Hidden Variables and Relativistic Tachyons
By Eric Dennis
(September 28, 2001)

[OBJECTIVESCIENCE.COM] Miguel Alcubierre's solitary wave solutions of the Einstein field equations offer the unexpected possibility that general relativity may prove consistent with the experimentally verified non-locality of quantum mechanics. This assuages the fear that quantum non-locality would ultimately require abandoning the mathematical structure of relativity.


Relativity and quantum mechanics are the two fundamental theories constituting modern physics. On one hand the standard model of particle physics, although incorporating aspects of relativity, is a quantum mechanical theory at root. On the other hand our understanding of the large-scale structure and evolution of the universe is basically relativistic. But the different perspectives from which the two theories are formulated have made it difficult to put them on one unified basis.

Despite impressive advances in mathematical machinery, a basic problem remains unanswered and usually even unmentioned. Quantum mechanics predicts, and experiments have well verified, that non-local (i.e. faster than light) influences are possible under certain conditions [1]. However, it is generally held that relativity precludes anything propagating faster than light, as this would seem to imply situations in which effects precede their causes.

Physicists have generally shied away from this apparent inconsistency because the superluminal predictions of quantum mechanics are somewhat obscure in its conventional interpretation. Nevertheless, the careful analysis of John Bell has shown that, whatever interpretation one wishes to employ, superluminal action is unavoidable.


The one place where this issue has been squarely addressed is in connection with so called "hidden variable" theories, i.e. alternative interpretations of quantum mechanics that deal fundamentally with what is and not merely with what is observed. The welcomed concreteness of these theories brings out superluminal causation more clearly, and so gives a starting point for analysis [2].

In the de Broglie-Bohm theory (dBB), for example, the "hidden variables" turn out to be precisely the variables that we do observe, namely the positions of particles. Here, superluminal influences take the form of forces exerted from one particle to another, instantaneously across space, analogous to Newtonian gravity. From the physicist's perspective it thus seems likely that dBB, if successful, will ultimately be found to arise from some deeper theory explaining its superluminal influences in more detail, presumably in terms of new entities propagating faster than light through a universal medium.

It seemed, therefore, that this deeper theory could not be relativistic at heart, but must yield relativistic predictions more fortuitously. In other words, both quantum mechanics and relativity would have to be pulled out of the hat. That appeared an ominous task, as it would entail a quest for new and unknown starting points from which to relaunch physical theory.

Fortunately, a newer development within relativity casts light on the problem and may eliminate the underlying conflict with quantum non-locality.


The innovation derives from a crucial difference between the concept of locality in special relativity (SR) and that in general relativity (GR). In SR, the principle of locality implies that nothing can ever travel faster than light, between any two points, at any time. But In GR "space" itself changes in time [3]. Effectively, the distance between two points A and B can change from one instant to the next. So whether or not something traveling from A to B in a given time T constitutes superluminal action may depend on when the trip was taken. It may be that a trip from the Earth to Alpha Centauri would require many light-years if taken today, but could be done in minutes at some future date, if the "space" in between were then altered in just the right way.

This sounds like something that, although not ruled out by the mere equations of general relativity, is exceptionally unlikely given the peculiar astronomical phenomena necessary to alter space in between the Earth and Alpha Centauri in "just the right way." Yet in 1994, the physicist Miguel Alcubierre published a paper that brought this prospect within the realm of the imaginable [4].

Alcubierre realized that one need not alter the entire A-to-B distance in one instant--one merely need distort space locally, in a limited region around our hypothetical interstellar craft. In effect, one bunches space up in front of the craft, and stretches it out behind, as if it were a sheet of rubber. The figure below shows how to travel between two points A and A' by first bunching up the space in between A and A', then making the trek, and finally unbunching the space. The travel time at light speed when the space is bunched may be much less than that when the space is unbunched.


One can view Alcubierre's construction as a repetition of this process many times, so that large distances may be traveled, but only small regions of space need to be manipulated at any one time. The total travel time may thus be far less than that required by light if one assumes the light must traverse the entire unbunched distance.

Technically, what Alcubierre did was to exhibit a certain class of solutions to the equations of general relativity describing such a disturbance propagating cohesively (as a "solitary wave") in an otherwise flat (Euclidean) background space. He showed that an object residing in this kind of solitary wave would never locally surpass light speed [5], but that it would appear to do so from the perspective of an outside observer who fails to take into account the solitary wave's distorted geometry.

Indeed Alcubierre solitary waves exist that can carry an object at arbitrarily high speeds from the outside observer's point of view. Although these solitary waves have all the outward effects of superluminal propagation, characterizing them as literally superluminal glosses over the question of when it is that a given A-to-B distance obtains--at the actual time of transit via an Alcubierre solitary wave, or later, in the absence of spatial distortions.


The significance of Alcubierre's work in connection with quantum mechanics involves not so much the possibility of interstellar travel, but of sub-microscopic spatial disturbances that could interact with regular particles. In the form of little Alcubierre solitary waves, or some other kind of super-fast disturbance, such a mechanism presents the startling possibility that effectively superluminal particle interactions are consistent with the basic structure of general relativity. So, for example, the inter-particle forces of dBB could ultimately arise from a sea of tiny Alcubierre solitary waves intermediating between particles, in a way described by a purely GR-type theory [6].

However seriously or unseriously one takes the prospect of maintaining many-particle quantum entanglements through the specific mechanism of Alcubierre solitary waves, his results are certainly provocative. Indeed, they should reinvigorate fans of Einstein in particular, who would welcome a reversal of the well-worn dogma that general relativity must be sacrificed on the quantum altar.

About the Author:

Eric Dennis received a B.S. in physics (1998) from Caltech and M.S. in physics from UC Santa Barbara (2000). He is currently a visiting graduate student at Princeton, where his work involves numerical simulation of quantum systems via a novel algorithm motivated by Bohm's version of quantum mechanics. Past areas of research have included quantum computation theory (topological error-correcting codes) and experimental condensed matter physics (electron spin coherence in semiconductors).


[1] I take "local" as implying the absence of not only instantaneous action-at-a-distance, but of any superluminal action. Whether or not there is a difference is a matter for physics to decide.

[2] Ironically this has been taken as evidence against "hidden variable" theories and for conventional interpretations, completely failing to confront the central result of Bell's work.

[3] By "space" I really mean a universal medium, a "plenum" or "ether". The "ether" concept is usually looked askance at by physicists, although conventional treatments of general relativity effectively reintroduce the concept by reifying "space" (and "space-time"). Certain physicists have even been more explicit about this connection (e.g. Sakharov's view of gravitation as elasticity; see Misner, Thorne, Wheeler, Gravitation, ff. 426), especially in the context of quantum field theories.

In the ether picture, changes in the distance between two points are understood as arising from the changing physical properties of the ether in between them. This would be analogous to changing optical path lengths in a crystal with time-dependent index of refraction.

[4]The warp drive: within general relativity, Class. Quantum Grav. 11-5, L73-L77 (1994). http://mentor.lanl.gov/abs/gr-qc/0009013 Abstract: "It is shown how, within the framework of general relativity and without the introduction of wormholes, it is possible to modify a spacetime in a way that allows a spaceship to travel with an arbitrarily large speed. By a purely local expansion of spacetime behind the spaceship and an opposite contraction in front of it, motion faster than the speed of light as seen by observers outside the disturbed region is possible. The resulting distortion is reminiscent of the "warp drive'' of science fiction. However, just as it happens with wormholes, exotic matter will be needed in order to generate a distortion of spacetime like the one discussed here."

[5] In fact, it turns out that locally, it would appear to someone in the disturbance that he is totally stationary. E.g. he would experience no inertial forces even though it appears to an outsider that his craft is accelerating.

[6] A more radical proposal might involve not Alcubierre solitary waves but sub-microscopic wormholes. Given the difficulty of integrating such a proposal into a (topologically simple) plenum-type formulation of GR, one tends to take it less seriously.

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