# Additivity of mass: provable from Newton's laws?

Within the context of Newton's laws, it seems implicitly understood that mass is an additive property of objects.

My question is: should this be considered "just" an experimentally confirmed fact, or is is provable from Newton's laws?

The following is a more precise context for this question.

Without attempting to be axiomatic in a mathematical-logical sense: whenever I physically combine two given objects into one, I "know" that this combined object will again obey Newton's second law with a mass term that is the sum of the mass terms from the two original objects. Note that I treat the law here as a description of real world situations, as precisely as allowed by the imperfection of my experimental setup (and within "classical" limits).

My question is then: under these circumstances, is it provable from Newton's three laws that mass is additive in this sense, or can this fact be concluded from experiments only?

I realize that, formally, much is missing from the above in terms of exact definitions. Briefly, I assume force, location, time (and thus also velocity and acceleration) to be concepts that actually apply to reality and that are objectively measurable in terms of certain units (which themselves belong to the "measuring view", not to the measured reality); furthermore, I assume Newton's laws to hold for any measurements I may take from any experiment, and in particular, I assume that the force applied to an object is equal to the vector sum of all forces acting upon it from other objects, according to the law of gravitational attraction and to no other influences. In other words I assume that my measuring methods define an inertial system to describe the world, and I only consider gravitational forces.

I do know that things may be different in a relativistic session, but this does not currenctly concern me since I am inquiring to the structure and consequences of the Newtonian world view.

I hope I phrased my question clearly. Can anyone shed light on this?

## Answers and Replies

vanhees71
Science Advisor
Gold Member
2021 Award
I guess what you mean is the conservation law following from symmetry of any closed system of Newtonian mechanics under Galilei boosts, i.e., the transformation
$$\vec{x}_{\alpha}'=\vec{x}_{\alpha}-\vec{v} t$$
is a symmetry of the system.

Now the most elegant way to formulate the symmetry is to use the description in form of Hamilton's canonical formalism. Then the generator (in the sense of a canonical transformation) of the Galilei boost is defined by
$$\vec{K}=\sum_{\alpha} m_{\alpha} \vec{x}_{\alpha}-\vec{P} t=M \vec{X}-\vec{P} t, \quad M=\sum_{\alpha} m_{\alpha}, \quad \vec{P}=\sum_{\alpha} \vec{p}_{\alpha}$$
since then
$$\{x_{\alpha}^j,K^k \}=-\delta^{jk} t, \quad \{p_{\alpha}^j,K^k \}=-m_{\alpha} \delta^{jk},$$
i.e., for an infinitesimal boost we have
$$\delta \vec{x}_{\alpha}=-\delta \vec{v} t, \quad \delta \vec{p}_{\alpha} =-m_{\alpha} \delta \vec{v}.$$
This means that the finite transformation is given by
$$\vec{x}'=\vec{x}-\vec{v} t, \quad \vec{p}_{\alpha}'=\vec{p}-m_{\alpha} \vec{v},$$
as expected.

Now Noether's theorem tells us that ##\vec{K}## is conserved, and this means that
$$\dot{\vec{x}}=M \dot{\vec{X}}-\vec{P}+\dot{\vec{P}} t=0.$$
Since now a closed system is also translation invariant and the corresponding generators of the canonical symmetry transformation are ##\vec{P}##, also ##\vec{P}=\text{const}##. This implies finally that
$$\vec{P}=M \dot{\vec{X}},$$
i.e., the center of the mass moves like a free particle with mass ##M=\sum_{\alpha} m_{\alpha}## with the total momentum of the system ##\vec{P}=\sum_{\alpha} \vec{p}_{\alpha}##, i.e., in linear uniform motion with the constant velocity ##\vec{V}=\vec{P}/M##. In this sense masses in Galilei-Newtonian physics is additive.

This is, as you mentioned, not the case for the Poincare group of special relativity, and indeed, for a relativistic system a composite bound object has a mass not given by the sum of the masses of its constituents. An example are atomic nuclei which are composite objects consisting of nucleons (protons and neutrons) bound together by the strong force. The mass of the nucleus is always smaller by the binding energy (divided by ##c^2##, with ##c## the speed of light in vacuo).

The deeper reason for this difference between Galilei-Newton and Einstein-Minkowski mechanics is by analysing the symmetry groups (Galilei vs. Poincare groups) for the quantum case. It turns out that the mass play a significantly different role for these groups. In the case of the Poincare group mass is a Casimir operator of the corresponding Lie algebra, defined by ##m^2 c^2=p_{\mu} p^{\mu}##, where the four-momenta generate space-time translations as in classical mechanics. The Galilei group's Lie algebra is more complicated. It turns out that the true unitary ray representations of the classical Galilei group do not even lead to any physically sensible quantum mechanics at all. What one has to take instead is a central extension of this group. In the case of the Galilei group, there is a non-trivial central charge, which is the mass of the system. The Poincare group has no non-trivial central charges, and thus mass is just a Casimir operator of the Lie algebra in any representation.

Delta2
Nugatory
Mentor
Within the context of Newton's laws, it seems implicitly understood that mass is an additive property of objects.

My question is: should this be considered "just" an experimentally confirmed fact, or is is provable from Newton's laws?
It's provable from Newton's laws in the sense that everywhere mass appears in Newton's laws it is raised to the first power (no ##\sqrt{m}## terms, no ##m^2## terms, no other weird powers of ##m##). Thus (for example) the gravitational force on an object of mass ##M=m_1+m_2## is equal to the sum of the gravitational force on an object of mass ##m_1## and on an object of mass ##m_2##. However, this just begs the question of why Newton's laws would be expected to take a form that basically assumes additivity of mass; that's a much more subtle question, one for which vanhees71 has the best answer.

None of this means that additivity of mass is not "just an experimentally confirmed fact" - it's an experimentally confirmed fact that Newton's laws describe the universe we live in. We might not live in a universe in which mass is additive (as you point out, when relativistic effects are taken into account we don't) and then experiments would tell us to use something other than Newton's laws.

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First of all, I want to thank all who commented on my question.

Based on these answers I am satisfied that, indeed, Newton's laws imply that mass is "additive"; no additional postulate is needed.

Furthermore, I learned that it would be a good idea for me to study Hamiltonians and Noether's theorem for a complementary view on classical mechanics.

And finally, I realized that my question was not as well-posed as I thought.

Being now satisfied that the "addition rule" follows from Newton, and recognizing the value and elegance of the proof given by vanhees71, I have been searching for a way to understand it without having to rely on advanced insights, developed long after Newton. What follows is what I have come up with.

I realize that not everyone will have the same desire for a simple proof. Those who don't are probably best advised to stop reading here. For those interested, I outline my line of reasoning.

Objects will be point-like masses. The situation is as follows: I have two objects O1 and O2 with masses m1 and m2, respectively. I bring these objects in close proximity. Do I now have an object with mass m1+m2?

Perhaps this is an ill-posed question, because I have no rules that tell me how to make one object out of two objects. In fact, the notion of "object" is somewhat undefined. And Newton's laws do not speak (directly) of things like "a force working on a conglomerate of two objects". I therefore choose to answer the pragmatic question: is the world, now that I have brought the two objects close together, essentially different from the world that would have a single object O3 of mass m1+m2 at exactly this location (and with the same speed, etc.), and that is otherwise the same? My answer will be: these worlds differ only insofar that we have two objects in close proximity as opposed to one object in essentially the same location. The motion of all other objects, will be the same; also, the combined objects O1 and O2 will stay together and will both move in exactly the same manner that the single object O3 would move. This I will consider sufficient to conclude that, loosely speaking, "mass is additive".

Once this form for the question has been found, the reasoning is simple. Briefly, additivity of forces working on a single object guarantees that all other objects will move in exactly the same way with O1 and O2 combined present, as they would with O3 present. To see that O1 + O2 will move exactly as O3 would, it is sufficient to prove that each of O1 and O2 will do so. But this follows immediately from Newton's second law, together with the fact that in the law for gravitational attraction, mass occurs in the first power, as noted by Nugatory.

This concludes the reasoning. There are certainly philosophical and epistemological undertones here, which I have not gone into. Any comments will be greatly appreciated.

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...everywhere mass appears in Newton's laws it is raised to the first power (no √mm\sqrt{m} terms, no m2m2m^2 terms, no other weird powers of mmm).

Newton's Law of Gravitation contains a mass squared term.

Nugatory
Mentor
Newton's Law of Gravitation contains a mass squared term.
They're two different masses, and it's linear in each one of them. What's the gravitational force between the earth and an object of mass ##M=m_1+m_2##? It's ##GMM_E/r^2=G(m_1+m_2)M_E/r^2=Gm_1M_E/r^2+Gm_2M_E/r^2##.

Point well taken. The constant of gravitation has mass raised to the -2 power... whatever that means!

Jano L.
Gold Member
Within the context of Newton's laws, it seems implicitly understood that mass is an additive property of objects.

My question is: should this be considered "just" an experimentally confirmed fact, or is is provable from Newton's laws?

... whenever I physically combine two given objects into one, I "know" that this combined object will again obey Newton's second law with a mass term that is the sum of the mass terms from the two original objects.

My question is then: under these circumstances, is it provable from Newton's three laws that mass is additive in this sense, or can this fact be concluded from experiments only?

Newton's laws operate with down-to-earth concepts derived from experience. The 2nd law

The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

which uses the concept of mass in "motion" ##m\mathbf v##, is a physical law based on experience only if acceleration, force and mass have meaning independent of the law itself. In present-day parlance this law says that given force, mass and acceleration are measured independently, they are related to each other via the equation

$$\mathbf a = k \frac{\mathbf F}{m},$$
where ##k## is constant of proportionality depending on the choice of units for the other quantities but independent of the particular values of the other quantities.

The common meaning of mass ##m## referred here is the number that quantifies amount of matter. It can be integer number of weights that balance the body on a scale or number of components used to assemble the body. Additivity of this concept of mass is definitoric and implicit in common thinking.

Trying to derive additivity of this mass from Newton's laws is a bad idea because we knew this already before Newton's laws.

On the other hand, if we consider technical concept of mass such as inertial mass, things are different. Inertial mass is defined via Newton's second law in terms of force and acceleration as the coefficient ##m_i## in the equation

$$m_i\mathbf a = \mathbf{F},$$

where ##\mathbf a## is acceleration of the body and ##\mathbf F## external force acting on it. Additivity of inertial mass does follow from Newton's laws - explanation follows.

I'll show that composed body's inertial mass is sum of inertial masses of the parts. Let us consider body that was composed of two bodies of inertial masses ##m_1## and ##m_2## and moves under action of the same forces, without any rotation. Based on the definition of ##m_1##, the external force acting on the body 1 and its acceleration obey the equation

$$m_1 \mathbf a_1 = \mathbf F_1+\mathbf F_{2-1}$$

where ##\mathbf F_1## is external force acting on the body 1 and ##\mathbf F_{2-1}## is force due to body 2 acting on the body 1.

Similarly,
$$m_2 \mathbf a_2 = \mathbf F_2+\mathbf F_{1-2}$$

Let us add both sides of the two equations; the sums are equal:
$$m_1 \mathbf a_1 + m_2 \mathbf a_2 = \mathbf F_1+\mathbf F_2+\mathbf F_{2-1}+\mathbf F_{1-2}$$

Because the two bodies were assembled into one, their accelerations have the same direction and magnitude so let us introduce ##\mathbf a## as the common acceleration of all bodies. We can also introduce total external force ##\mathbf F = \mathbf F_1 + \mathbf F_2##. Then we can rewrite the last equation into

$$(m_1 + m_2) \mathbf a = \mathbf{F}+\mathbf F_{2-1}+\mathbf F_{1-2}$$

This has almost the form of the equation that defines inertial mass of the composed body, but there are two additional force terms. However, provided Newton's 3rd law of motion is valid, ##\mathbf F_{1-2} = -\mathbf F_{2-1}##, so we indeed have

$$(m_1 + m_2) \mathbf a = \mathbf{F}.$$

Since this is the relation that defines inertial mass of the composed body, we conclude that inertial mass is additive. But I'd like to stress that this is a consequence of the definition of inertial mass and is not so interesting. What is interesting is that Newton's law is valid irrespective of the kind and composition of body and source of the forces. It is also interesting that it was discovered to be inaccurate when speed gets comparable to speed of light. Mass (rest mass) is still additive even in such extreme situations, but inertial mass may not if the parts strongly interact, as the 3rd law does not hold in theory of relativity. For example, atom has slightly lower inertial mass than the sum of inertial masses of nucleus and electrons, due to EM interaction holding them together.

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vanhees71
Science Advisor
Gold Member
2021 Award
Well, in special (and also general) relativity mass is not additive. E.g., the mass of a nucleus is not the sum of the masses of its constituents, the protons and neutrons, but it's smaller by the binding energy (divided by ##c^2##).

An even more drastic example is the mass of the protons and neutrons, making up most of the mass of the matter around us. It is almost completely dynamically generated by the strong interaction, i.e., due to the confinement of quarks and gluons.

CWatters
Science Advisor
Homework Helper
Gold Member
Ok so shoot me down in flames but...

On larger scales like galaxies 2+2 is somewhat more than 4 and we have to invent dark matter to explain how some stars move so fast :-)