Editing Alcubierre drive (section)

==Mathematics==
Using the [[ADM formalism]] of [[general relativity]], the [[spacetime]] is described by a [[foliation]] of space-like [[hypersurface]]s of constant coordinate time {{mvar|t}}, with the metric taking the following general form:

:<math>ds^2 = -\left(\alpha^2 - \beta_i \beta^i\right) \,dt^2 + 2 \beta_i \,dx^i \,dt + \gamma_{ij}\,dx^i\,dx^j,</math>

where
* {{mvar|α}} is the lapse function that gives the interval of proper time between nearby hypersurfaces,
* {{mvar|β<sup>i</sup>}} is the shift vector that relates the spatial coordinate systems on different hypersurfaces,
* {{mvar|γ<sub>ij</sub>}} is a positive-definite metric on each of the hypersurfaces.

The particular form that Alcubierre studied<ref name="Alcubierre" /> is defined by:

:<math>\begin{align}
 \alpha &= 1,\\
 \beta^x &= -v_s(t)f\big(r_s(t)\big),\\
 \beta^y &= \beta^z = 0,\\
 \gamma_{ij} &= \delta_{ij},
\end{align}</math>

where

:<math>\begin{align}
 v_s(t) &= \frac{dx_s(t)}{dt},\\
 r_s(t) &= \sqrt{\big(x - x_s(t)\big)^2 + y^2 + z^2},\\
 f(r_s) &= \frac{\tanh\big(\sigma (r_s + R)\big) - \tanh\big(\sigma (r_s - R)\big)}{2 \tanh(\sigma R)},
\end{align}</math>

with arbitrary parameters {{math|''R'' > 0}} and {{math|''σ'' > 0}}. Alcubierre's specific form of the metric can thus be written:

:<math>ds^2 = \left(v_s(t)^2 f\big(r_s(t)\big)^2 - 1\right)\,dt^2 - 2v_s(t)f\big(r_s(t)\big)\,dx\,dt + dx^2 + dy^2 + dz^2.</math>

With this particular form of the metric, it can be shown that the energy density measured by observers whose 4-velocity is normal to the hypersurfaces is given by:

:<math>-\frac{c^4}{8 \pi G} \frac{v_s^2 \left(y^2 + z^2\right)}{4 g^2 r_s ^2} \left(\frac{df}{dr_s}\right)^2,</math>

where {{mvar|g}} is the determinant of the metric [[tensor]].

Thus, because the energy density is negative, one needs [[exotic matter]] to travel more quickly than the speed of light.<ref name="Alcubierre" /> The existence of exotic matter is not theoretically ruled out; however, generating and sustaining enough exotic matter to perform feats such as faster-than-light travel (and to keep open the "throat" of a [[wormhole]]) is thought to be impractical.{{Citation needed|date=October 2012}} According to writer Robert Low, within the context of general relativity it is impossible to construct a [[warp drive]] in the absence of exotic matter.<ref>{{Cite journal |last=Low |first=Robert J. |date=1999 |title=Speed Limits in General Relativity |journal=[[Classical and Quantum Gravity]] |volume=16 |issue=2 |pages=543–549 |arxiv=gr-qc/9812067 |bibcode=1999CQGra..16..543L |doi=10.1088/0264-9381/16/2/016 |s2cid=6540037}}</ref>