The “Most Famous Equation”

In 2014 I posted a rather light-hearted blog piece about energy and mass in special relativity. Recently, in response to a talk I gave at the IHPS conference in June, and a discussion that arose during that conference, I have written a more serious piece about the same topic.

E = mc² may well be “the most famous equation”, but what does it actually mean? In particular, what does special relativity say about the relationship between energy and mass? Scientists and philosophers do not all agree, among themselves or with each other, about this. In this short piece I will critically appraise one recent study, and suggest that, while it is welcome, there is still more work to do on this topic.

In his paper Interpretations of Einstein’s Equation E = mc², Francisco Flores identifies six distinct positions on this question, which he believes constitute “all of the leading and influential interpretations of Einstein’s equation in the literature”. He then applies to each of the six three tests, which, he argues, a successful interpretation ought to satisfy. These tests then rule out five of the six, and Flores concludes that the remaining interpretation is the only viable one.

The Interpretations

Flores divides the six interpretations into two sub-groups. The first four he describes as property interpretations; the last two are ontological interpretations.

The property interpretations are possible answers to the question “do energy and mass describe the same, or distinct, properties of a body?” If the latter, a subsidiary question comes into play: “can one of these properties be converted into the other?” Flores identifies positions he describes as “same-property”; “different-properties, no-conversion”, and “different-properties, conversion”. He also highlights one view which he describes as “one-property, no-conversion”, in which only mass is regarded as a property (and hence there can be no conversion).

The last two interpretations, which Flores does not distinguish from one another very clearly, are described as “ontological” because he says they concern “the fundamental stuff of modern physics”. Both depend on the assumption that “there is no distinction between mass and energy as properties”, and hence they belong, in a sense, to the “same-property” camp.

My first worry about this contest is that he does not include an option analogous to “None of the Above” or “RON” (Re-Open Nominations) as sometimes used in elections and surveys. If one were to ask a random sample of physicists “how would you interpret the equation E = mc²”, one would be likely to find the response “I wouldn’t” among the most popular. Since the equation is usually presented to physics students as itself an option – and, furthermore, one that modern physics generally avoids – it would be difficult to know how else to answer the question. Luckily, the interpretations that Flores discusses will equally well do duty as interpretations of the rôles of, and relationship between, mass and energy in special relativity – a topic on which all physicists and philosophers of physics are likely to have a view.

The Tests

The first test Flores applies is described as “the familiar goal of philosophical interpretations of physical theories”, namely, that the theory is required to answer the question “what would the world be like if this theory were true?” In other words, the theory must make predictions about the world (and, though Flores does not say as much, we would surely hope that they were testable predictions). Secondly, he requires that “an interpretation I of a given physical theory T does not appeal to hypotheses outside T or theories other than T”. The third criterion tests the theory for what Flores calls “philosophical uniformity”: it asks whether the theory treats “elements in the mathematical formalism of T that are similar in type” similarly, or explains why these elements should be treated differently.

Rest Mass and Relativistic Mass

Flores is careful to establish at the outset that he “will focus exclusively” on rest mass as opposed to relativistic mass in the paper. This is a sensible decision, not least because no other type of mass is recognised by most physicists nowadays. However, it does restrict the applicability of the equation E = mc² to measurements made in the rest frame, and strictly speaking, as Lev Okun has pointed out, the equation should then be written in the form in which Einstein introduced it:

E₀ = mc²

where E₀ is the energy measured in the rest frame, and m is the rest mass [Okun p31].Yet at least one of the interpretations that Flores investigates – that of Bondi and Spurgin – makes it equally clear that the authors are considering relativistic mass, not rest mass: they claim that the equation holds in “all frames of reference” [Bondi & Spurgin p62] and use the symbols m, m₀ to denote relativistic and rest masses respectively, so that, for them, E = mc² is an equation relating energy to relativistic mass [1]. Flores does not mention this.

Another interpretation – that of Wolfgang Rindler – also clearly treats the equation as relating energy to relativistic mass. Rindler opts for a definition of relativistic momentum to coincide with the non-relativistic case, in other words, p = mv. This choice of definition then commits him to relativistic mass. For instance, he states that “if a momentum of the form mu is to be conserved, then the mass m must be of the form m = γm₀” [Rindler p80]. This is somewhat more serious, since Rindler’s “different-properties, conversion” interpretation of the equation is the one singled out by Flores as the only viable one.

Units and Dimensions

One of the interpretations discussed by Flores – the “same-property” interpretation – rests heavily on an argument involving units and dimensions. This interpretation is associated with Arthur Eddington and R. Torretti. It is one of many claims made by certain physicists (among whom Eddington features prominently) on the basis of the assumption that, if a fundamental physical constant (such as c) is represented by a numerical value (and usually that value is 1) it becomes dimensionless. Clearly, if we make such a substitution in the equation E = mc², we appear to have equality between energy and mass, which Eddington and Torretti then interpret as evidence that energy and mass are the same property.

Since the philosophy of measurement is a field largely neglected by most physicists (and many philosophers), misunderstandings about units and dimensions abound in the literature, as I have shown elsewhere [see e.g. Grozier (2017)]. Eddington’s mistake is to assume that, when “working in” a particular system of units in which c has the value 1, we can write Einstein’s equation as simply E = m and hence deduce that they are the same property. Apart from a worry about what happens when we move from that particular system of units to a system in which c does not have the value 1 (does it make sense to say that two properties are the same, but only when we are working in a certain system or systems of units?) we might reasonably object that, if we want to use a particular system of units, we must give those units names. It is legitimate to say that, in a certain system of units, the speed of light is “1 einstein” (where the “einstein” is my name for the unit of velocity in Eddington’s system) but it is not legitimate to just say that c is a pure number with the value 1. Flores treats Eddington and Torretti’s argument somewhat uncritically; he rejects it, but for different reasons. He does not seem to be aware of the particular problem I have just outlined.

Constructive and Principle Theories

Flores rightly points out, following Einstein, that special relativity is a principle theory (a “top-down” theory based on certain principles concerning the behaviour of macroscopic bodies) as opposed to a constructive theory (a “bottom-up” theory based on a what we know about matter at the fundamental level) and that such theories “do not afford bottom-up explanations” [Flores p259]. However, he fails to appreciate the full implications of this distinction.

Special relativity derives certain conclusions from its two assumptions (the principle of special relativity and the principle of the constancy of the velocity of light) by means of thought-experiments featuring “rigid bodies” which, as Harvey Brown has pointed out, and even Einstein himself acknowledged, are structureless [see Brown p.S89; also Einstein, pp 59,61]. Since these bodies have no internal structure, the theory cannot make any predictions concerning such things as the nature of heat; yet assertions that, because of E = mc², a body gains mass when it is heated, are common in the literature [2]. Special relativity, as Einstein argued, ought to be deducible from first principles at the level of atoms, if not that of fundamental particles; but to do that would involve, at the very least, considerations of quantum mechanics: a simple model in which a photon is sent from one observer to the other and then reflected back to its source will not do when the “observers” are individual atoms or particles. Again, Flores does not seem to be aware of this problem, which, while it may not impact directly on the interpretation of E = mc², is surely relevant [3]. For instance, in one of the interpretations considered by Flores – that of Marc Lange, who argues that energy is not a real property because it is frame-dependent (hence “one-property [mass], no-conversion”) – Lange requires the fundamental mass-energy relation of special relativity

E² = p²c² + m²c⁴

to hold simultaneously at the macroscopic and microscopic levels, and concludes that mass is not additive – a result which he uses to back up his argument that there is no conversion of mass to energy because the “mass defect” is a fiction based on the assumption of the additivity of mass. But how do we know that the equation holds for fundamental particles? Flores rejects Lange’s interpretation, but for a different reason.

Mass, Energy and Pluralism

Does it actually matter, though, which is the “correct” interpretation of the relationship between energy and mass in SR? Presumably not; it was, after all, perfectly possible that more than one of the six interpretations might pass Flores’ test, or that none would pass it: he was not necessarily looking for a single answer. As someone who has taken a physics degree within the last 20 years, I am inevitably inclined to avoid the concept of relativistic mass and to favour the “interconvertibility” view, since that is now the orthodoxy, at least among physicists. However, I can see the benefit of sometimes appealing to relativistic mass – for instance, in order to explain the unattainability of the velocity of light, since if mass increases with velocity it becomes harder to accelerate a body the faster it goes, and ultimately, as one approaches c, it becomes infinitely hard. Maintaining a pluralistic approach, and selecting a model that is appropriate to the problem in hand, seems a sensible, pragmatic way for philosophers and physicists to behave. And instead of worrying about which model is “true”, a better use of one’s time and energy would perhaps be to challenge the many inconsistencies and contradictions about mass and energy that one finds, not only in such likely places as press articles and popular science books, but also in physics textbooks, and even in some of the philosophical literature (and, indeed, on these “relativity mugs” on sale at the Science Museum shop! The small print says both “this formula suggests that tiny amounts of mass can be converted into huge amounts of energy” (conversion) and “matter and energy are really different forms of the same thing” (same-property).)

References

Bondi, H. & Spurgin, C. 1987. Energy Has Mass. Physics Bulletin 38, 62-63.

Brown, H. 2005. Einstein’s Misgivings about his 1905 Formulation of Special Relativity. Eur. Jnl. Phys. 26, S85-S90.

Eddington, A. 1929. Space, Time & Gravitation. (Cambridge UP)

Einstein, A. 1969. Autobiographical Notes. in Schilpp, Albert Einstein: Philosopher-Scientist, vol.1, pp 1-94. (Open Court).

Flores, F. 2005. Interpretations of Einstein’s Equation E = mc². Int. studies in the Phil. of Science 19, 245-260

Grozier, J. 2017. Should physical laws be unit-invariant? Studies in the History and Philosophy of Science (under review)

Lange, M. 2002. An Introduction to the Philosophy of Physics. (Oxford UP)

Okun, L. 1989. The Concept of Mass. Physics Today 42 (6) 31-16.

Rindler, W. 1977. Essential Relativity (Springer-Verlag)

[1] Hence Bondi and Spurgin do not even appear to be restricting this statement to inertial frames, which is usual when considering special, as opposed to general, relativity.

[2] These predictions remain mere conjectures, since they cannot be tested using current technology.

[3] This is not to suggest that the behaviour of fundamental particles is not consistent with SR; on the contrary, one of the first and most convincing verifications of SR was the observation that the lifetimes of muons from cosmic rays (and hence travelling at near light-speed relative to the observer) are significantly greater than those of muons moving slowly in the laboratory. However, a theory describing the behaviour of macroscopic objects cannot explain this fact.