Alcubierre metric

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Template:Expert The Alcubierre metric defines the so-called warp drive spacetime introduced by Miguel Alcubierre in 1994. 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. Even more striking, inhabitants of the bubble feel no awkward inertial effects, and travelers making a round trip inside a warp bubble experience no time dilation of the kind known from the famous twin paradox from special relativity.

The Alcubierre metric may be written

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

where

<math>v_s(t)=\frac{dx_s(t)}{dt}</math>

and

<math>r_s(t)=\sqrt{(x-x_s(t))^2+y^2+z^2}</math>.

Alcubierre chose a specific form for the function <math>f</math>, but other choices give a simpler spacetime exhibiting the desired "warp drive" effects more clearly and simply.

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. Natario has suggested a significantly different kind of warp bubble metric which does not feature expansion.

All known warp drive spacetimes violate various energy conditions. It is true that certain experimentally verified quantum phenomena, such as the Casimir effect, when approximated in the context of relativistic classical field theories, lead to stress-energy tensors which also violate the energy conditions. This initially led some to hope that Alcubierre type warp drives could perhaps be physically realized by clever engineering taking advantage of such quantum effects. Unfortunately, it turns out that a rather general class of warp drive spacetimes also violates certain quantum inequalities, and such violations are much harder to dismiss. The energy conditions in general relativity were never more than rough rules of thumb, but the quantum inequalities are generalizations of the uncertainty principle and seem to stand on solid ground in quantum field theory. Another problem with a large class of warp drive spacetimes is that even if the violations of the quantum inequalities were acceptable, the energy requirements may be absurdly gigantic, e.g. the mass-energy content of a star might be required to transport a small spaceship across the Milky Way galaxy. Counterarguments to these apparent problems have been offered, but not everyone is convinced they can be overcome. (If they could be, it has been suggested that an intense flux of blue shifted starlight would fry any inhabitants of the bubble.)

Perhaps the most troubling objection of all is that 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.

At present it seems fair to say that the consensus among physicists is that warp drive spacetimes of type considered here are probably impossible to realize. Some general counteraguments have been offered to the construction of any warp drive (not just ones of the Alcubierre or Natario forms). Nevertheless, a few physicists continue to hold out hope that some kind of warp drive might after all be physically possible, at least in principle.

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See also

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References

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