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    When we use logic to deduce features of Reality we must take special care to avoid the trap of assuming that a thing exists simply because we can conceive the idea of it. The fact that we know about that trap and believe that we can avoid it makes that particular trap especially insidious; after all, we all make a clear distinction between fantasy and Reality, even if some of us don't agree on precisely where we should draw the line. We don't, for example, believe that our ability to conceive the idea of unicorns necessitates the existence of unicorns: unicorns belong to the realm of fantasy. Nonetheless, we still have concepts that can trap us into believing that they refer to real things. To illustrate that point I have in mind the concept of "faster than light".

    We can imagine that concept easily enough, though we must perforce imagine it in symbolic form. We simply say that some thing occupies a certain place and that one second later it occupies a place more than 299,792.458 kilometers away. In a more cartoonish version of the concept we see in our mind's eyes a pulse of light, perhaps made visible to us by the small amounts of light scattered from it by dust in its path, flying through space and, flying in the same direction, a spaceship that overtakes and passes it. That image is substantially similar to the one that inspired Einstein to work out the theory of Relativity in the first place, so we may well ask Does it refer to some real thing?

    We know now that it does not. In order to move faster than light an object must occupy an inertial frame that moves faster than light relative to some other inertial frame. But in deducing the postulates of Relativity we showed that space and time have such a geometry that no point in space can move faster than light relative to any other point. The speed of light is not a barrier that we may somehow circumvent; rather, it is a boundary between existence and non-existence. And just as we can have no place beyond the boundary of space, so we can have no velocity beyond the speed of light.

    But for a very long time now physicists have conceived the speed of light as a barrier and not as a boundary. That fact leads me to wonder whether the Lorentz Transformation is yet incomplete or whether the physicists' belief in the reality of superluminal velocities (even if we cannot reach them) reflects the kind of misunderstanding that leads to the twin paradox. To answer that question I want to consider an hypothesis that a physicist first presented in 1967, one that I found appealing at the time and one that still attracts interest. The author of the hypothesis postulated the existence of particles, called tachyons, that exist always moving faster than light.

    Gerald Feinberg presented the tachyon hypothesis in the 1967 Jul 25 edition of The Physical Review in a paper in which he considered "...the possibility of describing, within the special theory of relativity, particles with space-like four-momentum, which therefore have velocities greater than that of light in vacuum." Dr. Feinberg then proceeded to describe "(a) quantum field theory of non-interacting, spinless, faster-than-light particles", noting also that "(t)he limiting velocity is c, but a limit has two sides."

    To be more specific, I should say that in his paper Feinberg did not discuss the possibility of boosting subluminal particles to superluminal speeds; rather, he discussed the existence of particles that are inherently superluminal. Feinberg acknowledged in the introduction to his paper that a subluminal particle accelerated to lightspeed would gain infinite kinetic energy and that the necessity for such a gain makes lightspeed an effective barrier to the achievement of superluminal motion from a subluminal start. He then went on to say, "The possibility of particles whose four-momenta are always spacelike, and whose velocities are therefore always greater than c is not in contradiction with special relativity, and such particles might be created in pairs without any necessity of accelerating ordinary particles through the 'light barrier'."

    In discussing the standard relativistic arguments against superluminal motion, Feinberg inferred two important properties of his putative tachyons. He noted that when relative velocities exceed lightspeed the corresponding Lorentz factor becomes purely imaginary (in the mathematical sense of a number multiplied by the square root of minus one), so, in order to ensure that the energies and momenta of tachyons are represented by real-valued numbers, he postulated that the tachyons all possess imaginary mass; that is, that the mass of a tachyon must be represented by an imaginary-valued number. In any calculations involving tachyons, then, the square root of minus one attached to the number representing the tachyon's mass and the square root of minus one attached to the Lorentz factor cancel each other out, leaving a real-valued number. And what does it mean for a particle to possess imaginary mass? Feinberg finessed the question by pointing out that tachyons can never be brought to rest and, thus, that they can never be subjected to the manipulations that explore the nature of mass.

    The second property that Feinberg inferred for tachyons offers a bit more subtlety. It turns out that an observer in some subluminal inertial frame might see a tachyon flying in one direction with positive energy and yet an observer in another subluminal inertial frame might see the same tachyon flying in the opposite direction with negative energy. Feinberg dismissed that objection by noting that it represents an interchange of the roles of particle emission and particle absorption between the frames. Though he didn't express it in such terms, he essentially claimed that the Lorentz Transformation so tilts the Feynman diagram of a tachyonic process that the emission of a particle in one frame becomes the absorption of the corresponding antiparticle in another frame.

    Having dismissed the more obvious relativistic objections to the existence of superluminal particles, Feinberg then turned to the relativistic quantum theory, the branch of physics in which physicists encode the dynamic properties of particles as aetherial waves of probability propagating in complex arrays called wave packets. To obtain a description of the simplest tachyons, Feinberg solved the Klein-Gordon equation for a scalar field with imaginary mass. The Klein-Gordon equation is one of the relativistic analogues of Schr÷dinger's equation, the fundamental equation of the nonrelativistic quantum theory, and it applies in particular to particles that carry zero inherent spin. In addition to assuming that his tachyons carry no spin, Feinberg sought to simplify them further by asserting that they also carry no electric charge or magnetic moment. Yet, for all the simplicity that Feinberg tried to build into them, those uncharged Klein-Gordon tachyons, if they actually existed, would display traits stranger than flying faster than light.

    Because they have no spin, we should expect those tachyons to obey the rules of Bose-Einstein statistics, the rules that organize particles that carry even halves of the fundamental unit of spin. Those rules tell us that we should be able to cram as many tachyons as we want into a single energy-momentum state. Not so, according to Feinberg. Large numbers of tachyons must be organized in accordance with Fermi-Dirac statistics, the rules that organize particles that carry odd halves of the fundamental unit of spin. Tachyons would thus obey the Pauli Exclusion Principle, the rule that makes electrons, each carrying one half unit of spin, occupy different orbits about an atomic nucleus.

    Feinberg also discovered what might be called "tachyon bloat", a phenomenon that becomes evident when we calculate the frequencies of the waves comprising the wave packet that defines the position and momentum of the tachyon (insofar as Heisenberg's indeterminacy principle will allow them to be defined). Because the tachyon has imaginary mass, we must subtract the square of that mass from, rather than add it to, the square of the wave number in the calculation of the frequency. But that means that the frequencies corresponding to small wave numbers will come out imaginary in our calculation, which means in turn that the waves with those frequencies, instead of oscillating forever as waves should do, will decay at various rates. Thus, a freshly created tachyon would have a wave packet comprising both oscillating and decaying waves. At first, before the decay has set in, the wave packet would look perfectly normal, very much like those associated with electrons or protons, but the elapse of time would make the packet smear out in a most unusual fashion as the transient waves decay, thereby spreading the tachyon over a wider volume of space.

    Weirdest of all, the tachyon that exists for me might not exist for thee. As Feinberg put it, "A state which contains no tachyons according to one observer will be seen by another observer to contain a large number of particles." Such a state manifests exactly the kind of contradiction that compels us to dismiss from our logical scheme one or more of the premises that led us to it. At this point, then, I call "reductio ad absurdum" and return to my original question: do we have in the Lorentz Transformation a complete understanding of Euclidean space and time or do we have more that we must learn?

    We have deduced the Lorentz Transformation, as Einstein did in 1905, from Einstein's two postulates and we deduced those postulates from a model of space and time that explicitly excludes superluminal inertial frames from existing. Feinberg didn't have the benefit of the latter deduction, but he did have the Lorentz Transformation and its deduction from Einstein's postulates. Now I want to ask whether the Lorentz Transformation encodes the exclusion of superluminal motion. Does the Lorentz Transformation permit the existence of tachyons or did Feinberg found his theory upon a misunderstanding of Relativity?

    Our observation that calculating the Lorentz factor for superluminal relative velocities yields an imaginary number shows us a red flag warning us that we may have our train of thought running on the wrong track, that we should proceed to consider superluminal frames with caution. Recall that we originally derived the Lorentz factor from our deduction of time dilation using a Feynman clock. Physicists knew of that particular form of Einstein's original derivation by the late 1960's, so Feinberg almost certainly knew of it. Consider how that derivation would look if we were to assume that one observer's clock occupies an inertial frame that moves faster than light relative to our frame. In the observer's frame we expect that the clock would work normally, with a pulse of light flittering back and forth between the clock's mirror and laser. But in our frame the clock, flying faster than light, would leave its pulse behind: the pulse would leave the clock and the clock simply would not count time at all, even though the observer moving with it would report that it works just fine. We thus devise the contradiction that invalidates one or more of our premises. From that contradiction we infer that superluminal frames, if they exist, cannot have any relationship with the space we occupy. Those putative superluminal frames must exist outside of space and, thus, not exist at all.

    So what went wrong in Feinberg's derivation of his tachyon hypothesis? It looks to me as if Feinberg "cookbooked" the Lorentz Transformation; that is, he used the formulae without reference to their roots. And, more to the point, he used only the dynamical formulae and ignored the kinematic equations of the transformation itself. If he had looked at those four equations in light of his hypothesis, he might have seen that they require one observer to represent distance and duration with real numbers while obliging another observer to use imaginary numbers to represent the very same observations. Seeing the absurdity implicit in that fact, he might have abandoned his hypothesis. Thus, we may reasonably conclude that the Lorentz Transformation does, indeed, encode the theorem that superluminal inertial frames do not exist and that Feinberg based his theory of tachyons upon a misinterpretation of Relativity.

    (I don't mean to disparage Dr. Feinberg, by the way. Soon or later all physicists make this kind of mistake. In other essays in this treatise I will show you some of mine. You will see beautiful derivations that I conceived, nurtured, and polished only to have some minor facts that I overlooked pop up and blow them all away with all the panache of a Clint Eastwood character.)

    But underlying that fact we find a semantic error that we will not easily correct. Somehow many physicists, such as Dr. Feinberg, have failed to distinguish between deformations of space and time (Relativity) and deformations in space and time (Ătherdynamics of the kind Lorentz hypothesized). Instead of seeing the Lorentz Transformation as a reflection of the proposition that space admits no inertial frames moving faster than light, those physicists see it as reflecting more of a dynamic limitation of space containing matter. Some modern theories of cosmology contain the assumption that space exists beyond the boundary defined by the speed of light moving away from the instant of Creation and some modern theories of physics contain the assumption that space can move through space at superluminal speed; the warped space theory of Miguel Alcubierre Mora gives us one example of such. That means that we (the cultural we) still have difficulty conceiving space as a dynamic entity in itself. We still tend to believe in Isaac Newton's absolute space and merely superimpose the relativistic space of Einstein and Minkowski upon it. And in so believing in space as a container of inertial frames, rather than as the manifestation of inertial frames, we subconsciously leave our minds open to accept the idea that space may also contain frames that move faster than light.

    Again we appear to have stumbled over a semantic block. When we conceive the idea of an inertial frame of reference, we do as I recommended in the essay on inertial frames and envision a ghostly grid floating in space. Therein lies the problem; we forget that the grid represents space itself, that the inertial frames do not actually exist as objects in an otherwise Newtonian space but constitute interpenetrating manifestations of space themselves. We get this problem from the fact that space is invisible and, thus, that when we do try to visualize space by means of our grids, we then intuitively reconceive those grids as objects in space.



Feinberg, Gerald, "Possibility of Faster-Than-Light Particles", Physical Review, Vol. 159, No. 5, Pages 1089-1105, 1967 Jul 25.

This is an easily readable paper, largely because Feinberg used very little of the advanced mathematics needed to express in full detail the quantum mechanical ideas that are the basis for his hypothesis. Though he refers to concepts that a lay reader will find unfamiliar, he explains his ideas clearly enough that the lay reader is unlikely to get lost.

Ford, Lawrence H. and Thomas A. Roman, "Negative Energy, Wormholes and Warp Drive", Scientific American, Vol. 281, No. 1, Pages 46-53, January 2000.

The authors provide a brief description of the ideas behind the Alcubierre warp drive.


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