Of Particular Significance

The Two Definitions of “Mass”, And Why I Use Only One

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 massat all, much less giving it a name.

The two interpretations of  what "mass" means.  The first (in green) is that mass is something that does not change with speed -- often called "invariant mass" or "rest mass", it is used by particle physicists.  The other, "relativistic mass", is just energy divided by c-squared. Note the two are almost identical at small velocities and so are essentially equal in daily life.
The two interpretations of what “mass” means. The first (in green) is that mass is something that does not change with speed — often called “invariant mass” or “rest mass”, it is used by particle physicists. The other, “relativistic mass”, is just energy divided by c-squared, and grows with speed. Note the two are almost identical at small velocities, and so are usually equal in daily life.

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 “reddishblue”. If I insisted on describing a raisin as “reddishblue”, 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.

38 Responses

  1. There appears to be a typo in the document where it says:

    In general, the relation between E, p, m and v is given by the two equations
    v = pc/E
    E² – (pc)² = (mc²)²

    The first should say
    v = pc²/E

  2. Not sure why relativistic mass is an unhelpful concept. First, it gives simple prescriptions for doing relativistic kinematics: use E=Mc^2 and P=MV, where M is the relativistic mass. Second, even conceptually, mass is a measure of inertia, i.e .how easy/hard it is to change the motion of an object. And in that respect, it is much harder to accelerate something moving close to the speed of light. Sure, mass is no longer invariant, but so what? This is relativity.

    1. The simple answer: mass should be a measure not of E and P but of response to a force F — after all, it is supposed to be a concept of inertia. The problem is that F = M a is not true relativistically; instead, the relation depends on the angles between F, v and a. This led people to start defining “longitudinal mass” and “transverse mass”, which is fine… but the point is that things are not as simple as they seem when you just look at E and p. See the fourth section of https://en.wikipedia.org/wiki/Acceleration_(special_relativity).

  3. All in all, it’s interesting. Really, the mass is a relative concept. That is, it is always manifested only in relation to another mass. After all, it is a property of a particle (if we can call a wave a particle at all) at low speeds to come into interaction with other particles or with the vacuum-aether, or it is a space distortion, from measuring the force of this interaction we can determine the mass. Therefore, what is mass and what the units to measure it – the question remains open I think until the nature of this property will be disclosed.

  4. Thanks very much for this. In his book Quantum Space page 82, Baggott says “In fact, elaborate QCD calculations demonstrate that about 95 percent of the mass of a proton or neutron is derived from the energy of massless gluons that carry the colour force between the quarks…. We see that Einstein’s great insight was indeed that m=E/c2 the mass of a body is a measure of its energy content, and we conclude that mass is not something that an object has. Instead, the mass of an object is something that it does.” I had terrible understanding what was meant by “something it does” until I read your article. As one of your amateurs, I really did not understand invariant mass.

  5. Absolute nonsense- energy is just a different form of matter and, as such, is attracted to matter and contributes to the total gravitational mass of the universe (ie, dark matter), but because of its immense velocities, also comprises “dark” energy like a compressed gas in a box.

    1. My response was to Pro Strassler- not to the other thoughtful responders- James Richard Spears MD

  6. More chaff than wheat on this thread and occasional name-calling (‘squirley’). Objectors to relativistic mass should read: http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html and respond to the following passage which cites length contraction near the speed of light: ” There is no argument in the literature about the uses of rest length versus moving length, so why should there be any argument about the uses of rest mass versus moving mass?

  7. Very useful article.Actually it help those who interested and enthusiastic in Particle Physics…….

  8. Mass can be defined only interms of universl energy concept , I.e unknown concept, presence of dark energy or unkown energy arround us & universal mass

  9. In reality in my book you see that total mass is invariant by force and variant by photon adding and then in photon adding process we have m=m_0 \gamma but by force, the relativistic total mass is invariant and static mass is transfered to kinetic energy as m_0=m/{\gamma}.
    Then fields are not energy carrier and force is transformer.

  10. I have to say that I disagree with this denigration of ‘relativistic mass.’ Two arguments against the usage are cited.

    Argument 1: Two uses of ‘mass’ leads to confusion.

    I don’t find invariant mass versus relativistic mass confusing at all; it is just harder to type out. And this is hardly the only double meaning in the lexicon of physics. ‘Mass’ can mean a quantity (“5 kg”) or it can mean an object that extends in space (“that mass is very hot, don’t touch it!”). [Note: when we use ‘energy’ we mean a quantity; when we want to designate energy as something quantized and occurring in time (a photon) we use ‘action’ (or ‘quanta’). Nothing similar is available for ‘mass;’ it means both quantity and object-extended-in-space.]

    Argument 2: Relativistic mass is an unnecessary concept:

    You are conflating convertibility with identity. Energy and mass may be quantitatively convertible, but they remain different entities with different forms (the photon is pure energy and it is an entity). There is also a deep and profound parallel between mass and energy that all great physicists have recognized (Einstein, de Broglie, Bell). One such parallel is storage: mass can store energy and energy can store mass. That which stores is kinetic, that which is stored is potential. Kinetic (rest) mass stores potential energy; kinetic energy stores potential (relativistic) mass. If you deny the reality of relativistic mass then you have to say that kinetic energy stores itself (or doesn’t store anything) which doesn’t make much sense and breaks the parallelism that E = mc2 implies.

    The preferred and precise terms for physics are as follows:
    Kinetic mass (rest mass)
    Potential mass (relativistic mass)
    Kinetic energy
    Potential energy

    Why would nature give us two energies but only one mass? Of course if your field is particle physics then your focus tends toward energy (kinetic interactions) and quantitative measures and you can do without potential (relativistic) mass. It is the visceral dislike of relativistic mass of some physicists that puzzles me.
    Mass or energy that is stored provides the element of probable release to physics, hence probability. This must be even-handed between mass and energy. There is probable release of potential (relativistic) mass as in absorption; and there is probably release of potential energy as in emission.
    Further support for potential (relativistic) mass would require a complete essay.

    1. Your understanding of relativity theory is squirley! Read Lev Okun. It makes no sense at all to talk of the relativistic mass of the photon.

      Professional particle physicists know better!

    2. And there is NO such thing as “pure energy” as Alan Guth (and many other scientists) has pointed out.

  11. As I understand it: Neutrinos have ONLY left hand chirality, It is therefore impossible to overtake them and see right hand chirality looking back at them. Therefore the upper bound of their speed must be c . This implies the lower bound of their mass is zero. However neutrino oscillation implies a non zero upper bound of their mass.
    Does this mass variability imply that neutrinos have only virtual mass as photons do: Or else acquire their mass by interaction with a vacuum uncertainty/virtual particle (maybe a virtual instance of its own anti-particle) causing neutrino to acquire Heisenberg mass uncertainty and thus Heisenberg-like speed uncertainty.

  12. First, Matt, let me say that this blog is the best place I’ve come across for clear explanations of particle physics. However, there’s one particular issue that has puzzled me for many years, and, judging by many of the questions you are asked, has vexed a lot of other non-experts too.

    Photons are massless particles, but this seems incompatible with the key equations they are meant to obey. If m=0, and E=mc^2, zero time c squared is still zero, so we have the ridiculous solution E=0. Likewise, if we take the special case equation for massless particles, E=pc , there is the same problem, since we learn that the equation for momentum is p=mv, therefore for light-speed photons p=mc. Substituting m=0, again we have E=0. And that momentum equation also gives a silly result, p=0.

    Please confirm that all these absurdities disappear if you treat the “mass” here as relativistic and follow your advice to substitute E/c^2 for this type of ‘m’: E=mc^2 becomes E= Ec^2/c^2, cancelling out gives the obviously correct E=E. E=pc becomes E= E x c x c /c^2, and again it cancels out to E=E. And momentum p=mc turns out to be p= Ec/c^2 = E/c. By the way, does that mean that the momentum of a photon is simply the ratio of its energy to its speed?

    If this is all correct, I realise that for experts like yourself it is all implicit in the definition m=E/c^2, but for dummies like myself it has to be hammered out like this until it feels right, and I hope I have done that here!

  13. There is a rest mass to the stationary reference.
    “Presence or absence of the internal structure of the particles” and “mass to increase with the addition of motion mass to rest mass” is another problem.
    So, the claim that there is no mass to the photon is a self-convenience of particle physics.

  14. fine, there are two definitions of mass.

    “relativistic mass” is E (multiplied by a constant).

    but what is the definition of “invariant mass” ?

    of course, there is another questions which begs to be asked (what is the definition of E), but for now, i would like some discussion on what is invariant mass.

  15. Prof. Strassler,
    Thank you for your clear explanation regarding mass. Indeed, we should abandon the concept of relativistic mass to simplify the picture.
    What I still do not understand, and cannot find in other texts you present or in discussions about them, is the following: Given the following: 1) an elementary particle obtains its mass from its interaction with the Higgs field; 2) momentum and energy are constants when moving an object along the space resp. time axes. 3) lengt and time as measured from a moving frame become transformed, qualitatively described by Lorentz’ transformations and quanitatively by the value of this constant, c. 4) Due to this space/time relation, also these properties ‘energy’ and ‘momentum’ are also related into a fourvector en the equation of E^2 = m^2p^2 + m^2c^4 holds. 5) Though we should not express kinetic energy in term of ‘relativistic mass’, it can be expressed as (gamma-1) times invariant mass, so the energy depends on invariant mass. Now, what puzzles me most is why momentum and energy require a relation with this property ‘invariant mass’. When we observe an electron from a moving frame, the electric charge for instance is not involved in the energy equation, while the mass is. (happily, there is no ‘relativistic charge’ confusion). What is so special about the way particles interact with the Higgs field that this effects the properties we call energy and momentum? Why are there no other conserved quantities such as energy and momentum that do relate to the interaction of a particle with some other field, such as the one that gives it electric charge of colour? Could you please explain this? Or am I asking ‘a wrong question’ and if so – why?

  16. I agree that relativistic mass is a useful concept of mass. Invariant mass is a mass at rest, but it contains a lot of small moving masses. This means the energy of invariant mass of a certain object consists of a lot of relativistic masses which have different velocity.

  17. Hmm… this explanation is very helpful.

    Follow-up question: I have read a few different places that heating up an object increases its mass (ever so slightly, of course). Was this based on the definition of relativistic mass? If so, I understand that this is just another way of saying that the object gained energy.

    And would this also apply to the mass defect associated with nuclear explosions? It is said that mass is lost in a nuclear explosion, but from what I can tell, what is meant is that *energy* diffuses away. So again, is this a statement based on a relativistic definition of mass?

    Thank you!

    1. Prof. Strassler mostly only discussed the implications for single particles like electrons and photons, but the concept of invariant mass can be applied to any system viewed at rest relative to its center of momentum frame. The atoms in an object gain momentum individually as you heat it, yet the sum of all those momentum vectors is still zero. When you weigh an object on a scale, the scale is necessarily not moving relative to the object’s center of momentum and you’re measuring the intrinsic mass, and yes it would increase slightly as you heat it.

      This is also what lets us talk about the invariant mass of protons, neutrons, and other composite particles too.

  18. Prof. Strassler,

    While I am struggling to evict the idea of “relativistic mass” from my mind, I am thinking about a non-relativistic example: Heat up a cathode, at 0 volts. Let some of the freed electrons accelerate toward a plate at +100 volts, where some of them pass through a hole in the plate.

    I calculate the speed of the electrons using KE = (1/2) mv^2
    100eV = (1/2) (mass of electron) * v^2
    v =~ 5930 km/s (about 0.2 c, much faster than I expected, )

    In that example, is it correct to say I am obtaining some 100eV electrons,
    or do I add the mass of the electron and say they are (510 + 0.1) KeV electrons?

  19. Professor Strassler and some other contributors last year had a very interesting and significant exchange on the subject of mass and energy with me. I think the conversations deserve to be known to visitors of this page. Unfortunately, the conversations occurred on a page that wasn’t on the subject of mass and energy. One of my points that I never got a satisfactory reply to was my point that if it moves (other than a shadow) it is matter and has mass. Yet current physics apparently doesn’t support such an self-evident notion when it come to photons (and energy). I could not get a satisfactory reply to this point from anyone even though I offered an explanation for observed phenomenon regarding light that is consistent with my point that anything that moves and interacts with matter is also matter.

    We also carried this conversation to other pages on Matt’s site and for those who might be interested, the conversations can be found by googling “Vincent Sauvé of Particular Significance”. Below is the first conversation:

    profmattstrassler.com/ 2012/08/02/at-a-workshop-and-higgs-papers-are-out/

    P.s. I agree that mass does not vary with velocity.


    1. Vincent — I’m sorry, but I have to ask you to please not use my site for blatant self-promotion. There’s a limit, and you’re pushing it. You ought to recognize this is not a fair fight; you seem to have plenty of time to argue with me, but you should realize how little time I have to discuss things with any one reader (you got more than your share), and especially a smart one like yourself who requires a more careful reply than I usually have time for.

    2. I know how about instead of trying to promote your views in a comment section, how about doing some real science with coroberating math to prove your theories.

  20. Thanks for this article, I recently finished watching your (video) lecture on the Higgs and this was one of the topics I wished you had taken some time to explain. Particularly, on that lecture, the fact that mass doesn’t simply “add up” (the mass of a proton is higher than the sum of it’s quarks) didn’t come out as something notable or counterintuitive (and I think members of your audience might have been interested in this surprising weirdness of the universe).

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