This was one of Alexander Pope’s truest adages (as long as you remember that the emphasis is on the word* little*: there is nothing dangerous in being well-informed, the trouble arises when one extrapolates from an inadequate knowledge base). Unfortunately, in this hyper-specialised world, all any of us can manage on most topics is a *little* learning; we can specialise in one, or perhaps two or three things, but for the rest we have to just accept what we are taught and what we read in books and newspapers, or hear on TV or radio.

I am frequently reminded of this by examples I come across, in blogs, popular science articles, and even some books, of the misuse of Einstein’s famous equation *E = mc*^{2}. It seems to me that these are often founded on a very inadequate understanding of the equation, what it means and where it comes from.

Of course, one cannot always judge the precise background of the people who write these things. Out-and-out crackpots are easy to spot, even without the help of John Baez’s crackpot index. And make no mistake – there are plenty of them out there. But nowadays, even for people who have studied physics, *E = mc*^{2} is just a bit of standard material you have to uncritically digest and regurgitate for the exam, without dwelling too deeply on the underlying concepts. If you do end up working as a scientist, you are unlikely to be engaged in researching relativity; it is regarded as just a part of what Thomas Kuhn called “normal science”, whose function is to back up the real cutting-edge stuff. Worse still – if you do question any of the standard dogma, you will be labelled a crackpot yourself. Best to steer clear of it then.

But there is undoubtedly a problem here, even if hardly anyone in academia ever discusses it. Let’s start by looking at where the famous equation came from.

Einstein derived his Special Theory of Relativity from two principles – the Principle of Special Relativity (that the laws of physics appear the same to all observers moving with constant speed relative to one another) and the Principle of the Constancy of the Velocity of Light (the speed of light has the same value for all such observers, regardless of their relative speeds). This led him to conclude that two observers moving relative to each other would disagree on their measurements of lengths and time intervals. It was a short step from there to realising that if one of the fundamental laws of physics, the conservation of linear momentum, held in the reference frame of one of these observers, it would not, in general, hold in the other. This must have been rather embarrassing, since it immediately conflicts with the Principle of Special Relativity on which the theory was based.

But Einstein had already turned the whole concept of absolute space and time on its head, so he was not unduly fazed by this. Instead he looked for some quantity that might be conserved in all reference frames moving at constant speed relative to one another (aka *inertial* reference frames), and which might become indistinguishable with classical momentum at low speeds. And he found it.

Classical momentum is just mass times velocity, or *mv*. The more mass a speeding object has, the more momentum; and the faster it’s going, the more momentum. If someone throws a heavy medicine ball at you it might knock you over; but a tennis ball travelling at the same speed will not. Nevertheless, a tennis ball travelling much faster will hurt you a lot more. The more momentum the thing has, the more you will feel it.

In special relativity, the distances and time intervals that different observers measure are related by a factor called gamma, which depends on the relative speed of the two observers:

gamma = 1/(1-v^{2}/c^{2})^{1/2}

It has the value 1 when the two observers are at rest relative to each other, but increases very gradually as the relative speed (*v*) increases. When this speed approaches that of light (*c*), gamma becomes very large. Now, if you multiply the classical momentum by the same factor gamma, you find that the resulting quantity, *mv *x gamma, is conserved in all inertial reference frames, and becomes equal to the classical momentum at low speeds. So the conservation of linear momentum is not infringed – it’s just that we have to modify our definition of momentum a little.

However, Einstein did not, for some reason, want to redefine momentum; he wanted it to still equal *mv*. Hence he took the only other option available: he redefined mass. He called the new mass *m* x gamma*, *where *m* on its own became known as the “rest mass”. To avoid confusion he re-labelled *m* as *m*_{0}; so *m = m _{0 }*x gamma

*.*Hence momentum (

*p*) becomes

*m*

_{0}

*v*x gamma or

*mv*, as before.

This new “relativistic mass” had some strange properties. Measurements of it varied according to the speed of the observer; if this was a large fraction of the speed of light, the mass would be correspondingly greater – for instance, a body travelling at 80% of the speed of light would appear 67% more massive. Even stranger, it seemed to be linked to the energy of the body in some way. Einstein derived the formula *E = mc ^{2} *in which the

*m*is this new relativistic mass. It seemed to have a 1:1 relationship with energy, suggesting that mass was perhaps no more than a

*form*of energy; from that, of course, flowed the idea that mass might be converted into energy, and hence the concepts of atomic power and the atomic bomb.

Later in life, however, Einstein had second thoughts about this. In 1948 he wrote to a friend that “it is not good to introduce the concept of [relativistic mass] of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the `rest mass’”. And indeed, special relativity as it is taught today – and as I learnt it, for instance, in 1997 – does not speak of relativistic mass at all, except as a historical curiosity. Momentum is defined as simply *mv* x gamma, where *m* is now the same as the rest mass – in other words, the classical, non-relativistic mass. The energy equation then becomes

*E ^{2} = p^{2}c^{2} + m^{2}c^{4}*

which, if you substitute *p* = *mv *x gamma, simplifies to *E = mc ^{2}* x gamma, which is consistent with Einstein’s original energy equation if we reinstate the concept of relativistic mass.

That much is understood by anyone who has done a course of relativity. What is not so widely appreciated is that the two approaches – Einstein’s original one and the modern one – give rise to two different, and incompatible, versions of the relationship between mass and energy.

The concept of relativistic mass leads to *E = mc*^{2} and an identification between energy and mass. This gives rise to what is known as the “concomitance view”: energy and mass are *concomitant*, they are *the same thing* (give or take the constant *c ^{2}*). But if you reject relativistic mass and take the modern approach, this leads to the “interconvertibility view”, in which mass and energy can be converted into one another: mass to energy in nuclear fission or fusion reactions, energy to mass in particle colliders.

To advocates of the concomitance view, nuclear explosions take some explaining. They cannot, after all, say that mass is being converted into energy, since in their interpretation both mass and energy are conserved at all times. All they can say is that the mass/energy is being “rearranged”. On the other hand, concomitance, which entails relativistic mass, is useful for explaining the impossibility of travelling at the speed of light. This is forbidden by the form of the equation – when *v=c* the denominator is zero – but for a more “physical” explanation we might want to say that as the object accelerates its mass increases, so it gets harder and harder to accelerate, or, what amounts to the same thing, it takes more and more energy to accelerate it by a given amount.

Although the interconvertibility interpretation is favoured nowadays, and is the version taught at first degree level, what we often find is that the two are very frequently mixed up and may appear together in the same textbook or newspaper article, or even – in the case of the “Relativity Mugs” sold in the Science Museum shop – on the same piece of crockery!

So far, so good. I see nothing wrong with there being two rival “paradigms” regarding mass and energy in SR (although the muddling up of the two concepts is a bit unfortunate). Thomas Kuhn would no doubt have been upset by this state of affairs, because he said that old paradigms disappear when their adherents die out and textbooks get rewritten. What he failed to take into account is that university teaching is not always done by the book; lecturers can choose to set students selected parts of textbooks, or even none at all, and may even just regurgitate the physics they themselves learnt decades earlier. Also that whatever happens at undergraduate level, new postgraduates tend to enter a research world dominated by older academics, and there is a very strong pressure to conform to those older academics’ ideas and prejudices – witness the way postgrads often have to “unlearn” the SI units they studied at school and degree level and adopt outdated units instead. Kuhn would have to fall back on the argument that the relativity revolution has yet to run its course.

This is not the end of the story, however. The odd stratification of the two camps – with the interconvertibility view being popular among physicists, while concomitance is the favourite among non-physicists – means there is little communication or debate between them. In particular, grandiose claims made by the comcomitance school have pointed up serious weaknesses in special relativity itself. I will say more about that in my next post on this matter …

Michael Weiss

said:Maybe I should wait for the next post, but isn’t this a case of a shift in terminology, rather than a paradigm shift or Kuhnian revolution? It’s not like the predictions differ.

And of course, people are sometimes sloppy with terminology, especially on coffee mugs …

Jim Grozier

said:I’m sure special relativity constituted a Kuhnian revolution; quite what the “paradigm” is not so clear but I’d say that how we understand the relationship between mass and energy is pretty central to it. I’ve now added two extracts from the same page of a textbook, which illustrate graphically how these concepts are muddled up.

Michael Weiss

said:I thought the two “paradigms” here were both special-relativistic, the difference being whether “mass” means “rest mass” (and thus is invariant but not conserved), or gamma times the rest mass (and thus is the time-component of the energy-momentum 4-vector). I don’t know if Kuhn would have regarded this shift in terminology as a revolution, but I sure don’t.

Of course, going from Newtonian invariant and conserved mass to the special relativity viewpoint, however expressed, is a Kuhnian revolution.

Jim Grozier

said:I’m not suggesting it constitutes a separate revolution – just not sure how detailed a paradigm is supposed to be. In the “original” Scientific Revolution, was the paradigm just the idea of a heliocentric universe, or was it heliocentric universe + circular orbits and epicycles, or was it heliocentric universe + elliptic orbits, or was it heliocentric universe + elliptic orbits determined by Newton’s laws?

Is it OK to say that the SR paradigm is just “the special relativity viewpoint”, or do you have to spell it out, and if so, to what degree? If you say SR means “inertial observers do not agree on measurements of distance and time”, is that enough or do we also need to know whether “inertial observers do not agree on measurements of mass” (which is the point of contention here)?

To get from a Ptolemaic universe to the fully specified and explained Newtonian one took over a century (1543-1666). Maybe revolutions really do last that long, and we have yet to come out of the SR one?

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