Velocities can be a tricky thing in general relativity. A cool concept seemly consistent with the Einstein field equations) is an Alcubierre drive, described by the Alcubierre metric.
However, I think I need some clarification of how it is supposed to work. As I understand it, it requires exotic matter to bend space around it to "move faster than light." Exactly what here is referred to be "faster than light?" And how does this work? Are you clumping together a punch of exotic matter at the front/rear and that bends spacetime?
What kind of energy levels are we talking about here? Can we even produce a such power needed to bend spacetime that much today, with current technology? Are the any rest products (exhaust/waste), and how are those handled? In a basic chemical rocket, chemical energy is released (often between kerosene and an oxidizer) and then released as then released as kinetic energy in exhaust molecules in the back of the rocket. The point is, a constant input of energy is needed for a traditional rocket to accelerate. How is this energy provided in a warp drive, and in what form? Does it have to constantly put in more energy to maintain the curvature, or can it just "cruise along" the bubble?
The local speed of light is, as far as I know it, always $c$. So what exactly is that Alcubierre drives implies "FTL travel?" By an observer stationary on Earth? If so, how does that work? If a light beam was sent alongside the warp drive, which would arrive first as measured by observers on the Earth, in the warp drive and at the destination?
Answer
The Alcubierre drive is like an escalator. Space expands behind the ship, and this is like the moving stairs coming out of the floor behind you, while space contracts in front of the ship, like the moving stairs disappearing into the floor in front of you. This is not, however, a description of the actual Alcubierre process.
We live in de Sitter space, characterized by positive vacuum energy which is powering the expansion of the universe. Negative vacuum energy is characteristic of anti-de Sitter space, which is not in our universe, so far as I know, although Alcubierre argued that the Casimir effect (http://math.ucr.edu/home/baez/physics/Quantum/casimir.html) due to quantum vacuum fluctuations of the electromagnetic field could be a source of negative energy.
As the Casimir effect is a quantum phenomenon, and as general relativity does not incorporate quantum effects, the development of a viable theory of quantum gravity may make the Alcubierre drive more likely.
As the Alcubierre wave is a fluctuation of space time, riders on the wave would not be moving through space and time in their local vicinity. There would be no inertia effects unless the riders moved within the bubble. Because of this, there would be no conflict with general relativity.
Calculations of the amount of negative energy required range from more mass than exists in the entire observable universe (Alcubier's original calculation), down to three solar masses, down to the mass of Jupiter, and down to ~700 Kg. Another possible difficulty is that riders in the bubble may be unable to send signals to the warped space in front of the bubble, perhaps making control of speed and direction impossible. What would be the coordinates through which one would steer in "hyperspace"?
Yet another problem is the interaction of classical phenomena with the warp drive when it is switched on and off, which might be fatal. Hawking radiation might provide extreme thermodynamic conditions (http://journals.aps.org/prd/abstract/10.1103/PhysRevD.79.124017).
The Wikipedia article seems like a good resource, but this may be a more readable introduction: http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20110015936.pdf. Also, this paper investigates some ancillary areas of interest using the Alcubierre metric: http://www.earthtech.org/publications/davis_STAIF_conference_2.pdf.
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