Recently, a remarkable article was published in this journal by A. Walstad,1 suggesting that the correct expressions for the momentum (p) and kinetic energy (K) of a particle moving at high speed could have been available to pre-1905 scientists, before the advent of special relativity.2 The idea that allowed Walstad to obtain the well-known expressions (p = γmv and K = (γ – 1)mc2, with ), without Einstein's two postulates, is the application to material particles of Einstein's box gedanken experiment, originally conceived for photons.3 This approach has an interesting pedagogical appeal, but it is important to first correct its conclusion that the above expressions for p and K were1 “already implicit in physics going back to Maxwell.” Instead, an extra postulate not belonging to pre-relativistic physics is implicitly added in Ref. 1, allowing for the introduction of the limiting speed c.
It might be interesting to analyze this problem from the point of view of an imaginary pre-1905 scientist who, after discussing the photon-in-a-box problem4 along the guidelines of Sec. II of Ref. 1 wants to extend the analysis to the particle-in-a-box case. He assumes that kinetic energy has inertia content for the particle as it was for the photon. He also realizes that the particle kinetic energy, once it is deposited on the right-hand-side of the box and dissipated here as heat, is indistinguishable from the photon energy (as reported in Sec. III of Ref. 1). However, our pre-1905 scientist had no reasons to write the extra inertia of the particle as δmparticle= K/c2, as he had no guiding principle (like Einstein's second postulate) providing a special status to the speed of light. Therefore, he could not guess that he has to divide the kinetic energy K of the particle by c2 to account for its extra inertia, as in the case of the photon. Instead, he might have reasoned that if the extra inertia of the photon was written in terms of the speed c of the photon (δmphoton = K/c2), then the extra inertia of the particle has to be written using the speed v of the particle: δmparticle = K/v2. From the latter expression, working out the same calculations as in Sec. III of Ref. 1, he would have obtained5 p = 2mv and K = mv2. He might have then interpreted these results with the idea that the experimentally measured mass, mexp = 2m, is composed of an intrinsic inertia and an extra inertia induced by the kinetic energy, both equal to m. However, this “mass-renormalization” would have appeared to him as purely semantic, as no new theory comes out: the final expressions for p and K would have assumed the same form, p = mexpv and . We incidentally remark that Galilean relativity is respected in this example (the extra inertia is given by the same constant in any inertial frame).
In the above discussion of pre-1905 physics there is no such concept as a limiting velocity and we cannot properly define a low-speed or a high-speed limit, because there is no other speed to compare v with. Of course, this is what makes the difference with the theory of special relativity, where the limiting speed c, appears. This is also why Ref. 1 must have implicitly gone beyond pre-1905 physics, as here the limiting speed c shows up as well in the final expressions for p and K. From the above analysis, it is clear that the limiting speed c has been injected in the theory of Ref. 1 through the hypothesis that extra inertia has the same form for photons and particles: δmparticle = δmphoton = K/c2.
Moreover, even if our pre-1905 scientist had a reason to admit the expression δmparticle = K/c2 for the extra-inertia, then he would have been in trouble with Galilean invariance. The expression δmparticle = K/c2 must be valid in any inertial frame. This implies that p is a nonlinear function of v in any frame. Such a result, associated to Galilei's linear composition law for speeds, implies that momenta do not sum up. For example, if we had a particle of mass m, speed v1 and momentum in the inertial frame R1 and a second particle (with same mass m) with speed v2 and momentum in the inertial frame of the first particle, then a direct application of Galilei's transformation gives, in R1, that the second particle has speed v = v1 + v2. The expression δmparticle = K/c2 leads to a momentum p = γvmv (for particle 2 in R1). And yet p ≠ p1 + p2, as . In the end, our pre-1905 scientist would have concluded that the hypothesis δmparticle = K/c2 is false, or (in an extreme act of faith towards δmparticle = K/c2) that the speed addition of Galilean relativity is not true (as we know today!). The latter finding, however, is quite far from pre-1905 physics.
From the above discussion, it is clear that the hypothesis δmparticle = E/c2 is an extra postulate, not deducible from pre-1905 physics. It appears as if it has the same physical content as Einstein's postulate on the invariance of the speed of light, both leading to the appearance of the γ factor. It has however a few peculiarities that might be worthwhile to underline: (1) it shows that the correct expressions for p and K can be deduced in just one frame (as already highlighted by Walstad in Ref. 1), without the need of invoking the principle of relativity for other inertial frames. (2) As only one frame is involved, it constitutes a direct demonstration that the expressions of dynamical variables, such as p and K, are independent of the synchronization procedure of different inertial frames (e.g., Einstein's synchronization2 or Tangherlini's synchronization6). (3) The previous point, conversely, shows that the postulate δmparticle = E/c2, even if coupled to the principle of relativity is not be sufficient to recover Lorentz' transformations (the synchronization procedure needs to be defined as well).
In conclusion, I think that Walstad's approach is a valid alternative pedagogical route to the dynamical concepts of special relativity (though the expression of the total energy is missing). However, the role of the hypothesis δmparticle = E/c2 as an extra postulate, not derivable from pre-1905 concepts, should be clearly stated.
References
We remark that, even if historically posterior to 1905, this gedanken-experiment does not require any concept from the two postulates of special relativity, as clearly shown in Ref. 1.
Of course, a pre-1905 scientist would not have used the word “photon.”
A remark: even if we wrote δmparticle = E/(αv2), with α a dimensionless quantity, to extend the analysis, the solutions would have had non-linear v-dependence, not in keeping with the experiment. We would have obtained , if α ≠ 1/2 (and γ an integration constant) and p(v) = 2mv(γ – lnv), if α = 1/2. For α → 1 we recover the value given in the main text p = 2mv.
