Alcubierre warp drive

[lwcamp]

Originally Posted by acrosome

Would such a ship have the same ability to acquire an enormous true velocity by warping just enough to hold steady over a planet, then destroy the planet by turning off the drive and impacting it at an arbitrarily large fraction of c?

I am going to have to get technical here, and lay out some terms, some theoretical results of general relativity, and the consequences that follow from them. Bear with me - understanding these will help you understand the restrictions on warp drives.

In general relativity, energy and momentum and angular momentum are well defined and local in the weak field limit. The weak field limit is basically Newtonian gravity (with a few extra bells and whistles, such as gravitational waves). So when you are in a region where gravity is approximately Newtonian, then general relativity predicts that energy, momentum, and angular momentum are not only conserved, but conserved locally. That is, if you take any arbitrary closed surface, the energy & etc. inside that surface only changes by the amount of energy & etc. that goes in or out of that surface.

But what about warp drives? Warp drives are definitely not in the weak field limit. They distort space-time so much that you are far beyond Newtonian gravity. Do you still have to worry about the conservation laws? Well, the weak field result still holds if you can surround the strong field region with a closed surface where everywhere on the surface remains in the weak field limit (even if the surface is very far away). Inside that surface, energy & etc. might not be localized but it will not change without passing through the surface. So lets look in sequence at what this means for energy, momentum, and angular momentum (with increasingly depressing results as we go on).

Energy first. Usually when we think about a warp drive, we envision a spacecraft that is not yet warping sitting in a region of normal space-time (say, around a planet or star, or even around a black hole or neutron star since if you get far enough away you can enclose the black hole or neutron star with a surface that is everywhere in the weak field limit). Then the spacecraft does its warping thing, moves someplace else, and turns off its warp to again become a normal, non-warping body in a region of normal space-time. Now, if we enclose the entire path taken by the spacecraft in one of our imaginary surfaces, far enough away that any warping effects have damped away to the weak field limit, then we can see that the energy of the spacecraft is unchanged minus any radiation it produced in the operation of the warp drive. This is probably no surprise. By playing around with these surfaces (such as a co-moving surface surrounding the warp bubble but out in the weak field limit) you can probably convince yourself that the energy travels with the warp bubble. Since mass is equivalent to energy that isn't associated with motion, you can think of the warp bubble as having mass.

Next, momentum. Momentum has a direction as well as magnitude, and that direction is also conserved (mathematically, we say that it is a conserved vector). A force is a rate of change of momentum, and corresponds to an exchange of momentum between two things. Classically, an object's momentum is its mass times its velocity. So let's go back to warping from empty space to empty space. There are no forces, so since momentum is conserved the spacecraft ends up with the same momentum before and after the warp. Since the mass is unchanged by the above argument (neglecting radiation) then the spacecraft ends up with the same velocity as before. So now we can answer your question about hovering over a planet. There are two ways of looking at this - the Newtonian viewpoint of the planet exerting a force on the warp bubble's mass (increasing the warp bubbles momentum toward the planet) or the Einsteinian version of the spacecraft being in free fall so that it keeps warping forward in such a way to keep its distance from the planet constant. In both cases, the spacecraft's momentum, and hence its velocity when it finally turns off its warp drive, are constantly increasing in the direction of the planet. If this continues the spacecraft can build up arbitrarily large velocities. Note, though, that since the spacecraft's kinetic energy is also constantly increasing and energy is conserved, this energy has to come from somewhere such as the fuel to run a nuclear reactor. Once the energy in that fuel is exhausted the spacecraft cannot continue to hover since doing so would increase its energy further and there is no energy left. This ultimately limits the available velocity - although with nuclear fuels available you can still get nuclear bomb size explosions. Note also that this means when you go away from a planet you must use power and when you warp toward a planet you gain power from your warp drive.

Now for the really depressing one - angular momentum. Angular momentum is also a vector (classically, anyway - relativity adds some additional components that are hard to visualize, but I'll just stick with the classical 3-vector version). Angular momentum is defined as the magnitude of the momentum times the perpendicular distance from the observer to the object (in a direction mutually perpendicular to the momentum vector and the direction from the observer to the object). If something isn't moving, it has no angular momentum (assuming no internal motions - a rotating object has angular momentum but not everything with angular momentum needs to be rotating). If something is moving but is headed straight toward you or straight away from you, it also has no angular momentum. The further its distance of closest approach becomes (assuming a straight line trajectory) and the faster it is moving and the more massive it is, the greater its angular momentum. So, consider an observer moving toward the spacecraft, and then have the spacecraft warp away in a direction perpendicular to the direction to the observer. From the observer's point of view the spacecraft starts off with non-zero momentum (it is moving in his reference frame) but zero angular momentum (it is going straight toward him). After the warp we know that it must be going the same speed and direction from conservation of momentum, but now it is not headed straight toward the observer. Consequently it has non-zero angular momentum. Since you can always find a frame of reference that does this, free-form warping violates the conservation of angular momentum and is thus disallowed in general relativity.

How do we get around this? I can think of three ways. First, the spacecraft can dump all of its energy when it turns on its warp drive. Without energy it has no mass and thus neither momentum or angular momentum. It leaves the energy behind in some form - matter or radiation (if it is in the form of radiation, you get the release of 20,000 megatons of energy per ton of spacecraft mass). Then, when it gets to its destination it absorbs as much energy as needed to precipitate the spacecraft out of warp (that is, a 100 ton spacecraft would need to have its warp bubble gobble up 100 tons of mass at its destination in order to release the spacecraft). The spacecraft will end up with the velocity of the mass the warp field gobbled up. Until the spacecraft precipitates out of warp it cannot exchange energy with the rest of the universe (doing so would add or subtract mass to the warp bubble, which would then violate angular momentum conservation). This probably means the spacecraft is flying blind and will have a difficult time finding the mass it needs to exit warp.

Second, you can have a prepared pathway of highly curved spacetime, and all warp travel is along this path. Since the spacetime geometry of the pathway is highly curved, we are outside of the Newtonian limit and angular momentum (and energy and so on) is not localized along the path. This lets you go from one point on the path to another, but you can't warp to places off the path.

Third is that the warp drive emits some sort of radiation which never decays away to the weak field limit. This last possibility is not very good for adventure fiction since strong field gravitational radiation smashing through the universe out to arbitrary distances is rather hard on the setting.