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.

August 17th, 2014 at 21:52

“The last post garnered 100 times the total number of views I had previously”

You’re welcome. That was because I (Ree81) posted it to reddit’s futurology section, where it got over 1000 upvotes (which means it became popular).

http://www.reddit.com/r/Futurology/comments/2dc64l/a_solid_summary_of_the_impossible_space_drive/

As for conservation of momentum I’m just a layman, but couldn’t it be possible that the EmDrive simply spends more energy in one direction, kind of overriding what usually causes momentum, that is, a kind of principal of equality?

Like… the universe likes to “do stuff” with the minimum (optimal) amount of energy expended, right? Well what if momentum arises in the same way? Simply because “it’s easiest to do it that way”. That’d mean that if you create a system (EmDrive) where energy is expended more in one direction, that energy has to “go” somewhere, take some form, and the universe is kind of forced to convert it into momentum as a sort of last resort.

I know it sounds like gibberish, and it probably is. But it sounds reasonable enough to me.

August 17th, 2014 at 21:56

Oh and what I meant by “easiest to do that way” was this.

Let’s say you’re floating in space with your twin (of equal weight), in a space suit, and you push your twin away. Now you’re going to start drifting away from each other at the same speed relative to where you started off.

The reason one twin wouldn’t gain a higher speed is because this is simply the easiest way of doing things, the most energy efficient.

August 18th, 2014 at 11:33

Thanks for the share. That’s probably the most exciting day my blog will every see. And thanks for continuing to read even after the deluge is over. I’ll have another two posts up over the next couple of days. The first is about the quantum vacuum, and the next is a special relativity estimate of the maximum thrust efficiency one of the drives could hope to have. Spoiler alert: The Cannae drive exceeds the maximum thrust efficiency by a bunch, leading me to be skeptical of the results.

I’ll have to respectfully disagree with your ideas about momentum. There are a few reasons.

1) Conservation of momentum in particular is one of the most foundational pillars of physics, and has been extensively tested in almost every conceivable context. While quantum mechanics technically allows its violation over very short distances (as in 1E-40 m, or smaller than a billionth of a billionth of billionth of a millimeter for something like the drive), it is the cornerstone of even our understanding of virtual particles. If it’s not true, then I’m not sure we can even talk about virtual particles any more because the theory behind them would be broken.

2) Momentum has direction associated with it. It is what we call a vector quantity. That way, when you and your twin push each other apart, one has a momentum one way, and the other has a momentum the opposite way, and when we add the vectors, they cancel out. Energy is what we call a scalar quantity — it has a value, but it has no direction associated with it. It doesn’t make physical sense to talk about expending energy in a certain direction.

3) There is a specific quantity that is minimized in mechanical situations that determines exactly how objects move. It is called the “action”. It uses some heavy math, but the basic idea is something like this: you try out every possible way an object can move through a situation (given forces on it and potentials and such), and see which path and momentum function minimizes the difference between the kinetic energy and potential energy added up over the time it takes. The path and momentum which minimize the action is the path that actually gets taken. That’s the “easiest way” for the system to evolve, although it sure isn’t easy to calculate a lot of the time. If you want to look this technique up, it’s called Lagrangian mechanics, and we use it for both classical and quantum physics.

August 18th, 2014 at 16:23

Thanks for your input.

About doing calculations, I read on reddit that someone tried to calculate how much thrust it could generate (or maybe it was efficiency?), but ultimately failed because the equations don’t allow for conservation of momentum to be broken.

Oh, here’s the thread: http://www.reddit.com/r/Futurology/comments/2djinf/greg_egan_calculates_emdrive_microwave_cavity/

August 18th, 2014 at 20:50

Thanks for the link! In particular, the Greg Egan link looks very good. I’ll study it more when I have the time. But, it appears to be very different from the calculation I have. That’s because Greg Egan is generously working within the framework of the EM drive theorists. They explicitly say that momentum *is* conserved for theirs, but is some rarely-recognized way. Greg Egan is (it seems at first reading) showing that their assertion doesn’t work.

It’s interesting that the Chinese experiments, the EM drive, and the Cannae drive keep being lumped together as replications of an experiment, because they are all somewhat different, and are only marginally consistent. Further (not that this *really* matters), the asserted mechanisms behind how they might work are all very different.

Now, there are *no* physics equations that exist that allow for conservation of momentum to be broken (except for the uncertainty relation in my previous comment, over very short distances).

Does that mean that the drive experiments are definitely wrong? No. But it does mean that they have to be stronger than the three centuries of mountains of evidence that conservation of momentum holds across all scales.

That is: The Cannae and EM drive could potentially work, but the experiments to date are not nearly enough evidence to show that they do. People keep insisting that the Chinese experiment, the EM drive experiment, and this experiment is plenty of evidence, but there is enough opportunity for unaccounted systematic error to ruin their results, whereas there are loads and loads of experiments that confirm conservation of momentum, as well as very fundamental “proofs”^1 that I’ve listed in the article. That’s why stronger experiments must continue to be performed before this claim is validated.

It would be awesome for physics if it were shown that there are any circumstances in which conservation of momentum could fail! But I’m not holding my breath at this point.

^1 I use proof in quotes, here. Physics is an experimental discipline. If enough evidence amasses that conservation of momentum does not hold, particularly in some specific contexts, then that’s it — it doesn’t hold in those cases, and the axioms of our proofs must be flawed. But, to reiterate, thus far the centuries of evidence favor conservation of momentum holding at all scales.

August 19th, 2014 at 04:15

Agreed. I’ve been saying across reddit that at this point, this amount of evidence isn’t proof, and no one should blindly believe we’ve broken conservation of momentum. But there’s nothing wrong with being hopeful, and the experiments *are* highly interesting even if somewhat lacking *right now*.

Evidence will pile on one side or the other.

As for why we have so much evidence of conservation of momentum is probably because we haven’t even tried breaking it earlier. It’s kind of like the lobster and the ocean metaphor.

If I go to the ocean and scoop up some water and don’t catch a lobster, are there no lobsters? What if I do it 100.000 times? Absence of evidence is not evidence of absence, after all.

Now if NASA could just build another, more powerful version that could also withstand a vacuum…

August 20th, 2014 at 04:06

The most interesting option of course is that we don’t fully understand the quantum vacuum. And of course there is ample evidence of that already: Dark Matter and Dark Energy being immediately first to mind for me at least. Exciting times! We are at the beginnings of a revolution in our understanding of the universe (again!). The quantum world is still so poorly understood.