Relationship between mass and matter

Question

I've read this book "An Introduction to the Philosophy of Physics" by Marc Lange.

In the book, Marc Lange introduces the reader to E=mc^2 and does very, VERY careful job to avoid any confusion between matter and mass. He constantly repeats that the inertia of a body (relativistic mass) is what changes, not "real" mass (he means a quantity of matter). In his explanation, relativistic mass is not "real" (material) mass. It has no physical meaning in terms of quantity of something, but has meaning when we try to exert force on the body. He goes on and on criticizing many textbooks and physicists for giving mass-energy conversion ANY physical sense. My "fresh eyes" pov sides with him.

This got me thinking. If Newton defined mass as a quantity of matter (density divided by volume), then does the word (and concept) mass cover two concepts at the same time "quantity of matter" and "inertia of a body"? What is matter, as we can only measure quantity of it through force? $F=ma$ in classical physics

Answer 1

does the word (and concept) mass cover two concepts at the same time "quantity of matter" and "inertia of a body"?

In modern usage there are two distinct concepts of mass: invariant mass and relativistic mass. If the word mass is used without a description it is usually intended to mean the invariant mass. It is the mass that is reported in tables of particle masses and is a property of the object being measured.

The “quantity of matter” is rather ambiguous. It could refer to the number of moles rather than mass. Similarly, the “inertia of a body” could refer to momentum rather than mass. I don’t think that either of those phrases are sufficiently unambiguous to serve as a definition of mass.

However, the invariant mass would better represent both phrases. It would be unusual to think that the “quantity of matter” is different for the same object in different reference frames. And the “inertia of a body” would be best represented by the invariant mass under the four-vector generalization of Newton’s second law.

He constantly repeats that the inertia of a body (relativistic mass) is what changes

The relativistic mass is not the inertia of a body. If you use the four-vector generalization of Newton’s second law then relativistic mass isn’t involved. If you do not use that generalization then there is a different inertia for forces parallel and perpendicular to the velocity. The relativistic mass is only the parallel inertia. This parallel and perpendicular inertia is one strong reason to use the four-vector generalization.

Answer 2

He goes on and on criticizing many textbooks and physicists for giving mass-energy conversion ANY physical sense.

Experiments, that are now routine in particle physics, have repeatedly shown that it is possible to convert two photons into particle-antiparticle pairs, i.e. convert energy into matter, and vice versa. You can also convert three photons into particle-antiparticle pairs. The number of particles has literally changed.

Invariant mass = rest mass is, with just the conversion factor of $c^2$, is just energy. There is no way to deny this.

The person you are reading, is simply just wrong.

He constantly repeats that the inertia of a body (relativistic mass) is what changes

The concept of relativistic mass is just bad teaching. Yes, the inertia of a body will change when it is moving; but if you want to start defining inertia for moving bodies, then you will have to start discussing transverse mass and longitudinal mass, neither of which is relativistic mass, and this is a dead end. The mathematical handling of these things only get more and more complicated when you insist upon using them.

The only acceptable conceptually clear way forward, is to only ever use invariant rest mass.

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Let's consider things a bit deeper. What properties that masses have that can be used to define mass? There is inertia and there is gravity, corresponding to the inertial mass and gravitational mass concepts from Newtonian mechanics. But for the same number of molecules of water, you heat it up, they gained energy, then both the gravitational pull due to the water and the inertia of the water will increase in lockstep.

The fact is that every mention of mass in Newtonian mechanics is just hiding that it is measuring the absolute value of energy in a body, as opposed to the usual assertion that only energy differences are important. You can replace every appearance of mass in Newtonian mechanics by energy of some form. There is simply no good definition of mass other than "energy of a system when the system in a frame of reference that sees the system to be not moving, on overall, and then divided by $c^2$"

You can make a black hole purely from light. You can change the inertia = gravity of an atom by giving it a photon. Sometimes philosophers of physics can just be wrong.