Alcubierre drive

Contents

1 Introduction

2 Mathematics of the Alcubierre Drive

3 Physics of the Alcubierre Drive

4 See also

5 Reference

6 External link

Introduction

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.

Ordinary matter distorts the geometry of spacetime, causing the effects of 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 of flat space. Since the ship is not actually moving within this region, 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.

Mathematics of the Alcubierre Drive

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

$ds^2 = -\left(\alpha^2- \beta_i \beta^i\right)dt^2+2 \beta_i dx^i dt+ \gamma_{ij}dx^idx^j$

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 $γ i j$ is a positive definite metric on each of the hypersurfaces. The particular form that Alcubierre studied (1994) is defined by:

$α = 1$

$\beta^x=-v_s(t)f\left(r_s(t)\right)$ , $β y = β z = 0$

$γ i j = δ i j$

where

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

and $f(r_s)=\frac{\tanh(\sigma (r_s + R))-\tanh(\sigma (r_s - R))}{2 \tanh(\sigma R)}$

with $R > 0$ and $σ > 0$ 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

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

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' (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. 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 usual rules of special relativity, with its Lorentz contraction, mass increase, and time dilation, the Alcubierre metric has some rather 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.

Note that the term "Warp drive" is 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.

Reference

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

External link

Miguel Alcubierre. "The Warp Drive: Hyper-Fast Travel Within General Relativity". Class. Quantum Grav. 11 (1994), L73-L77. (download)
Journal of Advanced Theoretical Propulsion
Problems with Warp Drive Examined

Categories: Spacecraft propulsion

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01-04-2007 01:21:04

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