K^2

You got that backwards. What makes people think that it is an elaborate joke? Because an elaborate joke would have a solution. A dumb one would be just random symbols. But you always hope that you got intrigued and pulled into wasting your time by someone intelligent, which would imply an elaborate joke. I don't think anyone seriously entertains the idea that this is anything but a joke, though.

50-something, apparently.

Is there a translation to anything machine-readable? Or is anyone willing to write one up? There are a few techniques I wouldn't mind trying, but I don't have the time for transcription.

I'm willing to bet that there is a phase transition in there somewhere, but I couldn't tell you exactly where. Off the top of my head, I would classify any stable arrangement where center of mass is at the center of the parent body to be a ring. This is never going to be true for a "few" moons. Certainly not for two. But how many you'd need before it turns into a "ring", I don't know.

In flat space-time. Space around star/planet is not flat. That's why we have gravity. In curved space-time that's sufficiently flat locally for Alcubierre Drive to operate, this is going to work differently. This is basic GR.

How did that even happen? Surely, an asteroid that small does not have a Roche limit for anything remotely rigid. Could these form as a result of an impact?

Yup. Thanks for catching that.

You are making a standard mistake, assuming that energy conservation works regardless of choice of coordinate system. That is simply not true. Energy is only conserved in an inertial frame of reference. If you consider everything from perspective of the rocket, you are in an accelerated frame of reference, and energy is not conserved. So your analysis is faulty. If you want to analyze it from perspective of energy, you should be considering motion relative to the star/planet, as that's, effectively the relevant inertial system. In that system, you are moving faster when you are closer to star/planet, and therefore, you get more work out of the same amount of fuel due to traveling a longer distance while burning it.

Question of "original direction" in curved space-time is not a trivial one. Would you say that an object in orbit ever changes its speed and direction? Not so, according to General Relativity. It continues "in a straight line", because the covariant derivative is precisely zero along a geodesic. You are going to get precisely the same effect with a warp bubble. The difference is that you are going to go along a spacelike geodesic, allowing you FTL travel. When you arrive, your velocity relative to the bubble will be exactly the same, but bubble's own trajectory is going to change because you've been moving in curved space-time. As a result, when you exit, your velocity, in Newtonian sense, is going to be very different. Different how? Well, that's a more complicated question. You'd have to work out what the spacelike geodesics actually are going to be. I suspect that patched Schwarzschild solutions are going to be a good enough approximation, just like patched conics are a good enough approximation for KSP to use. But I'm not familiar enough with Schwarzschild time-like geodesics to say what it's going to be like right off the bat. Except that you can't get 4N of thrust with a 2MW supply. You need about 1.2GW for 4N of thrust with quantum thrusters. The early experiments in the field have had errors. Theoretical analysis of quantum thrusters suggests that it cannot exceed efficiency of a perfect photon drive.

If the tower was light in original, it is not a fair replication. But that doesn't mean that the original experiment is reliable. Ice does not necessarily make a good frictionless surface. Air table experiment has the right idea. Somebody just needs to do that with a light tower to show that center of rotation is going to be near the gyro. In short, you still don't have a claim, and the source you have provided still refutes your claim, albeit, weakly, since it's not a perfect replication.

It's not that these people don't know Kutta-Joukowski theorem, or understand how angle of attack contributes to circulation. It's that they somehow don't realize that Bernoulli Principle isn't applicable to pressure differential on an airfoil. After all, technically, an air flow over a foil with no camber and positive angle of attack is still faster than flow bellow such a foil. Faster flow? Bernoulli! The misconception of "Lift is caused by Bernoulli" is hammered from so early on, that they have hard time overcoming it. And I don't think it's the case of not being familiar with D'Alembert paradox (which, admittedly, I never knew by name). It just seems like the fact that there is no lift for same reason that there is no drag isn't as obvious as it maybe should be. But then again, they probably learn the concept long before they start thinking about lift in these terms, so that might be the reason for it. Still, I find it really funny that rather than just connecting and causing a "Doh!" moment, it causes more of a reset that I've mentioned. And yeah, I've always been able to get through. Just never on the first try. P.S., it should be noted that it's entirely fair to approximate lift via Bernoulli Principle at low mach numbers, since low viscosity and low compressibility approximations are fair, so the boundary layer is thin and pressure gradient across it is very small. Meaning pressure differential in flows a small distance away from the wing surface is a fair approximation for a pressure differential at the wing surface. But distinction is important to understand, as well as the fact that this breaks down at higher mach numbers, when compressibility of the flow becomes a significant factor.

That really shouldn't be how a warp drive operates. I mean, it doesn't cancel conservation of energy. If your potential energy has changed, something has to give. There is a very "natural" sort of trajectory that an Alcubierre Drive can follow in a solar system to conserve energy and momentum, but it's nothing like what you're describing. I've been meaning to work out the actual math behind this for a longest time, but I never get far in it before I'm starting to develop a headache. I don't think I'm built for tensor calculus. I don't mind the abstract notions of it, but dealing with all of the coupled equations you get is just blargh.

We sort of know. Dark Energy contributes to the pressure terms in the stress energy tensor, which is distinct from energy term, which is the source of gravity in most contexts. So it's not going to "block" or "cancel" gravity in general. But it might be close enough for some specific purposes.

It's not outright impossible. In principle, a ship under warp can still send and receive messages. However, since it is moving FTL, it cannot send messages forward, and it cannot receive messages from behind. That should be pretty self-evident, but there are some papers that derive that rigorously for an Alcubierre Drive, in case anyone has any doubts. And the limitation here is precisely in that the ships have to be heading roughly head-on to break causality, which is exactly the direction the ships can't communicate in. In contrast, there is a specific direction in which messages can be sent without significant blue/red shifts, so a ship under warp can, in fact, briefly communicate with stations, outposts, or other places of interest as it passes by.

I can give you a full solution for a gyro of arbitrary mass, with arbitrary angular solution, attached to a base of another arbitrary mass on a frictionless surface. This might be slightly above undergraduate level, but for a graduate student in Physics, this should be peanuts. In fact, I think I've seen something similar on a candidacy exam one year. Anyways, let me write up some equations, and I'll post a general solution. Edit: Actually, that's way simpler than I was even thinking about. So long as the surface really is frictionless, and we can ignore the mass of the arm connecting gyro to base, you don't need anything complicated. Say you have gyro of mass m, carrying angular momentum L. (L ~ Émr²/2 for a solid disk gyro, and L ~ Émr² for a ring with spokes gyro). If the arm has length d, then the equation for rate of change of angular momentum is trivially dL/dt = mgd. Therefore, precession angular frequency is Ω = mgd/L. And, of course, center of rotation, given base of mass M, is going to be located dm/(M+m) from the center of the base. On video, the gyro goes through half rotation in 10.5 frames at 15FPS. So that's 21/15 seconds per turn, or Ω = mgd/L = 4.5s-1. It's harder to judge the sizes, but if the gyro is about 5cm across, then the arm is about 20cm long. Since this is just a very rough estimate, lets go with that. I'll also take gyro to be a solid steel disk, about 1cm thick. It's probably a bit off, but again, we just want to get an idea for what's going on. At typical steel density of about 9g/cm³, that gets me about 175g of gyro, which would be dead in the middle of the range you've suggested. We can use that to get L = mgd/Ω = 0.076 kg m²/s. And that lets us estimate É = 2L/(mr²) = 1,390s-1 or something like 13,300 RPM. This is a bit on the high side, so some of these size estimates are probably a little off. One possibility is that the gyro is actually a bit larger and has a ring shape, rather than solid cylinder, which at the same mass, would bring the above down by about a half. Still, this can definitely be ballpark. If we take the base to also be a steel disk, 1cm thick, and I'd say it's about same diameter as the arm, so about 20cm, we get 2.8kg. That's actually too heavy. It might also be an aluminum alloy. Duralumin has density of 2.8g/cm³, which would put the mass of the disk at 880g. If that is the case, center of rotation is going to be 3.3cm from the center of the base. That's definitely about consistent with the video. So there is absolutely no reason to say that something weird is going on. If you were to obtain precise measurements, we can verify them precisely.

Local causality is consequence of more general local properties of space-time, but what it really means is that no matter how weird things get, around any point, you can find at least a small, finite neighborhood where causality is preserved. This is equivalent to saying that two particles can't pass through the same point at the same time moving faster than speed of light with respect to each other. It doesn't sound like a significant limitation until you consider the fact that all real particles are delocalized, and their probability amplitudes overlap quite frequently. Local causality violations would break the entire field theory. It's total anarchy, and you can't build a self-consistent theory, at least anything like what we have, without causality working at least locally. Global causality violation is future events causing past events. The most extreme case is time travel, and the reason why people don't like it is that it causes things like Grandfather Paradox. Which would be a problem in classical setting, but it isn't a problem with field theory. It doesn't mean that it's necessarily possible, but if there is some grand cosmic reason why it can't be so, we don't know it yet. That'd make for an interesting paper. If spectra and duration of these bursts is available, it'd be possible to compare to dropping out of Alcubierre warp. I know people have written some general analysis papers on what happens to matter trapped in the warp bubble at the exit.

Forget about it. I've had to argue with some aeronautical engineers about the topic for hours. Trying to explain to them that Bernoulli effect on the wing surface is zero, because air is static at boundary layer usually results in response, "But we can approximate it as inviscid flow, so that boundary layer is infinitesimal." At which point I remind them that in inviscid flow, Kutta Condition and the turbulent layer go away, so there is no circulation, and lift is exactly zero. That usually gets me a few minutes of blank stare, at which point they seem to reset and go back to reciting Bernoulli principle. Consider that my bug report for the universe. If you ever figure out a fix, let me know. Just the opposite. Quantization of fields relies on them being fields over real numbers. Lattice QCD has to jump through some hoops to get that stuff to work in a simulation. And even there, you usually end up with some errors because it's impossible to get exact quantization.

I have to stop you right here. Alcubierre Drive does not allow for time travel in flat space-time. While it's true in general that a pair of FTL ships make up a time machine, it requires communication between them mid-voyage. This is possible, in general, because we picture the two FTL ships pass arbitrarily close to each other. For Alcubierre Drive, it requires the two ships to exchange information mid-warp, while heading on courses that prevent direct communication with each other without warp bubbles overlapping. And later is impossible to achieve without disrupting the warp. In principle, there exist space-time configurations that allow for not-quite-CTCs which are traversable to a warp ship. (Effectively, making it a CTC from perspective of ship proper.) But I'm not even sure if such geometries exist in nature. At a minimum, we are talking about flying deep under event horizon of a super-massive black hole with extreme amount of angular momentum. If Kerr Metric is a valid interior solution for these, there should be some near-CTCs available, which are accessible to an FTL ship. And even that's a big 'if'. In short, even if there is some fundamental problem with violating causality, the basic principles of warp drive are not a violation. That said, as I've pointed out earlier, we only have real restrictions on local causality. There is no fundamental principle in modern physics that says that global causality must be observed. Local causality is sufficient to build an effective field theory for whatever global structure of space-time you happen to have. And thanks to the principle of superposition, there are no history contradictions in such a system even if one happens to traverse a CTC.

Not so much a plot as a MacGuffin to kick it off, but yeah, I seem to recall something like that as the motivation for the Destiny and the gate seeding ships.

On the scale of observable universe? Yes, we can. Can it be sentient on a larger scale? Yes. But by that point, we don't even care. There are a whole bunch of things that we used to classify as exotic matter that is now routinely observed at accelerators. For the rest, it depends on what you mean. There is a list of theoretical and hypothetical stuff that can be classified as exotic matter and at least a couple that can be classified as exotic energy. I'm not at all sure if in this context you are talking about exotic matter and energy as they relate to Alcubierre Drive theory or universe expansion.

You can't break local causality. Global causality is far more flexible.

I have no idea where you are getting this number from. If it's in reference to the gyro on ice experiment, we don't know sufficient details about it to say that it really was frictionless. If you mean the video from your link, then you simply have no sense of scale. The gyro there is much, much lighter than the base. Again, the PAGE YOU ARE QUOTING insists that the point of rotation is center of mass. And based on the video, I would agree that it's very likely. The base is a thick metal disc, significantly larger than the gyro. It can easily be up to 10x heavier.

It's entirely possible to propel canoe by rocking it forward and back. Water acts as a reaction mass here. And yes, your best bet for pendulum test is to build a trapezoid out of wires, so that there is no ability to tilt, but only move across. You also want much longer vertical distance, if at all possible.

Water will already boil at 2.3kPa and 20°C. So if you drop pressure in the flask to just above that, it will dramatically reduce the centrifugal effect you need for this to work. In fact, if this is going to work at all, that sounds like you'd have to drop the pressure.

Honestly, you're on the right track with your methods. You are re-inventing the wheel, so to speak, and it'd be much more useful for you to try and understand core concepts behind momentum conservation, which is basically why gyros work in the first place. But that aside, your approach is scientific. You've been trying to eliminate the variables as you find them, and your experiments tend towards the critical kind, rather than self-affirming kind.