Energy from Mass, Mass from Energy

I probably ought to get a start on the big pile of grading I have waiting for me, but I just finished a draft of the problematic Chapter 7, on E=mc2, so I’m going to celebrate a little by blogging about that.

One thing that caught my eye in the not-entirely-successful chapter on momentum and energy in An Illustrated Guide to Relativity was a slightly rant-y paragraph on how it’s misleading to talk about the energy released in nuclear reactions as being the conversion of mass into energy, because what’s really involved is just the release of energy due to the strong force. It struck me as oddly hostile, but on reflection, there’s a sense in which it’s absolutely right.

When you say that a nuclear reactor or a nuclear explosion works by converting mass into energy, that creates the impression that some of the stuff you had before the reaction has vanished, turned into energy. If you count up the total number of quarks and leptons, though, that’s not true– you still have exactly as many quarks as you did at the start. Depending on what process you’re talking about, some of them may have changed from one type to another (the chain to fuse hydrogen into helium involves converting a couple of protons into neutrons by changing up quarks to down quarks), but the total number of material particles does not change as you go through the reaction.

Where does the energy come from, then? Well, you’ve got the same number of particles, but they’re in a different arrangement at the end of the reaction than when they started. The new configuration has a different energy than the old, which means some energy has been freed up to become kinetic energy of the reaction products. In that sense, the reaction isn’t really any different than the emission of light by an atom when one of its electrons drops from a higher energy state to a lower energy one– the interaction giving rise to the energy is different, but the fundamental process is just a re-arrangement of stuff that was already there.

Why do we talk about mass changes in the nuclear case and not the atomic one? Well, because the energy scales are very different. An ordinary electronic transition in an atom changes the energy of the atom by at most something like 10 eV, while the mass of an atom is several times the mass of a proton, which is 938,000,000 eV. Even for a really light atom like hydrogen, that’s a change in the mass of less than one part in 108, which is completely insignificant.

The energy released in fission or fusion reactions, on the other hand, is on the MeV scale, which is a few tenths of a percent of the mass of the reactants. Which isn’t terribly large, but is significant enough to be worth noting.

So, does that means that it’s illegitimate to talk about nuclear reactions as examples of converting mass to energy? That’s going too far in the other direction. While it’s true that the processes people talk about when discussing nuclear energy are only releasing energy by reconfiguring the particles that are already there, the whole point of the equivalence of mass and energy is that it’s every bit as valid to call the interaction energy “mass” as it is to call individual particles “mass.” And, in fact, 99% of the mass associated with everyday objects comes from exactly the same source as the energy released in nuclear reactions.

If you look up information on the proton, you’ll find that it’s made of two up quarks and a down quark. Looking up information on quarks will tell you that the up quark has a mass of 2.4 MeV/c2, while the down quark has a mass of 4.8 MeV/c2, for a total quark mass of 9.6 MeV/c2. Which is fine, but the first link in this paragraph will also tell you that the mass of a proton is 938 MeV/c2.

Where does the rest of that energy come from? It comes from the strong nuclear interaction, which binds the quarks together into the proton, and also binds protons and neutrons into the nucleus. There’s energy associated with the strong force, and that energy accounts for 99% of the mass of the proton. You can find that energy described as a couple of different particle types– sometimes pions, sometimes gluons– but whatever name you give to it, the energy comes from the strong interaction.

So, while it’s true that the energy released in nuclear reactions is freed up just by changing from one configuration of quarks and leptons to a different configuration of quarks and leptons, the vast majority of the energy associated with the reacting particles is also due to the strong nuclear interaction acting on a particular configuration of quarks.

So, as is usual when I run across a source taking some very strong position, my take on swung from “That’s weirdly hostile” to “That’s a good point,” and ended up somewhere in the wishy-washy middle. It’s certainly true that the creation of particles in high-energy collisions is a cleaner example of conversion between energy and mass than nuclear power is, so as much as I’m tired of hearing about particle physics, I still spent more time on that than on nuclear power in the E = mc2. Given that 99% of the mass of everything has its origin in strong interactions, though, I can’t get all that worked up over the distinction. It’s a decent enough hook for a blog post, though…

18 thoughts on “Energy from Mass, Mass from Energy

  1. …I will be sodding buggered. That’s really interesting to learn, that most of the “mass” in the world is really from force particles.

  2. So from a really stupid chemist’s point of view, would it be right to say that force and mass are inexorably linked, much like the particle and wave nature of matter? So that whenever you talk about one, you are really talking about the other at the same time?

  3. I’d go a little further: the quarks are very far from being little billiard balls of specific weight that interact with each other strongly. The whole picture is inherently relativistic, and there is really no operational way of separating which part of the proton mass comes from the quark masses, and which comes from their interactions.

    The masses for quarks you quote have precise operational definition which has to do with how they scatter with each other at high energies, and losses its meaning as you go to low energies. For example, if you had exactly massless quarks, I don’t think the proton mass would be that much lighter.

  4. @andre3

    Energy and mass are equivalent. Not force. Their relationship is different from the wave/particle duality, but it is true that they are inexorably linked.

  5. I read in an article about Wilczek that there are hundreds of virtual quark-antiquark pairs coming into existence and annihilating quickly (as virtual particles often do … ‘cept around black hole event horizons) around each proton. Is this true, or yet another science journalist gone astray?

  6. I will add that talking about the mass of an electron or a quark also begs the question of what that mass actually “is”. It might (some people hope) be the excitation energy of some sort of ‘string’. And I’ll add that you can create matter (positrons and antiprotons, for example) from energy and that this matter is stable if you keep it in the right place.

    However, the important detail about mass and energy is that the person who got screwed out of a Nobel Prize for discovering fission (Lise Meitner) was the only person who put 2 and 2 together. She knew the masses in the periodic table and could easily calculate that there was some mass left over if Uranium turned into the lighter elements Hahn had identified. She also knew Einstein’s work, and she worked out with her nephew (Otto Frisch) that the mass difference was the same as the 200 MeV energy you should get when the fission products repel each other.

    Frisch quickly provided the experimental proof that the mass difference of 0.2 amu showed up as 200 MeV of energy in an ion chamber.

  7. I think it’s a good point, because often people have the notion that nuclear reactions are somehow mysterious and different from chemical reactions. As you say, the only real difference is the energy scale, which is MeVs rather than eVs, and if you were perverse you could talk about ordinary chemical reactions as converting mass into energy.

  8. Hmmm, since I progressed beyond my bachelor’s in physics, I’ve become increasingly dissatisfied with the language of “converting mass into energy.” If we consider the mass-energy equivalence in special relativity, it isn’t that mass and energy can be converted from one to another, but rather that mass is a form of energy.

    In special relativity, specifically, mass is the total energy of everything that is going on inside the object in question. With a proton, that energy is the masses of the quarks, as well as the energy that stems from the strong force that binds them. With a potato, I can change its mass by changing its temperature (though granted, the amount I can change its mass through this effect is pretty well negligible).

    In special relativity, we distinguish this “internal energy”, which we call mass, from the energy of motion (specifically, the overall motion of the object as a whole, not the vibrations of its components: those vibrations are internal, and count towards the mass). And what is usually done in what is often called “mass-energy conversion” is actually a conversion between this internal energy and energy of motion.

  9. The old practice was to refer to the relativistic mass γm_0, where m_0 is the rest mass, as “the mass.” In recent years many treat “mass” as invariant and that expression always refers to the rest mass (which of course makes for confusing notation for those used to the old practice.) In the predictive sense it’s the same physics, but the new conceptualization is misleading because the effective mass really is the relativistic mass. One way to look at it, is having a ring rotate very fast on a platform (fast enough to have discernible extra “relativistic mass”, and the ring is made to avoid Ehrenfest/Herglotz circumferential stress issues from Lorentz contraction standard of its rim.) Let’s say the sum rest mass of the ring is one kg, and it rotates at the stupendous rim rate of rω = 0.6c. So γ = 1.25.

    Let’s say we try to accelerate the platform. I think we’d all agree, that the extra force to accelerate the ring is 1.25 Newtons per kg of rest mass (for consistency with other ways to define or get energy, momentum etc. from this arrangement and its interactions), so for f = ma (in the rest frame of the platform) the effective mass of the ring is indeed “1.25 kg” and not the sum of proper “mass” of the ring constituents.

    Yes, you could say the new view implies the object as a whole in its own center of momentum frame etc. but even then, why treat parts differently? What if something is a system where object-demarcation etc. is ambiguous or distracting? Indeed, consider that the things we handle, as even for the binding energy etc., already include some adjustment (or could) and so intrinsic rest mass is rather undefinable and misleading. I think practice should return to using “mass” to mean the relativistic mass, which is also what we can actually determine directly.

  10. (Correction, I forgot to put it as 1.25 N per kg of summed rest mass per m/s² of acceleration. The inertia of the spinning ring is 1.25 kg, hence that is how we should refer to it’s “mass” and that of its constituents as well – all in one consistent way.)

  11. All this is very interesting, but it boggles the mind to see someone having actually pulled it off. An Italian is said to have achieved this in Bologna. I dont speak much Italian to understand the video, but the discussion that follows is intreaguing (

    I am going to assess myself – getting out my old theory books out as it seems too good to be true – and get back to this post with more



  12. “In special relativity, specifically, mass is the total energy of everything that is going on inside the object in question.”

    Not quite. Mass is the total energy blah-blah-blah in the rest frame (the center of momentum frame) of the object in question. That is, mass is the rest energy. And the invariance of mass is true in all of physics. Mass never changed when you chose a different inertial frame in Newton’s book.

    Rephrasing with the lingo of thermodynamics, it is essentially the total internal energy of the object in units of J/c^2, although we need to tweak the definition slightly because internal energy only accounts for the thermal (kinetic) energy of the constituents of an object in the object’s rest frame.

    In recent years many treat “mass” as invariant

    I would agree only if you think that the 1950s are “recent”. The quantity mass (symbol “m”) used by particle and nuclear physicists has been the invariant quantity ever since relativistic problems had to solved. The Mandelstam variables, with their emphasis on invariant quantities, followed soon after. That was a half century ago.

    The “relativistic mass” served one (essentially pedagogical) purpose: write p = mv to keep 19th century physicists happy by using the preferred frame language of Lorentz. It does not work in K = p^2/2m, for example, while K = E – mc^2 works just fine at both high and low velocities.

    @ the argument in comment 9:
    A rotating object is not an inertial frame. You jump into it at your peril. I would also add that there is excellent evidence from nuclear physics that the “rotational” kinetic energy associated with metastable states of high angular momentum is part of the mass of an object that gets measured with a spectrometer.

  13. Wonder what happends to the energies and hence mass ( via E = mc2) at molecular/atomic/nuclear energy levels due to heating.

  14. But CCP, you are basically just restating the newer definition as a matter of convention (which may include some contextual advantage not good for logical appreciation and wider integration, not e.g. the case of CGS versus MKS units), not adequately defending that choice. My counter-argument is that it makes more sense to correlate “mass” to the energy we actually observe in our own frame, not the COM frame of the object. Why treat them differently, especially when we refer to “mass-energy” and note that the distinction is not always clear. So would you have us sum up the constituent “invariant proper masses” of a composite body (not much isn’t, you know) to get the correct “mass” of the composite, even though (as I and Greg noted) there are internal processes making for a greater collective, effective “mass” in terms of inertia etc?

    That way, it really is always true that E = mc². And I’m not “getting into” the rotating frame of reference, I’m taking the rotating system in *my frame as a “given” just so I *don’t* have to think about the “invariant masses” of the specific constituents. The rotating ring example just goes to show that mass is not invariant, because rotating a bunch of them (their being connected together doesn’t change the logical concept. Furthermore, rotational energy of nuclei becoming part of the “mass” we measure just goes to support my point that the energy from motion be included, not your point that it isn’t.

    BTW folks, you don’t need to use things like ^ much anymore because fine characters read equally by almost all browsers are available (except for good subscripts.) You can also type them in my using Alt + (number) but the results may not correlate to MS Word and other applications. Go to and find plenty of good stuff. Here’s a sample:
    ħ (from Alt +295) ↖ ↗ ↘ ↙ a|1⟩ + bθ|2⟩ ℍ ∂ ∯ ∇

  15. Sorry, but there is nothing “new” about defining mass as an invariant quantity. You are free to eliminate KE in favor of calling the energy of motion “mass”, but you will need to rewrite all of freshman physics to do so and have a hard time justifying it.

    My position is that it is silly to replace E^2 = p^2 + m^2 with M^2 = p^2 + m^2, where the second m is invariant in both expressions, when we have a perfectly good name (energy) and symbol (E) for the first one. And you can’t just pretend that momentum doesn’t appear in a statement about the energy-momentum 4-vector.

    I’m taking the rotating system in *my frame as a “given”

    That means you are frame jumping, because you cannot use a Lorentz transform to define a proper mass in the accelerated coordinate system of the ring itself. You cannot use gamma to connect them like you did.

  16. One note about the “same number of material particles” business:

    p + p –> d + e+ + nu

    doesn’t represent conservation of particles. Sure, there are six quarks on both sides, with one up having converted to a down. (By “d”, I mean deuterium nucleus, a p and a n bonded together.) But there are also two leptons (well, one lepton and one antilepton) on the right side.

  17. My understanding is that by the usual accounting, one lepton plus one anti-lepton is zero leptons total, so that reaction doesn’t really increase the total number of particles. I could’ve made that clearer, though.

  18. Well, we are quickly descending into semantic territory, so, YOU’RE A NAZI, I mean, when you say “counting particles”, it depends on whether you view an antiparticle as a “negative particle” or any such. I’d argue that in this context, you’re talking about converting between mass and (other forms of) energy, so you should count particles regardless of whether they’re antimatter or matter. Mass is always positive, regardless of matter/antimatter status.

    As you note, it’s all really more subtle than that. But I think it’s more obfuscatory to say that nuclear physics is just about rearranging particles and that a lepton plus an antilepton counts as zero particles than it is to say that particles are “created out of energy”. Sure, *lepton number* is conserved, but particle number is not. (This is in contrast to chemistry, where particle number generally is conserved… unless you count photons as particles, in which case sometimes it’s not. However, as you note, in chemistry, you’re also “converting between mass and energy”, in that the rest mass of atoms on one side (including internal energy in rest mass) is different from the other side. It’s just there, the difference is something like a part in a billion, so chemists can talk about “conservation of mass” and not be too wrong.)

Comments are closed.