Albert Einstein taught us that c, the speed of light in vacuum, is nature's ultimate speed limit, the highest speed at which matter, energy, and information can travel through space-time. In several AV columns I've discussed ways for getting around this annoying natural law, the law that SF writers and fans most wish to violate. Two AV columns discussed the possibility of getting around the lightspeed limit by popping through a trans-spatial wormhole shortcut. See [ Analog-6-89, "Wormholes and Time Machines"] and [Analog-5-90, "Wormholes II: Getting There in No Time"]. Two others discussed the possible use the faster-than-light nonlocal handshaking implicit in quantum mechanics for FTL messages and travel. See [Analog-11-86, "The Quantum Handshake"] and [Analog-9-88, "Paradoxes and FTL Communication"]. Both of those methods, however, are beyond our present technology, and the latter is probably impossible.
But now we may have new possibility. No serious physicist, to my knowledge, has ever suggested that it might be possible to make photons travel faster than c through empty space. Faster-than-light light seems a contradiction in terms. But this changed last February when K. Scharnhorst of the Alexander von Humboldt University in East Berlin published some new calculations indicating that, under the right conditions, photons of light can be induced to break the light-speed barrier.
Before I bring down on myself a flood of objections from readers knowledgeable about wave velocities, let me say that I am not talking about the difference between the group velocity and phase velocity of light waves. I'll pause, however, to explain that distinction first. It's well known in physical optics that a "wave packet" may travel at a different speed from the individual waves from which it is formed. The crests and troughs of a light wave with frequency n and wavelength l travel through space with a speed called the phase velocity and given by the product (nl). If, however, you want to use light to transmit a packet of information or a burst of energy, you have to put together a band of such waves of differing frequencies. The wave packet containing the energy and information travels with a speed that depends on how frequency varies with wavelength over the transmission band. This quantity is called the group velocity and is given by the derivative dn/d(1/l), the rate at which the wave frequency n changes when the reciprocal of wavelength 1/l is varied. In free space the group and phase velocities are identical and are both equal to c, but in a dispersive medium where the transmission speed depends on wavelength, the group and phase velocities can differ both from each other and from c.
For example, microwaves are high frequency radio waves (or low-frequency light) with wavelengths of a centimeter or so. They are used in applications ranging from aircraft radar to police speed measuring devices. The phase velocity of microwaves travelling inside a waveguide (a square pipe used in microwave plumbing) is considerably larger than c. This is because there is a cavity resonance in the waveguide at a slightly lower frequency which elevates phase velocities on its upper frequency-slope. But a wave crest is not a signal. Nature, while permitting FTL wave crests, consistently refuses to allow signals or energy to travel FTL. It's always the case, even in wave guides, that the group velocity is less than or equal to c.
The effect that Scharnhorst's paper discusses is independent of frequency and wavelength, and so is not the result of dispersive (wavelength-dependent) effects. Phase and group velocity are equal and increase together. Scharnhorst's effect is a consequence of quantum electrodynamics (QED), the quantum theory of light and electromagnetism. The theory of quantum electrodynamics tells us that empty space, when examined on a very small distance scale, is not empty at all; it seethes with the fireworks of vacuum fluctuations. Pairs of "virtual" (energy non-conserving) particles of many kinds continually wink into existence, live briefly on the energy credit extended by Heisenberg's uncertainty principle, and then annihilate and vanish when the bill for their energy debts falls due a few picoseconds or femtoseconds later. The seemingly smooth passage of a photon or electron through space at the QED distance scale, is revealed to be a punctuated chain of interactions and transformations involving virtual particles.
For example, a travelling photon may briefly be transformed into a virtual electron-positron pair, which moves forward less than one photon wavelength before annihilating to create a new photon indistinguishable from the old one. During the photon's brief existence as a pair, one of the virtual particles may initiate a "game of catch" using a virtual photon as the ball, tossing it one or more times to itself or its partner. These QED complications of the smooth passage of photons through space have the effect of making the photon travel more slowly. In part, this is because the photon spends a fraction of its existence as an electron-positron pair which can only travel at sub-light velocity.
There is, however, a way of suppressing part of the vacuum fluctuations, of making "empty" space more empty, using a technique based on the Casimir effect. If two conducting parallel plates with no electrical charge are placed very close together, the presence of this pair of conducting walls suppresses all virtual photons with wavelengths larger than twice the plate separation distance, because the wave structure of such a photon would intercept the walls in less than half a wavelength and the electric field of the wave would have a forbidden non-zero value within the conductor. Thus, the more closely the plates are spaced, the broader becomes the spectrum of virtual photons that are suppressed, and the vacuum between the plates becomes "emptier" of vacuum fluctuations and lower in energy density.
Since the energy density of normal vacuum is defined to be zero, the vacuum between the metal plates actually becomes a region of negative energy density. The amount of negative energy increases as the plates are brought closer together, so there is a physical force pulling them together which can do work (like lifting a weight) on an external system. This force, though very weak, has been measured in the laboratory. The experimental observation of an attractive force between two electrically neutral metal plates is an experimental confirmation of one of the more bizarre predictions of QED and is called the Casimir effect.
Scharnhorst has given a new twist to the Casimir effect by considering the velocity v of a photon travelling across the gap between the plates. If the plates are separated by a gap d, the Casimir effect suppresses all virtual photons with a wavelength of 2d or greater. Because these virtual photons are absent, they cannot participate in games of catch between virtual particles. Therefore a real photon travelling between the plates spends less time as an electron-positron pair because the QED vacuum fluctuations are suppressed. For this reason, the photon travels faster across the gap. Its speed of travel through normal vacuum is c, so its speed v in the negative energy vacuum between the plates is greater than c! Einstein's lightspeed barrier has been broached by a photon!
OK, that's the good news. The bad news is that in reasonable experimental situations, the Scharnhorst effect is not very big. In fact, it's abysmally small. With a plate gap of d, v/c = 1 - (1.6 × 10-60 × d-4). If we make d as small as experimentally possible, say 1 nanometer (= 1 × 10-9 m) or about ten atomic diameters, we find that (v-c) = 1.6 × 10-24 c.
This is an unmeasurably small change in the velocity of the photon, and only for a very small travel distance at that. But even such as small boost in speed comes as a surprise to those of us who had considered c as the ultimate speed limit. Moreover, special relativity says that if in one inertial reference frame an object travels only one part in 1024 times faster than c, one can find another reference frame in which the departure and arrival times of the object are simultaneous and therefore the velocity is infinite.
And, with heroic measures, one might boost the effect. Scharnhorst's calculation raises some interesting questions about limiting situations. Light in normal space is "slowed" by quantum fluctuations which cause it to spend part of the time as an electron-positron pair. It travels slightly faster in the space of lower (and negative) energy density between the Casimir plates, where part of the quantum fluctuations are suppressed. In circumstances achievable in the laboratory, this increase in speed will be very small. But what about outrageously extreme circumstances? The problem of doing better than the laboratory situation is that normal metals are made of atoms which become very lumpy and non-planar at the nanometer scale. So let's use something else, something smoother, something non-atomic. Suppose, for example, that we make a pair of Casimir plates from superconducting neutronium. Or perhaps the two-dimensional equivalent of cosmic string, a "cosmic wall". Cosmic walls, if they exist at all, are supposed to be smooth down to Planck-scale dimensions (10-35 m) and also are perfect superconductors. Suppose that between two such plates we makes a gap on the order of nuclear dimensions, about a femtometer (10-15 m). If one takes Scharnhorst's equation for index of refraction at face value, c/v goes to zero and a photon travels at infinite speed when the gap between the plates is decreased to about 1.13 × 10-15 m, or about the diameter of a proton. Of course, the approximations used in the calculation may not be valid because of higher-order effects at such small distances.
Nevertheless, such calculations serve to illustrate the point that once the lightspeed barrier is breached, physics becomes very strange. And perhaps there are there other, simpler ways of suppressing vacuum fluctuations and creating a region of space with a negative energy density. Suppose that we could create a large bubble of space inside which there exists the same negative energy density and suppression of vacuum fluctuations that is present in the above example, where the pair of Casimir plates is separated by a femtometer. Put such a bubble around a space ship and the ship can presumably travel within the bubble at any desired speed. And now suppose that we can move the bubble along so that it paces the ship ... It would seem that we have a FTL drive that is consistent with special relativity and quantum electrodynamics.
One should not get too excited about these prospects until Scharnhorst's calculations are verified and expanded. But at least it appears that a small chink has appeared in Einstein's previously impervious lightspeed barrier.
H. B. G. Casimir, Proc. Kon. Ned. Akad. Wetensch. B51, 793 (1948);
K. Scharnhorst, Physics Letters B236, 354 (1990);
Marcus Chown, New Scientist, 32 (7 April, 1990).
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