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Warp Theory: the Conservation

In the previous section warp theory, the theoretical science of `warp drive' was briefly discussed. The methods imposed showed the most attractive issues of the theory, however warp drive has one major drawback. The warp drive, does not agree with known conservation laws and hence defies the laws of physics as understood in the late 20^th Century. Here I will present these drawbacks in order to present an equal ended forum on the warp drive.

Fact: Warp Drive is only at best a theoretical science.

The Weak Energy Condition

The initial problem of the warp drive was recognized by Alcubierre in his breakthrough paper[1] . The problem being a weak conservation principle in General Relativity (GR) is broken, known as the Weak Energy Condition (WEC). The formula for a non-trivial space is given by T_abV^aV^b=>0. The problem which arises is that it causes the flat spatial metric within the zero Christoffel space to obtain a positive curvature. Thus violating relativity by a local expansion of spacetime, such an effect was predicted much earlier by Einstein. This effect however, was applied to explain an expanding universe, thus when applied locally contradicts Einstein's invention, which is known as the Cosmological Constant.

Expanding and Contracting Spacetime

The idea of spacetime expanding and contracting is not at all new, it should be noted that once again Einstein predicted such an effect. From the local metric g_ik acting on a Riemannian surface geometry, the curvature connection of spacetime is derived directly from a metric with a signature of rank 2. Thus the curvature connection would be given in a linear form do to the presence of the metric, hence space would be required to expand and contract from the metrics signature. This expansion and contraction of spacetime is called gravitational radiation, which is assumed to propagate at the speed of light c. The key between this idea and Miguel's proposal is that the Alcubierre paper requires two sperate sets of gravitational waves. However, do to the spatial metric proposed by Alcubierre one set of the gravitational waves would be required to defy the WEC.

The Energy Conditions

Violating certain conservation laws is to be expected in GR however, this is a consequence of the geodesic nature of spacetime. No path within such a space can be considered linear in its own accord, and hence at some arbitrary scale is expected to violate certain conservation principles. Such a defiance is commonly associated with another Energy Condition in GR, known as the Strong Energy Condition (SEC). A paper by Van Den Broeck [2], takes a similar notion in principle, since it does not completely remove the "negative energy" condition required by Alcubierre.

By shrinking the metric by an amount determined by B²(rs), the violation of the WEC is lowered by some 28 orders of magnitude. However with an energy requirement ranking over a hundred trillion trillion fold, this can hardly be considered a minute and expected perturbation induced by GR. This is a real problem, a problem grave enough to allow many physicists to believe such a problem will never be overcome, and that the warp drive should only exist within the realm of science fiction.

Quantum Mechanics

While such a "negative" energy scares most with a background in GR, it does not send chills up the spines of Quantum physicists. Negative energy is required by the laws of probability, because of Hesienberg's Uncertainty principle, where one is expected to find negative energy of some form some where.

Under certain conditions one may even expect a peak in the negative energy far above the norm, while this is expectable in quantum theory, it plays a much to little role in GR. Currently, there would be no way to obtain the conditions necessary for the minimum negative energy which is required by the model proposed by Ford and Pfenning[3] (except for some momentary fluke, not likely to be experienced in GR). These problems are no laughing matter, in fact a second paper by Van Den Broeck [4], considers the implications of such conditions. Where I now quote:

"...it is clear that warp bubbles present, enormous practical difficulties, which may never be overcome." --Broeck.

More Practical concerns

While many difficult issues have been addressed, such as the negative energy requirement, some rather simplistic problems have been ignored (or rather left to be assumed by the reader). Such as obtaining the mass necessary to curve spacetime in order to induce the warp drive. From the stand point of GR, this is not an initial concern, however it is for body which obeys conservation laws. An object of several tons can not generate a mass several times its own magnitude through any known scientific process. And if it were possible, one runs into the very severe risk of generating a singularity in spacetime, i.e. a black hole. Again, even under the best of circumstances, it appears that negative energy densities are required for the warp drive Thereby from a physical point of view this makes the warp drive look even more unrealistic.

Furthermore, a paper by Olum [5], considers a spatial metric of form:

ds²=(1-4t²x²)dt²-4tx(1-t²)dxdt+(1-t²)dx².

From the above equation, one may have a space which appears to yield superluminal travel. However, the apparent faster than light motion is just an allusion which appears upon an observers orientation to the metric. The velocity of an object within this metric thus never travels faster than light, at first this may appear to validate the premise of the warp drive. However, it is shown that once again, for apparent FTL motion to occur negative energy densities are required.

References

[1] Alcubierre M. The Warp Drive: Hyper-Fast Travel Within General Relativity. Class.Quant.Grav. 11 (1994), L73-77.

[2] Broeck C. A `warp drive' with more reasonable total energy requirements. Class.Quant.Grav. 16 (1999) 3973-79

[3] Pfenning M. and Ford L. The unphysical nature of "Warp Drive." Class.Quant.Grav. 14 (1997) 1743-51

[4] Broeck C. On the (im)possiblity of warp bubbles.

[5] Olum K. Superluminal travel requires negative energies. Phys.Rev.Lett. 81 (1998) 3567-70

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