# Matter and mass

Matter is anything that has mass and occupy space.
Yet what is mass?
Mass is the quantity of matter.
This is a roundabout way of saying that you do not know what is mass.
Mass is not weight.
What is mass?

Weight is a measure of how much force is acting on an object due to gravity. Mass is the measure of how much substance is in the sample. Matter is just a word saying anything that occupies space.

HallsofIvy
Homework Helper
Mass is a measure of inertia.

I have a problem with this dynamical definition of mass. It implies that in a static universe, there is no mass. And yet we know that all chemical substance has mass. For eg, we can say I want 4 grams of salt or 3 grams of coal, without having the salt or the coal moving. That is why mass is usually stated as the quantity of matter. It is also the reason why we don't say mass is weight because we can still have 4 grams of salt or 3 grams of coal even in the absence of gravity. Also, inertia is the resistance to force, and yet force is usually defined in terms of mass, so if mass is inertia, this will be yet another roundabout circle.
What is mass?

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'What is mass' can have many answers depending on the context of the question. The latest particle experiments are probing precisely this, exploring the relationship between mass and gravitation and whether or not there's a mediating particle responsible. In terms of basic classical mechanics - which is largely formulated in terms of forces as fundamental objects - mass can be called a measure of resistance to acceleration - i.e. inertia. Classical mechanics, as far as I am aware, doesn't give mass a fundamental definition. The mass-energy relationship E=mc^2 comes from relativity and is a mathematical truism, not merely an experimental fact - it is derived purely from the mathematics of special relativity and turns out to be accurate.

Then in classical mechanics, force is the fundamental quantity.
Force is defined as a push or a pull.
:)
Then mass is defined in terms of force.
And matter is defined in terms of mass.
Rather strange indeed that everything is defined in terms of a push and a pull.
In fact mass can also be defined thermally. A bigger mass of water is able to absorb more heat energy before having one degree rise in temperature. No push or pull is needed, just hot or cold! This is the definition that the chemists and other branches of science will be happier with. The dynamical properties might just be - properties only.

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ZapperZ
Staff Emeritus
Then in classical mechanics, force is the fundamental quantity.

Not necessarily. If you look at Lagrangian/Hamiltonian mechanics, there isn't even a "force". So no. It isn't a "fundamental" quantity.

What a "mass" is, if you look at the Standard Model, is more of a "interaction" of objects that couple with a background field. If the Higgs bosons are detected, then, then we know that this way of looking at it is valid.

Zz.

Not necessarily. If you look at Lagrangian/Hamiltonian mechanics, there isn't even a "force". So no. It isn't a "fundamental" quantity.

Sorry, I meant to make this clear by saying
basic classical mechanics

To imply pre-lagrangian areas of study - that which you study in high school and the first couple of undergrad years.

ZapperZ
Staff Emeritus
Sorry, I meant to make this clear by saying

To imply pre-lagrangian areas of study - that which you study in high school and the first couple of undergrad years.

I noticed that. However, the OP took what you wrote and generalized it to all of classical mechanics, which isn't correct.

Zz.

Then why do they mention the four fundamental "forces" if there are no forces to begin with?
Lagrangian or Hamiltonian or Newtonian is the same. They described the motions of masses, but do not tell you what a mass is , and its relationship with matter - that is the heart of the problem.

jtbell
Mentor
I think Newton's definition of "mass" is the best we can do, conceptually: "quantity of matter" or more colloquially, "how much stuff" an object contains. To this day our standard unit of mass is a physical artifact: a certain platinum-iridium cylinder in a basement in Paris. We measure the mass of any object by (ultimately) comparing it with the mass of this cylinder.

Also, inertia is the resistance to force, and yet force is usually defined in terms of mass, so if mass is inertia, this will be yet another roundabout circle.

It doesn't have to be that way, conceptually. Think of "force" as "that which makes an object accelerate." A simple example is the force produced by a stretched or compressed spring. Hypothetically, we could define our unit of force as the force exerted by a standard spring (made in a specified way out of a specified material) compressed by a standard distance. Then we would have to introduce a proportionality constant into Newton's Second Law: F = kma.

Now the Second Law becomes something to verify experimentally, rather than a definition of force or mass. If we take our standard spring, compress it by the standard amount, and apply it to the standard mass, we observe that the standard mass accelerates by a certain amount. If we take two identical copies of the standard spring, compress them by the standard amount, and apply them simultaneously to the standard mass, we observe that the standard mass accelerates by twice the original amount. And so forth.

I think Newton's definition of "mass" is the best we can do, conceptually: "quantity of matter" or more colloquially, "how much stuff" an object contains.

This presupposes certain properties of the mass-gravity-field entity, whatever form it may have, does it not? We make the assumption that all matter 'feels' the gravitational force equally - a perfectly reasonable assumption, but very possibly completely wrong. It is conceivable that in fact all mass is not created equal, and that the gravitational and inertial effects are not simply a linear function of mass. Of course, we'll have no way of verifying this until the results of gravitational wave, graviton and higgs boson experiments are very thoroughly examined.

ZapperZ
Staff Emeritus
Then why do they mention the four fundamental "forces" if there are no forces to begin with?
Lagrangian or Hamiltonian or Newtonian is the same. They described the motions of masses, but do not tell you what a mass is , and its relationship with matter - that is the heart of the problem.

No one said there's no "forces". It is just that the concept isn't "fundamental" in Lagrangian/Hamiltonian mechanics, because a "force" is simply another "level" of operations.

Furthermore, these "four fundamental forces" deals more with the type of interaction. These are the QFT-type interactions, which if you look at the mathematics, have NO classical forces that you can recognize out of classical mechanics. Do not confuse the words being used to describe such a thing to the general public with the actual mathematical formalism.

If you think the Hamiltonian/Lagrangian mechanis is no different than the Newtonian mechanics, then would you like to show me where in the Lagrangian of, let's say, a simple pendulum, is there a "force"?

Zz.