K^2

Which is silly, because Europa has no better odds of holding life, but even if we find some, it could still have the same origin as Terrestrial life. So mere existence of life on Europa would be fairly useless information. If you are going to put a lot of money on a long shot, might as well go with the one that gives you way better results, no? Oh, and mission to Titan would be way cheaper, seeing how you wouldn't need to drill through a crap load of ice to get to the water. Titan's relatively easy to explore. The sort of equipment we've delivered to Mars would do fine.

Titan should be very high on priority list for research. The conditions on that moon allow for a possibility of a life form that uses methane for solvent and catabolizes acetylene and hydrogen to produce methane. And we see that something is breaking down a hell of a lot of acetylene, releasing methane into atmosphere. It would be an extremely simple life form, limited by energy supply from evolving into anything complex, but it would be extraterrestrial life that's entirely distinct from anything on Earth. Even if we find life on Mars, for example, we'd need to spend a lot of time to convince ourselves that it's not cross-contamination with Earth. That life on Earth and Mars don't have the same origin. Titan environment is too different for this to be possible. We find life on Titan, and that's check and mate. It's the second data point. Two origins of life in the same star system would make it a statistical impossibility that universe isn't filled with life. Of course, we might find nothing. We might find a sterile cold world with some natural mechanism for breaking down acetylene. Which would still be pretty cool, because no known naturally occurring catalysts allow for that sort of conversion rates at temperatures so low. But the very possibility that life is there, given what we would learn from it, makes it worth all the risks. We need another lander on that moon. If we are going to send a robotic mission past asteroid belt, Titan has to be the first choice.

Take a look at this article. Don't get hung up on math. The main point is that when you get close to the black hole, you end up with weird trajectories. Things can spiral in, or do a pretzel sort of twist before escaping. And, of course, any trajectory that takes you to the event horizon takes you down to the center of a black hole, never to escape again. So as people said, assumption a) is wrong for trajectories near a black hole.

Indeed. The low order correction to orbital motion around spheroid is orbit precession. With the right choice of orbit, you can match rate of precession to the Earth's motion around the Sun, resulting in a Sun-synchronous orbit. But this still primarily works because Earth is a fairly symmetrical object. One of the cool things about General Relativity is that it completely covers effects of non-inertial frames of reference. That includes centrifugal force. The magnetic field's contribution in neutron star is primarily to the total energy. Yeah, these fields are strong. Ultimately, though, these are effects of General Relativity. Gravity just doesn't work the way you normally expect when you are that close to something that dense. All orbits end up wonky, and the way tidal forces and gravity vary with radius is a bit different. Worse, the pull of gravity depends on how you are moving. It takes an infinite amount of force to keep an object at event horizon. The tidal force, then, is also infinite. Yet, if you free-fall into a black hole of sufficient size, tidal forces can be quite manageable. Unfortunately, almost any source I can point to assumes that you can work out gravity from the metric, and only list metrics. But just as an example, if you start with Schwarzschild Metric, which is valid for any spherical object of mass M with no significant charge, magnetic field, or angular momentum, you can derive surface acceleration due to gravity to be: GM/(R3/2 Sqrt(R - 2GM/c²)) The quantity RS = 2GM/c² is the Schwarzschild radius. That's where event horizon of the black hole is going to be. It's easy to see that if R = RS, surface acceleration is infinite. However, if R is much, much larger than RS, then the quantity under the square root is essentially just Sqrt®, and formula reads GM/R², which is your typical Newtonian acceleration due to gravity formula. The interesting things happen when R > RS, but not too much greater. Surface of a neutron star is such a place. If you add angular momentum, electrical charge, or strong magnetic fields, things get even more interesting.

No. The pull is proportional to the mass, but so is the amount of force required to accelerate a heavier object. So the acceleration of a falling object is exactly the same regardless of the mass. No, unfortunately, you can't just assume an average radius. When you are far enough away, from the mass distribution, it behaves a lot like a point object + some corrections. But on the surface, any changes in elevation will cause differences. The gravity's pull doesn't even have to be "straight down" everywhere, either. Center of the rock can be right bellow your feet, yet you might be pulled sideways because there is a larger lump of mass to the side. You really have to sit down and carry out the integration to compute the field at each point. At the surface of the neutron star, the gravity is strong enough for Newtonian estimates to be far off. You have to use formulas from General Relativity. So long as the source is spherical, the formulas aren't that much more complicated, but there are some additional factors. A rotating neutron star will have a different gravitational pull than a static one. Magnetic field also plays a huge difference. In fact, magnetic field is the only reason some massive neutron stars don't collapse into black holes. So that should give you some idea of how much difference that makes.

And the fact that this is a completely different mission, with completely different sail design doesn't bother you at all? Oh, but it does. If you go to the pages of universities that were responsible for the equipment, you'll note that they were trying to make them absolutely tiny. There are a whole bunch of ion drive designs. You can take a look at the Wikipedia page for a start. It's all going to depend on what's more important for the task. But limitations for modern drive are up to about 50km/s of specific impulse and up to about 1mN/kg of specific thrust. You can compute everything else from that. They are all rather efficient (except for electrostatics) in 70-80% ranges. You can easily compute specific power from that as well. So lets take SunJammer's actual mission. Travel to L1 from Earth after being launched on escape trajectory. It can only brake at 70% efficiency, so that puts it at 7.6mN of total thrust. That can be matched with less than 8kg of hardware. With a 16kg payload, that would still leave you with 8kg for the propellant. Even if we consider that ion thrusters don't scale great, and we happen to use only 20km/s one, that still gives you 5km/s of delta-V for this particular mission. While the entire mission to L1 is going to be done in under 2km/s. That's less than 3kg of propellant, putting total mass in under 27kg. Now, for the actual sun dive, yeah, if I take 16-to-16 mass, I'd have to scale down on ion drive's thrust and settle for a much slower dive. But as I pointed out, we're nowhere near that on the actual SunJammer. Equations of motion in polar coordinates with gravity and thrust thrown in. Then I simply get Mathematica to NDSolve.

Note, these equations assume a roughly spherical body of reasonable density. Essentially, larger moons, planets, stars. That sort of thing. If you need to compute surface gravity on an asteroid of irregular shape or surface of a neutron star, you have to get creative.

Have you read about the payload. It's a very tiny magnetometer and a very tiny mass spectrometer. If they make up 2kg combined, I'd be surprised. Almost the entire mass of the ship is the support structure for the sails. ´ What, for the 100W I'd need to power the 10mN ion thruster? That can be done with well under 1kg of solar panels. Or an even lighter RTG. Everything I found points to the only part of considerable weight being support structure. Would you like to quote your numbers? But even at 16kg total, ion thruster still wins. That's for a rigid structure. Not a fold-out, like on the SunJammer. That might be the source of confusion here. Which is why the question is how much delta-V you need. And for any reasonable mission in the Solar System, ion thruster gives you lighter weight for the same thrust. And yes, I'm counting fuel, tanks, and payload here. Would you like to show math on that? Given the inverse square law, you end up with a rather short run up. Even if you do a two stage mission with a sun dive, you still end up with absurdly small velocity on escape. Oh, and don't forget that once you hit heliopause, you'll be generating more drag with that thing than you do thrust. So you have to jettison or fold away the sail by then. Not that thrust is at all significant at that point anyways. I wrote Mathematica code to do the trajectory in about half an hour. I was able to match it with ion thrusters down to sub-Mercurial orbit. Which is about as far as you'd want to bring any sensitive equipment anyways. But give it a shot. Maybe you'll get something different.

Say you get a 60 fps camera and drop it at 10m/s. If your markings are less than 20cm apart, they'll start to blur together. And that's assuming a stable enough fall that no vibration or tilting cause problems. And then you have to process the image, which is going to be tough to do with time constraints using an MCU. It's an out of the box idea, but I don't think it's workable in this particular case. Now, if you have an external high speed camera connected to a decent computer that will radio lander's altitude in, that could really work. Nobody said you couldn't use some telemetry, right? Might not even need the board with markings if you calibrate it in advance.

Don't consider this a substitute for learning Calculus (and subsequently Analysis) properly, but as a quick patch, and to get you started, watch . It will give you some idea on what's going on.To derive the actual rocket formula, you need to know a few more things, but the important bit is that instead of working with acceleration/forces and integrating over time, you consider how the momentum changes with respect to fuel consumed (specific impulse) and integrate over mass of the ship. You do this right and you will get Tsiolkovsky's rocket formula. A lot of orbital mechanics can be learned without knowing Calculus, though. You'd be missing out, but if you understand Conservation of Energy and Conservation of Angular Momentum, you are off to a good start. That formula is derived from equation he wrote down, though. You just have to do a change of variable (|T|dt = - g ISP dm) and then integrating from m_wet to m_dry.

Platinum's expensive. You can make a good catalyst for H2O2 from potassium permanganate. One of the products of heat decomposition of it is a strong catalyst for hydrogen peroxide decomposition. Soda bottles are very unreliable. The 2L ones can usually hold 4-5atm even without reinforcement, but some will burst at under 2. If you think you know what you're doing, go for it. Just be careful. And I would consider using something a bit more reliable than a soda bottle. There are a lot of options, of course. The limitations are price, weight, and power consumption vs range and precision. I'm not aware of any son/rad/lidars that would give you significantly better range at acceptable reliability that don't come with a price tag and weight of a commercial aircraft ground radar. Doesn't mean they don't exist, of course. And yes, having a base unit does open up some additional possibilities. If base station is located a bit to the side, for example, you can turn a simple radio receiver into a VOR receiver. Building a mini VOR station is also not that hard. But it's a project all in itself. There are two more cheap options that you can throw in to get a rough estimate on your altitude. Maybe use it to disconnect the parachute and kick on the engines at low power a bit early, and then throttle up as you get within sonar's range. If you drop your velocity to 15m/s or less with a chute, and come in gently, you should have enough altitude/thrust reserves to fix any problems. So if you don't mind some work on extra wiring and coding (consider ordering a custom circuit board; it's not that expensive) you should probably throw in a cheap GPS receiver and a good pressure sensor. If you calibrate pressure sensor prior to launch and manage to make a good static port, with a good sensor you can get surprisingly accurate altitude readings. If you get two pressure sensors and a pitot tube, you can get air speed as well. Vertical air speed is a good indicator of your true vertical speed. Similarly, GPS can put you within a few meters on altitude. Not as reliable as pressure sensor, but you can get very good velocity data from it, and maybe you can come up with other uses for it. Naturally, you want to combine readings from both and pass them through your filter. (Like I said, I strongly recommend Kalman filter for this.) Even if GPS readings are way worse than pressure data, with the right filter, even a very bad reading from a secondary instrument can help you reduce error on your primary. With pressure sensor backed up by GPS, I think you can get to the parachute release and kick on the engines within ±5m. Then do corrections once you are within sonar range, and final corrections in IR range for a soft touchdown. That kind of leaves stability as your weakest link. Don't go cheap on accelerometers. Make sure you have enough of them to get good six-axis readings. Consider spending a bit extra on optical gyros. (Ring Laser or Fiber Optic) These things are crazy precise for measuring angular velocity. Also bulkier and significantly pricier. The rest is up to MCU, coding, and reliability of any mechanical parts. I guess, that's mostly going to be your pressure system and valve control. Unfortunately, mechanical parts are where I'm pretty much useless in terms of good advice. I don't have any better ideas than bolting micro servos on to valves, and I'm not sure that's fast/reliable/precise enough. If you'll want any advice on the code once you're working on it, though, I'll be happy to help out.

400psi can give you a roughly sonic stream. Of course, it all depends on volume of the gas you have on board. Since you haven't specified that, I'm not entirely sure how you are doing your math. It's also a considerable amount of pressure, which can cause serious injuries in case of failure, even if you avoid any metal parts that could turn into shrapnel. Pressure wave alone could do it. If you can get a composite pressure chamber, it might be worth considering. Otherwise, it sounds kind of insane. What sort of delta-V are you looking for, anyways? You'd be looking at something in the 30m/s range for impact speed. So that's minimal dV you need in suicide burn. Of course, acceleration isn't instant. In principle, an egg can survive up to about 50g, but with harness and safety margin, more than 10g would be pushing it. That gets you to stop in 0.3s at a cost of only about 3m/s more in dV. You'll also have to start braking within 4.5m of the ground. Looking at some affordable sonar options for robotics, you can start getting updates from about 6.5m at 50ms intervals. That means your first ground detection will be within 5-6.5m. That's enough, but leaves almost no margin for error. You'll probably want some reserve thrust here. Furthermore, you can count on at best 5-6 position updates while braking. Worse, voltage readings will lag. These are just the sort of hardware limitations you'll have to live with. So in terms of delta-V reserve, you don't have to worry about much. If you don't manage a soft landing in ~40m/s, you've already gone splat. And you can certainly do 40m/s with a water/air rocket. So it's all about control. Control is the hard part. Lets forget valves and lag on thrust for a moment. I'm going to pretend that you manage to design your valves, and they have perfect response time. We are back to our worst case scenario. Your sonar is reading something between 5-6m, the error is about 10% due to the fact that voltage hasn't adjusted yet, and you have 4 more readings to estimate your velocity and time of impact. You can Kalman filter that data, and you will want to, along with accelerometer data once rockets kick in. You'll also manually set uncertainty on terminal velocity before you do your first Kalman pass. Hopefully, you can get to within 10% on that as well. Your goal is to be within safe stopping distance until you hit about 80cm from impact. There you can start using IR range finder as backup. You'll want to get at least a few readings off of that before impact as well. You won't have enough to make major corrections on velocity, but it will give you smaller errors on the range. The actual control logic should assume worst case scenario. Take your velocity, go up a couple of sigmas, take distance and take away a couple of sigmas. You need to be able to stop within that. And you don't have direct thrust measurements. Only accelerometer data, which you'll also have only a few updates on. There is no time to PID this stuff. You'll need a look-up table of pre-calibrated values. Given x acceleration on each jet and y position on the valve, go to z position for correction. If you don't finish correction within 50ms between updates, your lander is probably already flipped over, and trying to correct further is moot. Naturally, this is also where lag corrections will have to come in. If you have to adjust little, you can assume corrections to be instantaneous. If you have to correct a lot, you need to over-correct. That has to be part of the look-up table as well. Again, calibration is the way here, and there is nothing you can do about that. Finally, MCU. You'll need six axis acceleration readings, at least two sonar readings, run Kalman in between, compute corrections, get valve servo positions, set them, and still have time for a break before the next 50ms wave of updates. I hope you are comfortable with assembly. So that's what this project looks like on the control side of things, rockets aside. The reason you might want to look at high delta-V options is to start slowing down way in advance and do a gentle glide, which is "easy" on the control side. If you want to do it with water rockets, you have to suicide burn it, and then the actual delta-V isn't your problem.

It's a nightmare of a challenge. Let me run through some options from worst to best, but honestly, if you aren't willing to "fake" thrusters with ducted fans, I wouldn't hold out a hope of actually getting it to work. With water rockets, short delta-V means timing and valve precision must be perfect. Slight delay on valves opening, or misjudging pressure and not adjusting in time - splat. And that's assuming you have a radar and MCU that can handle precision/timing to begin with. Give me a $1k budget, and I'm still not sure I'd be able to build a working machine for this task. Solid rockets would be your next best hope, but as people pointed out, no throttle control. With 4+ rockets firing, you could put them on servos and angle each rocket individually to get forces and torques to cancel on all axes. Algorithm for this one would be a mess. You still need a very good MCU, accelerometers, and really good servos. Being better with code and electronics than mechanics of the valve system, I'd go for this option and correct on the fly. I could build it cheaper this way, too. But it's comparable to the water rocket idea in complexity. Liquid bi-prop rockets. Just a big no. Not on any sane budget. Hybrid. Now we're getting into semi-realistic scenarios. Hybrids are pretty benign. You can throttle them a little. You can ignite them as needed. Radar + MCU + a bit of code and calibrated servos on the throttle needles, and you have yourself a working lander. Use single NOX tank with four solid rockets made out of something like dense rubber, and you have yourself a cheap-o hybrid sky crane. Of course, there will be explosions, fires, electrical problems, and you'll lose about a dozen prototypes before you get a working one. But if you know what you're doing, this is purely an exercise in patience. Pulse jet. Lots of noise, not a lot of thrust, but if you can figure out ignition, it could make it. And you just need a small propane tank on board for fuel. Same requirements as above, but way less likely to blow up and spray corrosive substance and shrapnel all over the place. You will need some retro parachutes to slow it down before you fire the jets, however, and this is cheating a little, this sort of approach works only in earth's atmo. You might as well go for the fans at this point. Liquid monoprop. Now we're cooking with gas. Or superheated steam, as the case may be. Naturally, getting your hands on some real space-worthy monoprop is difficult to put it mildly, but hydrogen peroxide is fairly accessible. You can order the concentrated stuff from a number of reagent stores. Highly unstable, super corrosive, and yet safe in comparison with hybrids. Also, doesn't require ignition, not likely to blow up, and makes for very easy to build engines. They won't last long due to oxidation, but they'd be easy to replace. Throttle control is as easy as turning the valve. The main drawback of H2O2 as monoprop is low delta-V, but that's compared to the good stuff. You'll have more than enough to land the egg. So if I had to build a lander for this challenge using rocket thrust specifically, hydrogen peroxide would be my first choice. That said, your best and easiest option is still ducted fans. Yes, it basically boils down (heh) to building a quadropter to land the egg safely after parachutes brought down the descend speed. On the plus side, you can make the whole thing super light using capacitors instead of batteries and only allowing the rotors to run for a few seconds, giving you performance similar to retro-rockets in almost every way. Fans give you the cheapest, safest way to do this. As well as you'd be able to get a lot of help designing the thing, because a lot of people do work with quadropters these days. If you want a challenge you can actually finish, go with that.

Easy! All you need is to make the star out of anti-hydrogen and wait until it goes supernova.

AngelLestat, when you show me a sail that can actually unfold at 6kg/1,200m², then we talk. Right now, the support structures weigh dramatically more. In fact, the total for the SunJammer is 32kg. See the problem? That brings the total specific thrust to 0.34mN/kg. Same as aluminum foil. So lets see, SunJammer's sail is 32kg generating 10.8mN of thrust at the most. A modern ion thruster can generate the same with a 10kg unit. That leaves me not 6kg, but 22kg for Xenon. What can a 10kg rocket with 22kg of xenon do? About 60km/s. Does SunJammer's mission involve total delta-V over 60km/s? I did not think so. Same probe with ion thruster could have done mission with a cheaper launch. And yeah, you can make up all sorts of future materials that would boost solar sail's specific thrust. Except that we aren't anywhere close to the limit on ion thrusters either. And ion thrusters have way more room to improve their specific thrust. Solar sails cannot physically catch up to ion thrusters on that. Yeah, except you have to spiral in at 70% efficiency (45° angle), killing most of your velocity at high altitude, simply because of how low the thrust is. Why don't you do the actual math on trajectory something like SunJammer would take on a Sun dive, and then compare it to an ion thruster. I can match that trajectory with ion thruster at a fraction of the total weight. That's a fact. It's not free if you had to launch a 32kg sail into LEO to do a job that could have been done with a much lighter ion thruster unit.

Only positrons are present in significant numbers. Trying to scoop up anti-protons from magnetosphere is not going to go any faster than manufacturing them at an accelerator.

If you were a smaller plane that had to hang around An-225, you'd wear shoulder pads too.

Heavy nuclei collisions. You want a whole mess of quarks to interact at pretty much the same time if you want a good chance at anything more exciting than a meson. Naturally, you'll need at least two GeV of energy as well. In practice, way more, since you are going to lose a bunch of energy to a shower of stuff you don't want. Again, mostly mesons. There are a handful of accelerators in the world capable of managing that. RHIC and LHC spring to mind, but there are at least a couple more, maybe a few. Beam-beam collisions are good for increasing CoM energy, but if you want to increase amount of antimatter produced, density is going to be important, so it's probably better to use a stationary target. Something like gold foil, probably. Beam would consist of some massive nuclei as well. Again, gold would work, but there could be better combinations. To be honest, I don't know how much it depends on structure of the nuclei involved. Some might be better than others, and energy you can put into the beam is also a factor, but this isn't my field. At any rate, you still need at least one heavy ion accelerator if you want a decent rate of production.

Cool. Never seen it painted like this. Makes sense, I suppose, all things considered.

The exterior curvature is going to affect the ship's velocity, still. In a nutshell, it's because there is no such thing as "same velocity" in two different places on curved manifold. It's a meaningless statement. There is, however, parallel transport, and that's what will define ship's velocity after exiting warp.

Idobox, a Sun dive can still be done better with an ion drive. Albeit, not by much. So maybe costs of drive would enter at that point. But good point on inclination change. A mission that might require several major inclination changes can still be done with a lighter unit on a solar sail. Not mission to Sedna, though. A bi-elliptic to Sedna can still be done in well under 50km/s with an ion drive. It's hard to imagine a specific mission that fits the bill, and it's only going to get worse as specific thrust of the ion thrusters improves. But I'll give you that there can, in principle, be such a mission in solar system.

The duration of the trip isn't a factor. Only the bubble geometry. In practice, you'll probably end up with losses you'll have to replenish, thereby creating an energy drain that's proportional to the time you need the bubble up, but these computations don't take it into consideration. It's just the amount of energy you need up front. I believe, Dr. White's computations were for 10c and a fairly small probe. Maybe a few meters in diameter. I don't know the exact geometry used for initial Alcubierre Drive estimate, but it doesn't matter all that much, given the scale of the result. In Alcubierre Drive, energy requirement scales as square of bubble's velocity. With other geometries, it could be a little different. On the other hand, Alcubierre's Drive is pretty insensitive to the bubble size. In other words, you don't need that much more total energy to make the bubble larger. I don't know how true it is for the Dr. White's drive.

At anything less than 50km/s delta-V, modern ion thrusters are still better. As in, lighter unit will accelerate payload to desired delta-V in less time. And that's at 1AU, where you'll never need delta-V in excess of 10-20km/s. Kuiper Belt? That starts at 30AU. So you'll have at best 0.1% of thrust using solar sails. To beat ion thuster there, you'd need target speeds of 400km/s. And again, by the time you'd build up a fraction of that, you'd be so far out of the Solar System that the sail would become absolutely useless. There is no niche for solar sails. They've become obsolete the moment we started experimenting with ion thrusters.

Long range problems seem veery counter-intuitive. You can end up with CTCs once you take Poincare to be a local symmetry. And that's the only change we make going from QFT in Minkowski to Quantum Gravity. So it has to root itself in non-renormalizability somehow. And as you say, that shouldn't have any but the short-range effects. I'll read up on it, but it seems very strange. Thanks for bringing it to my attention, though.

Can I generalize your statements to "Quantum theory of Gravity is non-renormalizable?" And yeah, it's a problem within the theory, but only at the level of us trying to make use of it. It's not that QFT requires nicely ordered spaces. It's that we can only apply QFT as a theory to such spaces. Breakdown of theory, in this case, is not an indication of a physical problem. Yet, there are ways to deal with it. The most direct thing is not to mix gravity and particle field theory. Solve for space-time geometry, and then try to do QFT in that curved space-time. It's fairly straight forward, and you can still end up with problems that you may or may not be able to solve in that specific geometry. That's essentially what you are talking about. A more general approach is to treat gravity as Yang-Mills Theory on Poincare group. Then you essentially have to do Quantum Gravity. As I've said above, it's not a renormalizable theory, but you can construct an effective theory which is valid up to a certain scale. Usually, Plank scale, because bellow Plank scale you're in trouble either way. So long as your space-time is "flat" on Plank scale, your theory then holds on the larger scale. There is a good paper on Arxiv that goes into more detail. Note that this approach might not be applicable in practice to a general problem you are trying to solve. The main point here is that physics doesn't actually break there. We just end up with insane math we can't do. P.S. By the way, general path integral formalism doesn't require time ordering. It's one of these crutches we rely on to make math easier. There are a number of theorems and equations in QFT that are derived without considering time ordering. Dyson-Schwinger Equations are an example. (They are often derived in texts with help of time ordering, but they can be derived making no such assumptions.) In General Relativity, a lot of limitations applied by Special Relativity globally are actually only enforced locally. It means that around any point in space-time, you can find a region of space-time for which the constraint holds. It can be a very small region, however. For example, speed of light is a global limit in Special Relativity. Nothing can go faster than light. In General Relativity, there is no such global limit. Hence the Alcubierre Drive, and the fact that Universe expands faster than light. However, in any sufficiently close proximity, two particles will never travel faster than light relative to each other. In most of the universe, that "neighborhood" spans many galaxies, but if you either stumble on some extremely curved space, or you create your own warped space with a warp drive, you can, in principle, break the light barrier. Causality in Special Relativity is a very strong statement. It says that if event A causes event B, then in every frame of reference B follows A. In general, sequence of events can change if you travel close to speed of light, but only casually unrelated events can actually swap places. So, for example, if I flip a switch and, just by chance, somewhere on the other side of the planet a light bulb went on at the exact same time, then depending on observer, one or the other might have appeared to happen first. But if me flipping the switch was the cause of the light coming on, then from every observer's perspective, switch was flipped first, and light went on second. That's causality. In General Relativity, such constraint is also local. So for typical space-time geometries, on the scale of individual particles you can still do time ordering. And that time ordering is frame-independent, which is very important for the field theory. But then you get to something like event horizon of a black hole, and our theory breaks down all together. On the other hand, large scale events no longer have to follow in sequence. You can go back into the past and kill your grandfather. That doesn't cause any grand cosmic paradoxes or problems with "continuity". The paradox is resolved on the quantum level, and while there are different ways to describe resolution, it's most obvious in Many World Interpretation, where you essentially end up with alternate timelines. Both of these are relevant to Alcubierre Drive, because, in principle, you can use AD as a time machine if you have suitably curved space-time. The second part is that space-time at the bubble boundary is extremely curved. And it wasn't clear for the long time what would happen at said boundary, and if it would prevent AD from being even a theoretical possibility. Now we know that so long as bubble thickness is greater than plank scale, there aren't any fundamental problems with it. For thinner bubbles, the theory isn't clear, but odds are, we'll run into problems.