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Sep 24 2013

Death to the relativistic mass!

In Newtonian physics, mass is an invariant, by which I mean that any observer would measure the same value for the mass of an object, irrespective of how fast the observer was moving. Also, the relationship between momentum (p) and mass (m) was given by momentum equals mass times velocity (v). i.e., p=mv.

When relativity came along, the relationship between momentum and mass became more complicated by the presence of an additional velocity dependent multiplicative factor denoted by the Greek letter gamma γ. Now the expression for the momentum was p=γmv.

But the old Newtonian form could be retrieved by introducing the concept of ‘relativistic mass’ in which the multiplicative factor could be tacked onto the old mass to give a new relativistic mass M=γm, thus resulting in p=Mv. This relativistic mass is now velocity dependent and this led to a spate of popular descriptions of how an object’s mass depended on its velocity.

The idea of relativistic mass created a mess because while it had a certain heuristic appeal, it coexisted along with those who deplored its use and retained the term mass to refer only to the old invariant mass m, which the advocates of relativistic mass now referred to as the ‘rest mass’. It was argued that which approach one adopted was largely a matter of taste.

The eminent Russian physicist Lev Okun argued strongly that the idea of relativistic mass should be abandoned because whatever heuristic value it had was strongly overcome by its lack of consistency and the misconceptions it created. (His 1989 article in Physics Today was one such effort.) He was so persuasive that I was convinced and never referred to it again. I also thought that the issue was now dead.

So I was surprised to be told about this 2005 paper by Gary Oas who did an exhaustive survey of physics textbooks and popular books and found that relativistic mass is alive and well and still being used by people who I feel really should know better. Like Okun, Oas argues that this is a mistake and that we would be far better off burying the concept of relativistic mass once and for all.

I totally agree.

8 comments

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  1. 1
    Rob Grigjanis

    I vaguely remembered some confusion on this point from an early relativity course I took, so I dug out the text we used;

    A.P. French, Special Relativity (in the M.I.T. Introductory Physics Series), Norton & co, 1968

    Lo and behold, on page 23 the author says that what one means by the word mass is “essentially a matter of taste”. Ironically, an earlier paragraph contains (where the m referred to is the ‘relativistic mass’)

    Note that K [kinetic energy] is not obtained by substituting into the expression (1/2)mv^2 the value of m calculated from Eq. (1-18) – a frequently made error, because the temptation to cling to the Newtonian form of the kinetic energy is very strong.

    m(v) is totally about Newtonian temptation, and is very, very ugly. Looks like a scalar, smells like a turd.

  2. 2
    Mano Singham

    Yes, that is precisely the kinds of errors generated by using relativistic mass.

  3. 3
    MNb

    Alas those two articles are behind paywalls; I would have like to read them.
    Anyhow I don’t remember being ever confused about mass and relativistic mass. So I looked it up in my textbook Elementary Modern Physics by Weidner and Sells, 1980 (third edition), where in the first place I found (and remembered correctly) that gamma is the Lorentz constant. Shame on you for not naming my compatriot. OK, stupid chauvinism aside, Weidner and Sells use m0 for rest mass and gamma times m for relativistic mass indeed.
    So it’s a pity I can’t read the two articles, because don’t really have an idea what the problem is.

    “misconceptions it created”
    My first, intuitive reaction is that this is due to bad explanation. RG above confirms this.
    So I am not convinced.

  4. 4
    Mano Singham

    The Oas article is on arXiv which should be free. Can you try again?

  5. 5
    Rob Grigjanis

    Yes, the Oas article is not paywalled.

    Maybe I misread you, but the problem isn’t confusion between the rest mass and the relativistic mass. The problem is that it’s too easy for beginners to infer, from the usage in some texts, that the relativistic mass is a Lorentz scalar. It’s actually the zero component of a 4-vector proportional to the four-momentum, i.e. it’s proportional to the energy. So it’s definitely not invariant under arbitrary Lorentz transformations.

  6. 6
    colnago80

    In addition, E/m(0) is the generator of static time translations and the statement of conservation of energy is equivalent to the statement that the laws of physics are invariant under static time translations (i.e. the laws of physics are the same today as they were in the past and as they will be in the future).

  7. 7
    Mano Singham

    I think that those who get to the stage that you describe can navigate through the process quite well. It is the students who work at more elementary levels who can get confused.

  8. 8
    MNb

    Ah yes – I suppose I was acting silly once again a few hours ago. I’ll read it tomorrow.

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