this is so in the two pages from Rosser's explanation of the Bucherer experiment (W. G. V. Rosser, Relativity and High Energy Physics (Wykeham Publications, London 1966), Sec. 2.2 "Variation of mass with velocity;" see also pp. 28-29 for the application and verification of relativistic mass in cyclotrons).

The same conclusion holds for all experimental evidence that confirmed the theoretical prediction that the mass of a particle depends on its velocity.

It is self-evident that anyone who questions the concept of relativistic mass must first address the experimental facts that proved the relativistic increase of mass; it would be sufficient to try to demonstrate that Bucherer's experiment would be possible if the electron mass were velocity independent. Although this is the way of doing physics, unfortunately "the most vigorous campaign ever waged against the concept of relativistic mass" ignored (!?) the experimental evidence. Instead, general statements, e.g., that the concept of relativistic mass might lead to confusions, are made. Or, theoretical arguments are advanced. Most are irrelevant, e.g., that rest mass is an invariant, whereas relativistic mass is the time component of the four-momentum (see [5], Ch. 4 or the three-page summary below).

A somewhat relevant argument appears to be the insistence that γ=(1-v²/c²)^-1/2 should not be "attached" to the mass, because it comes from the 4-velocity. That is, of course, correct - γ ensures that the velocity of a particle cannot exceed that of light; in other words, γ ensures that no timelike 4-velocity vector, which represents the state of motion of a particle of non-zero rest mass, can become lightlike or spacelike. But that is kinematics; it says nothing about the dynamics, that is, it says nothing about why a particle cannot exceed the velocity of light (i.e. why the particle's 4-velocity cannot become lightlike or spacelike), and particularly - what is the mechanism that prevents it from doing so. That mechanism is, in fact, suggested by Newtonian mechanics, where mass is defined as the measure of the resistance a particle offers to its acceleration - when Einstein postulated that the velocity of light c is the greatest velocity a particle (with non-zero rest mass) can achieve, it was almost self-evident to assume that a particle would offer an increasing resistance when accelerated to velocities approaching that of light, that is, a particle's mass will increase and will approach infinity when the particle's velocity approaches c, thus preventing the particle from reaching and exceeding the velocity of light. And that was repeatedly experimentally confirmed. However, increased resistance and increased relativistic mass are rather only naming the mechanism that prevents a particle from reaching the velocity of light; the origin and nature of the resistance a particle offers when accelerated (an open question in classical physics) and of the increased resistance a particle offers when accelerated to velocities approaching that of light (an open question in spacetime physics) constitutes one of the deepest open questions in spacetime physics.

See a three-page summary of Why relativistic mass should be studied in depth, not rejected.

² That open question involves not only the origin and nature of the resistance a particle offers when accelerated (an open question in classical physics) and of the increased resistance a particle offers when accelerated to velocities approaching that of light (an open question in spacetime physics), but also the very nature of proper (rest) mass. Starting from the trivial indication of the need to deal with the nature of proper mass (a hot macroscopic body have a greater proper mass than an identical cold one) one should deal with the nature of proper mass of composite "elementary" particles such as the proton and neutron. Exploring Wheeler's idea "mass without mass" [22], Frank Wilczek formulated the problem [23]:

How is it possible that massive protons and neutrons can be built up out of strictly massless quarks and gluons? The key is m=E/c². There is energy stored in the motion of the quarks, and energy in the color gluon fields that connect them. This bundling of energy makes the proton's mass.

Two more notes regarding Wilczek's quote:

• The "energy stored in the motion of the quarks" results in a relativistic increase of their masses demonstrating that some of the proper mass of the composite proton is relativistic (due to the relativistically increased quark masses). Given this, it looks inexplicable how a particle physicist could reject the concept of relativistic mass.
• one of the open questions is the mechanism of (how) "This bundling of energy makes the proton's mass" (while constantly keeping it explicitly in mind that there is no energy as such in physics; it is always energy of something as Wilczek points out).

References

1. F. Rahaman, The Special Theory of Relativity: A Mathematical Approach, 2nd ed. (Springer 2022), see particularly Sec. 9.2 Experimental Verification of Relativistic Mass

2. G. Tallents, An Introduction to Special Relativity for Radiation and Plasma Physics (Cambridge University Press, Cambridge 2022), Sec 4.3: "The electromagnetic wave reduced vector potential for nonrelativistic particles a₀ = eE₀/(m₀cω) takes a value a₀/γ for relativistic electron motion in the electromagnetic wave due to the relativistic increase of the electron mass m₀ to γm₀."

3. A. K. Raychaudhuri, Classical Theory of Electricity and Magnetism: A Course of Lectures (Springer Nature, Singapore 2022), Sec. 23.3 Variation of Mass with Velocity

4. S. Weinberg, Foundations of Modern Physics (Cambridge University Press, Cambridge 2021). On p. 110 Weinberg writes "Because of the presence of the factor γ in Eqs. (4.4.5) and (4.4.6), the quantity mγ is sometimes called the relativistic mass. I will not use this terminology, because it suggests that we can calculate the acceleration produced by any force just by replacing m in Newton's F = ma with mγ, which is not the case." One may only wonder whether this can be regarded as a valid argument against the concept of relativistic mass, especially given the overwhelming experimental evidence proving that relativistic mass is an experimental fact (it is rather the opposite - it is the consistent use of relativistic mass that clearly demonstrates that it cannot merely replace the Newtonian mass in Newton's second law as, for example, shown by Rosser [in the book quoted above] while discussing "the changes in other mechanical quantities, arising from the variation of mass with velocity"; Feynman [p. 15-10] also discusses "some further consequences of relativistic change of mass"). Moreover, Weinberg himself writes on p. 121 "Given the existence of the force exerted by electric fields, the force exerted by magnetic fields is an inevitable consequence of Lorentz invariance. It is a special feature of electromagnetic forces that the only change in the equation of motion introduced by special relativity is the replacement of the mass m in the momentum with mγ, which in this one case allows us to treat mγ as a relativistic mass."

5. V. Petkov, Seven Fundamental Concepts in Spacetime, SpringerBriefs in Physics (Springer, Heidelberg 2021), Ch. 4 Relativistic Mass

6. H. D. Young, R. A. Freedman, Sears and Zemansky's University Physics with Modern Physics, 15th ed. (Pearson Education 2020)

7. Ø. Grøn, Introduction to Einstein's Theory of Relativity: From Newton's Attractive Gravity to the Repulsive Gravity of Vacuum Energy, 2nd end. (Springer 2020)

8. W.-G. Boskoff, S. Capozziello, A Mathematical Journey to Relativity: Deriving Special and General Relativity with Basic Mathematics (Springer 2020)

9. A. Romano, M. M. Furnari, The Physical and Mathematical Foundations of the Theory of Relativity: A Critical Analysis, (Birkhäuser, Springer Nature Switzerland AG 2019)

10. H. Günther, V. Müller, The Special Theory of Relativity: Einstein's World in New Axiomatics (Springer 2019). See particularly p. 91: "The velocity independence of masses is incompatible with the Lorentz-transformation."

11. J. Allday, Space-time, an introduction to Einstein's theory of gravity (CRC Press, London 2019). After an excellent introduction and discussion of the concept of relativistic mass, Allday writes: "While this is a perfectly valid approach to the issue of mass, it is not the only possible view. Our colleagues in the particle physics community refer to rest mass as the mass of the particle, as the value is a characteristic property of the species concerned. For a particle in motion, particle physicists tend to work with the energy and momentum rather than worrying about the mass in such circumstances. We will follow their pattern." (p. 120) with a note at the end of the chapter: "I trained as a particle physicist and like to keep to the club rules..." (p. 136; the dots at the end of the one-sentence note are Allday's.) Amazing... should one assume that in physics "club rules" can overrule the verdict of the ultimate judge - the experimental evidence?

12. J. B. Kogut, Special Relativity, Electrodynamics, and General Relativity: From Newton to Einstein (Academic Press [Elsevier], London 2018)

13. R. D'Auria, M. Trigiante, From Special Relativity to Feynman Diagrams: A Course of Theoretical Particle Physics for Beginners, 2nd ed. (Springer-Verlag Italia 2017)

14. R. Cooke, It's About Time: Elementary Mathematical Aspects of Relativity (Providence, Rhode Island, American Mathematical Society 2017)

15. J. S. Walker, Physics, 5th ed. (Pearson Education, Inc. 2017)

16. C. Christodoulides, The Special Theory of Relativity: Foundations, Theory, Verification, Applications (Springer International Publishing Switzerland 2016)

17. P. Schmüser, Theoretische Physik für Studierende des Lehramts 2: Elektrodynamik und Spezielle Relativitätstheorie (Springer-Verlag Berlin Heidelberg 2013)

18. E. Rebhan, Theoretische Physik: Relativitätstheorie und Kosmologie (Springer-Verlag Berlin Heidelberg 2012)

19. W. Rindler, Relativity, Special, General, and Cosmological (Oxford University Press, Oxford 2006)

20. J. J. Callahan, The Geometry of Spacetime: An Introduction to Special and General Relativity (Springer, Heidelberg 2000)

21. J. A. Wheeler, Geometrodynamics (Academic Press, New York 1962), p. 25)

22. Frank Wilczek, Mass Without Mass I: Most of Matter, Physics Today 52, 11, 11 (1999)

23. Frank Wilczek, Mass without Mass II: The Medium is the Mass-age, Physics Today 53, 1, 13 (2000)

24. C. T. H. Davies, C. McNeile, K. Y. Wong, E. Follana, R. Horgan, K. Hornbostel, G. P. Lepage, J. Shigemitsu, and H. Trottier, Precise Charm to Strange Mass Ratio and Light Quark Masses from Full Lattice QCD, Phys. Rev. Lett. (2010) 104, 132003

25. M. Tanabashi et al. (Particle Data Group), Review of Particle Physics Phys. Rev. D 98, 030001 (2018) p. 632