# Alcubierre drive

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Depiction of a warp bubble. The space ahead of the ship contracts whereas the space behind it expands.

The Alcubierre metric, also known as the Alcubierre drive or warp drive, is a theoretical model for propelling a spacecraft faster than the speed of light.

The Alcubierre Drive is a solution of Einstein's field equations of general relativity. In this theory, matter distorts the geometry of spacetime, this curved geometry being interpreted as gravity. The physicist Miguel Alcubierre proposed a method of stretching space in a wave, causing the space "ahead" of a spacecraft to contract along the axis the spacecraft wishes to travel in 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 actually moving within this bubble, but rather being carried along as the region itself moves, conventional relativistic effects do not apply. There is no known way to induce such a wave, however, or to leave it once started; the Alcubierre drive remains a theoretical concept at this time.

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## Mathematics of the Alcubierre drive

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Using the 3+1 formalism of general relativity, the spacetime is described by a foliation of space-like hypersurfaces of constant coordinate time [itex]t[itex]. The general form of the Alcubierre metric is:

[itex]ds^2 = -\left(\alpha^2- \beta_i \beta^i\right)\,dt^2+2 \beta_i \,dx^i\, dt+ \gamma_{ij}\,dx^i\,dx^j[itex]

where [itex]\alpha[itex] is the lapse function that gives the interval of proper time between nearby hypersurfaces, [itex]\beta^i[itex] is the shift vector that relates the spatial coordinate systems on different hypersurfaces and [itex]\gamma_{ij}[itex] is a positive definite metric on each of the hypersurfaces. The particular form that Alcubierre studied (1994) is defined by:

[itex]\alpha=1\,[itex]
[itex]\beta^x=-v_s(t)f\left(r_s(t)\right),[itex]
[itex]\beta^y = \beta^z =0[itex]
[itex]\gamma_{ij}=\delta_{ij}[itex]

where

[itex]v_s(t)=\frac{dx_s(t)}{dt},[itex]
[itex]r_s(t)=[(x-x_s(t))^2+y^2+z^2]^{\frac{1}{2}}[itex]

and

[itex]f(r_s)=\frac{\tanh(\sigma (r_s + R))-\tanh(\sigma (r_s - R))}{2 \tanh(\sigma R)}[itex]

with [itex]R > 0[itex] and [itex]\sigma > 0[itex] arbitrary parameters. 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

[itex]-\frac{c^4}{8 \pi G} \frac{v_s^2 (x^2+y^2)}{4 g^2 r_s ^2} \left(\frac{df}{dr_s}\right)^2[itex]

where [itex]g[itex] 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' (Alcubierre, 1994). The existence of exotic matter is not theoretically ruled out and the Casimir effect lends support to the proposed existence of such matter; however, generating enough exotic matter and sustaining it to perform feats such as faster-than-light travel (and also to keep open the 'throat' of a wormhole) is thought to be impractical. Low (1999) has shown 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. Template:Clear

## Physics of the Alcubierre drive

For those familiar with the effects of special relativity, such as Lorentz contraction, mass increase and time dilation, the Alcubierre metric has some apparently peculiar aspects. Since a ship at the center of the moving volume of the metric is at rest with respect to locally flat space, there are no relativistic mass increase or time dilation effects. The on-board spaceship clock runs at the same speed as the clock of an external observer, and that observer will detect no increase in the mass of the moving ship, even when it travels at FTL speeds. Moreover, Alcubierre has shown that even when the ship is accelerating, it travels on a free-fall 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 g-forces. Enormous tidal forces would be present near the edges of the flat-space 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 Alcubierre drive and science fiction

Note that faster-than-light travel is often used in science fiction to denote a wide variety of imaginary propulsion methods, most of which have nothing to do with the Alcubierre drive or any other physical theory. In Star Trek, the Alcubierre theory has largely been accepted due to the similarity of the appropriate terms, in order to explain the apparent breaking of the laws of physics in most of the series. However, this is only a happy coincidence: Dr. Alcubierre's work on "warp drive" was published in 1994, well after the pseudophysics of the warp drive were set down by Paramount Pictures.

## Reference

• The Giant Leap: Mankind Heads for the Stars by Adrian Berry (ISBN 0312877854)

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy