Mass Is Just Energy You Haven’t Met Yet

How can colliding two protons give rise to more massive particles? Why do vibrations of a string have mass? And how does the Higgs work anyway?

There is one central misunderstanding that makes each of these topics confusing. It’s something I’ve brought up before, but it really deserves its own post. It’s people not realizing that mass is just energy you haven’t met yet.

It’s quite intuitive to think of mass as some sort of “stuff” that things can be made out of. In our everyday experience, that’s how it works: combine this mass of flour and this mass of sugar, and get this mass of cake. Historically, it was the dominant view in physics for quite some time. However, once you get to particle physics it starts to break down.

It’s probably most obvious for protons. A proton has a mass of 938 MeV/c², or 1.6×10⁻²⁷ kg in less physicist-specific units. Protons are each made of three quarks, two up quarks and a down quark. Naively, you’d think that the quarks would have to be around 300 MeV/c². They’re not, though: up and down quarks both have masses less than 10 MeV/c². Those three quarks account for less than a fiftieth of a proton’s mass.

The “extra” mass is because a proton is not just three quarks. It’s three quarks interacting. The forces between those quarks, the strong nuclear force that binds them together, involves a heck of a lot of energy. And from a distance, that energy ends up looking like mass.

This isn’t unique to protons. In some sense, it’s just what mass is.

The quarks themselves get their mass from the Higgs field. Far enough away, this looks like the quarks having a mass. However, zoom in and it’s energy again, the energy of interaction between quarks and the Higgs. In string theory, mass comes from the energy of vibrating strings. And so on. Every time we run into something that looks like a fundamental mass, it ends up being just another energy of interaction.

If mass is just energy, what about gravity?

When you’re taught about gravity, the story is all about mass. Mass attracts mass. Mass bends space-time. What gets left out, until you actually learn the details of General Relativity, is that energy gravitates too.

Normally you don’t notice this, because mass contributes so much more to energy than anything else. That’s really what E=m is really about: it’s a unit conversion formula. It tells you that if you want to know how much energy a given mass “really is”, you multiply it by the speed of light squared. And that’s a large enough number that most of the time, when you notice energy gravitating, it’s because that energy looks like a big chunk of mass. (It’s also why physicists like silly units like MeV/c² for mass: we can just multiply by c² and get an energy!)

It’s really tempting to think about mass as a substance, of mass as always conserved, of mass as fundamental. But in physics we often have to toss aside our everyday intuitions, and this is no exception. Mass really is just energy. It’s just energy that we’ve “zoomed out” enough not to notice.

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14 thoughts on “Mass Is Just Energy You Haven’t Met Yet

  1. duffieldjohn

    The photon has no mass, but photon energy-momentum is resistance to change-in-motion for a wave propagating linearly at c. And if you trap a photon in a mirror-box (see http://arxiv.org/abs/1508.06478) you increase the mass of that system. Then when you open the box, it’s a radiating body losing mass, just like Einstein’s E=mc² paper. You can liken electron-positron annihilation to opening one box with another, whereafter each loses all of its mass and then isn’t there any more. Couple that with Einstein-de Haas, electron diffraction, electron spin, etc, and IMHO it’s clear that electron mass is merely resistance to change-in-motion for a wave going round and round at c in a box of its own making.

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      1. duffieldjohn

        I meant that literally. See Einstein’s E=mc² paper ( https://www.fourmilab.ch/etexts/einstein/E_mc2/www/ ) and note “the mass of a body is a measure of its energy-content”. Six lines above that, Einstein said “body” and “electron” on the same line. The electron is a body. The mass of a body is either due to its energy content, or its interaction with the Higgs field. It can’t be both. See page 173 of A Zeptospace Odyssey by Gian Francesco Giudice.

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        1. 4gravitonsandagradstudent Post author

          I have no idea what your background is, so my apologies if the following is something you know already.

          “The mass of a body is a measure of its energy-content” isn’t mutually exclusive with “the electron gets its mass from the Higgs field”, in the electron’s case they mean the same thing. The mass of a body is a measure of its energy, and that energy includes energy of interaction: that’s why the proton’s mass comes from its interaction via the strong force, and it’s how the Higgs gives things mass. I think you’re misunderstanding Einstein to be talking about some sort of energy inherent to the mass, but that’s not a necessary feature of the argument. His setup just happens to be “zoomed out” enough that it doesn’t care where the mass comes from. In particular, Einstein didn’t know about the necessity of the Higgs field, that was figured out decades later.

          More generally, it’s extremely bizarre to give a page quote reference from a popular physics book when arguing with a physicist. It would be like if I cited a high school civics textbook to argue with ohwilleke here about legal matters. (In case that comparison didn’t make sense, ohwilleke is a former lawyer.)

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          1. duffieldjohn

            I’m an IT guy who’s always been into physics, but who developed a special interest ten years back when our teenage children gave up all their science subjects. Having done some writing I fancied myself as a science communicator, but I started finding significant differences between what you read in say the Einstein digital papers as compared to contemporary sources. With respect, I don’t think I’m misunderstanding Einstein’s E=mc² paper. Take a look at this http://arxiv.org/abs/1508.06478 by a guy called Martin van der Mark and another guy called (not the Nobel) ‘t Hooft. The Guidice book refers to the Higgs sector as rather arbitrary, frightfully ad-hoc, and the “toilet” of the standard model. Anyway, sorry to comment so much on your blog. Please don’t hesitate to say if you’d prefer to take it offline etc.

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            1. 4gravitonsandagradstudent Post author

              Since you’re an IT guy and not a humanities guy, I’m not sure this is quite the issue you’re having, but I should point out that “what Einstein wrote on the subject” doesn’t matter as much in physics as primary sources do in the humanities. Later people like Wheeler aren’t just interpreting Einstein, they’re working from the full mathematical framework and accumulated evidence and trying to come to their own interpretations. That doesn’t mean they’re necessarily right, but “Einstein didn’t view it that way” is never going to be the reason why.

              Looking at that paper, the authors appear to work at a corporate research campus focused on healthcare and lighting. They’re not particle physicists, and it shows, because the proposal they’re hinting at (that electrons are just configurations of light) doesn’t actually work. Electrons interact with the weak force, while photons do not. That interaction is the core of why you need something like the Higgs, and it doesn’t seem like the authors of that paper are even aware that that’s something they’d need to address.

              The Higgs sector is arbitrary, but not just by existing. The existence of the Higgs sector is pretty much required by the rest of the theory. It’s arbitrary, (the “toilet” of the standard model) because most of the details aren’t fixed by math alone. This is in contrast to the rest of the standard model, which to a great extent has to take the form it does in order to be mathematically consistent. The theory needs a Higgs, but the precise way the Higgs interacts with itself (the Higgs potential) is more arbitrary. If you’ve heard people worrying about whether we live in a metastable vacuum, the reason why they aren’t certain is because the Higgs potential could take any number of shapes, some of which leave our universe stable and some of which don’t, and the only way to distinguish is by experiment. This is something a lot of people find unsatisfying, that we can’t predict all that much about the Higgs from first principles, and my guess is it’s why Guidice views the Higgs sector with distaste.

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  2. ohwilleke

    “A proton has a mass of 938 MeV/c², or 1.6×10⁻²⁷ kg in less physicist-specific units. Protons are each made of three quarks, two up quarks and a down quark. Naively, you’d think that the quarks would have to be around 300 MeV/c². They’re not, though: up and down quarks both have masses less than 10 MeV/c². Those three quarks account for less than a fiftieth of a proton’s mass.”

    First, it is worth recalling the huge disparity in the precision of the experimentally determined values of the rest mass of the proton, 938.272,046(21) MeV, and the experimentally determined values of the rest masses of the light quarks which are known to approximately one significant digit accuracy. Calculations of the proton mass from first principles using lattice QCD have a precision on the order of 1%. Experimental measurements of the proton mass are roughly a million times more precise than the results of first principles QCD calculations of the same quantity made using the world average values for the fundamental constants of the Standard Model. (The estimated precision of the difference between the rest mass of the up quark and the down quark is much greater than the measurement of the absolute value, however.)

    Amplitudologists are absolutely the lagging turtle in this race so far, even if we’d like to think that they will eventually catch up somehow.

    Second, I recognize that this observation is true in a lies to children sense.

    But, if I recall correctly, there are several ways in which reality is more complicated than the quark-gluon proton model in which quarks acquire their rest mass from a Higgs field and most of the mass of the proton is the integrated energy strength of the gluon field of the proton generated by the three quarks.

    (Feel free to correct me if I’m all wrong in any of these respects, I’m trying to confirm that my understanding is correct in light of mainstream particle physics, not to expound speculative theories and welcome the change to improve my understanding by being told that I’m wrong).

    A proton is really a statistical blend of all sorts of stuff other than a simple three quarks bound by gluons model that is reflected in parton distribution functions. The uud model of the quark may be a plurality description of a proton statistically, but it is incomplete. This makes simple allocations of composite particle mass to its components profoundly more complicated than it would be otherwise.
    The mass of a quark if a function of the momentum scale implicated in the renormalization scheme that you are using, but, while the heavy quark masses are well defined at essentially all momentum scales where they can exist, and light quark masses can be defined in sensible ways at momentum scales as low as 1 GeV or so, the renormalized light quark masses cease to make sense and are ill defined at momentum scales on the other of the pion mass (ca. 140 MeV) and lighter. This seems like a really bad feature of a central tool of particle physics. But, maybe quark confinement makes this a purely hypothetical problem with no observable consequences that is really some sort of category error. Anyway, if you do blindly extrapolate the quark mass beta functions to low enough momentums the quark mass relative to gluon masses surges.
    One common way to describe mass generation from gluon fields in hadrons is to describe gluons as acquiring mass dynamically, even though theoretically, gluons have a rest mass of zero. Conceptually, this is quite a different concept than simply integrating the energy of the gluon field and localizing that mass to a centroid, and it is my understanding that to the extent that the observable predictions of dynamically massive gluons differ from those that see mass as arising from integration of a continuous strong force field, that the dynamically mass hypothesis prevails.

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    1. ohwilleke

      That last bit was supposed to be three numbered paragraphs. Don’t know why the formatting disappeared and apologies if it is a bit hard to parse out the three distinct concepts there.

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    2. 4gravitonsandagradstudent Post author

      First of all, amplitudeologists aren’t in the business of predicting the mass of the proton, since it’s not an amplitude. Agreed that lattice QCD isn’t as good at this as one would like, but it’s understandable for such a complicated system (and see below).

      For what appears to be your first bullet point, you’re right that a proton is in practice a PDF-shaped soup of a lot of different stuff. That’s not contradictory to the claim that the proton mass is due to the forces between three quarks, but complementary to it: for the strong force, that soup of stuff is what a high energy of interaction looks like.

      Your other two points appear broadly correct, though I’m not familiar enough with the technical details of dynamical mass vs. integration of a continuous force to comment. In both cases, it’s an elaboration of the story, but the central point, that masses don’t tend to be fundamental but are due to interactions (note for example that masses run in renormalization specifically because we’re dealing with particles that interact) still holds, and that’s most of what I’m striving for here. As you say, there’s a certain amount of lies-to-children involved, yes.

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      1. ohwilleke

        “First of all, amplitudeologists aren’t in the business of predicting the mass of the proton, since it’s not an amplitude.”

        The mass of the proton itself is not an amplitude, but as best as I can discern, there are a whole lot of amplitude calculations that go into that calculation and more importantly, it appears that the principal difficulty in the calculations is calculating the amplitudes associated with gluon propagation which are more difficult than in QED mostly because gluons interact with each other while photons don’t.

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        1. ohwilleke

          Gluons, of course, share this confounding property with our favorite hypothetical particle, the graviton, which makes the amplitude calculations for each difficult to calculate for very similar reasons, IIRC.

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        2. 4gravitonsandagradstudent Post author

          The impression I had was that most calculations of the proton mass are done using lattice QCD, where perturbative amplitudes aren’t especially relevant. While part of their uncertainty comes from uncertainties in various fundamental constants that amplitudes research could indirectly help clean up, I didn’t have the impression that that was their main bottleneck.

          If you’re referring to an alternate, perturbative approach to calculating the proton mass, that’s something I haven’t run into, and you’d know more about it than I. I would be curious how far in loop order they have to go to get something vaguely sensible.

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  3. Sean Corali

    Hi ohwilleke and Matt (von Hipple) 🙂

    Have you guys seen Matt Strassler’s article called “Massive Source of Confusion”? There are two interpretations of the mass issue. Professor Strassler (and all modern particle physicists) favors interpretation #1. Einstein himself, in later years, favored it , too, in his personal letters to Licoln Barnett.

    Also PBS Digital Studios (Gabe Perez) those physicists explain the meaning. Google :The Real Meaning Of E=mc2″.

    It is a scarlett red background with Einstein’s face. The video explains there is no mass to energy alchemy or conversion ever going on.

    Former Fermilab physicist, Suresh Emre also has a fabulous article about misconceprions regarding E=mc2 and he also stresses no mass to energy conversion is occuring, no alchemy.

    Sean Carroll also has a YouTube video demystifying Mass with Veritasium asking questions.

    Good stuff menrioned above, great sources. Have a look.

    Good blog as well. I enjoy it.

    Take care,
    Sean

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    1. 4gravitonsandagradstudent Post author

      Yeah, like Strassler, I should clarify that I’m referring to rest mass here. Come to think of it, that may be what’s confusing duffieldjohn, the Einstein paper he dug up may be one of the ones that use relativistic mass.

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