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The Alcubierre metric, also known as the Alcubierre drive or Warp Drive, is a speculative 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).

In 1994, the Mexican physicist Miguel Alcubierre proposed in the Journal of Classical and Quantum Gravity 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 slower-than-light object to accelerate to faster-than-light speeds. However, there are no known methods to create such a warp bubble in a region that does not already contain one, or to leave the bubble once inside it, so the Alcubierre drive remains a theoretical concept at this time.

Contents 1 Mathematics of the Alcubierre drive 2 Alcubierre Metric 2.1 Mathematical representation 3 Physics of the Alcubierre drive 4 The Alcubierre drive and science fiction 5 Criticisms 6 Expansion on the work of Alcubierre 7 See also 8 References 9 External links

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:

where α is the lapse function that gives the interval of proper time between nearby hypersurfaces, βⁱ 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 (1994) is defined by:

β^y = β^z = 0

γ_ij = δ_ij

where

and

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

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. Low (1999) 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.

The Alcubierre Metric defines the so-called 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[1].

The Alcubierre metric may be written

where

and

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.

For those familiar with the effects of special relativity, such as Lorentz contraction 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 (Alcubierre, 1994).

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 (Natario, 2002), 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 energy-momentum 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 (Krasnikov, 1998) 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. [2][3]

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.

Star Trek fans claim that, 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.^{[citation needed]} In fact, the physics of warp drive in Star Trek have never been defined specifically onscreen and none of the "technical manuals" based on the show has made any reference to Dr. Alcubierre's theory.

In a 1978 production memo, Dr. Jesco von Puttkamer, technical advisor for Star Trek: The Motion Picture, proposed a model of warp drive which bears some striking similarities to Dr. Alcubierre's later theory, employing the same principle of a distortion in spacetime moving a ship faster than light inside a pocket of spacetime within it. (The memo is reprinted on pp. 153-4 of the book The Making of Star Trek: The Motion Picture.)

However, later Star Trek technical advisors did not follow this model, and modern Star Trek productions tend to follow a warp-drive model based on the use of "subspace" as an alternate dimensional realm through which a ship may travel at hyperlight speeds, analogous to the use of hyperspace in much science fiction. However, the specifics remain vague enough that some consider it possible to reconcile Star Trek warp drive with the Alcubierre theory (for example, see Aftermath by Christopher L. Bennett in the Starfleet Corps of Engineers Ebook series). Another, more recent book called Warp Speed by Dr. Travis S. Taylor delves more into detail about the Alcubierre theory, as well as using it as the basis for the entire book.

Although it precedes Alcubierre drive, the anime version of Captain Future featured a similar mechanism, called undulating mode.

Alcubierre drive theory is also mentioned in Orbiter, a graphic novel by Warren Ellis.

Stephen Baxter mentions Alcubierre drive theory in his short story collection 'Vacuum Diagrams'.

• The pilot inside the bubble is causally disconnected with its walls. Therefore the bubble cannot be used for the first faster-than-light trip to a distant star. In other words, to travel to Vega (which is 26 light-years 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 (Krasnikov, 1998).

• 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 stress-energy 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 -10⁶⁷g might be required to transport a small spaceship across the Milky Way galaxy. Counterarguments to these apparent problems have been offered (Krasnikov, 2003), but not everyone is convinced they can be overcome.

• It has been suggested that an intense flux of blue shifted starlight would fry any inhabitants of the bubble.^{[citation needed]}

• If warp drives were genuinely possible, we would expect to see, even in toy models such as the Alcubierre warp drive, some indication of mass-energy being gathered up, transported, concentrated, reorganized and used to create and operate the warp bubble. But by their very nature, current warp drive metrics seem to have a very different character: the bubble appears (and may eventually disappear) "spontaneously". We can observe a kind of "circulation" of energy-momentum around the bubble, but nothing which suggests a phenomenon which can be created or controlled by physical means.

• Chris Van Den Broeck tried to address potential issues in a 1999 paper also published in Classical and Quantum Gravity. By contracting the 3+1 dimensional surface area of the 'bubble' being transported by the drive, while at the same time expanding 3 dimensional the 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[4].

• Work by González-Díaz resolved the problem of quantum instability for 2 dimensions. In his paper published in Physical Review D, Vol. 62, González-Díaz proposed considering closed, time-like curves. This refinement allows for multiply-connected spaces, closing the geodesic incompleteness and satisfying quantum instability requirements.^{[citation needed]}

• Exact solutions in general relativity (for more on the sense in which the Alcubierre spacetime is a solution).
• Spacecraft propulsion
• Faster-than-light
• Krasnikov Tube

• Lobo, Francisco S. N.; & Visser, Matt (2004). "Fundamental limitations on 'warp drive' spacetimes". Class. Quant. Grav. 21: 5871-5892.  See also the eprint. arXiv. Retrieved on 23 June, 2005.
• Natario, Jose (2002). "Warp drive with zero expansion". Class. Quant. Grav. 19: 1157-1166.  See also the eprint. arXiv. Retrieved on 23 June, 2005.
• Broeck, Chris Van Den (1999). "A `warp drive' with more reasonable total energy requirements". Class. Quant. Grav. 16: 3973-3979.  See also the eprint. arXiv. Retrieved on 23 June, 2005.
• Low, Robert J. (1999). "Speed Limits in General Relativity". Class. Quant. Grav. 16: 543-549.  See also the eprint version. arXiv. Retrieved on 30 June, 2005.
• Hiscock, William A. (1997). "Quantum effects in the Alcubierre warp drive spacetime". Class. Quant. Grav. 14: L183-L188.  See also the eprint. arXiv. Retrieved on 23 June, 2005.
• Pfenning, Michael J.; Ford, L. H. (1997). "The unphysical nature of 'Warp Drive'". Class. Quant. Grav. 14: 1743-1751.  See also the eprint. arXiv. Retrieved on 23 June, 2005.
• Alcubierre, Miguel (1994). "The warp drive: hyper-fast travel within general relativity". Class. Quant. Grav. 11: L73-L77.  See also the eprint version. arXiv. Retrieved on 23 June, 2005., and also at iop.org
• S. Krasnikov (1998). "Hyper-fast travel in general relativity". Physical Review D 57: 4760.  See also the eprint version. arXiv.
• L. H. Ford and T. A. Roman (1996). "Quantum field theory constrains traversable wormhole geometries". Physical Review D: 5496.  See also the eprint. arXiv.
• S. Krasnikov (2003). "The quantum inequalities do not forbid spacetime shortcuts". Physical Review D 67: 104013.  See also the eprint version. arXiv.
• Berry, Adrian (1999). The Giant Leap: Mankind Heads for the Stars. Headline. ISBN 0-7472-7565-3.  Apparently a popular book by a science writer, on space travel in general.
• T. S. Taylor, T. C. Powell, "Current Status of Metric Engineering with Implications for the Warp Drive," AIAA-2003-4991 39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Huntsville, Alabama, July 20-23, 2003

• The Alcubierre Warp Drive by John G. Cramer
• The warp drive: hyper-fast travel within general relativity - Alcubierre's original paper (PDF File)
• Problems with Warp Drive Examined - (PDF File)
• Marcelo B. Ribeiro's Page on Warp Drive Theory
• A short video clip of the hypothetical effects of the warp drive.
• Doc Travis S. Taylor's website
• The (Im) Possibility of Warp Drive (Van Den Broeck)
• Reduced Energy Requirements for Warp Drive (Loup, Waite)
• Warp Drive Space-Time (González-Díaz)