V

V. Alan White How Not to Read

‘The Most Famous Equation’

Abstract. Marc Lange has argued that E=mc² does not entail (among other things) that mass-to-energy conversions (or vice-versa) are real physical processes. I argue that while one example Lange offers in defense of this claim is at least consistent with it, another more important example at best only supports this claim in a trivial way, and at worst hides the reality of these conversions behind a lack of clarity about the nature of physical systems.

Marc Lange recently argued that a proper

interpretation of Einstein’s ‘most famous equation’—E=mc²—reveals that (i) it provides no conceptual or empirical ground for championing (as some have) an ontological priority of energy over mass [Lange 227], and much more, that (ii) ‘the conversion of mass to energy [or vice versa] is not a physical process’ [Lange 234]. While Lange extends his argument for (i) even further to state that ‘mass is a real property whereas energy is not’ [Lange 238], the following remarks will not directly challenge his reasoning for that extension of (i) or (i) itself (since it may well be that (i) as stated is correct). However, the following will show that his argument for (ii) fails in such a way as to cast doubt over his advocacy of anything beyond (i).

Lange’s argument for (ii) is made by way of two examples, each attempting to establish that an epistemic ‘shift in perspective’ [Lange 236] is the sole source of any claim that mass ‘is converted into energy or vice versa’. The first example, involving a ball of gas that is heated, posits a synchronic basis for a perspectival shift in which at once we may consider such a ball of gas as either having an increased mass as a single-body system^[1] due to the heating [Lange 234-35], or having that same system’s constituent molecules merely exhibit increased kinetic energy from the heating [Lange 235]. The second and clearly more physically interesting example, involving the spontaneous decay of a tritium nucleus into emission products (³He, an electron and antineutrino), posits a diachronic basis for a more complex set of perspective shifts. According to the usual perspective, we regard the decay products as having a ‘mass defect’ as compared to the predecay tritium nucleus (thus the equally orthodox interpretation that mass has been converted into energy [Lange 236-37]). Lange argues, however, that should we shift our perspective to consider the entire decay system (as opposed to the set collection of the decay product masses), we find no such ‘defect’. Lange then blames what he believes to be the widely-held but errant view that mass has in this case changed into energy upon failing to distinguish the perspective of the mass-equivalence of the predecay and decay systems from the perspective of the mass-asymmetry of the predecay and decay bodies (of the systems). Should we avoid conflating the two perspectives, Lange argues, we thus see that ‘[t]he ‘mass defect’ is not real’ and more generally that mass-energy ‘conversions’ are always merely perspectival in this way [Lange 237].

Note that as far as Lange’s thesis about the mere conventionality of mass-energy conversion is concerned (and, more strongly, his thesis (i)), his first example better serves as support than does his second, because it can hold the essential features fixed in time. It is not necessary to compare the preheated gas ball to the heated one (as Lange points out [Lange 235])—the perspectives on just the heated ball itself may freely shift (depending on interest), regarding the heated ball first as a mass unit, then as a collective of the energies of its constituent molecules (‘defectively’ summed to be of less mass than the ball itself), but equally referring to the ball or its constituents as the same physical system.

The second example, however, requires some comparison of (at least) two different physical systems across time. What is the difference between these? It is certainly not their mass-energy content as (logically compared) systems, which, due to mass-energy conservation, ultimately is the foundation for Lange’s train of thought. But clearly the predecay and decay systems are not spatiotemporally identical as is the heated gas ball at one moment, because the various decay products become increasingly spatiotemporally separated from both their original location in the nucleus and from one another in the decay process. Therefore, any decay system is distinct from the predecay system, and since the spatiotemporal volume of a decay system lies in the forward light-cone of the predecay tritium system, that difference is invariant—‘real’ as Lange would have it [Lange 225].

Lange does not deny the reality of tritium decay [Lange 236]; however, he does not attempt to reconcile this claim of reality of the decay with his denial that there is a real physical process of mass-energy transformation involved in the decay. So we are left wondering, what accounts for the real difference between the predecay and decay systems?

To answer this question we must attend to one other important distinction between Lange’s two examples—the energy sources of the systems studied. In the gas ball example, the heat-source is external to the ball in its initial state, although that fact is ultimately irrelevant to Lange’s examination of the ball in its final heated state. In the tritium decay example, on the other hand, the source of energy is the tritium nucleus itself—it is an isolated system in the beginning, and subsequent changes evolve in similar isolation from external (non-tritium) energy sources. Since change (work) only occurs through increased entropy— energy flow—any energy difference between the predecay and decay systems must obtain from within and between them, and given that energy flow is temporally asymmetric, the energy configuration of a decay system obtains from the predecay system. Since the products in a decay system possess kinetic energy not obviously present in the predecay system, the only possible source of the energy is in ‘hidden’ (kinetic?) energy in the predecay system, or alternately that some mass of the predecay system is converted into energy.

Lange must favor some version of the former over the latter, given (ii). He argues for that possibility with his principle that ‘[m]ass is not additive’ [Lange 237], according to which mass wholes need not be equal to the sum of the masses of their constituents, the difference residing in kinetic energy associated with their constituents in a zero-sum momentum frame [Lange 229-30]. This in turn leads him to regard mass not as a measure of amounts of matter ‘stuff’, but as a quantitative measure of a body or system’s resistance to force ‘as a unit’ [Lange 231]. Hence, for Lange mass is, of course, attributable to the predecay nucleus as a single body, which is also a spatiotemporal system. The decay products, however, cannot as a collection of bodies make up a spatiotemporal system, and to compare them thus with the predecay nucleus is, for Lange, illicitly to change perspectives on measurement in the attempt. In order rightly to compare system with system, the entire decay system—not merely the bodies of the system—must be considered [Lange 236]. Given conservation laws and the isolation of the predecay and decay systems, Lange then claims that comparatively the two must be equal in mass, exhibiting no ‘mass defect.’

In his attempt to declare a decay system a mass unit, Lange cites Einstein [Einstein 225]: ‘Every system can be looked upon as a material point as long as we consider no processes other than changes in its translation velocity as a whole’ [Lange 236]. But clearly any spacetime system containing the decay products has no such translational unity as itself a mass under special relativity, since the spacetime itself is merely a coordinate backdrop to mass-energy measurements.^[2] This is why Einstein reasoned that a translated system of separate bodies might be regarded ‘as a material [center of mass] point,’ because the spacetime system could be thought to ‘shrink’ to encompass just such a point without affecting measurable translational properties. But, if a such a ‘shrunk’ decay system is to be regarded as one mass point, as Lange maintains, then any energy imparted to constituents of the real, complex decay system (as opposed to the predecay system, for example) is necessarily included in the mass measurement of that point by E=mc², and is thus eliminated as a property separate from mass. Presumably it is the possibility of such elimination that moves Einstein to warn that any system may be regarded as a material point ‘as long as we consider no processes other than changes in its translation velocity as a whole’ [Einstein 225; my emphasis]. Therefore, Lange’s argument may indeed secure his perspectival thesis for tritium decay—but only by eliminating any question of physical changes of the distribution of mass-energy in a decay system, endangering the very reality of tritium decay that Lange professes to accept, and yielding his thesis (ii) as only a quite trivial consequence.

It is plainly not plausible under these circumstances to so regard the predecay and decay systems comparatively as mass units, because to do so constitutes a petitio of Lange’s thesis (ii) by compressing and eliminating any information about mass-energy states into a mere mass-point representing a decay system. Hence, apart from any spacetime volume enveloping the decay products, only the products themselves can be measured to provide nontrivial information about decay mass-energy states. In addition, while it is true that a decay system may be counted as part of another system’s mass—think of the earth’s entire mass as itself a unit subject to forces during the entire decay process—and in that way be regarded as containing mass, still, in no way can a real decay system (one not ‘shrunk’) be a mass unit itself.

The decay products themselves, on the other hand, behave as masses as defined by Lange. They can collide with other particles, have their trajectories bent in fields, and the like. Since these products, and the original tritium nucleus, are masses by Lange’s own definition, they are commensurable, yielding in comparison the logical judgment that the sum of decay masses does not equal the predecay mass as each mass is measured in its respective spacetime. The real predecay and decay systems, however, are not commensurable in this way (since the predecay system is a mass and an ‘unshrunk’ decay system is not).

Given that we can thus speak intelligibly of a comparative mass defect, and given that mass-energy conservation by E=mc² is correct in some sense, the only conclusion to be drawn is that, in comparison to the predecay mass measurement, the defect within a decay system is made up in real mass-to-energy conversion. We should note, moreover, that this account need not violate Lange’s own nonadditivity principle for mass, since we interpret the circumstances of judging differential mass only in terms of accounting for the change of mass measurement with respect to the two spatiotemporally related systems, not in terms of diachronically or synchronically comparing a whole mass and its mass-and-kinetic-energy-bearing constituents.^[3]

Considering both of Lange’s examples, we see at once (pun intended) the strength and the weakness of his argument for (i) and (ii): his mass-nonadditivity thesis leads him to make a claim about arbitrary perspectives focusing either on a mass whole or a ‘defective’ sum of that same system’s mass constituents (where ‘sameness of system’ is established synchronically, or perhaps diachronically and nontrivially in some way), with the comparative mass difference residing in kinetic energy of the constituents. However, that claim most convincingly holds in synchronically comparing these perspectives in some cases, such as the heated gas ball (and to that extent also favoring Lange’s thesis (i)). Diachronic measurements, however, need not regard masses in such a parts-to-whole relationship across some spatiotemporally related but distinct physical systems, due to the necessity to use mass-to-energy conversion (or vice-versa, as in nuclear fusion) to explain some such invariant changes of system. Lange’s use of perspectival shifts based on the thesis of nonadditive mass is not justified in these cases.

^[1]Lange does not define what he takes to mean by the crucial term ‘system,’ though context suggests that he means in most cases a spatiotemporal volume as opposed to a logical set collection of entities. In any case, use of the term here will be consistently in the spatiotemporal sense.

^[2]Significantly, Lange explicitly develops his arguments only within the scope of special relativity [Lange 219], though with minor adjustments my argument here would apply to a general relativistic account as well.

^[3]This point is carefully made—it should not be interpreted as an endorsement of Lange’s principle of nonadditivity (in part because I am not convinced that this principle is clearly enunciated to cover all physical situations). The point is that even given the terms of Lange’s own argument, (ii) as stated above does not follow because of the petitio attribution of mass-unity to a decay system.