An Alcubierre Warp Drive stretches spacetime in
a wave causing the fabric of space ahead of a spacecraft to
contract and the space behind it to expand. The ship can ride
the wave to accelerate to high speeds and time travel.
The Alcubierre drive, also known as the Alcubierre metric or
Warp Drive, is a mathematical model of a spacetime exhibiting
features reminiscent of the fictional "warp drive" from Star
Trek, which can travel "faster than light" (although not in a
local sense  see below).
The key characteristics of the application of Alcubierre warp
drives for time control and time travel are presented in the
picture below. This is followed by more detail describing the
effect below.
Alcubierre Warp Drive Description
In 1994, the Mexican physicist Miguel Alcubierre proposed a
method of stretching space in a wave which would in theory cause
the fabric of space ahead of a spacecraft to contract and the
space behind it to expand. The ship would ride this wave inside
a region known as a warp bubble of flat space. Since the ship is
not moving within this bubble, but carried along as the region
itself moves, conventional relativistic effects such as time
dilation do not apply in the way they would in the case of a
ship moving at high velocity through flat spacetime. Also, this
method of travel does not actually involve moving faster than
light in a local sense, since a light beam within the bubble
would still always move faster than the ship; it is only "faster
than light" in the sense that, thanks to the contraction of the
space in front of it, the ship could reach its destination
faster than a light beam restricted to travelling outside the
warp bubble. Thus, the Alcubierre drive does not contradict the
conventional claim that relativity forbids a slowerthanlight
object to accelerate to fasterthanlight speeds.
Alcubierre Metric
The Alcubierre Metric defines the socalled warp drive
spacetime. This is a Lorentzian manifold which, if interpreted
in the context of general relativity, exhibits features
reminiscent of the warp drive from Star Trek: a warp bubble
appears in previously flat spacetime and moves off at
effectively superluminal speed. Inhabitants of the bubble feel
no inertial effects. The object(s) within the bubble are not
moving (locally) faster than light, instead, the space around
them shifts so that the object(s) arrives at its destination
faster than light would in normal space.
Alcubierre chose a specific form for the function f, but other
choices give a simpler spacetime exhibiting the desired "warp
drive" effects more clearly and simply.
Mathematics of the Alcubierre drive
Using the 3+1 formalism of general relativity, the spacetime is
described by a foliation of spacelike hypersurfaces of constant
coordinate time t. The general form of the Alcubierre metric is:
where
α
is the lapse function that gives the interval of proper time
between nearby hypersurfaces,
β^{I}
is the shift vector that relates the spatial coordinate systems
on different hypersurfaces and
γ_{ij}
is a positive definite metric on each of the hypersurfaces. The
particular form that Alcubierre studied is defined by:
where
and
with
R
> 0 and
σ > 0
arbitrary parameters. Alcubierre's specific form of the metric
can thus be written;
With this particular form of the metric, it can be shown that
the energy density measured by observers whose 4velocity is
normal to the hypersurfaces is given by
where
g
is the determinant of the metric tensor. Thus, as the energy
density is negative, one needs exotic matter to travel faster
than the speed of light. The existence of exotic matter is not
theoretically ruled out, the Casimir effect and the accelerating
universe both lending support to the proposed existence of such
matter. However, generating enough exotic matter and sustaining
it to perform feats such as fasterthanlight travel (and also
to keep open the 'throat' of a wormhole) is thought to be
impractical. Low has argued that within the context of general
relativity, it is impossible to construct a warp drive in the
absence of exotic matter. It is generally believed that a
consistent theory of quantum gravity will resolve such issues
once and for all.
Physics of the Alcubierre drive

For those familiar with the effects of special relativity, such
as Lorentz contraction and time dilation, the Alcubierre metric
has some apparently peculiar aspects. In particular, Alcubierre
has shown that even when the ship is accelerating, it travels on
a freefall geodesic. In other words, a ship using the warp to
accelerate and decelerate is always in free fall, and the crew
would experience no accelerational gforces. Enormous tidal
forces would be present near the edges of the flatspace volume
because of the large space curvature there, but by suitable
specification of the metric, these would be made very small
within the volume occupied by the ship.
The original warp drive metric, and simple variants of it,
happen to have the ADM form which is often used in discussing
the initial value formulation of general relativity. This may
explain the widespread misconception that this spacetime is a
solution of the field equation of general relativity. Metrics in
ADM form are adapted to a certain family of inertial observers,
but these observers are not really physically distinguished from
other such families. Alcubierre interpreted his "warp bubble" in
terms of a contraction of "space" ahead of the bubble and an
expansion behind. But this interpretation might be misleading,
since the contraction and expansion actually refers to the
relative motion of nearby members of the family of ADM
observers.
In general relativity, one often first specifies a plausible
distribution of matter and energy, and then finds the geometry
of the spacetime associated with it; but it is also possible to
run the Einstein field equations in the other direction, first
specifying a metric and then finding the energymomentum tensor
associated with it, and this is what Alcubierre did in building
his metric. This practice means that the solution can violate
various energy conditions and require exotic matter. The need
for exotic matter leads to questions about whether it is
actually possible to find a way to distribute the matter in an
initial spacetime which lacks a "warp bubble" in such a way that
the bubble will be created at a later time. Yet another problem
is that, according to Serguei Krasnikov, it would be impossible
to generate the bubble without being able to force the exotic
matter to move at locally FTL speeds, which would require the
existence of tachyons. Some methods have been suggested which
would avoid the problem of tachyonic motion, but would probably
generate a naked singularity at the front of the bubble.
Difficulties
Significant problems with the metric of this form stem from the
fact that all known warp drive spacetimes violate various energy
conditions. It is true that certain experimentally verified
quantum phenomena, such as the Casimir effect, when described in
the context of the quantum field theories, lead to stressenergy
tensors which also violate the energy conditions and so one
might hope that Alcubierre type warp drives could perhaps be
physically realized by clever engineering taking advantage of
such quantum effects. However, if certain quantum inequalities
conjectured by Ford and Roman hold, then the energy requirements
for some warp drives may be absurdly gigantic, e.g. the energy
10^{67}gram equivalent might be required to transport a
small spaceship across the Milky Way galaxy. This is orders of
magnitude greater than the mass of the universe.
Counterarguments to these apparent problems have been offered,
but not everyone is convinced they can be overcome.
Chris Van Den Broeck, in 1999, has tried to address the
potential issues. By contracting the 3+1 dimensional surface
area of the 'bubble' being transported by the drive, while at
the same time expanding the 3 dimensional volume contained
inside, Van Den Broeck was able to reduce the total energy
needed to transport small atoms to less than 3 solar masses.
Later, by slightly modifying the Van Den Broeck metric,
Krasnikov reduced the necessary total amount of negative energy
to a few milligrams.
Krasnikov proposed that, if tachyonic matter could not be found
or used, then a solution might be to arrange for masses along
the path of the vessel to be set in motion in such a way that
the required field was produced. But in this case the Alcubierre
Drive vessel is not able to go dashing around the galaxy at
will. It is only able to travel routes which, like a railroad,
have first been equipped with the necessary infrastructure.
Miguel Alcubierre 
The pilot inside the bubble is causally disconnected with its
walls and cannot carry out any action outside the bubble.
However, it is necessary to place devices along the route in
advance, and since the pilot cannot do this while "in transit",
the bubble cannot be used for the first trip to a distant star.
In other words, to travel to Vega (which is 26 lightyears from
the Earth) one first has to arrange everything so that the
bubble moving toward Vega with a superluminal velocity would
appear and these arrangements will always take more than 26
years.
Coule has argued that schemes such as the one proposed by
Alcubierre are not feasible because the matter to be placed on
the road beforehand has to be placed at superluminal speed.
Thus, according to Coule, an Alcubierre Drive is required in
order to build an Alcubierre Drive. Since none have been proven
to exist already then the drive is impossible to construct, even
if the metric is physically meaningful. Coule argues that an
analogous objection will apply to any proposed method of
constructing an Alcubierre Drive.


