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Alcubierre-Broeck Warp Drives
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It is often said that special relativity implies that no information can travel faster than the speed c.
c = 299792458 m/s.
To the extent that we restrict our physics to special relativity and demand that the traditional order of causation is always correct, this is true. However, special relativity is only a special case of physics within a more general theory known as general relativity.
In general relativity the vacuum speed of light is locally invariant, but not globally invariant. The vacuum speed of light remote from a given observer need not be c. It can be greater than c and can vary with direction depending on the gravitational field involved. Because this is allowed starships that travel between the stars faster than c are not ruled out a-priori within the physics of general relativity.
In 1994 physicist Miguel Alcubierre of the University of Wales published how to use general relativity to consider a space-time geometry , actual Warp Drives, that would undo the special relativistic time dilation effects and allow for faster than c travel in the May issue of Classical and Quantum Gravity. His space-time geometry would also allow for acceleration of the ship without any internal forces crushing the crew. They would feel weightless as the ship accelerated. Unfortunately in the arbitrary manner that the metric was developed it was shown to require a lot of negative energy that would violate a few quantum mechanics energy conditions and a greater magnitude of it than all the mass of the known universe.
The space-time geometry that he had derived can be represented in the following equation
ds2 = dct2 - (dx - bfdct)2 - dy2 - dz2
f can be any function of the coordinates that is one at the location of the starship and zero far from it. Transforming coordinates to a particular choice of the starship coordinate frame and allowing it to travel faster than c (b > 1), coordinate singularities crop up in the equation corresponding to event horizons in front of and behind the ship enclosing it in a mathematical bubble. This is called the warp bubble. Even though the event horizons remain for any choice of ship frame, we show that there is a choice for star ship coordinates for which the singularities are transformed away.
June 1999 Chris Van Den Broeck of the Institute for Theoretical Physics at the Catholic University of Leuven, Belgium, came up with an alteration for this space-time geometry that would retain all of the desired warp drive qualities but reduced the negative energy requirements down to the order of a transversable wormhole. His space-time geometry can be represented by this equation
ds2 = dct2 - B2[(dx - b fdct)2 + dy2 + dz2]
B can be any function that is large near the starship and one far from it though he used a specific top hat function for his example calculations. That not only brought the negative energy requirements down to a hopefully one day reachable goal but it also solved one of the quantum mechanics energy condition violations.
The above metrics were designed to study linear motion and Alcubierre himself wrongly believed that the effect of the acceleration beyond c speeds of the Starship was a result of the space behind the ship happening to be in a state of expansion while the space in from of the ship is in a state of contraction. This isn't really the cause. There are other Warp drive spacetimes besides the one above. Consider the following warp drive loop.
ds2 = dct2 - dz2 - dr2 - (rdf - bfdct)2
Let b be any function of time. Let f be a function that is zero far from the warp loop and is at a maxima of one at (z = 0, r = R). This spacetime has a geodesic at the loop such that an object initially following the loop at its initial warp speed b will continue on accelerating according to however b is changed.
Notice that in this spacetime there is no diametric nature to the space immediately behind and in front of the ship.
The Van Den Broeck Warp Drive was actually a case of a more general type of Alcubierre-Broeck Warp Drive SpaceTime. A more general case of an Alcubierre-Broeck Type Warp Drive SpaceTime can be written
Where the Metric tensor reduces to that of special relativity far from the ship, and dx1 represents a coordinate distance displacement.
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