I almost certainly won’t keep up this momentum. The last post garnered 100 times the total number of views I had previously. I doubt this one will keep up, but I shall press onward.

Now: About rockets.

A conventional rocket works by throwing stuff backwards. If stuff is thrown backwards, other stuff must be thrown forward, just like a rifle kicks backward when the bullet flies forward. We call this the conservation of momentum.

For a rocket, the fuel that is burned doubles as “reaction mass”, mass that gets thrown backward to fling the rocket forward. The combustion supplies the energy (something *all* propulsion mechanisms must have), and the exhaust provides the push.

A drive that wouldn’t have to carry its own reaction mass would be very handy. First, you don’t have to carry dead weight into space, which is incredibly expensive. Second, without a limit to the reaction mass you can access, there is not nearly as much of a limit on how long you can provide a thrust. (Specifically, the limit is only the available energy).

How do we *know* the Law of Conservation of Momentum is true? Could the Cannae drive disprove it? This *is* the law that some say the Cannae drive violates. You can’t have an object sitting in space (or in a vacuum chamber, or whatever), and then suddenly move to the left. Something else had to use a force to move that object, and that something must experience a force to the right in the process.

The conservation of momentum is ingrained in physics — if it isn’t true, then we have a lot to redo. Like, the past 300 years. Fortunately, we have at least two good proofs^{1} of conservation of momentum: Newton’s 3rd Law and Noether’s Theorem

** Newton’s 3rd Law**

You know this by heart, right? “For every action, there is an equal and opposite reaction.” But what does that *mean*? I usually find that people can quote Newton’s 3rd Law (after you remind them which one it is), but they have trouble applying it.

What Newton’s 3rd Law means is that every force occurs between a pair of physical bodies^{2} and that every force has a friend. Say you’re standing facing a buddy, separated about about a foot. Each of you keep your feet together. Now, you put your hands on your buddy’s shoulders and push. What happens? You both fall over. Why? You pushed your buddy, right? Well, Sir Isaac says that when you pushed on your buddy, your buddy automatically, simultaneously pushed back on you, just as hard, in the opposite direction, without lifting a hand. In this case, the pushing occurs between the surface of your hands and your pal’s shoulders.

This happens for every force. The floor is holding you up, and you’re pushing down on the floor ^{3}. The Earth is pulling you down with gravity, and you’re pulling up on the Earth with gravity, too, just as much ^{4}.

What do forces do? They change the momentum of objects over time, as encapsulated in Newton’s famous 2nd law (in a less famous form), F = change in momentum / change in time.

Back to the rockets. By Newton’s 3rd Law, the stuff pushes back, and the rocket goes forward. That very simple idea is easy to get confused once you add more to it. My favorite example is a January 13th, 1920 science editorial in the New York Times about Robert Goddard’s rocket research. I read this to my students with the most sniveling voice possible:

“That professor Goddard, with his “chair” in Clark College and the countenancing of the Smithsonian Institution, does not know the relation of action to reaction; and of the need to have something better than a vacuum against which to react – to say that would be absurd. Of course, he only seems to lack the knowledge ladled out daily in high schools.”

This is hilarious, because the author is making fun of Goddard for misunderstanding the Newton’s 3rd Law while completely failing to understand it himself. The atmosphere isn’t needed, because the exhaust itself provides the reaction force.

Newton’s 3rd Law, if you integrate over time, gives you the Law of Conservation of Momentum.

Let’s go back to you and your friend. You push your friend and change their momentum in your forward (their backward) direction. Newton’s 3rd says that your friend automatically pushes back on you just as much (without raising a hand), changing your momentum in your backward direction. If the forces are equal magnitude, and the amount of time the forces are applied is equal, the the magnitude of momentum change is the same for each of you — only one is your forward direction and one is your backward direction. Therefore, the *total* momentum of you and your friend added together is the same before and after^{5}. That is, momentum is conserved!

**Noether’s Theorem**

I advise you to just go and read everything you can about Noether’s Theorem^{6}. It has been called the most beautiful result in physics, and I’d have a hard time disagreeing. But it’s a little more involved than Newton’s 3rd law. I’ll state it first, and then explain a simplified example: For every continuously differential symmetry of the system, there is a conserved quantity.

What the heck does that mean?

First of all, we’ve already shown that *between* particles, the total momentum is conserved, by Newton’s 3rd law. So, let’s focus on a specific particle and see what happens (arguably, this is closer to an analogy than an example).

If you’ve got a particle sitting in space (we’ll start in the frame of reference in which the particle is initially at rest), then it’s only going to move if a spot right next to it has lower energy, right? Like a particle resting on the slope of a bowl. It will experience a force toward the bottom of the bowl, and that’s because that direction has a lower (gravitational potential energy). We can think of the environment causing forces on objects by the shape of the energy profile: the forces go from higher energies to (nearby) lower energies.

Well, if there is a force applied to that particle, its momentum changes, right? So the momentum would not be conserved.

For momentum of the particle to be conserved, there would have to be no slope: the energy would have to be the same everywhere near the particle (this is translational symmetry). Then, whether the particle is initially rolling or initially staying still, it’ll keep on going with constant momentum.

That energy being the same everywhere nearby the particle? That’s the continuous symmetry. In this case, if there is a continuous spatial symmetry, then momentum is conserved.

But I cheated! The bowl is also a physical body! This really belongs to the Newton’s 3rd Law example!

So, we have to consider all particles in the universe to be, together, our “particle” here, and we’re comparing it to the energy due to the shape/geometry/etc of the Universe^{7}. That is, as long as physics is exactly the same if you pick up all the particles in the Universe, scoot it all over by a smidge, put it back down, and everything (forces, energy, particle motions, etc) acts exactly the same as before, then momentum is conserved in such a Universe!

Similarly, if you cut and paste all the matter in the Universe from one time to another, and everything is the same, then energy is conserved^{8}!

There are others, too. The symmetry of nature imposes these conservation laws, which is pretty sweet. The proofs are incredibly easy with the right mathematical framework, so if you are a physics student and have seen Lagrangian mechanics, you should really look it up.

**What this means**

Essentially, one of three things must be happening in the Cannae experiment:

1) It’s wrong.

2) It’s right, and momentum is not conserved. (In which case, I’m out of a job, because physics is broken).

3) It’s right, momentum is conserved, and *something* is carrying away the unmeasured momentum.

Let’s focus on (3). This *something* must be particles or radiation, both of which are, in principle, detectable. A falsifiable prediction! If the Brady et al group is right, then we should be able to detect these particles.

Now, why are physicists so skeptical that (3) is possible? The claim by the Brady group is (if one is a little generous in rephrasing it for them into something meaningful) that virtual particles in the vacuum are being given energy and promoted to real particles, and these real particles are carrying away momentum. What’s wrong with that (and/or, what does that mean)?

Find out next time…

1. Insofar as you prove things in science, right? Everything is conditioned upon further evidence. But you can prove things in math, and these proofs are mathematically solid. All that *could* be questioned is whether the model behind the math is supported by evidence in the real world, which it hella is.

2. Things get a little uglier in a field theory, where one considers the agent to be the field itself, but conservation of momentum still works out in the end.

3. This is not gravity, by the way. It’s just a contact force, which we call the normal force (normal in the math sense of perpendicular). The action/reaction pair of forces must always be the same type of force. The floor isn’t pushing up on you with gravity, is it?

4. But the Earth is way bigger than you, so the same amount of force doesn’t matter as much to it as it does to you.

5. Momentum is what we call a vector quantity — it has magnitude and direction. Some amount of forward momentum, plus the same measure of backward momentum add up to zero.

6. Did I really just link Wikipedia? As if my dear reader wouldn’t search there first, anyway? Yes. Yes, I did.

7. I’m sorry, but my analogy is getting gradually less precise, though I think the conceptual idea is adequate for most people. More technically, what we’re *really* doing is looking at the Lagrangian and/or Hamiltonian of our system of particles. The Lagrangian and Hamiltonian are both ways to account for the energy and motion of particles in the system. These are determined by the environment of our particles, i.e., everything that exists that isn’t our particles. This is the Universe itself.

8. This is probably not the best place to bring this up, but it turns out that energy is not absolutely conserved, because of expansion. However, for small volumes of space (say, our galaxy), or over small durations^{9}, the energy is conserved.

9. But not *too* small, because then quantum particles can borrow energy for short periods of time, so long as energy is conserved on average. More on this tomorrow.