Two definitions of mass, and why I use only one of them

Unfortunately, in the process of revolution in science, which occurred with the concepts of space, time, energy, impulse and mass, Einstein, among other things, left behind two different and contradictory definitions of mass. Because of this, everything that we say and have in mind can be interpreted in two very different ways. At the same time, there is no confusion directly in physics. Experts know exactly what they are talking about, and they know how to make predictions and use the appropriate equations. The whole question is only in the meaning of the word itself. But words are important, especially when we talk about physics with people who are not experts in this field, and with students for whom the equations are not yet fully understood.

In my articles, under the "mass" I mean the property of an object, which is sometimes also called the "invariant mass" or "rest mass." For us with my colleagues in particle physics, this is just a good old “mass”. The terms "invariant mass" or "rest mass" are used to clarify what you mean by "mass" only if you insist on introducing a second quantity, which you also want to call "mass", and which is usually called " relativistic mass. Specialists in particle physics avoid this confusion by not using the concept of “relativistic mass” at all.


The rest mass is better than relativistic in that the first mass is a property about which all observers agree. Objects do not have many similar properties. Take the speed of the object: different observers will not agree on the speed. Here comes the car - how fast is it? From your point of view, if you are on the road, let's say it travels at a speed of 80 km / h. From the point of view of the driver of the car, it does not move, but you move. From the point of view of the person traveling towards the car, it can move at a speed of 150 km / h. It turns out that speed is a relative value. It makes no sense to ask about the speed of the car, because you can not get an answer. You should ask what is the speed of an object relative to a particular observer. Each observer has the right to make this measurement, but different observers will get different results. The principle of relativity of Galileo already included this idea.

The dependence on the observer is applicable to both energy and momentum. It applies to relativistic mass. This is because the relativistic mass is equal to the energy divided by a constant — namely, c 2 — so if you define the mass as “relativistic”, then different observers will disagree about the mass of the object m, although everyone will agree that E = mc 2

But the rest mass, which I call simply “mass,” does not depend on the observer, so it is sometimes called the invariant mass. All observers agree on the mass of the object m, defined in this way. And all observers will agree that if you are resting relative to an object, its energy measured by you will be equal to mc 2 , and otherwise the energy will differ in a big direction. Total: with the definition of mass, used by me in articles,

• If the velocity of the object relative to the observer is v = 0, then the observer will measure that the object has E = mc 2 and the pulse is p = 0.
• If instead the object moves relative to the observer, then it will measure that E> mc 2 , and the momentum is also greater than zero (p> 0).
• In the general case, the relationships between E, p, m and v are given by two equations:
ov = pc / E
• which is consistent with the two previous statements, because if p = 0, then v = 0 and (therefore, E = mc 2 ), and if p> 0, then v> 0 and (since pc> 0) E must be greater than mc 2

These equations and their graphical representation are discussed in detail in another article .

I would like to let you understand the reasons why the specialists in particle physics use these equations and do not consider that the equation E = mc 2 is always satisfied. This equation refers to the case in which the observer does not move in relation to the object. I will try to do this by asking a few questions, the answers to which vary greatly depending on the choice of the meaning of the word "mass". This will help draw your attention to the big problems in the case of the existence of two competing definitions of mass and explain why it is much easier in particle physics to work with a mass that does not depend on the observer.

Does a particle of light, photon, mass or not?

If you use my definition of mass, then no. A photon is a massless particle, therefore its speed is always equal to the universal speed limit c. But the electron has mass, so its speed is always less than. The mass of all electrons is 0.000511 GeV / c 2 .

But if you mean relativistic mass, then yes, it does. A photon always has energy, so it always has a mass. No observer sees him massless. It has only zero invariant mass, also known as rest mass. Each electron will have its own mass, and each photon will have its own. An electron and a photon possessing the same energy will, by this definition, have the same mass. Some photons will have more mass than some electrons, while other electrons will have more mass than other photons. Worse yet, for one observer, the mass of a certain electron will be greater than the mass of a certain photon, and for another everything can be the other way around! Therefore, the relativistic mass leads to confusion.

Is the electron mass greater than the mass of the atomic nucleus?

If you use my definition of mass, then no, never. All observers would agree that the electron mass is 1,800 times smaller than the mass of the proton or neutron that makes up the nucleus.

But if by mass we mean relativistic, then the answer will be: it depends on the situation. The electron mass at rest is less. A very fast electron has more. You can even arrange everything in such a way that the electron mass will exactly match the mass of the selected nucleus. In general, one can only say that the electron rest mass is less than the rest mass of the nucleus.

Does neutrino have mass?

When using my concept of mass, the answer to this question was unknown from the 1930s, when the concept of the neutrino was first proposed, until the 1990s. Today we know (almost certainly) that the neutrino has mass.

But if we mean relativistic by mass, then the answer will be: naturally, we have known about this since the very first day of the existence of the concept "neutrino". All neutrinos have energy, so, like photons, they have mass. The only question is the presence of an invariant mass.

Do all particles of the same type — for example, all photons, all electrons, all protons, all muons have the same mass?

When using my concept of mass, the answer to this question will be in the affirmative. All particles of the same type have the same mass.

But if we mean relativistic by mass, then the answer will be: obviously not. Two electrons moving at different speeds have different masses. They have the same invariant mass only.

Is the old Newton's formula F = ma true, relating mass, impact, and acceleration?

When using my concept of mass, the answer will be: no. In the Einstein version of relativity, this formula is corrected.

But if by mass we mean relativistic, then the answer will be: it depends on the situation. If the force and motion of the particle are perpendicular to the vector, then yes; otherwise not.

Does the mass of a particle increase with increasing speed and energy?

When using my concept of mass, the answer will be: no. See the chart above. Different observers can assign different energy to a particle, but everyone will agree with its mass.

But if by mass we mean relativistic, then the answer will be: yes. Different observers can assign different energy to a particle, and, therefore, different masses. They agree only about the invariant mass.

So, at least we see the presence of a linguistic problem. If we do not denote exactly which of the definitions of mass we use, we will get completely different answers to the simplest questions of physics. Unfortunately, in most books for non-professionals and even in some first-year university textbooks (!), The authors switch back and forth between these terms without explanation. And the most common confusion among my readers is related to the fact that they are informed by two types of information about the mass that contradict each other: one is suitable for the rest mass, and the other for relativistic. It’s very bad to use one word for two different things.

This, of course, is just a language. With the language, you can do anything. Definitions and semantics do not matter. When a physicist is armed with equations, the language becomes an imperfect carrier. Mathematics is never confused, and a person who understands mathematics will also not get confused.

But for most people and for beginner students this is a nightmare.

What to do? One option is to insist on using all possible terms. But because of this, the explanations will be very confusing.

• Energy of a resting object = invariant mass multiplied by c 2 = relativistic mass multiplied by c 2
• Mass of a moving object = invariant mass, as before, but energy = relativistic mass multiplied by c 2 is greater than before, due to the energy of motion.

This is too verbose. My colleagues and I just say:

• At a resting object of mass m, the energy E is equal to mc 2 ,
• and for a moving object, the mass is still m, and the energy E is greater than mc 2 , exactly the energy of motion.

This method is no less meaningful, it uses fewer different concepts and definitions, it avoids two contradictory meanings of the word “mass”, one of which does not change with movement, and the other changes.

From the point of view of linguistics, semantics and concepts, it is necessary to avoid the concept of "relativistic mass" and remove the words "invariant" and "rest" from the definitions of "invariant mass" and "rest mass" because "relativistic mass" is a useless concept. This is just another name for particle energy. Using the concept of "relativistic mass" is the same thing as insisting on the term "reddish blue." If I begin to insist on using the term “reddish-blue” to describe raisins, you will argue: but we already have a word for this color: purple. What's wrong with him? And you can also say: “To say that the color of raisins is a kind of blue, is wrong and it is confusing. We can conclude that the color of the raisins is a bit like the color of the sky, but in fact they are different. ” In approximately the same way, the relativistic mass multiplied by c 2 is just another name for energy (for which we already have a suitable word), and to describe energy as if it were something like mass means to confuse the reader.

This is another reason why it is bad to call energy a form of mass. In Einstein's equations, space and time are connected together in the same way as energy and momentum. You can even remember that energy is conserved because of the independence of the laws of physics from time , and the impulse is due to the independence of laws from place. Therefore, if we say that mass is E / c 2 , then what is p / c? It should mean something. What exactly? But no one gave this value a name. Why? Because “impulse” is a good name for p, and the name is not needed for p / c. So why is "energy" not suitable for E? Why do we need a new name for E / c 2 ? Especially, if we consider that another value appears in the equation with E and p:

The value on the right clearly does not need a new name, since it is clearly neither E nor p - it is not saved as E and p, but it does not depend on the observer (unlike E and p!)

The concept of "relativistic mass" did not appear out of nowhere and out of some nonsense. It was introduced by Einstein himself, and for good reason, since he dealt with the relationship between the energy of a system of objects and the mass of this system. But although the concept of a relativistic mass was propagated and spread by other famous physicists of that time, Einstein himself, apparently, rejected such a way of thinking, and also for good reason. The community of modern specialists in particle physics did the same.

In articles and studies, I never use relativistic mass. I use energy instead, because for a particle the relativistic mass itself is simply energy divided by c 2 . And by “mass” I always mean “invariant mass,” or “rest mass,” on which all observers converge. The electron mass is always 0.000511 GeV / c 2 , no matter how fast it moves. The mass of any electron is less than the mass of the atomic nucleus. All photons in a vacuum are always massless. And the mass of the Higgs particles is 125 GeV / c 2 , regardless of their velocity. Particle physics specialists use such a linguistic and conceptual arrangement. This is not necessary, you can make another choice. But this approach allows us to avoid a lot of practical and conceptual problems, which I tried to show here.


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