The Quantum Pontiff


Physics and Physicists points to an article: “The Einstein formula: E_0=mc^2 ‘Isn’t the Lord laughing?'” by L.B. Okun on confusion about Einstein’s famous mass and energy formula.

Abstract:The article traces the way Einstein formulated the relation between energy and mass in his work from 1905 to 1955. Einstein emphasized quite often that the mass $m$ of a body is equivalent to its rest energy $E_0$. At the same time he frequently resorted to the less clear-cut statement of equivalence of energy and mass. As a result, Einstein’s formula $E_0=mc^2$ still remains much less known than its popular form, $E=mc^2$, in which $E$ is the total energy equal to the sum of the rest energy and the kinetic energy of a freely moving body. One of the consequences of this is the widespread fallacy that the mass of a body increases when its velocity increases and even that this is an experimental fact. As wrote the playwright A N Ostrovsky “Something must exist for people, something so austere, so lofty, so sacrosanct that it would make profaning it unthinkable.”

Reminds me of a (apocryphal?) story my Physics 1 TA told me. He described how a friend of his, who was rather geekly looking, you known glasses with tape holding them together stuff, was walking down the street one day when a group of local yokels drove by. The group, spotting the weak geek sought immediately went into yokel mocking mode and so shouted at him “Hey geek! E equals MC squared!” The geek thought for a few seconds and yelled back “Only in the rest frame!”


  1. #1 Ian Durham
    August 5, 2008

    Heh heh. I actually had a somewhat edited debate with Frank Wilczek about this in the Op-Ed columns of Physics Today a year or two ago. He was advocating for the elimination of the idea of conservation of mass on similar grounds. But if you take mass to be the magnitude of the four-momentum vector, it is always conserved in classical situations. Tom Moore’s awesome text Six Ideas That Shaped Physics (Unit R in particular) very elegantly describes this.

  2. #2 Dmitry
    August 6, 2008

    That’s a very strange article. I’ve just read it through, but somehow didn’t get the point. It’s absolutely clear that there are two different quantities: “rest mass” m_0 and “relativistic mass” m. The article seems to very passionately claim that the second concept should never be used. And I just fail to understand why.

  3. #3 Ian Durham
    August 6, 2008

    Dmitry, I have to admit I haven’t done more than glance at this article but here’s my conjecture as to why. The only self-consistent definition of mass is as the magnitude of the four-momentum vector. In SR (natural) units such that c = 1, m^2 = E^2 – p^2 where the relativistic energy E = gamma*m^2 and the relativistic momentum p = gamma*m*v. Gamma=sqrt(1-v^2). So, if v=0 the only term we have is the first, and thus m^2 = E^2 in SR units. Converting to SI units, this gives the usual E = mc^2. More appropriately this should be called the rest energy. If v is not 0, E increases but so does p in such a way that m always remains constant. For a less half-hearted description, see Moore’s book that I listed in my previous reply.

  4. #4 Dmitry
    August 6, 2008

    Ok, Ian, that’s clear. Of course I agree that this notion of rest mass is perfectly fine, clear, useful etc.

    What I don’t understand is what’s wrong with the defining M=gamma*m, calling this thing “relativistic mass” and having possibility to use usual p=Mv law. It makes some explanations simpler and that’s exactly why this definition is used in all the popular books on the subject. If Feynman (and all the others) used that way of speaking, why can’t we?

    Somehow Okun managed to write a whole article exactly about this issue without giving any convincing answer to this question.

  5. #5 Ian Durham
    August 7, 2008

    Ah, OK, I see what you’re saying. I think I would say that defining a gamma*m might give a false impression of what really happens. So, if mass is the magnitude of the four-momentum vector, it’s invariant under Lorentz transformations, i.e. it ought to be the same in every frame. But then if you stick a gamma on the front of it, you’re implying it’s not invariant.

    I think the confusing part comes from the fact that what Feynman and others were thinking of when they were speaking that way was what we sort of think as physical, tangible mass. In other words, accelerate a bowling ball to near the speed of light and it will be much more massive to observers not accelerating with it. What you’ve really done is converted some energy to “physical mass” but this total four-momentum magnitude never changes. You’ve just shifted stuff around.

    Here’s a better example (hopefully). Think of an atom. Some of its mass comes from the masses of the constituent electrons and nucleons, but some also comes from the binding energy both within the nucleus and between the nucleus and the electrons. But, in an accelerated frame more of its mass will come from the constituent particles than from the bonding.

    The other tough conceptual thing is that most writers do not make it clear enough that, the closer to the speed of light something gets the “more invariant it becomes” (so-to-speak). In other words, suppose something is moving at 0.75 the speed of light in one frame and 0.5 the speed of light in a different frame. And object moving at 0.999999 the speed of light in that same first frame will be moving at pretty close to the same speed in that same second frame.

  6. #6 Ian Durham
    August 7, 2008

    One more thing: my example with the bowling ball is a little deceptive since to actually do this in all practicality one would have to pump a great deal of external energy into the system in order to accelerate a bowling ball to near the speed of light and some of this injected energy will show up in the bowling ball’s “mass.”

  7. #7 Jonathan Vos Post
    August 8, 2008

    The last class that I taught as a Physics substitute teacher in a middle school and Pasadena was on Mass versus Weight. The first problem set was about bowling balls on Earth and on the Moon.

    The question that the most students got wrong was this:

    “What is the mass of a 1 kilogram bowling ball on the Moon?”


  8. #8 Dmitry
    August 8, 2008

    Ian, thanks for the examples, but I really do understand the arguments and am familiar with the theory. But you didn’t convince me.

    Here’s the essence of your argument (and as far as I understand, Okun is trying to make more or less the same point): “I think I would say that defining a gamma*m might give a false impression of what really happens. So, if mass is the magnitude of the four-momentum vector, it’s invariant under Lorentz transformations, i.e. it ought to be the same in every frame. But then if you stick a gamma on the front of it, you’re implying it’s not invariant.

    Yes, sure. “Rest mass” is invariant. “Relativistic mass” (gamma*m) is not. But what’s the big deal? Are you (and Okun) also going to ban using the notion of “length” because it’s not invariant?

    Length (“physical, tangible” length) is not invariant — and every special relativity textbook, popular or not, is explaining it. Well, mass (“physical, tangible mass” in your words) is also not invariant. Of course every textbook mentions that point. I don’t see any point in forbidding this kind of language, even if it’s a bit sloppy.

    That everybody should make a clear distinction between these two definitions of mass, goes without saying.

  9. #9 Ian Durham
    August 10, 2008

    I understand where you’re coming from, but I have two alternate thoughts.

    First, think, my view extends from my experience teaching undergraduates including non-physics majors. Certainly physicists should have no trouble with the distinction. But students, particularly those who are not used to this type of thinking, have a tendency to see “rest mass” and “relativistic mass” as being one and the same.

    Second, I personally don’t consider the magnitude of the four-momentum vector as a “rest mass” since there is no such thing as a truly universal rest frame. In that sense, I think they ought to ditch both names and just use plain old “mass.”

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