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The Tachyon Drive: Vex = infinity with Eex =0

John G. Cramer

Alternate View Column AV-61
Keywords: tachyon imaginary mass electron neutrino rocket reaction mass
Published in the October-1993 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 3/14/93 and is copyrighted ©1993 by John G. Cramer.
All rights reserved. No part may be reproduced in any form without
the explicit permission of the author.

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Light speed, c = 3 × 108 meters per second, is the ultimate speed limit of the universe. The well-tested physics orthodoxy of special relativity tells us that nothing can go faster than c. When any massive object with rest mass M (taken to be in energy units) has velocity v=c (or relativistic velocity ß = v/c = 1), the object's mass-energy becomes infinite. This is because the relativistic mass increase factor g = 1/(1 - ß2)1/2 has a zero in its denominator, and the net mass-energy E is given by E = gM. Therefore, it would require all the energy in the universe and more to accelerate the object to a velocity of ß = 1.

If the massive object could somehow be drop-kicked over the light-speed barrier so that v was greater than c, then both g and E would become imaginary quantities (like [-1]½ ) because ß would be larger than 1 and (1 - ß2) would be negative. This, says physics orthodoxy, is Nature's way of telling us that such quantities have nothing to do with our universe, in which all measurable physical variables like E must have real (not imaginary) numbers as values.

"Not so!" said Gerald Feinberg, the eminent physicist and SF fan who died last year at the age of 59. In a 1967 paper, Feinberg postulated a type of hypothetical particles with a rest mass M that also has an imaginary value (M2<0). Then E = gM, the observable mass-energy of these particles, becomes real and positive and is compatible with other energies in our universe. Feinberg christened his hypothetical particles "tachyons" (from the Greek word for swift) for their characteristic that they always travel more swiftly than c.

Normal particles (or "tardyons" in Feinberg's terminology) have a velocity of 0 when their mass-energy is smallest (at E=M). They have a velocity slightly less than c when their mass energy is very large compared to its rest mass (E>>M). Tachyons (if they exist) would behave in an inverted way, so that when their mass-energy is smallest (E=0) they would have infinite velocity (1/ß = 0) and when their mass energy is very large compared to their rest mass (E >> |M|) they would have a velocity slightly larger than c.

This can perhaps be seen more clearly by considering some equations of special relativity. When any particle (tachyon or tardyon) has rest mass M and mass-energy E, it has a momentum P (in energy units) given by E2 = P2 + M2. For tardyons (normal particles) it should be clear from this equation that E cannot be less than M and is always greater than P. For tachyons, however, we have the peculiarity that M2 is negative, so that the energy equation becomes E2 = P2 - |M|2 or P2 = E2 + |M|2. This means that E can be as small as zero (when P = |M|) and that P is always greater than E and cannot be less than |M|. These quantities are related to the relativistic velocity ß by the equation ß = P/E. This tells us that when a tachyon has its minimum momentum P = |M|, it will also have its lowest possible mass-energy (E=0) and will have infinite velocity.

The theoretical work on tachyons in the 1960's by Feinberg and others, particularly Sudarshan and Recami, prompted a "gold rush" among experimentalists seeking to be the first to discover tachyons in the real world. They studied the kinematics of high energy particle reactions at large accelerators, they built timing experiments that used cosmic rays, and they probed many radioactive decay processes for some hint of tachyon emission. Although there were a few false "discoveries" among these results, all of the believable experimental results were negative in the decade or so after the initial theoretical work. Some cold water was also thrown on the tachyon concept from the theoretical direction when it was demonstrated (by physicist and SF author Gregory Benford, among others) that tachyons could be used to construct an "anti-telephone" capable of sending information backwards in time in violation of the principle of causality, one of the most fundamental and mysterious laws of physics. Tachyons were therefore metaphorically placed on a dusty shelf in the museum of might-be particles for which there is no experimental evidence, and there they have languished for the past 25 years. But this may now be changing: a new and growing body of evidence from an unexpected direction supports the possible existence of tachyons.

There is great fundamental interest in the mass of the electron neutrino ([nu]e), because it is a leading "dark matter" candidate. Several very careful experiments have been mounted to measure its mass through its effect on the beta decay of mass-3 hydrogen or tritium. Tritium, with one proton and two neutrons in its nucleus, is transformed by the weak interaction beta-decay process into mass-3 helium (two protons and one neutron) by emitting an electron and an anti-neutrino (3H -> 3He + e- + [nu]e) with an excess energy of 18.6 keV. This is the lowest energy beta decay known, and therefore the one which is affected most strongly by the mass of the electron neutrino.

If the kinetic energy of the emitted electrons is measured for a very large number of similar tritium decays, one finds a bell-shaped "spectrum" of energies ranging from essentially zero electron energy to a maximum of about 18.6 keV. This maximum-energy tip of the electron's kinetic energy distribution is called the "endpoint", and is the place where the neutrino is emitted with near-zero energy and where the neutrino's mass will make it's presence known. When the endpoint region is made linear (using a plotting trick called a Kurie plot), then the straight-line dependence of the electron's kinetic energy takes a node-dive just before it reaches zero, displaying the effect of neutrino mass.

Because of the relativistic relation of mass, energy, and momentum (E2 = P2 + M2) it is the mass-squared of the neutrino that is actually determined by the tritium end-point measurements. The mass-squared is allowed to vary from negative values (too many electrons with energies near the end-point) through M[nu]2=0 (the expected number of electrons with energies near the end-point), to a positive mass-squared (too few electrons with energies near the end-point), and this variation is used to fit the experimental data. The resulting fit is quoted with the measured value of M[nu]2 plus-or-minus the statistical error in the measurement plus-or-minus the estimated systematic error in the measurement.

At least five experimental groups have made careful measurements of M[nu]2, and several of these groups have published their results in scientific journals. The two most recent published values are:
Zürich (Switzerland) M[nu]2 = -158 ± 150 ± 103 eV2 (1986)
Los Alamos (USA) M[nu]2 = -147 ± 68± 41 eV2 (1991)

As the numbers imply, both groups find an excess of electrons with energies near the tritium endpoint. There have also been recent informal reports (but no further publications) from these and other laboratories, particularly a group at a well-known weapons laboratory in California, of measurements which continue to give negative values to M[nu]2 with even more statistically meaningful error estimates. I was told by one of the experimenters that if the a similar result had been found with the same errors but with the positive of the determined value for M[nu]2, there would have been much publicity, with press conferences announcing the discovery of a non-zero mass for the electron neutrino.

OK, this is a SF magazine, not a scientific journal. We are not scandalized by the possibility that M[nu]2 is negative, indicating that the electron neutrino is perhaps a tachyon. In fact, we rather like the idea that a well known particle may routinely be breaking the light-speed barrier. Let us then suppose that the [nu]e is a tachyon with an imaginary mass of, say i × 12 eV. What are the physical consequences of this? The answer is disappointing. The tritium endpoint measurement is so difficult precisely because assuming a small neutrino mass (real or imaginary) has very few observable consequences. The "dark matter" implications are also nil. Since tachyons can have any mass-energy down to zero and are never at rest, they, like photons, cannot contribute to the excess of dark matter in the universe.

The above-mentioned "tachyon anti-telephone" with its violations of causality is also essentially impossible. Neutrinos are fairly easy to produce (using an accelerator to create beta-decaying nuclei) but very difficult to detect. The only successful neutrino detectors use either neutrino-induced nuclear reactions (the Homestake and Gallex experiments) or hard neutrino-electron scatterings (Kamiokande and SNO) to detect neutrinos with extremely low efficiency. But to use the possible tachyonic super-light speed of the electron neutrinos, they must have mass-energies comparable to or less than 12 electron volts. This is about 10-6 of the lowest neutrino energy ever detected, neither of the above detection schemes can be used in this energy range, and there is no known alternative method of detection. Thus, even if the [nu]e is a tachyon, the law of causality is safe from our tamperings for the foreseeable future.

This brings us our second question: What new SF gimmicks are suggested by the possibility of easy-to-produce tachyons? I have a delightful answer. We can make a tachyon drive.

Consider the central problem of rocketry: how can one burn fuel at a high enough exhaust velocity to provide reasonable thrust without an unreasonable expenditure of energy. This is the dilemma that plagues our space program, and the solutions we have developed are not very good.

So let's consider a device that makes great quantities of E=0 tachyons and uses them as the infinite velocity exhaust of a "rocket". Within the constraints of the conservation laws of physics, we can make all the tachyons we want for free, provided we make them in neutrino-antineutrino pairs to conserve spin and lepton number. Momentum conservation is not a problem because we want and need the momentum kick derived from emitting the neutrino-antineutrino pair. This leaves us to deal with energy conservation

The paradox here is that with a high-momentum exhaust of tachyons produced at no energy cost and beamed out the back of our space vehicle, the vehicle would seem to gain kinetic energy from nowhere, in violation of the law of conservation of energy. The solution to this paradox (as can be demonstrated by considering particle systems) is that the processes producing the tachyons must also consume enough internal energy to account for the kinetic energy gain of the system. Thus, a tachyon drive vehicle might be made to hover at no energy cost (antigravity!), but could only gain kinetic energy if a comparable amount of stored energy were supplied.

How could we arrange for an engine to produce great floods of electron neutrino-antineutrino pairs beamed in a selected direction? All I can do here is to lay out the problems and speculate. Neutrinos are produced by the weak interaction, which has that name because is much many orders of magnitude weaker than electromagnetism. Neutrino production of any kind is improbable. On the other hand, in any quantum reaction process the energy cost squared appears in the denominator of the probability, and if that energy is zero, it should make for abig probability. The trick might be to arrange some reaction or process that is in principle strong but is inhibited by momentum conservation. Then the emission of a neutrino-antineutrino pair to supply the needed momentum with zero energy cost would make the process go. A string of similar atomic or nuclear systems prepared in this way might constitute an inverted population suitable for stimulated emission (like light, correlated neutrino-antinuetrino pairs should be bosons), resulting in a beam from a "tachyon laser" that might amplify the process and produce the desired strong beam of tachyons.

That's about the best I can do at the moment, for providing the scientific underpinnings of a tachyon drive for SF purposes. I think it's a nifty idea to which I will devote more thought. I just hope it survives the ongoing experimental measurements of M[nu]2 for the electron neutrino. Watch this space for further developments.

References:

Tachyons:
"Particles That Go Faster Than Light", Gerald Feinberg, Scientific American, 69-77 (February-1970);
Tachyons, Monopoles, and Related Topics, E. Recami, ed., North Holland Publishing Co., (1978).

Neutrino Mass Measurements:
"Measurement of the Neutrino Mass from Tririum Beta Becay", E. Holzschuh, Rep. Prog. Phys. 55, 1035-1091 (1992).


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