K^2

It'd be very hard to pull off without the source. You can try and fake it by adjusting time and positions of things, plus maybe hijacking the shader, but it'd be iffy. To do it right, you need to replace a lot of core functions.

Tidal forces of large black holes are quite mild. Hawking Radiation, likewise, completely harmless. So if you can find a supermassive black hole with no significant accretion disk, you can safely orbit it at distances being discussed here.

Comes down to relativistic addition formula. For a classical rocket, if it's traveling at velocity v, and exhaust velocity relative to rocket is vp, then exhaust is traveling at v - vp. That is no longer the case for a relativistic rocket. So trying to exploit conservation of momentum becomes a bit complicated. Another complication is that there are two different velocities to be considered. From perspective of inertial observers, map velocity is the relevant quantity, and it's limited to c. So as you get close to c, your rocket's efficiency drops dramatically. The faster you go, the more fuel you need to burn for the same dV. On the other hand, if you care about how fast you get somewhere by ship's clock, then the relevant quantity is the proper velocity. And that's not limited to c. The reason for that is that while you can't go faster, by burning more fuel you make the distance you need to travel shorter, thanks to Lorentz contraction. And if all you care about how fast you get there, it's just as good. And from perspective of fuel efficiency, it's better. Once you're going fast enough, the hyperbolic nature of Lorentz contraction starts to compensate for the logarithm in the rocket formula. And that has some absolutely insane consequences, such as it becomes actually feasible to build a matter-antimatter rocket large enough to cross the galaxy in a matter of decades. By ship's time, of course. On Earth, tens of thousands of years would pass anyways. The actual derivations involve serious math. You can take a look at Wikipedia article on proper acceleration to get some insights.

The only way to use an object in perfect circular orbit to gain a perfect circular orbit is lithobraking. That said, if either of these two orbits is elliptical, it's entirely achievable. And this might be what they are talking about. Using Venus to achieve an orbit closer to circular, then burn fuel the rest of the way.

As I've been corrected in another thread, your lowest periapsis can actually be no lower than 3/2 of the Schwarzschild Radius. In other words, well above event horizon. But yeah, if you drop that low, you'll be moving almost at c. The Obereth effect gains aren't going to be nearly as insane as they would be classically, because rocket formula is quite different at relativistic speeds.

Hence, some people pushing to re-classify Earth-Moon system as a double planet.

With tweezers.

Yeah. Given the size of the system, all you need to do is loop over all bodies, then loop over all other bodies again inside that loop to add forces, making sure to skip "pairs" where indices match. Then run separate loop to update all velocities and positions. You want that in separate loop to minimize errors.

1/500th of AU is almost exactly 1 light second. The distance to the Moon is still much smaller. The semi-major axis is 0.00257AU, which is 1.285 of your units. But it still shouldn't have that effect. You do iterate through all possible pairs for which forces are applied, right? Sorry if I'm bugging you with too many questions, but your results for the Moon are a bit puzzling, given how well everything else seems to work.

So that vx for the moon is without adding the orbital velocity around Earth, then? Is that what you have on that screenshot? Does it still happen if you add it in? Also, why is the Moon so far away? I'm gathering you are using light-seconds as your distance units? Moon should be slightly over 1 light second from Earth, not 4.

a) Virtual particles do not require that energy. They are off-the-shell. Their energy could be "anything". Of course, the further the particle is off the shell, the less it's going to be around. Momentum has to be, eventually, transferred into real, on-the-shell particles. Energy for that has to come from somewhere. Virtual particles do not solve any problems here. They simply allow you to transfer energy/momentum to something else.

Could you post initial mass, ry, and vx for Sun, Earth, and Moon that you are using?

I don't know how many times I need to reply to this nonsense. If massless radiation is carrying momentum p, it has energy E = pc. That means, you need 300MW per 1N of thrust. I'm fine with that. If that's the power-to-thrust ratio they claimed, they would have had a photon drive, and nobody would care. But they claim a much lower energy use. So you can't have momentum carried out by massless radiation. If the mass was generated by the drive, add mc² onto the above. It becomes even less energy-efficient. The only way you can break 300MW/N is by using a reaction mass. That's the only way this math works out.

What of it? Well, yes, breaking local causality would be kind of bad. But QM already shows us that notion of locality gets a bit slippery on a small scale. And if our space happens to have a more complicated underlying geometry, say, similar to Holographic QM, you could actually end up with macroscopic light barrier violations via QM without ever truly violating locality. And it wouldn't be inconsistent with anything we've measured so far by any statistically significant margin. It's ugly, and Occam's Razor suggests that it shouldn't work that way, but that'd be in category of tweaks to the theory, rather than "start from scratch". Any momentum conservation violations in one frame are energy conservation violations in another. Ergo, no local time symmetry. Ergo, no causality. It's actually very straight forward. Fortunately, it doesn't go the other way for (global) causality. Local causality we'd still like to keep, with the above caveats in mind. Edit: Just to clarify, it did arise from a different discussion, but these things are quite related. ZetaX seems to suggest that both causality and symmetry violations are equally bad, and I disagree. Causality violations, even if they appear local at a glance, could simply mean that the rabbit hole goes deeper. Symmetry violations means that there is no rabbit hole, and we've been in the padded room all along.

No, because conserved currents. The momentum has to be carried away by stable particles, or it can't work in vacuum. That means mass poles. That means you have to have propellant. At best, Dr. White's hypothesis buys it a greater length scale, which could allow the craft to use low vacuum, such as that found in low orbit, as propellant. It would prevent the craft from operating in interplanetary space, but could still be useful for LEO operations. Of course, even this is a huge if.

Well, for starters, FTL neutrinos wouldn't really be nearly as big of a deal. Neither from theoretical nor practical standpoint. It seemed likely that there was an experimental error, but it wouldn't have uprooted anything fundamentally if it wasn't. Enough things appear to propagate FTL on both Quantum and Cosmological scales to merit merely a "huh" on the freaky physics scale. And there were no direct applications. If they could propagate slightly FTL through vacuum, maybe there would be some use. And even that's a stretch. If you have to send them through a rock, well, that's just a theoretical curiosity, then, with no practical use. EMDrive supposedly generates enough thrust to be practically applicable already. And at power/thrust ratio that blows away all competition. That's not just a curiosity. That's something a lot of people would genuinely very much like to have. And that's even if it turns out to be just an ion drive. It's not something with maybe some potential applications in the future. We can find fantastic uses for it right now. And on the flip side, it wouldn't be just a minor curiosity in theoretical physics. It's a big fat cross on the entire discipline. It's not another, "The theory wasn't quite right, here's a fix." It's a genuine square one. And square one here goes back to the Greeks. Not only that, but it leaves all of the questions of how theory so wrong could have given predictions so right all along completely open. Which kind of casts doubt on scientific principle overall. And then we're completely up the creek without a paddle. So if you're wondering why EMDrive generates so much more tension than superluminal leptons, it's because the later has not a tiny fraction of conflict between value to application and theory of the former. Whenever something has potential of providing immediate practical benefits, but casts under doubt the system that brought forward advancements before it, there is tension. And few things we've encountered have as much potential to do both as a reactionless drive. Fortunately, it's also a reason to be absolutely certain that nothing will come out of it, beyond, maybe, new applications in ion propulsion. We just can't be that wrong about foundations and only be finding out about it now.

So it's not a Hill Sphere problem, then. At least, not initially. It'd be interesting to see this with a longer trail. In fact, could you make it almost exactly a year long, and do a screen capture exactly a year after the simulation started? It could be very informative in terms of what else might be going wrong that might be possible to improve. Even if not, the orbit of Earth-Moon system around the Sun is an interesting shape to look at. They really do co-orbit the Sun, unlike moons of other planets. As I've indicated a number of times, I expect the Moon to drift away eventually in any case, but I feel like a "few years" of clean simulation isn't an unreasonable demand here. Have you played with the time step at all, by the way? It's more than a quarter hour long right now. Maybe that's a bit much?

Oh, you should be adding all of the forces together, then running a single step of Velocity Verlet on the total force. Not doing an update per source.

You need to keep an ax_old and ay_old separate for each body, like you do rx, ry, vx, and vy. Also, a few typos. a = F/m1, not m2. And you have vx in ry update.

He's kind of deriving fast and loose there. I haven't tracked it down to the exact assumptions he's making implicitly to make his claims, but I'm sure there are some. Just take his Symplectic Euler example, and expand it for some arbitrary pair of coordinates (p, q) and (P, Q). To be Symplectic, it must approximately satisfy p_{n+1}Q_{n+1} - P_{n+1}q_{n+1} = p_n Q_n - P_n q_n. That works out for potential U ~ O(q²), which gives me error O(h²). But with U ~ O(1/q), I still have O(h) terms that I can't make disappear. I do agree with his logic on Stormer-Verlet following from Euler. And, of course, for the special case of H(p,q) = T(p) + U(q), it reduces to Velocity Verlet. So as I've said above, for U = k|q_1 - q_2|², it is most definitely symplectic. I'll definitely dig further into this to see if I can find where an assumption is made that's not consistent with gravity, but you can also just try it out with a two-body simulation with highly elliptic orbit to confirm that energy steadily grows regardless of h.

Verlet is not symplectic with 1/r potential. No known method is. Verlet is symplectic with r² potential, such as Hooke's law used in collisions, which is why it's a favorite in game dev. But no method is symplectic for all potential types.

I've found that there are energy gains even with nice circular orbits over rime, but yeah, it is nowhere near as bad as elliptic ones. Maybe it is good enough. Yes, v is vx for x and vy for y. Same with acceleration and force.

Ok, this is called Forward Euler method, and it is the cause of it being as bad as it is. Try applying Velocity Verlet. The standard description is a bit confusing because of velocity update, but basically, you want following scheme. a_new = F/m v = v + dt * (a_old + a_new)/2 x = x + dt * v + dt * dt * a_new/2 a_old = a_new Do this for x and y for each time step. a_old can start out as 0, but then you need to maintain it from step to step. The beauty of this method is how simple it is, yet, it makes a huge improvement. I would still expect the moon to drift off, but it should take much longer. An even better improvement can be gained by going to 4th order Runge-Kutta (RK4) method, but it is also significantly harder. Real NASA-grade simulations usually use relatively low order implicit methods specifically designed for this. They involve some crazy math.

Not true at all. Total stress-energy is still conserved, because it is a conserved current of local Poincare symmetry. So energy conservation violations are still to within a coordinate system choice, which is true with or without wormholes.

Gravity is notoriously difficult to deal with numerically. Even if you have prior experience with numerical methods in differential equations, unless you studied numerical methods in gravity extensively, it is very unlikely that your simulation is up to the task. Don't take it the wrong way. It is a deceptively hard problem. Your description makes it sound even more unstable than it needs be, though. How do you update position and velocity in your simulation?