Conservation of Linear Momentum

We define the Universe as the set of everything that exists. One of the elements of that set, space, provides a continuum in which all of the other elements carry out their existence. Those two statements comprise a single axiom, from which I will deduce Isaac Newton's laws of motion, thereby taking the first step in deducing the Map of Physics. I begin with an apparently simple question.

What motion does the Universe have?

That question has two answers. Seemingly inconsistent with each other, those answers give us the means to parley that simple axiom and one other into the most basic rules of existence.

Taken as a whole, the Universe absolutely cannot
move. Motion can only occur __in__ space. No space exists outside the
Universe for the Universe to move in. Therefore, the Universe can have no motion
as a whole.

But space exists as part of the Universe and bodies move freely within it, even though space as a whole cannot move. Can we make those bodies' motions part of the Universe and, if so, do they give the Universe as a whole a motion?

In answer to the first part of the question we must assert that motion consists in more than a body going from one place to another. Bodies possess motion as a property, one that does not inhere in the bodies themselves, but nonetheless provides the body with something that it may give to or obtain from other bodies. We see the truth of that statement to Reality reflected in the collision of two billiard balls: both balls suffer a change in their motions as a consequence of the collision.

Before I answer the second part of the question I must introduce my other axiom, a slightly modified version of Euclid's fifth Common Notion ("The whole is greater than the part."). That common notion usually comes to us as "The whole equals the sum of its parts" and I want to use it in that form. But of what do we seek parts and the whole? We may say that we seek the parts and the whole of what we can observe or reasonably infer from observations. I then adapt Euclid by proclaiming that with regard to any property that we may divide into parts and give to bodies the amount of that property possessed by the whole equals the sum of the amounts of that property possessed by the parts. In that way I bring into play the rules of arithmetic and all of its logical extensions (e.g. analytic geometry, calculus, trigonometry, complex analysis, etc.).

Certainly, then, the motion of the Universe as a whole must equal the sum of the motions of all of the bodies in the Universe. If I have a number of bodies and I have assigned some numerical value to the motion of each of them, then the motion of any collection of those bodies equals the sum of the motions of the individual bodies. That statement applies even to the collection of all of the bodies in the Universe. After all, in defining a collection of bodies I have merely drawn an imaginary boundary around those particular bodies. That act of drawing can have no effect upon the bodies properties. That statement remains true to Reality even if I draw my boundary to enclose everything that exists.

Thus the Universe considered as a whole has a motion equal to the sum of the motions of all of the bodies contained within it. But the Universe as a whole can have no motion, so we must have as necessarily true to Reality at all times that the motions of all of the bodies in the Universe add up to a net zero.

Fortunately for my purpose I have no obligation or need to measure the motions of all of the bodies in the Universe. I need only assume the zero sum and ask what rules maintain it. In answer I get:

1) A body cannot change its motion unless another body compels it to do so.

2) If a body changes its motion, then the body that forced the change changes its own motion by the same amount in the opposite direction.

Those rules bear a strong resemblance to Newton's first and third laws of motion, but before I can claim that they congrue perfectly with those laws I must prove and verify that what I have been calling motion congrues perfectly with what physicists call linear momentum.

I want to show you how to relate the motion of a body to the body's velocity; more precisely, although we associate velocity with bodies moving, I want to show you that we need more than velocity to describe the amount of motion in a body. To that end I define dynamically equivalent bodies in the following way: two bodies are dynamically equivalent to each other if they possess equal but opposite amounts of motion when they move at speeds that are equal and oppositely directed.

If we have two dynamically equivalent bodies floating motionless before us and if some force pushes them apart, we know that the motions of the two bodies must add up to zero both before and after the force acted. Further, we know that those bodies move at the same speed, albeit in opposite directions. If we now reverse that scenario, we see the two bodies moving toward each other at the same speed, colliding and sticking to each other, and ending up as a two-body cluster floating motionless before us. Now imagine that we have a set of standard bodies, each dynamically equivalent to all of the others, and let's perform some imaginary experiments.

First, consider the scenario I described above, that of two standard bodies coming together at the same speed V and forming a motionless two-body cluster. How would that scenario look to an observer moving in the negative direction at the speed V; that is, with velocity -V? Initially one body floats motionless near that observer and the other body approaches it with velocity +2V. After the collision the two-body cluster moves past the observer with velocity +V. As that observer conceives the action, one body gained a certain amount of motion and the other body lost the same amount of motion in the collision, in accordance with Rule #2. More importantly, that observer can tell us that one standard body moving with a certain velocity carries as much motion as do two standard bodies moving with half that velocity.

With that fact in mind, let's now imagine that two standard bodies adhering to each other float motionless before us. We see a third standard body, moving with speed W, strike the pair and stick to them. What speed will we measure for the resulting three-body cluster?

Again we have a collision in which one object, the two-body cluster, gains as much motion as the other body, the striking body, loses and in which the resulting three-body cluster carries away all of the motion originally residing in the striking body. But now we know that the striking body will lose twice as much velocity as the two-body cluster gains and we know that the velocity w that the three-body cluster gains equals the velocity that the striking body has left over after the collision. Thus we can write an equation

W-2w = w,

(Eq'n 1)

which yields our answer, w = 1/3 W. Thus we find that a standard body moving at a certain speed carries as much motion as do three standard bodies moving at one third that speed.

We might believe that we discern a clear pattern here, but let's make certain with one more imaginary experiment.

Imagine that three standard bodies adhering to each other float motionless before us and that a fourth standard body, moving with speed S, strikes the trio and sticks to it. What speed s will we measure for the resulting four-body cluster?

Yet again we have a collision in which one object, the three-body cluster, gains as much motion as the other body, the striking body, loses and in which the resulting four-body cluster carries away all of the motion originally residing in the striking body. But now we know that the striking body will lose thrice as much velocity as the three-body cluster gains and we know that the velocity s that the four-body cluster gains equals the velocity that the striking body has left over after the collision. Thus we can write an equation

S-3s = s,

(Eq'n 2)

which yields our answer, s = 1/4 S. Thus we find that a standard body moving at a certain speed carries as much motion as do four standard bodies moving at one fourth that speed.

We have a clear pattern and we can see that it will continue indefinitely. For a single standard body moving at speed V striking and adhering to a motionless cluster of N standard bodies, which then moves at speed v, we have the equation

V-Nv = v,

(Eq'n 3)

which has the solution v = V/(N+1). We now know that the quantity that remains the same throughout these encounters (and any other collisions that we may conceive with different numbers of standard bodies) equals the product of the number of standard bodies comprising a cluster and the velocity of the cluster (with the proviso that the bodies all move with the same velocity; that is, that the cluster has no internal motions, such as rotation). That is a rather clumsy way of stating the proposition, all the more so when you see that we will repeat the phrase "number of standard bodies comprising a cluster" frequently in any suitably detailed discussion of moving bodies. To simplify further discussion let's replace the phrase "number of standard bodies comprising a cluster" with the phrase "mass of a body".

On first impression that definition of mass looks nothing like Isaac Newton's. In the first of the definitions with which he opened Philosophiae Naturalis Principia Mathematica Newton defined quantity of matter (mass) as a "measure of matter that arises from its density and volume jointly". Unfortunately Newton didn't define density. In his second definition, though, Newton defined quantity of motion (linear momentum) as a "measure of motion that arises from the velocity and the quantity of matter jointly". Elaborating, he noted that "if a body is twice as large as another and has equal velocity there is twice as much motion". That elaboration transforms Newton's definition of mass into a proposition equivalent to my definition of mass in terms of dynamically equivalent bodies. Thus, when we use the word "mass", Newton and I refer to the same thing.

But the product of the mass and the velocity of a body gives us the body's linear momentum (p = mv), so Rules #1 and #2 do indeed congrue with Newton's first and third laws of motion. Now what can I say about Newton's second law of motion?

So far I have described changes of motion as impulsive events, in which the motions of bodies change more or less instantly. But motion can also change gradually. Instead of having its velocity bumped abruptly from one value to another, a body can accelerate smoothly. We will encounter this kind of situation frequently, so we want to replace the phrase "the rate at which the body's linear momentum changes" with a more compact term. Conventionally we use "force" and we define it algebraically thus:

F = dp/dt = d(mv)/dt = ma + vdm/dt.

(Eq'n 4)

This gives us Newton's second law of motion in its common mathematical form, though we have it more as a definition than as a prescription for the behavior of bodies. Indeed we have equated cause with effect, defining the applied force (F) as the cause of a change of motion by equating it to the inertial reaction (dp/dt) as the effect upon the forced body.

Of the two terms on the rightmost side in Equation 4 we find the first the more familiar. If we push on a body, it accelerates at a rate proportional to the applied force and inversely proportional to its mass. And how do we know how much force acts on a body? In essence we calibrate some apparatus that exerts a force by having it push on bodies of known mass (standard bodies) and then measuring those bodies' accelerations. This tells us that the measure of force manifests itself to us as a ratio, just as mass does. As we must do with all of our measured quantities, we must refer force to some arbitrarily defined standard. Nonetheless, once we have defined that standard we may apply the concept of force to making useful statements about Reality.

Less familiar to us and ultimately a good deal spookier, the second term on the rightmost side of Equation 4 tells us to multiply a body's velocity by the rate at which the body's mass changes. How shall we interpret that instruction? Commonly a rocket motor provides an example, though the body in question is not the rocket itself, but its exhaust plume. In calculating the force (thrust) generated by the rocket motor we use the velocity of the plume relative to the rocket and the rate at which the motor feeds mass into the plume; the rocket's bumping the velocity of its propellant up to the speed of the plume and thereby gaining the necessary equal and oppositely directed reaction produces the force.

As for the spooky part, that second term plays the central role in the deduction of Einstein's famous mass-energy equivalence equation. But I want to discuss that topic in another essay in this series.

In the study of General Relativity we ultimately encounter some discussion of Mach's Principle, which expresses Ernst Mach's objection to the absolute space that Isaac Newton assumed in the Principia and described with reference to his spinning bucket experiment. Newton assumed that space exists independent of matter, acting as a stage on which matter plays out its role without altering the stage. In Newton's imaginary experiment the water in a bucket experiences centrifugal force when it rotates in that absolute space. Mach objected that the inertia that Newton imputed to the water has meaning only in reference to other bodies (e.g. "the fixed stars") in the Universe. More commonly put, the water has mass because other bodies exist.

Based upon what I have deduced above, I object to that usual interpretation of Mach's Principle and say that bodies don't have mass because other bodies exist: I say that bodies have mass because nothing, not even "outside the Universe", exists outside the Universe. I can't say that Mach got it completely wrong: certainly we can say that the distribution of matter in the Universe has some relation to the inertial property of matter. But Mach still saw space as a kind of absolute thing, in which one view of the distribution of matter provides the inertia of bodies. We have since learned that space is much weirder than Mach could have known. In later essays this topic will re-emerge and we will confront the issue of how the weirdness of space affects Mach's Principle.

Thus I have obtained Newton's three laws of motion. Chief among them we have the third, the law pertaining to conservation of linear momentum. It is an absolute law, admitting of no exceptions. Under no circumstance can linear momentum be created from nothing or destroyed to nothing. An appropriate equal and oppositely directed reaction must always accompany any action that changes the linear momentum of a body.

Some people dislike that law and have striven to break it. I remember that in 1962, when I was a sophomore in high school, I read in Popular Mechanics of a device called Dean Drive. In its essentials the Dean Drive comprised a pair of counter-rotating masses mounted in a sliding frame whose motion within a larger frame was controlled by a motor. For a short time I could fantasize about vessels, carrying cargo and passengers, being hauled into space and around the solar system by plutonium-powered steam locomotives not significantly more complicated than what Southern Pacific was running on its tracks ten years before. But nothing ever came of Dean Drive. In succeeding years I devised some of my own space drives, even when I should have known better (after all, I took my Bachelor of Science degree in physics (UCLA, 1969)). Nothing came of those, either, except a clearer understanding of the laws of physics. No, from my own experience I can say that Newton's third law of motion is absolutely unbreakable.

That fact makes Newton's third law a touchstone on which we can test other presumed laws of physics. It gave physicists in the first half of the Twentieth Century the rationale behind the discovery of the neutrino, for example. When they discerned from their measurements of certain beta decays that the decayed nucleus and the emitted electron (or positron) seemed to acquire together a net linear momentum, those physicists hypothesized the existence of a thitherto unknown particle, one that would carry the required equal and oppositely directed reaction to the momentum gained by the nucleus and the electron/positron. The physicists had to take Newton's third law on faith at that time and observations made in 1956 near Savannah, Georgia, justified that faith by showing effects that could only have come from neutrino bombardment of the detector. Having deduced Newton's third law, we no longer have to take it on faith and that fact makes the law an even better touchstone in theoretical physics, particularly in the development of the Map of Physics. Henceforth I will use logic to parlay that law into an ever-widening array of laws governing the motions of bodies in the Universe.

habg