When you look around on the internet, you can find a whole host of articles about the mystery of mass. For example there’s a Guardian piece by Ian Sample, who says the origin of mass is “one of the most intriguing mysteries of nature”. Or there’s Concepts of Mass by Max Jammer, who says ”the notion of mass, although fundamental to physics, is still shrouded in mystery”. There’s also the ATLAS article by Michael Chanowitz, who talks about uncovering “the deep mystery of the origin of mass”. There’s just one little problem with all that, and it’s this: Einstein solved the mystery of mass over a hundred years ago.
The mass of a body is a measure of its energy-content
Einstein didn’t say mass was a mystery. He said “the mass of a body is a measure of its energy-content”. That was in 1905, in his famous E=mc² paper does the inertia of a body depend upon its energy-content? The answer is yes. As it happens he used L to denote energy rather than E, but that doesn’t matter. What matters is this: “If a body gives off the energy L in the form of radiation, its mass diminishes by L/c²”. A cooling body radiates energy, and in doing so, loses mass. Then when you heat the body up, it gains mass. It’s all very straightforward when you understand why. Unfortunately on a variety of websites you can read that the mass of a body is not a measure of its energy content. For example on Space.com you can read that the mass of an electron is “the interaction between an electron and the Higgs field. And that’s all there is to it”:
Higgs simulation from What the Higgs is going on with mass? by Paul Sutter in Space.com, image credit Lucas Taylor/CMS
The idea comes from the electroweak sector of the standard model, where the weak force is said to be mediated by massive vector bosons. They have to be massive because the force is short range. See the Wikipedia Higgs mechanism article where you can read that “according to Goldstone’s theorem, these bosons should be massless”. If they aren’t, the bosons aren’t gauge bosons, and the theory isn’t a gauge theory. And since electromagnetism is, the result would not be the electroweak theory. So a fix was needed, and it was provided by a number of contributors. They “discovered that when a gauge theory is combined with an additional field that spontaneously breaks the symmetry group, the gauge bosons can consistently acquire a nonzero mass”. Interestingly enough Peter Higgs’s original paper was rejected by Physics Letters in 1964 as not having any relevance to particle physics. So Higgs added a sentence at the end saying it implied “the existence of one or more new, massive scalar bosons”, then he submitted it to Physical Review Letters. It was called Broken Symmetries and the Masses of Gauge Bosons. It’s only a page and a half. You can read all about it in Higgs’s 2017 article on the prehistory of the Higgs boson. The trouble with all that is this: Einstein didn’t say the mass of a body was due to some interaction with some space-filling field. He said the mass of a body is a measure of its energy content, not something else. So somebody who says mass is something else, is saying Einstein was wrong. They’re also saying E=mc² is wrong. It isn’t. There is clearly something amiss somewhere. Sheldon Glashow didn’t call it “Weinberg’s toilet” for nothing.
The toilet of the Standard Model edifice
See Matthew Chalmer’s Particle Headache in the 12 November 2012 issue of New Scientist. That’s where you can read Guido Altarelli saying this: “The minimal standard model Higgs is like a fairy tale”. You can read more in Gian Francesco Giudice’s 2009 book A Zeptospace Odyssey. On page 173 he says this: “The most inappropriate name ever given to the Higgs boson is ‘The God particle’. The name gives the impression that the Higgs boson is the central particle of the Standard Model, governing its structure. But this is very far from the truth”. He also says “the Higgs sector is rather arbitrary, and its form is not dictated by any deep fundamental principle. For this reason its structure looks frightfully ad hoc”. Giudice tells us the mass of a body is the intrinsic energy of a body at rest, and tells us about E=mc². No problem there. The problem comes when he says “the Higgs mechanism accounts for about 1 per cent of the mass of ordinary matter, and for only 0.2 per cent of the mass of the universe. This is not nearly enough to justify the claim of explaining the origin of mass”. He also says the Higgs boson is “like the toilet of the Standard Model edifice”. I wonder if somebody is talking in code here, and I’m reminded of a house that had been used as a squat – the toilet was full of sh*t and it stank to high heaven. By the by, Giudice wrote an essay in 2017 on The Dawn of the Post-Naturalness Era, which featured an “an imaginary conversation with Guido Altarelli”. He’s also head of theory at CERN.
Mass is ambiguous
So yes, there is clearly something amiss. Especially since, as Matt Strassler says, the Higgs boson doesn’t get its mass from the Higgs field. Being generous, I suppose the situation isn’t helped by the way the word “mass” is ambiguous. It’s qualified in different ways to mean different things. There’s active gravitational mass, passive gravitational mass, and inertial mass. Then there’s invariant mass, intrinsic mass, and proper mass. And then there’s relativistic mass, effective mass, and rest mass. But active gravitational mass is a measure of energy, not mass. As Einstein said in 1916, “the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy”. Passive gravitational mass is a measure of energy too. See the Wikipedia mass article, where you can read how “pure-energy” photons exhibit a behaviour similar to passive gravitational mass. Inertial mass is also a measure of energy. A photon is said to be massless, but it has a non-zero inertial mass. This is why as per the last line of Einstein’s E=mc² paper, radiation conveys inertia between the emitting and absorbing bodies:
Invariant mass is something you should forget about because the mass of a body varies with gravitational potential, which is why we have the mass deficit. Since intrinsic mass and proper mass are more of the same, you should forget about them too. Relativistic mass is another measure of energy – in Lev Okun’s 1989 Concepts of Mass you can read how Einstein cautioned against it. That leaves effective mass and rest mass. A photon exhibits an effective mass when you slow it down in glass. Let’s come back to effective mass later. For now, let’s focus on rest mass.
Unqualified mass is rest mass
When you see the word “mass” without qualification, what people usually mean is rest mass. That’s the mass of a body sitting there in front of you. It’s a measure of resistance to change-in-motion. A skateboard has a lot less mass than a car. It’s a whole lot easier to get a skateboard moving than it is to get a car moving. Then it’s a whole lot easier to stop the skateboard than it is to stop the car. Force equals mass times acceleration, and we know that the same force accelerates the skateboard aplenty but not the car. But Einstein said the mass of a body is a measure of its energy-content. So how does that work? Why is a body harder to push if it contains more energy? Like I said, it’s all very straightforward. That’s because of the wave nature of matter, and because most of the time, energy is kinetic energy, momentum is just another way of measuring it, and so is mass.
Kinetic energy and momentum are two aspects of energy-momentum
A 10kg cannonball is a fairly massive body. You can heft it in your hands and feel the mass of it. Let’s say this cannonball is in space, and so are you. You’re wearing a spacesuit, and the cannonball is coming towards you at 10 metres per second. You might say this cannonball has kinetic energy KE=½mv². You might also say the cannonball has momentum p=mv. Now, turn on your backpack thrusters, and catch that cannonball. Apply a constant braking force for as long as it takes to bring that cannonball to a halt:
Astronaut image from the Inverse NASA astronauts article by Danny Paez, cannonball top right added by me
Kinetic energy is looking at this in terms of stopping distance. Momentum is looking at it in terms of stopping time. We divide by c to convert from one to the other, because c is distance divided by time. You cannot remove that cannonball’s kinetic energy without removing its momentum, which is why kinetic energy and momentum are two aspects of the thing called energy-momentum. Like electricity and magnetism, they are two sides of the same coin. But if you’ve ever looked closely at a coin, you will notice that it doesn’t just have a flip side. It has an edge. That coin has a third side. So does energy-momentum, because for mass, you divide by c again.
The flip side of energy-momentum
You can perhaps begin to appreciate this if I tell you I made a mistake with the cannonball scenario. Let’s rewind that. You’re still in space wearing a spacesuit, but that cannonball isn’t moving towards you at 10 metres per second. Instead the cannonball is at rest, and you’re moving towards it at 10 metres per second. It doesn’t have any kinetic energy or momentum. You turn on your backpack thrusters and scoop up that cannonball. Apply a constant force for as long as it takes to bring that cannonball up to 10 metres per second. You have given that cannonball some energy-momentum. Kinetic energy is looking at this in terms of distance covered, momentum is looking at it in terms of acceleration time. And the moot point is this: when we say it was the cannonball moving, we say it has kinetic energy and momentum. When we say it’s you moving, we say that the cannonball doesn’t have kinetic energy or momentum. All it has is its resistance to change-in-motion. All it has is mass. This is why Einstein talked about moving in uniform parallel translation with respect to the system. That’s what led him to the conclusion that matter is made of energy. In one situation you talk of kinetic energy, or momentum, or energy-momentum. In the other situation you talk of mass. The distinction depends on who’s you say is moving. For example in Compton scattering where a photon interacts with a free electron, the electron receives a “bump” and is sent recoiling off at an angle. Meanwhile the photon is deflected and its wavelength is increased. But imagine you’re that free electron, and you think it’s you moving instead of the photon. Instead of delivering a bump, the photon would feel like it was a bump. It would feel like the photon was something that had inertia instead of momentum. It would feel like the photon had mass.
Photons are always moving
Of course when it comes to a photon in space, you would say it’s always moving at 299,792,458 m/s. This photon isn’t something you can slow down with your thrusters like the cannonball. Nor is it something you can speed up with your thrusters like the cannonball. It’s always moving at the speed of light. So it’s never at rest with respect to you. So rest mass doesn’t apply to a photon. That’s why the photon has no rest mass. Instead it has energy E=hc/λ and momentum p=h/λ, where h is Planck’s constant of action, c is the speed of light, and lambda λ is the photon wavelength. You cannot remove energy without removing momentum, so the photon has energy-momentum. And as per the Compton scattering example, the distinction between energy-momentum and inertia depends on who you say is moving. So how do you fix it so that the photon isn’t moving? You can’t accelerate to 299,792,458 m/s. You can’t chase a light beam and catch it up, as per Einstein’s thought experiment. That’s where Einstein said this: “if one were to pursue a light wave with the velocity of light, one would be confronted with a time-independent wave field. Such a thing doesn’t seem to exist, however”. However it does. Einstein didn’t know it in 1905, because the electron hadn’t been discovered for long. But you don’t pursue the light wave to be confronted with a time-independent wave field, you use another method. It sounds like a cheat, but it isn’t. Instead it’s simple. All you do is catch that photon in a mirror-box.
The photon in the mirror-box
There’s a handy little paper by Martin van der Mark and Gert (not the Nobel) ‘t Hooft called Light is Heavy. It talks about a balance scale, and explains why a box containing hot gas is heavier than a box containing cold gas. Imagine something like a gedanken spring steel box in space full of gedanken spring steel bullets ricocheting around inside. When you push the box, you have to push against the bullets bouncing off the side you’re pushing against. When I tap my magic wand and make those bullets move faster, you find it harder to get the box moving. So the box has more mass. Hence “although nothing at all is at rest inside the box, the gravitational mass is equal to the rest mass of the box as a whole”. Van der Mark and ‘t Hooft go on to replace the gas with light, whereupon the same principle applies. When you push the box, you have to push against the photons bouncing off the side you’re pushing against. When I tap my magic wand and increase the photon frequency, you find it harder to get the box moving. So the box has more mass. It’s heavier too, because gravity bends the light down so it exerts more force on the bottom of the box than the top:
Image from Light is Heavy by van der Mark and ‘t Hooft
The box is a body, and the mass of a body is a measure of its energy-content. So when you catch the massless photon in the mirror-box, it increases the mass of the system. The photon is still moving at the speed of light going around and around inside the box. But because its average location doesn’t change with respect to you, it’s effectively at rest with respect to you. You have effectively stopped the photon, and when you do so, its energy looks like inertia. And if that box is just the right size, the photon will look like a standing wave. Standing wave, standing field. It will look like a time-independent wave field. It will look like it’s standing still. Then when you open the box, what you’re holding in your hands is a radiating body that loses mass. The photon is out of the box in a flash. It departs at c from a standing start. But it wasn’t really a standing start. Because whilst the photon looked like a standing wave, it was always going around and around at c, all the time. Its path changed, that’s all. From a closed path to an open path. It all fits neatly with everything Einstein said about mass and energy. Apart from one little detail: there is no box. Because the electron is a photon in a box of its own making.
A photon in a box of its own making
The van der Mark and ‘t Hooft paper points out that elementary particles have non-zero spin, which “seems to imply that they all must have some sort of intrinsic dynamics”. You betcha. If the electron didn’t have any internal dynamics it wouldn’t have a magnetic moment. It wouldn’t go around in circles in a uniform magnetic field. So yes, “one could say matter is just ‘canned’ energy, a box with internal dynamics, and radiation is free energy”. That’s the size of it. That’s why the mass of a body is a measure of its energy content. The mass of an electron is not some measure of its interaction with distant objects, as per Mach’s principle. Nor is it some measure of its interaction with some kind of cosmic treacle. Because the electron isn’t a fundamental particle. It’s a 511 keV photon in a box of its own making. So is a positron, albeit with the opposite chirality. When you perform electron-positron annihilation it’s like opening one box with another, like each is the key to the other’s lock. Then each is a radiating body that loses mass. All of it. And then the boxes aren’t there any more:
Annihilation image from the CSIRO Australia Telescope National Facility
This is what E=mc² is all about, and it applies to an electron just as much as it applies to a cannonball or a car. I’ve bought the T-shirt, and it isn’t wrong. Einstein even referred to body and electron on the same line. And Einstein wasn’t even the first to think along such lines. JJ Thomson, who discovered the electron in 1897, called it a corpuscle. A corpuscle is a small body, and again: the mass of a body is a measure of its energy-content.
Are not gross bodies and light convertible into one another?
Take a look at the conceptual development of Einstein’s mass-energy relationship by Wong Chee Leong and Yap Kueh Chin. Note their introduction. In 1704, over three hundred years ago, Sir Isaac Newton knew that gross bodies and light were convertible into one another. That’s what pair production and annihilation do. We’ve known about pair production and annihilation since 1932, the thick end of a hundred years. They convert light into matter, and vice versa. Simply by changing the wave path from an open path to a closed path, and vice versa. We start with two high-energy photons, we perform gamma-gamma pair production such that a photon interacts with a photon, and the result is an electron and a positron. Because each photon ends up interacting with itself, and stays interacting with itself, wrapped and trapped as a spin ½ particle. The result is mass, and all we did was convert linear momentum into angular momentum. To do this we had to convert some other linear momentum into angular momentum, because of conservation of angular momentum, and because those photons are the only tool in the box. Not only that, but they are the box. The electron has mass because a photon interacts with itself, not with something else. When it does, we don’t call it momentum any more, we call it mass. We don’t call it a photon any more. We call it an electron. We don’t call it light any more. We call it matter. That’s why we have the strange theory of light and matter. How absurd it gets.
Matter is made of energy
As to why all this isn’t common knowledge I just don’t know. The wave nature of matter is common knowledge. So is electron diffraction. So is the Einstein-de Haas effect, and the Poynting vector. Why isn’t the flip-side symmetry between energy and mass common knowledge? Perhaps the underlying issue is a change of interpretation wherein people think mass is mysterious because they think of energy as some kind of abstract book-keeping thing. Or just the capacity to do work. Or just the property of a thing. Einstein didn’t think that. He thought of energy as thing in its own right. Energy is the one thing we can neither create nor destroy. The thing that’s really fundamental. In his E=mc² paper Einstein used an L for Lagrangian instead of an E for energy, but his meaning is clear. He said this: “If a body gives off the energy L in the form of radiation, its mass diminishes by L/c²”. Radiation is a form of energy. The photon isn’t some billiard-ball thing that has energy. The photon is energy. And matter is made of it. I think the best way to really grasp this is by taking a closer look at Compton scattering:
Image from Rod Nave’s hyperphysics
In Compton scattering some of the incident photon energy is converted into electron kinetic energy. The electron gets kicked aside and the photon is deflected. The photon changes direction and its wavelength increases because it loses energy. The electron is accelerated in the usual sense and the photon is decelerated in the vector sense. Do another Compton scatter on this self-same photon, and another and another ad infinitum, and in the limit there’s no photon left. You can take all the kinetic energy away from a fast-moving cannonball, and you’re still left with a cannonball. But it isn’t the same for a wave. When you take all the kinetic energy away from a wave in the surf, it just isn’t there any more. So the wave is energy. It’s the same for a photon. When you take all the kinetic energy away from a wave in space, it just isn’t there any more. So the photon is kinetic energy. Perform Compton scattering repeatedly with the self-same photon, and in the end it has been entirely converted into the kinetic energy of electrons. And yet, and yet: in pair production, you can make an electron and a positron out of a photon. So the electron is kinetic energy too. But now this kinetic energy is hidden. We call it mass-energy. Or potential energy. Or field energy. Only there’s angular momentum in this field, and a secret Poynting vector going around and around. Because of this, electrons have their dipole magnetic moment, and go round in circles, helixes, or spirals in a uniform magnetic field. Or drift in a non-uniform magnetic field. That’s why we have magnets. That’s why when you annihilate that electron with a positron you see 511 keV photons racing away at the speed of light. From a standing start. Because in atomic orbitals and everywhere else, electrons exist as standing waves. Standing wave, standing field. Standing waves that only look like they’re standing still.
Electron mass is resistance to change-in-motion for a wave going around and around
This is why the mass of a body is a measure of its energy content. If there is no energy content, there is no body. Photon energy or photon momentum is a measure of resistance to change-in-motion for a wave moving linearly at c. Electron mass is resistance to change-in-motion for a wave going around and around at c. That’s it. It’s as simple as that. That’s all there is to it. Matter is made of energy and that’s the only ingredient. The rest mass of an electron is a measure of how much energy is in there going around and around in circles at c, going nowhere fast. The electron’s energy is 511 keV and its rest mass is 511 keV/c², only it isn’t really at rest. The electron’s kinetic energy is a measure of the extra energy that makes it move. It tells you how fast this energy that’s going nowhere fast, is going somewhere. But it can never go as fast as light because it is light, going around and around. Accelerating an electron can be likened to stretching a helical spring. Getting the electron moving is like deforming an open circle of spring steel into one turn of a helix. The energy required increases as we attempt to deform it further. As we accelerate more and more, the helix is effectively stretched straighter and straighter. But a photon travels at c, it can’t travel in a straight line at c and still be going around in circles at c. It would have to go faster than light to do that. And light doesn’t go faster than light. So matter can’t travel as fast as light. Matter can’t go faster than the light from which it’s made.
Mass can vary
Not in vacuo, anyway. Cherenkov radiation is akin to a sonic boom, because the speed of electrons can exceed the phase velocity of light in water. Light is slowed down in a material such as water. When you slow down a photon it exhibits an effective mass. Some of its energy-momentum is effective as mass. When you slow it down further, more of its energy-momentum is effective as mass. When you slow it down to an effective speed of zero by trapping it in a mirror-box, all of its energy-momentum is effective as mass. Hence as Henri Poincaré said, “what we call mass would seem to be nothing but an appearance”. The lesson to learn is that there’s no substantial difference between effective mass and mass, there’s just a sliding scale between the two. So, if the speed of a photon varied a little as it propagated through free space, its mass would vary a little. Of course, we’re confident that this doesn’t happen for photons, but oranges are not the only fruit, and quark is not the only cheese. Talking of which, there is such a thing as the mystery of quark mass. A proton consisting of two up quarks and a down quark has a mass of 938MeV. A neutron consisting of an up quark and two down quarks has a mass of 940MeV. A π+ pion consisting of an up quark and a down antiquark has a mass of 140MeV. A ρ+ rho meson consisting of an up quark and a down antiquark has a mass of 770MeV. Something doesn’t add up here:
Image from Rod Nave’s hyperphysics
It sounds as if there’s something wrong here. I am reminded of quantum electrodynamics, the first quantum field theory. It suffered from the “problem of infinities”, so much so that some authors say most workers in the field doubted its correctness, whilst others say physicists were so overwhelmed by the problems that they believed a conceptual change was needed. It didn’t happen.
A problem in the foundations of particle physics
In their 1993 essay on the conceptual foundations and the philosophical aspects of renormalization theory, Tian Yu Cao and Silvan Schweber say “serious doubt has often been cast on the whole program”. They describe QED as a conceptually unstable theory, and say a stubborn historian might reject it. They also say renormalization “took the framework of QFT as given, and made no attempts to alter its foundations”. They go on to say that “what is hidden in the locality assumption is an acknowledgment of our ignorance of the structure of the electron”. Could a similar issue apply elsewhere? Yes. See asymptotic freedom and QCD – a historical perspective by David Gross dating from 2004. He says “by the 1950s the suspicion of field theory had deepened to the point that a powerful dogma emerged – that field theory was fundamentally wrong, especially in its application to the strong interactions”. And that “the feeling of most was that renormalization was a trick”, one that “simply swept the infinities under the rug”. He also said “quarks had not been seen, even when energies were achieved that were ten times the threshold for their production”. And that “the non-relativistic quark model simply did not make sense. The conclusion was that quarks were fictitious, mathematical devices”. That sounds interesting. Let’s investigate this. Let’s take a look at what the proton is, and what it is not.