Tachyons

Special relativity is often summarized as "nothing can go faster than light." In 1962 three physicists — Olexa-Myron Bilaniuk, V. K. Deshpande and E. C. George Sudarshan — pointed out that this is not quite what the theory says. ^[1]

What relativity forbids is accelerating an ordinary object up to the speed of light. As a massive object approaches $c$, the energy needed to push it further grows without limit. The light barrier is a wall that cannot be crossed from below.

But suppose something were created already on the far side of the wall — something that always moves faster than light and can never slow down to it. Relativity, they argued, does not obviously forbid that. In 1967 Gerald Feinberg gave such hypothetical particles a name, from the Greek tachys, "swift": tachyons. ^[2]

Tachyons have never been found. Yet thinking about them clarified two deep points in modern physics: faster-than-light signals would allow messages into the past, and a "faster-than-light" mass term in a field theory usually means something very different — that the vacuum is unstable.

The Far Side of the Light Barrier

Three kinds of particle

Relativity sorts objects by how they relate to the speed of light:

• Ordinary matter (electrons, protons, you) always moves slower than light. Speeding it up costs more and more energy, and reaching $c$ would cost infinite energy.
• Light and other massless particles always move at exactly $c$.
• Tachyons, if they existed, would always move faster than light.

The tachyon's behaviour is a mirror image of ordinary matter. When an ordinary particle loses energy, it slows down. When a tachyon loses energy, it speeds up. At zero energy it would move infinitely fast. To slow it down towards $c$, you would need to give it infinite energy. The light barrier would be a wall that cannot be crossed from above either.

The price: an "imaginary" mass

The strange behaviour comes from a strange ingredient. In relativity, an object's energy depends on its speed through a factor that involves $\sqrt{1 - v^2/c^2}$. For $v > c$ the quantity under the square root is negative. For the energy to be a real, measurable number, the object's rest mass must be imaginary — a multiple of $i = \sqrt{-1}$.

This sounds absurd, but it is less so than it looks. A tachyon can never be at rest, so its "rest mass" is never directly measured. What is measured — energy and momentum — remains real.

The Formal Description

Energy and momentum

For any free particle, relativity links energy $E$, momentum $p$ and rest mass $m$:

Here $E$ is energy, $p$ is the size of the momentum, $m$ is the rest mass, and $c$ is the speed of light. For a tachyon, set $m = i\mu$, where $\mu$ is an ordinary positive number, so that $m^2 = -\mu^2$:

The particle's velocity is $v = pc^2/E$. Since $pc > E$ for a tachyon, $v > c$. Writing energy and momentum in terms of velocity gives

These formulas show the mirror behaviour directly. As $v \to c$ from above, $E \to \infty$. As $v \to \infty$, $E \to 0$. ^[1,2]

Would tachyons be stable?

A charged tachyon moving through vacuum would be faster than light in that vacuum, so it would emit Cherenkov radiation — the electromagnetic analogue of a sonic boom, familiar as the blue glow in water around nuclear reactors. By radiating, it would lose energy and therefore speed up further. This offered a possible experimental signature, discussed below.

Why Faster-Than-Light Means Backwards in Time

Simultaneity depends on the observer

The most important consequence of tachyons has nothing to do with their mass. It comes from a basic feature of relativity: observers moving relative to each other disagree about which events happen "at the same time." This is the relativity of simultaneity, the same effect that underlies time dilation.

For events connected by a slower-than-light signal, all observers agree on which came first. For events connected by a faster-than-light signal, they do not. Some observers would see the signal arrive before it was sent.

The Lorentz transformation makes this precise. Suppose a signal travels a distance $\Delta x$ in time $\Delta t$ at speed $u = \Delta x/\Delta t$, as measured by one observer. A second observer moving at speed $v$ along the same line measures a time interval

If $u < c$, the bracket is always positive for any $v < c$: every observer sees the signal arrive after it was sent. If $u > c$, then any observer with $v > c^2/u$ finds $\Delta t' < 0$. For that observer, the signal arrives before it departs.

The tachyonic antitelephone

A single backward-in-time observation is strange but not yet paradoxical — one observer's "arrival" is another's "departure." The paradox appears when two observers cooperate.

Richard Tolman noted in 1917 that faster-than-light signalling would let an effect precede its cause. ^[3] In 1970 Gregory Benford, David Book and William Newcomb turned this into a sharp thought experiment, the tachyonic antitelephone. ^[4]

The scenario runs as follows:

1. Alice, on Earth, sends a tachyon message at speed $u > c$ to Bob, who is on a spaceship moving away from her at speed $v$.
2. Bob, the moment he receives it, sends a tachyon reply back at the same speed $u$, measured in his own frame.
3. Because Bob's "now" is tilted relative to Alice's, his reply, fast in his frame, can travel backward in Alice's time.

Working through the velocity addition rule shows that the reply reaches Alice before she sent the original message whenever

For example, if tachyon signals travel at ten times the speed of light ($u = 10c$), Bob need only be receding at about 20% of light speed. The faster the tachyons, the slower Bob needs to move.

Now suppose Alice has agreed to send her message only if she has not already received Bob's reply. If the reply arrives, she does not send; then Bob has nothing to reply to; then no reply arrives; then she sends. This is the same logical structure as the grandfather paradox, and the related bootstrap paradox, for closed timelike curves — but here no exotic spacetime geometry is needed, only faster-than-light signals in flat space. ^[4]

"A detector that receives messages from the future"

The antitelephone is the precise version of a popular idea: that if tachyons exist and we could build a detector for them, we could receive information from the future.

It is important to classify this correctly.

• Mathematical fact: within special relativity, any signal faster than light, combined with a moving relay, can produce a reply that arrives before the original was sent. This follows from the Lorentz transformation and is not controversial. ^[3,4]
• Speculation: that tachyons exist, that they can be emitted and absorbed at will, and that they could carry controllable information. None of these has any experimental support.
• Unresolved: how such a device could exist without logical contradiction. Most physicists regard the paradoxes as evidence that such signalling is impossible, not as a technology waiting to be built.

A tachyon detector by itself would not "see the future." In any single frame, a detected tachyon was emitted somewhere; the ordering puzzle only becomes a genuine message into one's own past with a moving relay, as in the antitelephone.

The reinterpretation principle

Bilaniuk, Deshpande and Sudarshan, and later Feinberg, proposed a way to soften the problem. ^[1,2] In frames where a tachyon appears to travel backward in time, it also appears to carry negative energy. They suggested that such an event should be reinterpreted: a negative-energy tachyon absorbed in the future is physically the same as a positive-energy anti-tachyon emitted in the future and travelling forward in time. Under this reinterpretation principle, every observer sees only positive-energy particles moving forward in time; they merely disagree about which end was the emitter and which the absorber.

The principle removes negative energies, but it does not dissolve the antitelephone paradox. Benford, Book and Newcomb argued that when signals form a closed loop with an agent who can choose whether to send, reinterpretation cannot turn the story into a consistent sequence of causes and effects. ^[4] Reinterpretation also makes the "sender" of a message frame-dependent, which is hard to reconcile with anyone deliberately sending information. Proposals in which only self-consistent histories are allowed — the Novikov self-consistency principle — could in principle avoid contradiction, but they do so by restricting what the participants can do.

The Modern View: Tachyons Mean Instability

When physicists moved from single particles to quantum fields, the meaning of an imaginary mass changed.

What a negative mass-squared does to a field

A quantum field is like a mattress of springs filling space. The mass term sets how stiff each spring is. A field with ordinary mass sits at the bottom of a bowl-shaped potential and vibrates around it. A field with negative mass-squared sits at the top of a hill.

For a free field with $m^2 = -\mu^2$, the wave equation gives waves whose angular frequency $\omega$ and wavenumber $k$ satisfy

For short waves ($k$ large), $\omega$ is real and the waves oscillate normally. For long waves ($k < \mu c/\hbar$), $\omega^2$ is negative, $\omega$ is imaginary, and the amplitude grows exponentially in time, like $e^{|\omega| t}$. The field does not send signals faster than light. It falls off the hill.

In 1969 Yakir Aharonov, Arthur Komar and Leonard Susskind showed that in such theories the apparently superluminal modes are small oscillations about an unstable equilibrium, and that disturbances do not in fact propagate outside the light cone. ^[5] The "tachyon" is a symptom of standing in the wrong place.

Tachyon condensation and the Higgs field

What happens after the field falls? It rolls down to the bottom of the potential, where the curvature is positive and all excitations have ordinary, positive mass. This process is called tachyon condensation.

The best-known example is the Higgs field. Its potential, in the Standard Model, has the form

where $\phi$ is the Higgs field, and $\mu^2$ and $\lambda$ are positive constants. The minus sign in front of $\mu^2$ is a negative mass-squared at $\phi = 0$: the symmetric, empty-looking state is unstable, exactly like a tachyonic field. The field therefore settles at a non-zero value (about 246 GeV in energy units) throughout space. Excitations around this new minimum are ordinary particles — the Higgs boson discovered at CERN in 2012 has a real, positive mass of about 125 GeV. ^[6,7]

So the Standard Model contains a "tachyonic" mass term, but no faster-than-light particle. The imaginary mass is not a property of a particle; it is a sign that the vacuum rearranges itself.

Tachyons in string theory

The same reinterpretation dominates string theory. The original bosonic string contains a tachyon in its spectrum, signalling that its vacuum is unstable; this was one reason the theory was abandoned as a description of nature (see 13-Dimensional Models). In the 2000s Ashoke Sen showed that tachyons on unstable D-branes describe the decay of those branes: as the tachyon field "rolls" down its potential, the brane disappears, leaving behind a different, stable configuration. ^[8,9] Here too, the tachyon is a tool for understanding instability, not a particle that outruns light.

Evidence

Direct searches

Cherenkov searches. In 1968 Alväger and Kreisler at Princeton looked for charged tachyons produced by gamma rays in lead, using the Cherenkov radiation such particles should emit in vacuum. They found none, and set an upper limit on the production rate. ^[10]

Cosmic-ray claims. In 1974 Clay and Crouch reported possible signals arriving just before the main particle front of cosmic-ray air showers, which they interpreted as possible tachyons. ^[11] The result was not confirmed by later experiments and is not regarded as evidence for tachyons.

Tachyonic neutrinos. In 1985 Chodos, Hauser and Kostelecký proposed that neutrinos might be tachyons. ^[12] Direct measurements of the neutrino mass since then have not required this. The KATRIN experiment's 2022 result found a best-fit mass-squared value consistent with zero or a small positive number, and an upper limit of 0.8 eV on the neutrino mass. ^[13]

The 2011 OPERA anomaly

The most publicized case came in September 2011. The OPERA experiment, which detected neutrinos sent from CERN to the Gran Sasso laboratory in Italy, about 730 km away, reported that the neutrinos arrived about 60 nanoseconds earlier than light would have, corresponding to a speed exceeding $c$ by about 25 parts per million. ^[14]

The collaboration presented the result as an unexplained anomaly and invited scrutiny. The response illustrated how science handles extraordinary claims:

• Theory: Andrew Cohen and Sheldon Glashow showed that neutrinos this fast would rapidly lose energy by emitting electron–positron pairs, stripping the beam of its higher-energy neutrinos before it reached Gran Sasso; they argued this was incompatible with the beam OPERA actually observed. ^[15]
• Astrophysics: neutrinos and light from Supernova 1987A, which travelled about 160,000 years to reach Earth, arrived within hours of each other, limiting any difference between neutrino speed and light speed to about 2 parts in a billion (at lower neutrino energies). ^[16]
• Independent measurement: the neighbouring ICARUS experiment, using a specially bunched beam in 2012, measured neutrino arrival times consistent with the speed of light. ^[17]
• Instrumental cause: in February 2012 OPERA identified two faults in its timing system, a badly connected fibre-optic cable bringing the external GPS signal to the experiment's master clock, and a problem with the oscillator used to time-stamp events between GPS synchronizations. ^[18,19]

OPERA's corrected, final analysis found the neutrino speed consistent with $c$: an arrival time difference of $6.5 \pm 7.4\,(\text{stat.})\,{}^{+8.3}_{-8.0}\,(\text{sys.})$ ns, compatible with zero. ^[14] The "faster-than-light neutrino" was a loose connector.

Summary of the evidence

No tachyon has ever been observed. Every test of particle speeds, including those with neutrinos, is consistent with nothing travelling faster than light. Tachyonic mass terms do appear in accepted physics — most importantly in the Higgs potential — but always as signals of vacuum instability that resolves into ordinary particles.

Limitations and Open Problems

Quantum theory of tachyon particles. Attempts to build a consistent quantum field theory of freely propagating tachyon particles face serious difficulties. The instability described above means there is no stable vacuum around which tachyon particles could exist, and it is hard to maintain both Lorentz invariance and a well-defined notion of particles. ^[2,5] This is a major reason the particle interpretation has faded.

Causality. Even at the classical level, faster-than-light signals combined with relativity lead to the antitelephone paradox. Proposed escapes — reinterpretation, self-consistency constraints, or a preferred reference frame that breaks relativity — each carry costs. ^[4]

What would change the picture. A reproducible observation of any signal or particle outrunning light, with a demonstrated physical cause and not an instrumental one, would force a major revision of physics. The standard of evidence for such a claim is high precisely because, as OPERA showed, a subtle instrumental error is far more likely than new physics.

What the Failure Taught Physics

Tachyons began as a question about the letter of relativity: does it really forbid faster-than-light motion, or only crossing the light barrier? The answer turned out to be more interesting than either "yes" or "no."

No faster-than-light particle has ever been detected, and precision tests from supernova neutrinos to the corrected OPERA result agree with the speed of light as a universal limit. The negative mass-squared terms found in accepted theories describe unstable vacua, as in the Higgs mechanism, rather than particles outrunning light.

Special relativity alone still does not rule out particles that are always on the far side of the light barrier. If they could carry signals, some arrangement of senders would receive replies before sending them. Whether that possibility is merely unrealized or genuinely forbidden remains unresolved, though the antitelephone paradox and the instability of tachyonic fields give physicists strong reasons to suspect the latter.

The tachyon's lasting legacy is therefore not a particle but a lesson: whenever a theory seems to contain something faster than light, the first question to ask is not "how do we build a detector?" but "what is unstable here?"

References

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[2] Feinberg, G. (1967). "Possibility of Faster-Than-Light Particles." Physical Review, 159(5), 1089–1105.
https://doi.org/10.1103/PhysRev.159.1089

[3] Tolman, R. C. (1917). The Theory of the Relativity of Motion. University of California Press, Berkeley.
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[4] Benford, G. A., Book, D. L., & Newcomb, W. A. (1970). "The Tachyonic Antitelephone." Physical Review D, 2(2), 263–265.
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[5] Aharonov, Y., Komar, A., & Susskind, L. (1969). "Superluminal Behavior, Causality, and Instability." Physical Review, 182(5), 1400–1403.
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[6] Higgs, P. W. (1964). "Broken Symmetries and the Masses of Gauge Bosons." Physical Review Letters, 13(16), 508–509.
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[7] ATLAS Collaboration (2012). "Observation of a New Particle in the Search for the Standard Model Higgs Boson with the ATLAS Detector at the LHC." Physics Letters B, 716(1), 1–29.
https://doi.org/10.1016/j.physletb.2012.08.020

[8] Sen, A. (2002). "Rolling Tachyon." Journal of High Energy Physics, 2002(04), 048.
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[9] Sen, A. (2005). "Tachyon Dynamics in Open String Theory." International Journal of Modern Physics A, 20, 5513–5656.
https://doi.org/10.1142/S0217751X0502519X · arXiv:hep-th/0410103

[10] Alväger, T., & Kreisler, M. N. (1968). "Quest for Faster-Than-Light Particles." Physical Review, 171(5), 1357–1361.
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[11] Clay, R. W., & Crouch, P. C. (1974). "Possible Observation of Tachyons Associated with Extensive Air Showers." Nature, 248, 28–30.
https://doi.org/10.1038/248028a0

[12] Chodos, A., Hauser, A. I., & Kostelecký, V. A. (1985). "The Neutrino as a Tachyon." Physics Letters B, 150(6), 431–435.
https://doi.org/10.1016/0370-2693(85)90460-5

[13] KATRIN Collaboration (2022). "Direct Neutrino-Mass Measurement with Sub-Electronvolt Sensitivity." Nature Physics, 18, 160–166.
https://doi.org/10.1038/s41567-021-01463-1 · arXiv:2105.08533

[14] OPERA Collaboration (2012). "Measurement of the Neutrino Velocity with the OPERA Detector in the CNGS Beam." Journal of High Energy Physics, 2012(10), 093. (Original 2011 claim: arXiv v1.)
https://doi.org/10.1007/JHEP10(2012)093 · arXiv:1109.4897v1 (2011)

[15] Cohen, A. G., & Glashow, S. L. (2011). "Pair Creation Constrains Superluminal Neutrino Propagation." Physical Review Letters, 107, 181803.
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[16] Longo, M. J. (1987). "Tests of Relativity from SN1987A." Physical Review D, 36(10), 3276–3277.
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[17] ICARUS Collaboration (2012). "Measurement of the Neutrino Velocity with the ICARUS Detector at the CNGS Beam." Physics Letters B, 713(1), 17–22.
https://doi.org/10.1016/j.physletb.2012.05.033 · arXiv:1203.3433

[18] Reich, E. S. (2012). "Flaws Found in Faster-Than-Light Neutrino Measurement." Nature (news).
https://doi.org/10.1038/nature.2012.10099

[19] Cartlidge, E. (2012). "Loose Cable May Unravel Faster-Than-Light Result." Science, 335(6072), 1027.
https://doi.org/10.1126/science.335.6072.1027