What is mass? Part 2
Mass in pre-relativity physics
[This is the second entry in a series of posts on the topic “What is mass?” beginning with Part 1 and continuing in Part 3.]
Here are three definitions of mass.
Mass measures how much matter there is in an object.
Mass measures the ability of an object to resist being accelerated.
Mass is the thing that gravity pulls on.
While different, these three definitions are all also basically true in some important way. Still, don’t let the fact that they are all statements about the same thing trick you into thinking these are different ways of framing the same definition. Conceptually, these definitions are quite distinct! Each one contains a major concept (“matter”, “acceleration”, “gravity”) that does not appear anywhere in the other definitions. We will now investigate each definition in turn in greater detail.
Definition #1: Mass measures how much matter there is in an object. This definition has the advantage of being intuitive. Nonetheless, if we had to explain it to an alien that hadn’t evolved an intuitive understanding of the concept “matter,” we would be in trouble: It’s hard to supply a definition of “matter” without immediately looping back to mass1. It is also hard to imagine an experimental procedure to measure the “amount of matter” in an object. Obviously, the Earth is made out of more matter than I am, but what about a box of air and a box of lead that are the same size? For the time being we can content ourselves with the (genuinely ridiculous and impossible for so many reasons) thought experiment of a scientist with a superpowerful microscope who can individually count all the atoms in two objects to determine which contains more. Yes, we have to pretend these atoms are atoms in the original Greek sense2 and have all the same mass as one another.
Definition #2: Mass measures the ability of an object to resist being accelerated (i.e. its inertia). This definition has the advantage that it supplies a far more tractable experimental procedure for measuring the mass of an object than definition #1. To understand this experiment, however, we will have to briefly review the concept of inertia.
Everyday life is full of friction. As a consequence, if I push on the copy of Principia Mathematica lying on the desk next to me, essentially as soon as I stop pushing on it, it stops sliding. If I pushed particularly hard I could get it moving quickly enough that it would slide perceptibly after I stop pushing, but it would still come to a stop very quickly. We are so used to this kind of thing that it is very easy to think that this is something natural or preferred about the state of “rest” and that all objects inexorably tend to come to rest. This is not so, and this kind of thinking has been an enormous drag on progress in physics3.
For example, suppose I pushed my book just as hard while it was resting, not on my desk, but on a well-greased bowling lane or on an air hockey table. Obviously it would travel farther before coming to rest. But nothing about the book or how hard I pushed it has changed. Therefore, the fact that the book comes to rest—and how quickly it does so—is a result instead of interactions between the book and the rest of its environment—the air in the room and the surface it rests on. If I pushed the book in the distant vacuum of interstellar space where there is no air friction or other forces acting on it, the book would continue on at the same speed (after I stopped pressing) forever.
The “natural” state of the book (that is, the state of the book when no one is pressing on it) is no more to be at rest than to move in a straight line at a constant speed. In fact, the whole problem with asserting that the natural state is “rest” as opposed to “motion” is that one cannot give a rigorous definition that will distinguish between the two.
Suppose that from your perspective, the book and I are drifting through space alongside one another at precisely the same speed and in precisely the same direction. From your perspective, both the book and I are moving. You can tell we are moving because we get farther away from you over time4. From my perspective, however, the book never draws nearer to me nor moves farther away. I say the book is not moving. If I look back at you, I see that you are getting farther and farther away from the book and I. Therefore, I assert that the book and I are at rest while it is you who are moving at a constant speed and in a constant direction away from us.
Both descriptions of the situation are completely consistent with one another, and there is no experiment we can do that will tell us which of us is “right”; therefore, there is no physically meaningful distinction between being “at rest” and moving in a straight line at a constant speed. It is merely a matter of relative perspectives. This fact is called the Galilean principle of relativity.
There is, however, an absolute difference5 between constant speed, straightline motion of the kind we have been discussing and motion at a changing speed and/or along a curved path. Motion at a constant speed and in a straight line is called inertial, while motion at a changing speed or along a nonstraight path or both is called noninertial. (The reasons for this naming convention will be more clear shortly.) The law of inertia states that an object only moves noninertially when there is some force acting on it. In the absence of forces, the motion of the object is inertial.
Imagine you are bombing down some particular cop-free section of the interstate at 90 miles per hour. And also imagine that this section of the interstate was recently repaved during the Great American Infrastructure Revitalization of the 2030 Green New Deal. In fact, the asphalt was paved to the highest specifications of smoothness that modern lasers can possibly measure. It’s so smooth it makes Charmin toilet paper look like the surface of the Moon. So, even though your car is bombing along at 90 miles per hour, because the asphalt is so smooth there are absolutely no jiggles. If you are traveling in a straight line, do you feel like you are moving? No! For all you can tell, you are sitting still in a parking lot. You can reach over into the passenger seat and fish out a french fry from the fast food bag just as if you were sitting still. You tip your head back and take a big gulp of your milkshake just as casual as you please. No problem. There is no difference as far as the objects in your car are concerned between moving and sitting still. This is Galilean relativity.
Now—meep-meep—a roadrunner suddenly runs out into the road ahead. You slam on the brakes. You feel yourself lift up out of your chair a little. Your upper body pulls away from the seatback. Your seatbelt extends then abruptly tightens, catching you. The milkshake flies out of your right hand, forward, and smacks against the windshield, splattering everywhere. The french fries—which a moment before were sitting sedately in the passenger’s seat—also fly forward off the seat and hit the glovebox. Your car skids to a halt, and the roadrunner, mercifully spared, zooms away. What just happened?
Your inertial motion—bombing along in a straight line at 90 miles per hour—was relative. There was no experiment you could do in the car to tell if you were “moving” or “sitting still”. But the noninertial motion required to change your speed in a short amount of time—that was very easy to detect. How did you detect it? You saw that everything in your car—your body, your milkshake, your french fries—flew forward. This happened because of inertia.
Recall that the law of inertia states that objects continue to move inertially until some force acts on them and makes the move noninertially. By braking, you forced your car to start changing its speed, to move noninertially. But at first there was no force acting on any of the objects inside the car. Your body continued to move forward at 90 miles per hour as the car around it slowed down. This continued until the seatbelt caught you and began to exercise a force on your body that caused your body to move noninertially, to slow down with the rest of the car. The same thing happened with the drink and the fries: both moved forward inertially as they had been until they physically contacted some part of the car—the windshield or the glovebox—that could exert a force on them and cause them to move noninertially.
Definition #2 adds one more quantitative detail to this picture. Not only is inertia the ability of all objects to persevere in inertial motion unless a force forces them to move noninertially, but also: Some objects have more inertia than others. Definition #2 treats the fact that some objects have more inertia than others as its definition of mass: If an object has more inertia (viz. more ability to resist being accelerated) then, definitionally, it has more mass.
Your body has more ability to resist acceleration than the milkshake. The glass of the windshield was strong enough to bring the milkshake to a (messy) stop. But if you had not been wearing a seatbelt, the glass would not have been strong enough to supply the force necessary to bring you into noninertial motion with the rest of the car: Instead, you would have smashed through the windshield6. This is why seatbelts in cars must be incredibly strong—in general they can exert up to several thousands of pounds of force on a human body before snapping.
Now that we have an intuitive understanding, let’s express these ideas more clearly. We have said that objects with larger masses have a greater ability to resist acceleration. Implicitly, we are saying they have a greater ability to resist acceleration by some force. I can express this in an equation:7
where m is the mass, F is the force, and a is the acceleration.
Let’s check that the right hand side of this equation correctly reproduces our concept of inertia: If it takes more force to produce the same acceleration (if we increase F and hold a constant), then the RHS increases, so m increases. This conforms with our notion that a body which requires more force to accelerate has more inertia.
Incidentally, this equation is known as Newton’s second law of motion. In our understanding, this can be thought of as supplying definition #2 of mass.8
Wheew! This definition certainly required the most explanation, but it was worth it. Along the way we introduce two of the most important and foundational statements in all of pre-relativity mechanics: the Galilean principle of relativity and Newton’s second law of motion. Also, as a result of this discussion, we are now qualified to design an experiment by which we can measure the mass of an object: We apply some known, constant force to an object9, measure the acceleration that results, divide the former by the latter, and declare that the result is the object’s mass. This experiment is certainly much more tractable that counting all the atoms as in definition #1!
The main difficulty with this definition is that we have to already know what force is10.
Definition #3: Mass is the thing that gravity pulls on. Some objects experience a greater pull due to gravity than others. A simple bathroom scale is sufficient to illustrate this. Units like “pounds” are in fact units of weight, and weight is a force—the force of gravity.
An even more primitive scale illustrates the effect more clearly. Consider a simple spring, fixed at one end to the ceiling and with a hook at the other end from which we can hang various objects. The farther a spring is stretched from its equilibrium position, the stronger the force that the spring exerts, trying to pull itself back to its original shape. Also, if an object is not accelerating, then, but Newton’s second law, there must be zero total force on it: All forces acting on it in all different directions must precisely balance. Suppose we hang two masses from two identical strings. One is a bag of feathers and the other is an identical bag of sand. The sandbag stretches its spring farther, but eventually both bags are suspended—stationary—from their springs. Since neither bag is moving up or down, the downward force the bags experience due to gravity must equal the upward force they experience due to the spring. Since the spring affixed to the bag of sand is stretched farther than the spring affixed to the bag of feathers, the gravitational force experienced by the bag of sand is greater than the gravitational force experienced by the bag of feathers.
This much thus far is merely a record of experimental facts. Definition #3 uses these observations as the basis to define a “property” of objects called mass which is the thing that gravity pulls on. The greater the mass of the object, the stronger gravity pulls on them. By this definition the bag of sand is more massive than the equally-sized bag of feathers because it experiences a greater force due to gravity.
As with definition #2, we can use math to be more precise. The essence of definition #3 is to assert
That is, the mass is (linearly) proportional to the (gravitational) force experienced by the body. We can make this proportionality into an equation by introducing a constant of proportionality called the gravitational field g.
Actually, calling the gravitational field a constant of proportionality is misleading. In everyday conditions on the surface of the earth the strength of the field is approximately constant, but in general the gravitational field depends on one’s position relative to sources which generate the gravitational field. It would be better to write it as a scalar field
that is, something which takes on the value of a single number (a “scalar”) depending on one’s position in space. But for our purposes here, since the gravitational field is approximately constant everywhere near the surface of the earth, we can just think of it as a constant number.
Like definition #2, definition #3 has the advantage of supplying us with a straightforward experimental procedure for measuring the mass of an object. This is the procedure that is used for almost all everyday purposes. In particular, since g is a constant on Earth, I can just relabel the units on my scale (a device which measures weight) from units of force (eg. Newtons) to units of mass (eg. kilograms) by dividing all the numbers by g. I now have a “mass-scale” that gives me units of kilograms directly. It is still worth bearing in mind, however, that such a “mass-scale” is specific to the value of the gravitational field on earth. It will not work on Mars unless I can figure out the strength of the gravitational field there and relabel again.
An equivalence coincidence?
We have discussed these definitions in such (perhaps excruciating) detail to underscore just one point: It is not at all obvious a priori that these three definitions would be definitions of the same thing. And yet they are.
Conceptually, so-called “inertial mass” (the thing which measures a body’s ability to resist being accelerated) has nothing to do with so-called “gravitational mass” (the thing which measures how strongly the gravitational field exerts a force on a body). And yet experiments11 show these two are always the same. This fact is known as the equivalence principle, although from the perspective of pre-relativity physics we might call it the equivalence coincidence.
(The fact that both of these are in some sense also equal to the definition of mass as the amount of matter does is not, to my knowledge, a similarly named principle.)
I think it is illustrative to go all the way back to the big man himself, Isaac Newton, who basically created all of pre-relativity mechanics by writing a single book, Mathematical Principles of Natural Philosophy12 aka Philosophiæ Naturalis Principia Mathematica. This monumental work remains remarkably lucid to this day. Newton begins with a brief section where he defines the important terms and concepts, and then establishes his famous axiomatic laws of mechanics. After this, the work begins in earnest. Books I and II are devoted to solving specific types of mechanics problems, and Book III connects these results to the experimental and astronomical evidence of the day.
A convenient result of Newton’s axiomatic approach is that the theoretical meat and potatoes of his mechanics appears at the beginning in his definitions and laws of motion. The very first definition Newton gives is supposed to tell us what mass is:
DEFINITION I.
The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.
Thus air of a double density in double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders … It is this quantity that I mean hereafter everywhere under the name of body or mass.
One difficulty with Newton’s writing is that he sometimes uses different terms to refer to the same concepts we talk about in physics today, but it is usually not too hard to figure out what he means13. We see that in this definition, the words “quantity”, “body”, and “mass” all mean the same thing, and this thing is precisely the mass we are trying to define here. He also uses the word “bulk” to mean the volume of an object.
So Newton is just claiming that mass is volume times density, and gives the example that twice the volume of matter of twice the density is four times the mass. Likewise three times the volume of matter that is twice the density is six times the mass. This is his effort to get at the same intuitive understanding of mass that appeared in our own definition #1.
People have criticized Newton for using density in his definition of mass on grounds of circularity, since it is hard to see how to come up with a definition of density that doesn’t reference mass. We too are familiar with how hard it is to avoid circularity when it comes to definition #1, but we understand what Newton means: matter is the amount of “stuff.”
Within this very first definition, however, Newton notes that mass
is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter14.
Even as he is defining mass as the amount of matter, Newton tells us that this mass is also always equal to a conceptually-distinct definition of (gravitational) mass. The “amount of stuff” is always equal to the “thing that gravity pulls on”. This fact is not something that Newton has deduced from pure thought but learned through experiments with pendulums.
Newton’s Definition III is the equivalent of our definition #2:
DEFINITION III.
The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, endeavors to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line.
This force is ever proportional to the body whose force it is; and differs nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita, may, by a most significant name, be called vis inertiæ, or force of inactivity. [...] Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished; nor are those bodies always truly at rest, which commonly are taken to be so.
[The bolded emphasis here is mine.] If we can look past unfamiliar Latin terminology, we find here a clear definition of inertia (the “vis inertiæ”) as a “force of inactivity” “by which every body, as much as in it lies, endeavors to persevere in its present state”.
Newton points out that we often think of inertia as “resistance” when we are trying to put an object into motion (“accelerate” it in the colloquial sense). But when we try to slow an object down (“decelerate” it in the colloquial sense), we experience an “impulse” or force from the body onto us (for example, the impulse of a milkshake splattering onto a windshield). But Newton makes clear that such notions of “motion” and “rest”, “impulse” and “resistance” are merely relative, in the Galilean sense.
He also identifies “inertia” with “mass” (inertia “differs nothing from the inactivity of the mass” in his words), although he is slightly obscure about how he thinks about this. One feels, perhaps, that Newton connects the amount of matter with inertia in a pseudo-philosophical way. There is a gesturing, if incomplete, at the idea that it is reasonable to expect matter to behave in this way. Why else would Newton first define the concept of an innate force of matter, vis insita, apart from inertia, only to later assert that that this innate force is identical with a force of inactivity, vis inertiæ? There is also the sentence “A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion” in which the parenthetical “from the inactivity of matter” is included almost as if to suggest a supra-physical causal principle by which “the inactivity of matter” causes there to be a law of inertia. (The way Newton talks about this also stands in contrast to the way he talks about the connection to gravitational mass, which is justified on experimental grounds.)
Whatever Newton may or may not have meant here, I cannot condone the perspective that the concept of matter would lead us to “predict” the existence of inertia except in this sense: Perhaps our implicit notion of matter (unlike our implicit notions of force and acceleration) relies too intimately on a notion of mass to stand on its own. In fact, there are good reasons for supposing that definition #1 is not really a definition in the same sense as #2 and #3—which we shall discuss in more detail at the beginning of Part 3 of “What is mass?”
This way to relativity theory
We have arrived at a rather surprising conclusion, a conclusion which was already noted, if peripherally, in the foundational text of pre-relativity physics: that the innate force of inactivity of matter, as Newton called it, is always equal to the thing which gravity pulls on. From the perspective of pre-relativity physics this amounts to something like a cosmic coincidence with no deeper principle to explain why these two are equal. It also has some rather strange consequences.
Consider, for example, two different (gravitational) masses, which we write
These are subjected to the same gravitational field g and as a result will experience different forces:
What accelerations will the masses experience as a result of these forces? Well, they will each resist falling with the following inertias.
Therefore, we may write
and
But of course we know that
Therefore, the masses cancel out in both force equations and we are just left with the following.
All bodies, whatever their masses, fall with the same acceleration when subjected to the same gravitational field15. More massive bodies are pulled harder, but they also have more innate force to resist that pull in precisely the same measure and so the two effects exactly cancel. This is one hell of a coincidence!
You also may have noticed something interesting about the field from the final equation. Since the two accelerations are equal to the strength of the field, the gravitational field must have units of acceleration. This is very different from the fields of other known forces. The analogous expression for the electric field, for example, would be
where E is the electric field and q is an electric charge. As you can see, the electric field has units of acceleration times mass divided by electric charge. It is the unique “quirk” of gravity that its “charge” is mass and so the factor multiplying acceleration becomes just equal to 1.
As we noted earlier, acceleration is the most fundamental kind of motion that, unlike inertial motion, can be absolutely rather than merely relatively defined. All fields can give rise to acceleration, but only gravity is in some sense identical with it. This is the principle of equivalence, which points to the deep structure of gravity, not as a force after all, but as something about the structure of space and time themselves, a topic which will be continued in Part 3 of “What is mass?”
Indeed, in the same high school science class where you may have heard definition #1 you probably also learned that matter is “that which has mass and takes up space.”
A-témnō-os (“uncuttable”), viz. they are not made out of any smaller structures. One could even argue that if such uncuttables really did exist (they don’t), they would necessarily all have the same mass by definition #1, but I leave this to the theologians.
It is, in fact, another example of something Aristotle got wrong so convincingly that it took European scientists over a thousand years to correct course.
Note that concepts like “speed” and “motion” are breaking down into positions and times, as we expect from the discussion of measurables in Part 1.
By “absolute difference”, I mean a difference that all observers can detect and will agree about if they do the right experiments.
As the example of your body smashing through the windshield illustrates, your body’s inertia really can be thought of as a force, although some people insist on calling these inertial forces “fictitious forces”. I once had a physics professor who strongly objected to this nomenclature. After all, in the example just considered, you are not merely fictitiously dead.
The perspicacious reader may suspect there is some cheating here, but really there is not too much. The essential problem is that words are far less precise than mathematics. In our linguistic explanations we have said things to the effect of “when the force required to accelerate an object increases, its inertia increases”. But this does not uniquely define how inertia increases with the force required. In giving this equation, we are specifying one relationship out of the infinite set of monotonic relationships which could correspond to the words. The superior specificity of mathematics is the whole point of using math to express physical ideas in the first place. The only alternative is to use language in some highly formalized, specific way that is in fact quite different from how we use language in everyday life. This still happens today but was even more common in the past before modern algebraic and calculus notation was invented (cf., eg., Philosophiæ Naturalis Principia Mathematica). The obvious downside to this approach is that it turns reading a physics text into an almost lawyerly task.
Some people will object to this, and with reason. Properly, Newton’s second law has physical content outside of any notion of mass. We could say that the essence of Newton’s second law is the statement that the acceleration of a body (a) is linearly proportional to the force it experiences (F). This is a nondefinitional observation about how the world works. The definition arises when we introduce a constant of proportionality m and give it the name “mass”.
Supplying a “known, constant force” is actually much harder than one might at first think: We can’t use gravity because its strength depends on the mass (!!!). We can’t use a spring because it depends on the extension of the spring (which will presumably be related to the object’s position). One of the better approaches is probably to imagine supplying “hit” over a short period of time so that there isn’t time for the force to change appreciably. Since the object only accelerates while being actively acted upon by a force, we simply take the change in speed before and after the hit and divide by the length of the small time interval during which the hit occurred, and declare the result to be the object’s acceleration.
Supplying a sensible definition to force is probably even trickier than supplying a sensible definition to mass, which is why I won’t even begin to touch this topic here. The difficulty of the problem is obscured by the fairly common misconception that Newton’s second law is a definition of force. It is not. The Feynman Lectures on Physics, Vol. 1, ch. 12-1 has a good treatment of why.
The most precise experimental test of the equivalence of inertial and gravitational mass to date was completed in 2022 by the MICROSCOPE satellite experiment. The experiment measured that the two masses are identical to one part in one thousand trillions (1,000,000,000,000,000).
Of course, Galileo and Descartes and Kepler and Brahe et al. laid the foundations, but Newton really did erect the whole self-consistent structure atop those foundations, a structure which would leave plenty of room for others to innovate but which would remain fundamentally unchallenged for hundreds of years.
Another potential difficulty is that when Newton wants to talk about how much of something there is, he never (before Book III) just quotes a number measured in some units. Instead he always speaks of ratios between things. This approach is commendably honest to the fact that this is actually the only way to talk about amounts. When I say something has a volume of 2 cubic feet, this is just a shorthand for saying its volume is twice the volume of some other reference volume (the “cubic foot”), and I am assuming that you know what that volume is. Newton recognizes that the physics exists in the ratio between the two volumes regardless of whether or not we are both standardized to the same reference volume and so he drops the “units” and just talks about “double space”.
Newton describes these experiments in Book III, Proposition VI, Theorem VI.
This is a wonderfully unintuitive fact because the world is such a frictionful place. See this classic video of a hammer and a feather dropped at the same time on the moon (where air resistance is negligible).
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“There are more things in heaven and earth, Horatio, / Than are dreamt of in your philosophy.”