On the Derivation of E = mc2
| Scientific Paper | |
|---|---|
| Title | On the Derivation of E = mc2 |
| Read in full | Link to paper |
| Author(s) | Vesselin C Noninski |
| Keywords | special relativity, mass-energy equivalence, Lorentz transformation, longitudinal mass, Einstein 1905, principle of relativity |
| Published | 2003 |
| Journal | PhilSci-Archive |
| No. of pages | 10 |
Read the full paper here
Abstract
Analysis is presented of the derivation in [1] of what is popularly known as E = mc2. It is emphasized that once a relationship, describing a phenomenon in the stationary system, is known exactly and with certainty, any theory that would derive a different relationship regarding the same phenomenon in terms of the same stationary system should be rejected out of hand.
Overview
This short paper is one of a series by Vesselin C Noninski — alongside companion pieces on the Lorentz Force, on the physical consistency of the theory, on Simultaneity, on the Lorentz invariance of Maxwell's Equations and on the Michelson-Morley Experiment — that attempts to falsify the special theory of relativity on internal grounds alone, using what the author calls a "pencil and a paper". The target here is §10 of Einstein's 1905 electrodynamics paper (cited throughout as [1], in the Dover The Principle of Relativity pagination), where the equation of motion of a slowly moving electron is transformed between frames and integrated to give the energy of motion — the result the paper describes as "popularly known as E = mc2".
Noninski's claim is not that the algebra of §10 contains a slip, but that its conclusion is inadmissible before any algebra is done. If a relationship holding in the stationary system K is "known exactly and with certainty" at the outset, then no theory may afterwards deliver a different relationship between the same quantities in the same system K. Since Einstein's transformed equation carries an extra factor of β3 relative to the starting equation, Noninski holds that STR convicts itself. The paper's recurring epigram for this is that special relativity is "the determination to present the seeming as real" — the theory, on his reading, mistakes what an observer appears to see for what is actually the case, in the way that ships on the horizon appear as specks without anyone concluding that the ships have shrunk.
The argument
The two equations of motion
The starting point is the law of motion of a slow electron in the stationary system K:
m d2x/dt2 = εX
where x is the electron's coordinate, X the x-component of the electric field, m the mass and ε the charge. By the First Postulate — the Principle of Relativity — the same law must hold in the moving system k, the electron's own system, in that system's own variables:
m d2ξ/dτ2 = εX '
Noninski explicitly grants this step: "So far all is well and good — once the First Postulate is assumed to hold good there could be no objection." The objection begins at the next step, where the Second Postulate is applied by transforming the k-equation back into the coordinates of K by the Lorentz Transformation. Einstein's result (p. 62 of [1]) is
m β3 d2x/dt2 = εX
with β Einstein's 1905 shorthand for 1/√(1 − v2/c2). This is the quantity later textbooks call the longitudinal mass.
Why the paper rejects the β3 factor
The whole case rests on the claim that m, d2x/dt2, ε and X in the second equation denote exactly the same physical quantities, "fixed and as defined at the outset", as in the first. If so, the two equations are contradictory statements about K unless β3 = 1. Noninski presses this with an arithmetical analogy: to say that 2 + 2 may "appear as 5" from a moving system is no defence, because 2 + 2 = 4 is settled in the stationary system beforehand.
He anticipates and answers two replies. The first — that d2x/dt2 in the transformed equation is only how the k-observer sees things — he rejects on the ground that the k-observer sees it that way "only due to STR", so the reply presupposes what is in dispute; a theory with physical content must have the moving observer report truthfully what the laboratory expression actually is. The second — that the transformed equation is merely the k-equation re-expressed in K-coordinates and so need not obey the First Postulate — he rejects because the symbols written down are not arbitrary letters but the mass, acceleration, charge and field of the stationary system.
The methodological parallel with Maxwell's equations
The paper's strongest structural move is to borrow Einstein's own procedure from p. 52 of [1]. There, Maxwell's Equations are written in the moving system twice — once by invoking the First Postulate, once by Lorentz-transforming them — and the two results are then set equal, which is how Einstein extracts what he calls the Lorentz Force. (Noninski notes that a companion paper of his disputes even that derivation.) Applying the identical comparison to the electron's equation of motion, the two expressions "must express exactly the same thing", from which β3 = 1 follows immediately, "since in the initial equation there is no additional function of velocity v".
The consequence for the energy expression
Addendum 2 reproduces the §10 integration so the consequence can be seen. With W = ∫F dx and F = εX = mβ3 d2x/dt2, the integral becomes ∫mβ3v dv, and integrating from 0 to v gives
W = mc2/√(1 − v2/c2) − mc2
The point of reproducing it is that the β3 is exactly what makes the integral come out this way: set β3 = 1, as the author's argument demands, and the relativistic energy expression "cannot be derived". In a footnote he adds a suspicion about the cubic power itself — "one may wonder how the cubic power appeared, exactly fitting the requirement for the final derivation (any other form will not do)" — and observes that on p. 54 Einstein is willing to discard terms in the second and higher powers of v/c to recover the familiar Lorentz Force, but declines any such approximation here.
The experimental evidence
A closing discussion asks how the reputed experimental confirmations might be "explained away". Noninski points to his earlier papers for the claim that the Michelson-Morley Experiment and the µ-meson lifetime results are of questionable evidential value, then turns to particle accelerators. He suggests that relativistic corrections may be only one engineering concern among several of comparable priority, and that "sheer historical tradition" may explain their prominence in the textbooks — while conceding in the same breath, "These are, of course, all speculations." Bertozzi's 1964 American Journal of Physics measurement of the speed and kinetic energy of relativistic electrons is dismissed on the strength of a private communication from V. V. Petrov, to the effect that its five experimental points "have not been acquired at comparable conditions". Finally he speculates that if E = mc2 is confirmed empirically it may reflect "the peculiar properties of light" and the relation E = pc assumed for photons, "and has nothing to do with STR".
Assessment
What is attractive here is the discipline of the method. The paper attempts no alternative theory, introduces no aether and asks for no new experiment; it takes Einstein's own text, page by page, and asks whether the two routes from K to k and back are being applied consistently. The comparison with the p. 52 treatment of Maxwell's Equations is a legitimate and pointed piece of textual analysis, and the algebra the paper does reproduce is correct: ∫mv(1 − v2/c2)−3/2 dv from 0 to v does give mc2[(1 − v2/c2)−1/2 − 1], so the observation that β3 is precisely what the derivation needs is arithmetically sound rather than rhetorical.
The difficulty is that the identity of the two equations is asserted, not established, and the paper's own Addendum 2 shows why the assertion fails. There the electron is stipulated to be "stationary versus K" and stationary versus k at the same time. But in §10 of [1] the electron is at rest in k and moving with velocity v relative to K; the two frames are related by exactly that v, and the transformation is meaningful only because it is non-zero. If the electron were genuinely at rest in K, then v = 0, β = 1, and the two equations would agree trivially — which is the whole of the paper's β3 = 1 conclusion, obtained by having quietly removed the relative motion. The uncontroversial reading is that m d2x/dt2 = εX is the law for an electron instantaneously at rest in K, while the β3 equation describes a different electron, one moving at v; the symbols coincide but the physical situations do not, so no contradiction with the Principle of Relativity arises. The First Postulate requires the form of the law to be the same in every frame for correspondingly-prepared situations, not that a single accelerating body have the same coordinate acceleration in all frames — the latter is Galilean, and assuming it is assuming the conclusion. Addendum 2 also contains a slip of wording: it "imagine[s] there is a magnetic field in the laboratory frame K with components X, Y and Z", then immediately treats the resulting force as the Coulomb force εX; X, Y, Z are Einstein's electric-field components, and the argument requires them to be electric.
A second problem is one of identification. The result actually derived in §10 and reproduced in Addendum 2 is the relativistic kinetic energy, mc2(β − 1). It is not E = mc2, which comes from Einstein's separate September 1905 paper on whether the inertia of a body depends on its energy content, by an argument from light emission and momentum balance that involves no equation of motion for an electron and no β3. Even if the objection to §10 were granted in full, the mass-energy relation would survive it. This matters for the title, which promises a refutation of a derivation the paper does not in fact examine.
The treatment of measurement is the weakest part. The 1964 Bertozzi experiment is precisely the kind of direct test the argument needs to defeat — it measures electron time-of-flight against calorimetrically determined kinetic energy and finds velocity saturating at c while energy grows without bound — and it is set aside on an unpublished remark rather than analysed. Nothing is said about the storage-ring measurements in which the time dilation of unstable particle lifetimes is read directly off the decay rate (the CERN muon storage ring result, quoted to the sub-percent level), nor about the fact that synchrotron RF programmes are computed from E = γmc2 and would put beams out of phase within a few turns if the relation were wrong. The author's candour — "these are, of course, all speculations" — is to his credit, but a paper claiming that no experimental testing is necessary must still account for the tests that exist. On its own terms the paper is a clear and honestly argued piece of textual criticism; its central premise, that the two equations of motion refer to one and the same physical circumstance, is the step that does not hold.