K^2

The No-Communication Theorem is a theorem. That's why we call it that. That means that it's not something we don't think can be done. Or that we have strong evidence that it can't be done. It's that it absolutely cannot be done, with an actual mathematical proof. Entanglement does not work this way. It's possible that there is no such thing as entanglement, and we are seeing a completely unrelated phenomenon which we have not been able to distinguish from entanglement experimentally so far, but one that works by an entirely unknown mechanism which allows for causality violations, because the whole of Quantum Mechanics is absolutely wrong. But a) don't hold your breath for it, and b) even if it turns out to be the case, it means that absolutely all of physics is wrong, and anything is just a guess. Entanglement, though, cannot be used for communication. That is a mathematical fact. That doesn't burry the entire concept of FTL; we have several known loopholes in theory that allow for completely different ways to send signals FTL, and there can be other, undiscovered mechanisms for superluminal communication. But it's not entanglement. I can't stress it enough.

Nitpicking, but thinking of it as changing state of the particle is very confusing. It's better to think of it as changing state of observer. Mathematically, entirely a moot point, which is why both interpretations exist, but when you think about measurement changing state of remote particle, you get into a bad intuitive state where you expect information transmission to be possible. If you think of it as observer's state changing, then it's perfectly clear why no communication takes place. That said, you can actually use entanglement for teleportation. That's exactly how quantum teleportation works. The catch is that it requires a classical side-channel to operate. So it's real teleportation, and we've done it in a lab, but it's still speed-of-light limited. On the topic of wormholes and warp drives, math says it's possible. Math also says it's going to be very difficult. The crux is that FTL of any kind is equivalent to time-travel in another frame of reference. If you can time travel in some frame of reference, you can construct a closed time-like curve. And all known stable configurations that include closed time-like curves have regions of negative energy density. In other words, to have a stable wormhole or stable warp, you have to have something with negative energy. There's a lot of debate on whether such a thing is even theoretically possible, and whether something like Casimir Effect qualifies. Needless to say, nothing remotely the required magnitudes has ever been observed or even predicted to exist. But the important keyword above is "stable". There are known unstable configurations that allow for time travel and FTL. Notably, Kerr Metric, which correctly describes space-time near a rotating black hole, when taken to its extreme, produces what is known as naked singularity. A type of ring of finite radius and zero thickness, possessing immense mass and absolutely insane amount of angular momentum, which does not have an event horizon like a conventional black hole, but does have a small region of space around it which allows a particle to exit it before it enters. It is a valid configuration of space-time and does not require anything exotic, like negative energy, but it is known not to be stable. That means we'll have no luck finding one that exists naturally, but it still remains unknown whether such structures can be briefly created artificially. If they can, theoretically, this gives us ability to do all sorts of insane things, and FTL will be just a cherry on top. Unfortunately, all of this is ridiculously complex computationally. Exact solutions are almost non-existent in GR, and numerical simulations are very expensive. We know all of the equations, but we know almost nothing about possible solutions. And it may take us centuries to run through various possibilities to find something useful. Or we could get lucky, and one of our particle accelerators will produce naked singularities by accident. Stranger things have happened. There just isn't much point in guessing on something that's that far beyond the boundaries of well-understood. There's also an entire topic of sub-light warp and time-like wormholes. These have no time-travel implications, and so are free of exotic requirements. If there's going to be a way to get to the edge of Solar system in hours instead of decades, I'll take it. FTL may be the holly grail, but simply going light-speed without expending insane amounts of fuel would be fantastic. But even that remains very poorly studied at the moment.

When you are looking at surface of the water as equipotential surface, you have to consider frame of reference where water is at rest. It's the only case where it makes sense to talk about equilibrium. And since the Earth is rotating, the relevant frame of reference is a rotating one. In a rotating frame of reference, potential energy picks up the mω2(r sinθ)2/2 term. So the total specific (per unit mass) potential energy of fluid on the surface of a rotating sphere is given by U = ω2(R sinθ)2/2 - μ/R For Earth, θ is the latitude, so r has to be smaller near poles for the same potential energy. Formal derivation is via Canonical Transformation of the Hamiltonian. It's a lot easier to work with cylindrical coordinate system here, so I'm going to ignore θ. The generating function for azimuth angle is then given by F(φ, L, t) = (φ - ωt)L This generates a contribution to Hamiltonian (total energy) equal to K - H = ∂F/∂t = ωL For an object at rest in a rotating frame, kinetic energy is exactly ωL/2, so we have U - U0 = K - H - ωL/2 = ωL - ωL/2 Finally, for an object at rest in a rotating reference frame, L = ωr2, so we can simplify in one of two ways U - U0 = L2/(2r2) = ω2r2/2 Which is exactly the effective potential of a unit mass in a rotating frame, and with r = R sinθ it matches up with the equation at the top.

I'm having trouble looking up at what voltage capacitors will start to leak due to field electron emission, as that's going to be limiting factor on charge. Second limiting factor is mechanical strength. That one's easier. You can get steel wire loop going at about 500m/s regardless of loop radius and wire diameter. That isn't much. Gut feeling is that you'll get stronger current with superconducting wire.

I don't know if that's what we're getting, though. In video I've linked above, authors of the paper appear claim that their rough estimates on thrust experienced by loop of wire is roughly what they're getting. Now, I have not run these numbers, nor have I read the paper, so I'm going purely from the claim as reported in the video, but that sounds like tether is exactly what they got. Don't get me wrong, I'm not about to knock tethers, but I don't see an indication of a path to a more compact design. It's still a wire carrying current. You either increase its length, or you increase the current to get the thrust you need.

Au contraire. Landing gear is going to remain firmly in the static friction regime. It's what it's designed to do. A slipping wheel cannot provide lateral forces allowing any sort of control. And while rudder does assist at all in early stages of landing or late stages of takeoff, you rely on wheels for much of steering during runup. If wheels began to even just noticeably slip during takeoff before you are ready to rotate, you would not be able to keep the plane on the runway. This I can tell you from first-hand experience. The second piece of relevant information is that I'm aware of no plane that has landing gear that will survive more than twice the takeoff speed. In other words, the wheels will be rolling at most twice the speed at which they are designed to operate routinely, before they fail. At these speeds you are not suddenly going to develop slipping, excessive friction, etc. The wheels will continue to operate as wheels, providing static friction with the ground, with minimal resistance to rolling, and almost no slipping, save for that necessary for deformation at contact point. We have race cars that operate at higher speeds than landing gear failure point. Other than additional reinforcement, their physical properties are not that different. If what you were suggesting was remotely true, these cars would not be able to keep themselves on the race track. The landing gear will fail within seconds. We don't need to be talking about relativistic speeds. We can deal with real materials and real vehicles here. And real airplanes have very real limitations. Their landing gear is not designed for very high speeds, because at these speeds the aircraft is not normally in contact with the ground. Your statement that belt has to fail first has zero ground. There are just two places where stress is necessary. Centrifugal stress on rollers themselves, which are likely to be solid metal and can far, far outlast landing gear tires, and the belt itself as it bends around at end-points. Belt has numerous advantages. It can be made from better materials than rubber. It can have more reinforcement. It does not experience the maximum stress in the same place as maximum curvature. And it does not experience heating. If that's not enough, I can always increase the diameter of the rollers, reducing curvature at end points, minimizing stress. None of these mitigating factors apply to the landing gear. I can easily design a treadmill that can survive transsonic speeds, but real landing gear fails at much lower speeds. And if I were to try and design better landing gear, I'd still be unable to make one that's going to both perform its primary function sufficiently well and survive at higher speeds as the treadmill. Heck, even if we use the same metal for landing gear hubs as treadmill rollers, same rubber for main surface on both, and same cables for reinforcements on both, the treadmill is still going to be under far less stress and survive higher speeds. And at this point, we've morphed landing gear into something monstrous, and the treadmill is still just a treadmill. Again, completely unfounded. Real landing gear will fail way earlier, and one that's specifically designed to last as long as possible, made from same materials, merely slightly earlier. In both cases, in matter of seconds or less, I might add. And this one's a plain impossibility. We are talking about thrust of the plane times hundreds of meters per second for this scenario, which is enough power delivered to wheels to melt not just the rubber, but the metal hubs of the landing gear. Meanwhile, the belt remains cool, since that's not where energy is released, and there is quite a bit of it, allowing cooling along the entire surface. If somehow you came up with magical materials that allowed landing gear not to fail in scenario A), this is where it fails. You simply cannot have sufficient slippage for takeoff without generating all that heat.

All of these factors make it easier to pull the plane back. Think what happens if you apply brakes on a plane that's already sitting on top of a moving treadmill. Does the plane go forward or backward as a result? You keep making up arguments, but it sounds like you have not taken the time to draw force diagrams and consider refernce frames. You have to be able to do this to analyze dynamics of a system. This problem is good practice. Accelerating belt can keep in place a plane at full throttle EVEN if wheels are frictionless. Friction makes it a LOT EASIER. If full brakes are applied, treadmill doesn't even need to move. Under your analysis, applying brakes should be part of takeoff procedure, as it reduces ground's ability to slow down the plane. If your analysis fails trivially under a change of coordinate system, you made a misstake. Always check. A car has more sources of drag and higher moment of inertia attached to wheels. That makes it easier for the belt, but that is the definition of quantitative difference. Finally, while there are always non-constant sources of friction, for landing gear, they just aren't significant enough to consider at speeds it can possibly reach before failing. Otherwise, planes would have a lot of trouble taking off. If rolling resistance increased significantly, as load decreases, wheels would fail to roll during large chunk of takeoff, that means dragging the gear on every takeoff. The wear would make next landing hazardous. So landing gear is designed to have minimal folling resistance in wide range of speeds.

They all have the same problem. They don't consider the entire stress-energy flow. Best analogy that should appeal to everyone here, it's like trying to understand lift without considering how air flows away from the wing. Now, I'm not saying that it's definitely absolutely impossible to generate thrust without exhaust. There are two loopholes in conservation of momentum that I'm aware of, and there could, of course, be more. 1) It's not strictly necessary for an object that has momentum to be moving. So in principle, a linear gyro does not violate conservation laws, and in vicinity of a massive body, a linear gyro is just as good as reactionless drive. Not to mention that linear gyro allows for generation of artificial gravity on a ship, which is all kinds of nifty. Unfortunately, conditions under which you have excess momentum in the system not associated with movement of center of mass are scarce, and building a device that can function as a linear gyro is definitely an unsolved problem. 2) Mössbauer Effect allows for recoil-free transfer of momentum in a periodic lattice. If our universe happens to have large scale periodic structure, such as if the universe has T3 topology ("doughnut" shaped universe), then momentum is technically quasi-momentum, and reactionless drives are at least theoretically possible. I can tell you significantly less about how one of these would work than I could tell you about building a practical warp drive, but it's one of these "doesn't contradict basic physics" things. Every time people talk about reactionless drive, I allow for a tiny possibility that somebody accidentally stumbles onto one of these loopholes. But there is absolutely no doubt that it'd be entirely by accident for the kind of explanations we've been seeing for EMD, QT, and MET. You'd have the same chance of accidentally inventing reactionless drive by playing with magnets or literally trying to pull yourself up by your own hair. In terms of anything like a forecast, if we ever break free from tyranny of rocket equation, the most likely candidates are linear gyros and sub-light warp, likely working together. A ship equipped with both can take off and land on planets, establish or break orbit, and go anywhere within Sol in a matter of hours, all within comfort of 1g for all on board and without requiring enormous amounts of energy. And we're not going to get there with the help of crackpots playing with resonance chambers. It would be through a lot of very clever applications of field theory, involving some paradigm shifts in how we interpret the math, and an absurd amount of computational power. We're honestly making progress on warp. Just not something anyone should be expecting anything from for many decades.

Flippin' finally. This goes for both EM Drive and Mach-Effect Thruster. Physics 2:0 Sci-Fi.

That's not how it works. That's not how any of it works. Rolling resistance in landing gear is low enough that a human being can pull an airliner. Compare that with traction, which is roughly equal to airplane's weight. More importantly, rolling resistance doesn't really increase with speed by any significant amount. It just isn't a factor. We can take wheels to be frictionless, and it will be a fair approximation. The only reason the belt can apply a significant force on the plane, is because wheels are not massless, and making them turn requires energy. That energy has to come from work done by plane, conveyor belt, or some linear combination (as usual, depending on your choice of coordinate system). Either way, it means that there is a force applied by the belt on the aircraft, and that force is directly proportional to acceleration, as previously derived in this thread. For a car, you have added rotational inertia of the transmission and the engine. That can be rolled into the moment of inertia I, so that doesn't really change the physics, but does dramatically reduce acceleration of the belt required to keep the car put. In addition, there are viscous forces in the engine associated with air movement through manifolds. This is why a car is capable of engine-braking. As well as reduction in engine efficiency at high RPMs. This further makes it easier for the belt to keep the car in place at higher speeds, but the car on the belt easily exceeds maximum RPM for both the engine and the tires. So we are still back to the situation where there is no qualitative difference between vehicles placed on the belt. The critical factor remains the amount of mass that has to be turning for the vehicle to roll forward. It's pretty small, but not insignificant, for an airplane, and much higher for a car. And that's that.

That's called making stuff up. There is absolutely nothing in the physics of a tire that allows for a significant reduction of traction at high speeds, until you start to experience equally significant deformations, which leads to failure under load pretty much instantly. If anything, heating up the tires will slightly improve traction. The reason aircraft have lower traction at high speeds is due to air speed, which reduces loading factor on the landing gear, and traction is always proportional to load. Stationary plane on a conveyor belt will experience no such reduction in traction. @Reactordrone Yes, skids are a different story. They experience an almost constant friction regardless of speed, which a plane at full thrust should be able to overcome. They have a whole lot of other problems when used as landing gear for an airplane, but none of them are relevant to this particular question. So if we are talking a plane on skids on a conveyor belt, it's definitely impossible to keep it fixed in place by varying speed of the belt. Still possible to destroy it, though, as heat generated by friction is still proportional to the relative speed of the surfaces.

Who says that it needs to be the same materials? The wheels usually fail due to overheating and centrifugal stress on the tires. In this setup, nothing is stopping us from using a kevlar belt running over titanium drums. That stuff can get to mach 1 before there is a problem. And I'd probably use linac drive to just pull on the surface of the belt, since this whole thing just needs a few seconds of operation before the plane is completely destroyed.

But you're not living 200k years ago. You are living now. And the probability that you'd be living 200k years ago is vanishingly small, as vast majority of human population has lived or live much after that point. Of course, the argument can be wrong for us as it was for the few unfortunate who got to live back then. That's the whole point of a statistical argument. It's only right most of the time. Or, in this case, for majority of humans, whenever they happen to live. We can cross our fingers and hope that we're still in the tiny fraction of all humans that will live millennia from now across the stars, and that we just pulled a short straw to live in this era. There are two alternatives, however, that are far more likely. Statistically, either population will continue to grow, and then it's likely that we'll face massive extinction in very near future. Or population is going to level off, and we're about to enter millennia of stagnation. Of course, you hardly need to understand statistics to say that these two are the most likely outcomes.

A PID loop on position of the plane OR wheel speed will result in constraints I've given within reasonable error. You need to keep in mind that what you really have direct control over is torque applied to the belt's driving axles, not the speed. So this is inherently a second order differential, which is exactly what PID is designed to deal with. And any errors that build up in PID loop decay exponentially, so long as parameters are correctly tuned. Correct on all points. The caveat I'm trying to point out is that all of this applies to a car on the belt as well. The "thrust" output of car's engine does drop off with wheel speed, and there is an upper limit in theory, but it's way beyond the red line. Since a car is designed to overcome significant air resistance at cruising speeds, which isn't going to be present in the belt setup, it will still take an accelerating belt to keep a car in place if the driver is gunning it, and either the RPM limiter will kick in or something will mechanically fail long before this system reaches any sort of a steady state. So for both a gas engine car and an airplane, if full throttle is applied, unless the belt constantly and very rapidly accelerates, the vehicle will be able to move forward. And if the belt does accelerate, within seconds mechanical limits of either vehicle will be exceeded. Electric cars are a more interesting case, as their motor speed is inherently limited by battery voltage, and their top no-load RPM tends to be at much more reasonable speeds. You probably could keep something like a Nisan Leaf on a belt/dynamo with the driver pressing accelerator into the floor without anything breaking.

To build up takeoff speed in this case, would be equivalent to taking off with brakes set. If, somehow, you manage to gain speed, your tires will burst long before you can manage it. More realistically, since traction far exceeds thrust, and any slipping is due to deformation, the forward motion you are talking about will not even reach taxi speeds.

Demagoguery. I've given equations of motion for this setup. They clearly show finite acceleration. If you are going to argue with that, you need to point out an error in equations or do your own derivation. Your reasoning is on par with Xeno's paradoxes. I hope, I don't have to explain why these don't work. P.S. Earliest mention of the question is on a Russian forum. Some of the translations have been iffy. But that's not really relevant, so long as we agree on which constraints we are considering. If belt is matching wheel speed, or alternatively, tries to keep vehicle in place, acceleration of the belt is finite, albiet, very high.

The answer to both of these is that there is a rolling constraint between the wheel and the surface on which it is rolling. Whether the axle to which wheel is attached moved forward a meter, or the surface was dragged the opposite direction underneath, the wheel will have to turn a certain amount. And because the wheel has mass, making it rotate faster requires an input of energy. That energy has to come from somewhere, and that means there is a sort of "drag" between surface and vehicle that depends on relative acceleration between the two. And that's what letting us apply a force to the vehicle via the belt. That force simply has to counter whatever propulsion system is in place, whether it's propeller or torque delivered directly to wheels from a car engine. I've outlined the relevant equations in the post above. The formal way of deriving this is via Lagrangian. L = mvvv2 / 2 + I ω2 / 2 - T xv - Fb xb + λ1(rθ + xb - xv) + λ2(xb + rθ) Euler-Lagrange equation (∂L/∂xi - d/dt(∂L/∂vi)) and differentiated constraints give equations of motion. mvav = T - λ1 - λ2 I α = (λ1 + λ2)r Fb = λ1 + λ2 av = rα + ab ab = -rα The lambdas here are the undetermined multipliers that take care of the constraint physics. And once you used the 3rd equation to substitute for sum of lambdas in first and second, you'll see that this becomes the same equations I've marked above. In terms of the control system for the conveyor belt / treadmill, it doesn't really matter. If you hook it up to a sensor that's trying to keep a vehicle centered, if any force is applied to the vehicle, this will result in the belt accelerating. This is equivalent to keeping track of the wheel velocity, because the rolling constraint guarantees us that any relative velocity between vehicle and belt is the rolling velocity of the wheel.

Of course not. The question, as originally stated, is defined fully by constraints. It doesn't have anything to do with means of propulsion. There are two constraints. First, is the rolling constraint of whatever vehicle. vv = rω + vb Where vv is forward velocity of the vehicle with respect to ground, vb is the same for surface of the belt, and ω is angular velocity of the wheels, r being their radius. For simplicity, lets just deal with wheels of equal size. Second constraint comes from the problem. "Belt matches speed of the vehicle." There are several ways to interpret that. 1) Surface of the belt moves with respect to ground in reverse at the same speed as the vehicle moves forward: vb = -vv In this case, the vehicle is free to move forward. We can solve for ω = (vv - vb) / r = 2vv / r. In other words, if the car is moving forward at 20mph, the belt is moving in reverse at the same speed, and the car's speedometer shows 40mph. The car has no trouble just driving off the belt in this case. But in the original thread, where this whole story starts, it is said that "in a way that would keep car stationary." Which leads to a second popular interpretation of this constraint. 2) Surface of the belt matches speed of rotation of the wheel. That is, for a car, matching speedometer speed. vb = -rω Again, it is easy to solve for the missing variable. vv = rω - rω = 0. The vehicle remains stationary. And it doesn't matter at all what kind of propulsion we're dealing with. The very statement of the problem demands that the vehicle is stationary. If the vehicle moves, we have not satisfied the conditions of the problem. Is it possible to move the belt in such a way? Well, I'll get back to that in a moment. But first, there is one final interpretation, which I wouldn't even have thought of, but it's what Mythbusters have gone with, so it bears mentioning. 3) Surface of the belt matches some constant reference speed. vb = v0 Mythbusters considered two cases. Electric toy car and airplane. For electric car, v0 = rωmax of electric motor, which lead to the condition vv = rωmax - vb = 0 preventing the electric car from going up the belt. This does not generally work the same with internal combustion cars, by the way, but they never went full scale with this. And for an airplane, they chose v0 = vtakeoff and so simply had airplane wheels spinning at twice the rate they were meant for. That can cause wheels to fail, but not on a tiny plane they used in the show, so naturally, they had no problems seeing the plane fly. What they were trying to demonstrate with all of this, I have no idea. Ok, so the first interpretation is straight forward enough. Neither a car nor a plane have any trouble making it off the treadmill if it simply matches the vehicle's forward speed with respect to ground. So is the third. No matter how fast the belt moves, that either exceeds some mechanical limitation of the vehicle, or it doesn't. Plane has advantage, as its propulsion system is decoupled, so the only way to prevent the plane from taking off is by moving the belt fast enough to shred the wheels. But what about second interpretation? Can the belt be moving in such a way as to match airplane's wheel speed always, thereby, preventing it from taking off? This is where we take our equations of constraints and add them to equations of motion. I'm not going through conventional derivation from Lagrangian, as that is needlessly mathy, but go with Newton's laws instead. First, we need to reframe constraints in terms of acceleration. (By taking derivatives on both sides.) av = rα + ab ab = -rα Moreover, from Newton's first. mvav = ΣF= T - Fb Here, T is thrust and Fb is the force with which the belt pulls on the wheel. The wheel is free to spin, and if it was massless, that force would be zero. But the wheel does have some rotational inertia. So we have the following. Iα = rFb Here, I is combined moment of inertia of the wheels. And putting it all together, we can now solve this mess. Fb = T - mvav = T - (rα + ab) = T - (rα - rα) = T ab = -rα = Fbr2 / I = Tr2 / I So since T, r, and I are finite numbers, there is definitely the right amount of acceleration with which the surface of the belt can be moving, such that due to "drag" generated by the pull of the belt as it tries to spin up the wheels of the plane, the thrust completely cancels out, keeping the airplane put. For a real airplane, that works out to something in the broad neighborhood of 100g. There are two things I can say about that number. First, it's doable. You could construct a treadmill that will provide you with ~100g of acceleration at a sustained force matching thrust of the airplane. Second, this situation can't last for very long. Putting aside difficulties of maintaining these conditions on the treadmill itself, within about a second, almost any airplane's landing gear will disintegrate due to combination of heat and centrifugal forces. On the net, we have a problem that's fully resolved in constraints before we even have to talk about physics. If constraints are matched, the behavior does not depend at all on what kind of vehicle is placed on the belt. The only place where the choice of vehicle matters is in how spectacular the failure will be when the constraints can no longer be physically maintained. For a car, if the driver's gunning it, the RPM limiter's going to kick in pretty quickly, and probably prevent anything bad happening. For an airplane, most likely a big fireball.

Because everybody thinks they understand what's going on, while not being able to write down the equations of constraints or equations of motion relevant to the problem. Once you define these in mathematical terms, this is a trivial problem. But very few people can do either, so we end up with people who still think that there is a difference between a car and a plane in this problem. What's worse, most think that their fallacy is backed up by physics.

Nope, can't be done under modern Field Theory. The problem is that stress-energy tensor is the gauge charge of the local symmetries of space-time, which means there is an associated conserved flow. To put plainly, you can't just increase energy/mass of an object. That energy has to flow towards or away in a continuous manner, and the flow itself has inertia according to the energy/mass it carries. Because people's brains tend to swell and overheat when trying to reason about mechanics of quantum fields, there is a fantastic simplification here. The conserved current acts just like any other current. Even a liquid flow. So lets forget all about relativity and quantum mechanics. Try to build a classical propulsion system where you change the mass of the oscillating object. Lets say, a bucket of water. We'll pump water in, push the bucket away, pump water out, and pull the bucket back in. Wait, that can't possibly work! But why? Because the water flowing into/out of the bucket is the thing that carries mass. As it flows through whatever plumbing back towards you so that you can fill the bucket again, it cancels out all the thrust you may have generated. A relativistic "thruster" on the same principle suffers from exactly the same problem. The energy that flows into the charged particle before you push it away has to flow out at the far end. Then that energy has to flow back to the front end, presumably, as electrical current. And that electrical current will have exactly the same momentum as what you got out of the push in the first place, because it's exactly the same amount of energy just sloshing about inside your "thruster". Now, just like in the classical sense, you can chose to abandon that water. If you can fill the bucket from a store, and simply expel the water in the back before pulling bucket in, you will get net thrust. But it will be exactly the same net thrust as if you just expelled the water out the back. If you don't reclaim energy in the relativistic version, with charges, the energy is expelled in the only way massless energy can exist, as photons. So you ended up building a photon drive. Best case scenario, 300MW per 1N of thrust, but odds are, it will actually be extremely inefficient, and you'll get a lot less thrust than that.

You're out of date by at least a couple of decades. I'm not going to say that listening for boat noises is completely irrelevant now, I mean, when it works it works, and hunting for cargo ships is still a big part of sub's job. But there has been so much improvement in masking noises, that if you are only aware of things that make noise, you're going to be very dead. Modern passive sonar systems use the same techniques in looking for submerged objects as passive radar. They use noise made by other things in the ocean reflecting off hulls to try and identify things that are trying to stay very, very quiet. Acoustic cross-section is more relevant now than it has ever been.

Yeah, but they're mostly necessary for passive sensing. A civilian active sonar will give you the same sense for the acoustic cross-section of a target as a military passive sonar would. Sure, you're not going to detect the target from as far away, and you are certainly not going to do it without completely giving yourself away. But the cross-section is the same regardless of sensing equipment.

This is what Google Image Search is for. Here is a large version.

US did it first with Skylab, so it's American Roulette, if anything.

That shouldn't matter. At infinity, your orbital speed is zero, so direction and eccentricity become irrelevant. Id est, no circularization at infinity. It's the same reasoning for why escape velocity is direction-independent. So your claim of double the delta-V requirement and mine of sqrt(2) increase are at odds. I'll check the math later with a simulation, because I'm curious. I just wanted to see if you can spot any errors in my derivations, perhaps.