Matt Strassler [July 10, 2013]
It’s most unfortunate that in the process of revolutionizing our notions of space, time, energy, momentum and mass, among other things, Einstein left a legacy that included two different and contradictory definitions of the term `mass’, leading to two very different interpretations of what he said and meant — conceptually speaking. Don’t worry: there’s no confusion about the physics itself. Trained physicists know exactly what is being spoken of, and know how to make predictions and use the relevant equations. It’s purely a question of what the words mean. But words are important, especially when talking about physicists to non-experts, and even to young students, for whom the equations are still opaque.
Throughout this website, what I mean by `mass’ is a property of an object is sometimes called its “invariant mass” or “rest mass”. For me and my particle physics colleagues, it is just plain-old “mass”. The terms “invariant mass” or “rest mass”, used to clarify what you mean by the word “mass”, are necessary only if you insist on having a second, different quantity that you might call “mass”, which is more generally called “relativistic mass”. Particle physicists avoid any possible confusion by never using the concept of “relativistic mass” at all, much less giving it a name.
The nice thing about using mass instead of mass is that mass is a property that all observers agree on, whereas mass is not. There aren’t that many properties of an object that are like this. Take the speed of an object; different observers won’t agree on what it is. For instance, there goes a car; how fast is it moving? Well, from your point of view, standing by the road, perhaps it is moving at 80 kilometers per hour. But from the point of view of the car’s driver, the car isn’t moving at all; you’re moving. And from the point of view of someone in a car going the other direction, the first car may be moving at, say, 150 kilometers per hour. The bottom line? Speed is a relative quantity; you cannot ask, what is the speed of the car, because it has no answer. Instead you must ask, what is the object’s speed relative to a particular observer. Every observer has an equal right to make the measurement, but different observers will get different answers. Galileo’s principle of relativity (long before Einstein’s principle of relativity, which involved adjusting Galileo’s principle) already incorporated this idea.
This dependence on the observer also applies to energy, and to momentum. And it applies to relativistic mass. That’s because relativistic mass is simply the same as energy, divided by a constant — namely, c² — and so, if you define mass to be “relativistic mass“, then different observers disagree about an object’s mass m, though all agree that E=mc².
But rest mass, or, as I would call it, “mass“, does not depend on the observer, which is why it is also called invariant mass. All observers agree on an object’s mass m, with this definition. And all observers agree that if you were stationary with respect to the object, you would measure its energy to be mc², and otherwise you would measure its energy to be larger. To sum up: with this definition of mass, which I use throughout this website,
- If an object, relative to an observer, has speed v=0, then the observer will measure it to have E = mc² and zero momentum (p = 0.)
- If instead the object is moving relative to the observer, then the observer will measure it to have E > mc² and non-zero momentum (p > 0.)
- In general, the relation between E, p, m and v is given by the two equations
- v = pc/E
- E² – (pc)² = (mc²)²
which agrees with the previous two statements, because if p=0, then v=0 and E² = (mc²)² [and thus, since E and m are both positive, E = mc²], while if p>0, then v>0 and (since pc > 0) E has to be bigger than mc² for the second equation to be true.
The above equations, and a simple geometric interpretation of their form, are described in more detail in this article.
What I want to do here is give you a sense of the reasons why particle physicists use these equations, and do not view the equation E=mc² as always true, instead relegating E=mc² to a statement which is true only for an observer who is not moving with respect to the object. And I’ll do this by asking a few questions whose answers, depending on which “mass” you choose, are quite different, in order to focus attention on the big problems with having two conflicting definitions of the word `mass’, and to suggest why in particle physics the definition of mass as something that is observer-independent is much, much easier to work with.
Is a photon (a particle of light) massless? or not?
If you use “mass” as I do, the answer is YES. A photon is massless, and that’s why its speed is always at the universal speed limit c. Meanwhile an electron is not massless, and that’s why its speed is always below c. All electrons have a mass of 0.000511 GeV/c2.
But if by “mass” you mean relativistic mass, the answer is NO. A photon ALWAYS has energy, so it always has a mass; no observer will ever see it as massless. The only thing that is zero is its invariant mass, also known as its rest mass. Each electron has its own mass; each photon has its own mass. An electron and a photon with the same energy have (in this definition) the same mass. In fact some photons have larger mass than some electrons, and other electrons have larger mass than other photons. Worse still, one observer might see a certain electron as having a larger mass than a certain photon, but a different observer may see it the other way round! So relativistic mass is a relative mess.
Or how about this: does an electron have a larger mass than an atomic nucleus?
If you use “mass” as I do, the answer is NO, NEVER. All observers agree that an electron has a mass about 1800 times smaller than the mass of a proton or a neutron, which are the things that atomic nuclei are made from.
But if by “mass” you mean relativistic mass, the answer is IT DEPENDS. An electron at rest has a lower mass. A super-fast electron can have a larger mass. You can even get an electron’s energy set just right so that its mass is identical to that of a given nucleus. The only thing that we can say in general is that an electron’s rest mass is smaller than the rest mass of a nucleus.
Do neutrinos have mass or not?
If you use “mass” as I do, you would say: the answer was not known from the 1930’s, when the first neutrino was proposed, until the 1990s. But today we know (with almost complete certainty) that neutrinos DO have mass, YES.
But if by “mass” you mean relativistic mass, the answer is OBVIOUSLY, as we always knew, from the day the possibility of neutrinos was proposed. All neutrinos have energy, so, like photons, all neutrinos have mass. The only question was whether they have invariant mass — a question recently answered in the affirmative.
Do all particles of a given type — say, all photons, or all electrons, or all protons, or all muons — have exactly the same mass?
If you use “mass” as I do, the answer is YES [up to subtleties, from the uncertainty principle, for particles that decay rapidly — but that’s a sidelight, so let’s not get distracted.] All particles of a given type share the same mass.
But if by “mass” you mean relativistic mass, the answer is OBVIOUSLY NOT. Two electrons moving at different speeds have two different masses. They just have the same invariant mass.
Is Newton’s old formula F = ma, relating force, mass and acceleration, true?
If you use “mass” as I do, the answer is NO. The formula is modified in Einstein’s version of relativity.
But if by “mass” you mean relativistic mass, the answer is IT DEPENDS. If the force and the particle’s motion are perpendicular to one another, then yes; but otherwise, no, it is modified.
Does a particle’s mass increase when a particle’s speed and energy increase?
If you use “mass” as I do, the answer is NO (see the figure above). Different observers, viewing the same particle, may assign to it different energies; they will all agree on its mass.
But if by “mass” you mean relativistic mass, the answer is YES. Different observers, viewing the same particle, may assign to it different energies, and therefore different masses; they will only agree it has the same invariant mass.
So what you see here is that, at minimum, we have a very, very bad linguistic problem. If we do not make precisely clear which definition of mass we are using in a sentence, we get completely different answers to very basic physical questions. Unfortunately, most books for lay-persons and even first-year university textbooks (!) switch back and forth without being rigorous about what they mean. And the most common confusions I hear from my readers have to do with their having been told two contradictory things about mass — one of which is true of rest/invariant mass, the other of which is true of relativistic mass. It is very bad to have one word for two very, very different things.
Of course, it’s just language. We can do whatever we want with language. It’s definitions; it’s semantics. It doesn’t matter. For a physicist armed with equations, language is an imperfect medium anyway. The math never gets confused, and the person who understands the math isn’t going to get confused either.
But for the public, and for the beginning student, this is kind of a disaster.
What should we do about this? One option, of course, is to insist on using all of these possible terms. The price is that you end up having to say things in complicated ways.
- A stationary object has energy = invariant mass times c2 = relativistic mass times c2.
- For a moving object, mass = invariant mass as before, but energy = relativistic mass times c2 is larger than it was before, due to the energy associated with its motion.
This seems unnecessarily wordy. I (and my colleagues) simply say:
- for a stationary object with mass m, energy E equals m c2, and
- if the object is moving, its mass is still m and its energy E is greater than m c2, with the excess due to motion-energy.
This way of saying it has no less content, has fewer concepts and definitions to keep track of, and avoids having two contradictory meanings of `mass’, one of which is unchanged by motion and the other of which does change with motion.
A strong linguistic, semantic and conceptual reason to avoid “relativistic mass” altogether, and to drop the “invariant” or “rest” from “invariant mass’‘ and “rest mass”, is that “relativistic mass” is a completely unnecessary concept. It’s just a particle’s energy, renamed. Using the notion of “relativistic mass” is a little bit like insisting on the term “reddish–blue”. If I insisted on describing a raisin as “reddish–blue”, you’d complain. You’d say: “but we already have a word for the color of raisins: `purple.’ What’s wrong with that word?” And you’d say: “calling the color of raisins a form of blue is very confusing and misleading; it implies that the color of raisins is something like the color of the sky, whereas in fact they’re very different.” In a vaguely similar way: relativistic mass times c2 is just another word for energy (for which we already have a perfectly good word) and describing energy as being the same as mass is actually misleading and leads to great confusion.
Here’s another reason why calling energy a form of mass is deeply problematic. Just as time and space are linked together in Einstein’s equations, energy and momentum are linked together. (You may even recall that the reason energy is conserved has to do with the time-independence of physical laws, and the reason momentum is conserved has to do with the location-independence of physical law. [I’m ignoring subtleties that arise in Einstein’s theory of gravity, when gravitational effects are very large.]) So if we say that mass is E/c², what is p/c? It ought to be something. What is it?! Well, no one’s ever given it a name. Why? Because “momentum” was perfectly good for p, and p/c didn’t need its own name. So why isn’t “energy” perfectly good for E? Why do we need a new name for E/c²? Especially since there’s another quantity that appears in the equation for E and p that we saw earlier,
- E² – (pc)² = [ (mc²)² ] .
The quantity on the right hand side of the equals sign definitely does need a new name, because it is clearly neither E nor p in a new guise — it is not conserved (as E and p are) but it is observer-independent (which E and p are not!) [This point is explained more fully here.]
The notion of “relativistic mass” did not come from nowhere, or arise from something stupid. It came from Einstein himself, and for good reasons, having to do with the relation between the energy of a system of objects and the mass of that system. I have partially explain what motivated this way of thinking here and here. But although the notion of relativistic mass was further promulgated and popularized by other famous physicists of his day, Einstein apparently abandoned this way of thinking, also for good reasons. And so has the community of modern particle physicists.
On this website, as in my research, I NEVER use relativistic mass. Instead I just use energy in its place, since for a particle on its own, relativistic mass is nothing but energy divided by c2. And I ALWAYS mean, by “mass”, that quantity “invariant mass” (or “rest mass”) on which all observers agree. The electron mass is always 0.000511 GeV/c², no matter whether or how fast it is moving; every electron always has smaller mass than an every atomic nucleus; every photon in empty space is always massless. And Higgs particles have a mass of about 125 GeV/c², no matter how fast they are moving. This is the linguistic and conceptual convention that particle physicists use. It isn’t necessary; you could make a different choice. But this approach avoids a lot of practical and conceptual problems, as I’ve illustrated here.