One of the challenges for a person trying to explain physics to the non-expert — and for non-experts themselves — is that scientific language and concepts are often frustratingly confusing. Often two words are used for the same thing, sometimes words are used that are fundamentally misleading, and often a single word is used for two very different but related concepts. You’d think we’d clear this stuff up, but no one has organized a committee dedicated to streamlining and refining our terminology.
A deeply unfortunate case, the subject of today’s post, is the word “mass”. Mass was confusing before Einstein, and then Einstein came along and (accidentally) left the word mass with two different definitions… both of which you’ll see in first-year university textbooks. (Indeed, this confusion even extended to physicists more broadly, causing the famous particle physicist Lev Okun to make this issue into a cause celebre…) And it all has to do with how you interpret E = mc² — the only equation everybody knows — which relates the energy stored in an object to the mass of the object times the square of the universal speed limit c, also known as “the speed of light”.
Here are the two possible interpretations of this equation. Modern particle physicists (including me) only use the first interpretation. The purpose of this post is to alert you to this fact, and to point you to an article where I explain more carefully why we do it this way.
Interpretation 1. E = mc² is true only for an object that isn’t moving. For an object that is moving, E is greater than mc². Energy and mass are not at all the same thing; an object’s energy can change when its motion changes, but its mass never changes. This notion of mass is sometimes called “rest mass” (since it’s related to the energy stored in the object when it is “at rest”) or “invariant mass” (since it doesn’t change when it is moving.)
Interpretation 2. E = mc² is always true, for both stationary and moving objects. This can be viewed as saying energy and mass are essentially the same thing. [Recall that in interpretation 1, they are not at all the same thing.] Since the energy of a moving object is larger than when it is stationary, that means, similarly, that its mass is larger when it is moving than when it is stationary. This notion of mass is sometimes called “relativistic mass”, in honor of Einstein’s revolutionary notions of relativity.
To sum up — relativistic mass depends on how fast an object is moving, but invariant mass/rest mass is the same whether an object is moving or not; you can see this in the figure below. Which one of these should we call “mass”, with no modifier? Unfortunately, that’s up to the user.
Fortunately, in daily life, these two concepts are almost identical, because most objects we observe in daily life much more slowly than c, in which case their rest mass and relativistic mass are nearly identical, as you can see in the figure. But particle physicists and nuclear physicists and astronomers, among others, often have to be more precise. And when you’re reading an article or book about particles or nuclei or astronomy in which “mass” plays an important role, you will often need to know which of these two interpretations is being used by the author!
Einstein, in his early years, contributed to the second interpretation, perhaps inadvertently. But later he made clear statements (most notably in a letter to Lincoln Barnett, which I can’t find in full on the web, but which is quoted widely) in favor of the first interpretation. Not that Einstein’s opinion particularly matters; in science we respect our elders, but we do not slavishly follow them, the way people used to follow Aristotle. We come to a conclusion based on what we know, and often we know things that weren’t known to the previous generations. So why do particle physicists today choose interpretation 1? I’ll give you a couple of quick hints as to why, and if you want to learn more, you can read my article on the matter.
- If you use the first definition,
- If you use the second definition,
- no photons have zero mass (because all photons have energy)
- every hydrogen atom has a different mass, depending on how fast it is moving; and
- any particular electron may have a smaller or larger mass than any given hydrogen atom, depending on how fast each of them is moving. [For instance, an electron emitted by a decaying Higgs particle has a larger mass than the hydrogen atoms in your body (at least from your point of view).]
So with the second interpretation, you can’t even say which types of particles have larger masses than other types, and it is impossible for any type of particle to be massless. This is very inconvenient — one might even say, ridiculous — for doing particle physics.
Additional subtle but profound mathematical reasons [having to do with the ``hyperbolic geometry of space-time'', if you must know] support the first interpretation, as hinted at in this article on mass and energy, which shows how energy, momentum and mass are related by elegant equations if you use the first interpretation.
Anyway, as long as you are aware of the existence of these two different interpretations, you will usually be able to tell which one is in use by an author. On this website, the first interpretation is always used. If you would like to learn more about why particle physicists choose the first interpretation, click here for my more detailed article.