Alcubierre Warp Drive Mathematics

[Dominatus]

So, I decided to try and delve some more into this supposed Warp Drive being developed. After reading up on it some, I got the basics, about how it expands space behind the craft and contracts it in front... I get what it does. But I don't understand the how, or the why. Looking for an explanation on YouTube was useless- I kept getting alien-government conspiracy stuff. So I want to know how the formula works, why it works, how to make it work... You get the picture. Any help would be appreciated on this.

[peadar1987]

Well, the how, we don't know it it's even possible. The maths only works if you have negative mass, and it's hypothesised that perhaps there are exotic forms of matter for which this is true, in the same way that you can have negative charge.

[K^2]

Casimir effect results in a space-time region which has qualities of the negative energy density which you need for a Warp Drive or a traversable wormhole. People have been looking into it for a while. The net energy is still positive. It's a bit like having a bubble in a glass of water. The bubble behaves as if it has negative weight, but it doesn't make the glass lighter. But there appear to be ways to make it work for the warp drive. A lot of this is still in the air, because math for anything beyond the basic Alcubierre Drive is very hairy. But there is also a lot of progress being made. When Alcubierre's paper was published, the estimates for the energy were greater than mass of observable universe. With some improvements, people shaved it down to a large planet. Latest research manages to use oscillating field to drop this estimates to less than a ton, and that's a quantity of energy we can comprehend. It's still more than we can produce in a controlled fashion, but we're getting there.

The math, though... It's honestly not that bad, but it requires a lot of background. I mean, the fundamental equation is just ADM metric with some parameters. Once you know the metric, it's just algebra to compute required energy densities. And relationship between parametrization of the metric and trajectory of the ship has already been worked out for you. So it's just plugging numbers into the equations.

Understanding why all of that works, or being able to derive variations on the drive, requires thorough understanding of General Relativity. Without getting into specifics, what allows it to work is the fact that speed of light is a local limit. Two things close to each other cannot move faster than c with respect to each other. But once you have sufficient separation, this is no longer true. You can absorb superluminal motion into coordinate system choice, and this gives you space-time curvature with specific energy density distributions as a consequence. In other words, by achieving this specific energy distribution, you force the desired metric, and then your ship can go as fast as you like relative to remote objects. Understanding Yang-Mills and Noether's Theorem helps to see why these things are related in such a way, but not, strictly speaking, necessary. You can understand almost all of GR from perspective of differential geometry without diving into underlying field theory.

If you want to learn the actual math and understand how all of this works, you should start with Linear Algebra, learn Calculus, some Analysis would help, Differential Equations, Topology, and just the basics of Differential Geometry. Then if you want to understand the underlying physics, you need to learn Classical Mechanics, Electrodynamics, Quantum Mechanics, Group Theory, basic Quantum Field Theory, just a bit of Relativistic Quantum Field Theory, and get a good enough grasp of Yang-Mills to understand connection to Differential Geometry picture. That's where you start getting a general idea of how all of this works.

[LLlAMnYP]

If you want to learn the actual math and understand how all of this works, you should start with Linear Algebra, learn Calculus, some Analysis would help, Differential Equations, Topology, and just the basics of Differential Geometry. Then if you want to understand the underlying physics, you need to learn Classical Mechanics, Electrodynamics, Quantum Mechanics, Group Theory, basic Quantum Field Theory, just a bit of Relativistic Quantum Field Theory, and get a good enough grasp of Yang-Mills to understand connection to Differential Geometry picture. That's where you start getting a general idea of how all of this works.

...and having done all that you should be well on the way to finishing your doctoral thesis in theoretical physics:sticktongue:

[OrtwinS]

edit: This post used to contain a truckload of bollocks. And a significant amount of shameful behavior not befitting the balanced, thoughtful, and introspection-prone person I attempt (or pretended?) to be.

Edited December 11, 2013 by OrtwinS
unveiled foolish behavior requiring redaction for the sake of everyone

[K^2]

within 0.5 'local perspective years'

Map time matches ship's proper time under warp. That's the whole point of a warp drive. In "0.5 'local perspective years'," you can make the trip in completely flat space-time thanks to SR. You just need a really big rocket.

And the Alcubierre drive math is kinda flawed to begin with, since you have make quite a few a LOT of assumptions.

There are no assumptions beyond these of standard theory.

Source: my memory, but this was ~2 years ago. Algebra & multi-dimensional differential equations are the easy part, quantum mechanics is actually quite simple (which makes them the pure genius they are, and I use them in my work), but there is a chance I erred in the quantum field theory (Yang-Mills shizzle) (see what I did there ).

Because if you don't understand it, it's all just buzzwords to everybody else, right?

[Luis]

When Alcubierre's paper was published, the estimates for the energy were greater than mass of observable universe. With some improvements, people shaved it down to a large planet. Latest research manages to use oscillating field to drop this estimates to less than a ton, and that's a quantity of energy we can comprehend. It's still more than we can produce in a controlled fashion, but we're getting there.

Except that it needs to be negative energy. Which no one has found, made, or proved exists.

[DaveofDefeat]

Isn't negative energy just a mathematical concept created to make these equations work? Do we have any kind of theory that predicts negative energy?

[Winter Man]

Except that it needs to be negative energy. Which no one has found, made, or proved exists.

Read the whole post. It's relatively negative. Like having a floating ground for processing analog signals in a DC circuit, you can go to + or - 9V, but only because you floated the ground at +9V from an 18V source. Look up zero point energy (and try to avoid all the damned quacks trying to sell you chakra magnets or whatever!), there's a base energy to the universe which you can go below using casimir cavities.

[K^2]

Isn't negative energy just a mathematical concept created to make these equations work?

So is space-time curvature. Thing about modern physics is that questions of, "Is this really how it works, or is this just a mathematical trick," become kind of meaningless. How would you tell the difference?

Do we have any kind of theory that predicts negative energy?

More than one. But Casimir Effect is still the only one that's been experimentally verified, and even there we're far from understanding the whole thing. Winter Man mentioned zero fluctuations, and that leads to vacuum having energy. Unfortunately, when we try to predict how much, we end up with a value that is in disagreement with cosmological models by many orders of magnitude.

P.S. Actually, the more puzzling question is why we don't observe negative energy. The only requirement from field theory is that the stress-energy tensor is a conserved current. In non-relativistic limit, that means energy, momentum, and angular momentum are conserved. But it places no restrictions on the sign. In fact, we have no trouble giving momentum or angular momentum any direction we wish. It's only the energy component, which is associated with time-direction, that insists on being positive definite for some reason. A blind stab in the dark is that it is related to direction of time flow, but that only makes the whole thing more convoluted.

Edited December 11, 2013 by K^2

[OrtwinS]

Map time matches ship's proper time under warp. That's the whole point of a warp drive. In "0.5 'local perspective years'," you can make the trip in completely flat space-time thanks to SR. You just need a really big rocket.

Excuse me my poor choice of words, English is not my prime language. I meant a 4 ly trip which takes only 0.5 years from the Earth perspective (my local, not the vessel, which I probably should have named 'earth perspective time').

There are no assumptions beyond these of standard theory.

Assuming tech that creates space distorting fields and energy requirements that mis-match our current ideas of energy storage or production.

My criticism wasn't the assumptions in theory, it was the assumptions I made leading to any practical application. The thread is named "Alcubierre mathematics", sorry for straying of the path.

Because if you don't understand it, it's all just buzzwords to everybody else, right?

No, of course not.

After a good night rest and careful review I humbly admit that my memory nor my practice obsession have contributed in any meaningful way to the thread, and being a (supposedly) educated person I should have stopped myself from posting anything at all (except perhaps a note of interest).

I feel ashamed for letting myself getting carried away by the sense of "Oh yea, I once calculated stuff on that topic". 2 weeks of graduation student indulgence don't give one any authority on any topic.

I blame a remnant of 'youthful I-can-take-on-the-world-itis', in order to preserve my current sanity. Don't you ever do something, and the day after you want to slam your head on the desk thinking "**** I thought I had outgrown that particular behavior years ago"? Well this is one of those cases for me.

I apologize for the stress and the almost tangible dismissive passive aggression I seem to have awakened in you. Thank you for the notice though.

I'll edit my previous post to prevent further confusion and/or derailment, though I wouldn't if it could serve as a permanent landmark and reminder for myself to not behave as an arrogant dick.

Which is also the reason why I write this in the thread and not in a PM.

Peace K^2?

[kiwi1960]

If you want to understand the mechanics of it, then delve into the Star Trek theory, the show may be bogus, especially the way they get out of things (like firing a quantum torpedo at the event horizon of a black hole so they can collapse it and escape being pulled into it) but the science behind show is pretty much sound. Things like the warp drive, transporters and phasers are pretty much spot on as far as the science goes.

In fact, Star Trek communicators are nothing more than our cellphones... those large metal books thinks Kirk used to sign when a female ensign handed it to him is our I-pad... and a few years ago, an Aussie scientist was able to "beam" light a few inches... still too early to be of any practicable use though...

And NASA is developing the warp engine...

http://io9.com/5963263/how-nasa-will-build-its-very-first-warp-drive

The Star Trek Warp Drive

http://www.space.com/21201-star-trek-technology-explained-infographic.html

More from Nasa

http://www.newscientist.com/article/mg21929300.300-meet-the-nasa-scientist-devising-a-starship-warp-drive.html#.UqhN1H9kOBM

[ZetaX]

" Things like the warp drive, transporters and phasers are pretty much spot on as far as the science goes."

What¿! No! And reasons for that have been mentioned enough on this and other threads already.

[Winter Man]

What¿! No! And reasons for that have been mentioned enough on this and other threads already.

...no, no reasons for it not to work have been given. Only assumptions.

[Seret]

the science behind show is pretty much sound. Things like the warp drive, transporters and phasers are pretty much spot on as far as the science goes.

I'm trying to detect where the irony is in this post, but I'm concerned you might be serious.

[K^2]

...no, no reasons for it not to work have been given.

Not the way they have them on the show. I have a suspicion that all of their technology is antiquated salvage ran by wizards pretending to be engineers. Moreover, none of the wizards are aware that other engineers are also wizards, so they are constantly trying to "fake it", all the while making stuff happen with magic under the table. It's the only explanation I have for the techno-babble and inconsistency of explanations and effects.

[Scotius]

Sufficiently advanced technology and all that We are in position of a Victorian era man - accustomed with an idea of flying, existence of other planets and laws of physics. But did anyone came up with an idea of Space Shuttle? No - best idea for space exploration born then was a giant cannon. I'm pretty sure our grandchildren will laugh at our naive plans of sending people to Mars with chemical rockets.

[K^2]

Here is the thing, though. With very few non-critical exceptions, all of the technology we are using is based on physics that was already around during Victorian Era. I could explain absolutely every detail of operation of a Space Shuttle to somebody from Victorian Era in terms of science they could understand. Moreover, from a good popular description they'd be able to workout almost all of it for themselves. They wouldn't be able to build one, but they'd know how it works. Theoretical physics is way, way ahead of technology. Yes, if we start talking about what's possible and not, we'll probably be trying to figure out how to fire a rocket out of a cannon. But if somebody suggests a piece of future technology to us, we can analyze it pretty well.

And I don't know about you, but my computer doesn't run on wood gas. There is just way too much technology in ST that should have been displaced. This isn't even about science. It's just common sense.

[Seret]

Here is the thing, though. With very few non-critical exceptions, all of the technology we are using is based on physics that was already around during Victorian Era.

I agree with your post in general, a lot of stuff would be comprehensible to them. But stuff like electronics (including semiconductor devices like PV arrays), aerodynamics, and nuclear reactors would completely flummox them. The conceptual framework for that stuff only came around in the 20th century. Sometimes the engineering follows pretty hard on the heels of the theory, sometimes it takes centuries. That's why it's so hard t predict which ideas that are at the cutting edge now will actually find some practical expression in the future.

[K^2]

Ohm's Law - 1827. Faraday's Law of Induction - 1829. Kirchoff's Circuit Laws - 1845. Maxwell's Equations - 1861. Bernoulli's Principle - 1738. Navier-Stokes Equations - 1822. Reynolds Number - 1851.

Most of these predate Victorian Era. Rest are from early Victorian. They are sufficient to understand both electronics and aerodynamics. Now, they aren't sufficient to predict semiconductors, for example, because it's a quantum effect. But Kirchoff's Laws do describe transistors in a circuit as a non-Ohmic device. Similarly, the fact that flight is possible due to aerodynamics was absolutely clear. What was missing was understanding of circulation, which came about at the turn of the century.

There have never been a case when engineering followed theory closely. What we have are a few examples of lucky discoveries that were soon followed by theory. Radiation, semi-conductors, super-conductivity. Even flight, technically. Kutta-Joukowski Theorem, while already conceptualized, has not been available to Wright Brothers.

[Seret]

There have never been a case when engineering followed theory closely.

Nuclear was pretty quick. Rutherford was unlocking the structure of the atom in 1909, neutrons were only discovered in 1933, and we had reactors by 1942 and a-bombs in 1945. Sure, we knew about radiation before that, but we hadn't actually put it to any practical use.