Tachyons and Superluminal Particles

Special relativity forbids accelerating a particle past the speed of light. It does not, by itself, forbid a particle that was never below it. That distinction is the whole of this article, and it is sharper than it sounds: the equations stay perfectly consistent — real energies, real momenta, conserved four-vectors — if you postulate a particle always faster than $c$ . Such an object is a tachyon, and its rest mass is imaginary, $m = i\mu$ with $\mu$ real. The imaginary mass sounds like a contradiction and is not: nothing observable is imaginary. Put $m = i\mu$ into the standard formulae and the energy and momentum come out real for every $v > c$ , because the two imaginary factors cancel. What changes is the invariant. In $(+,-,-,-)$ a bradyon — an ordinary particle — has $p_\mu p^\mu = +m^2c^2 > 0$ , a timelike four-momentum. A tachyon has

p_\mu p^\mu \;=\; -\,\mu^2c^2 \;<\; 0 ,

a spacelike four-momentum. That single sign is the source of everything strange that follows, and it is why this article insists on re-deriving the metric rather than transcribing it. What follows is genuinely strange. A tachyon slows down when you give it energy; a zero-energy tachyon moves infinitely fast. Its energy has no invariant sign — a fast enough observer sees $E < 0$ — because that invariance is a property of timelike vectors only. And it would radiate in vacuum, Cherenkov-style, which for an object that accelerates as it loses energy is a runaway. This article owns the kinematics and one structural result: the light barrier is two-sided. Energy diverges as $v \to c^-$ and as $v \to c^+$ , and it diverges symmetrically, as $Mc^2/\sqrt{2|1-\beta|}$ from both directions. No finite-energy trajectory connects the two sectors. The wall between them is infinitely high on both faces. The causality question — whether tachyons let you telegraph your own past — is not settled here; it is owned by causal structure, the causality light-cone constraint and the reinterpretation principle. We state its two thresholds precisely, because one of them is very widely misstated — including by this article's own source — and then hand the argument over.

Key takeaways

A tachyon is a particle with $v > c$ always. Its rest mass is imaginary, $m = i\mu$ , but its energy and momentum are real: $E = \mu c^2/\sqrt{\beta^2 - 1}$ and $p = \mu v/\sqrt{\beta^2-1}$ , with $\beta = v/c > 1$ .
Its four-momentum is spacelike: $p_\mu p^\mu = E^2/c^2 - \mathbf{p}^2 = -\mu^2c^2 < 0$ , in the signature $(+,-,-,-)$ where a timelike vector has positive norm. Everything odd about tachyons is that one sign.
Energy decreases with speed: $\mathrm{d}E/\mathrm{d}v < 0$ . Take energy away and a tachyon speeds up; a zero-energy tachyon moves at infinite speed, carrying momentum $\mu c$ .
THE OWNED RESULT — the barrier is two-sided. $E \to \infty$ as $v \to c^-$ and as $v \to c^+$ , and it diverges symmetrically: $E \simeq Mc^2/\sqrt{2|1-\beta|}$ from both directions. No finite-energy trajectory connects bradyons to tachyons.
A tachyon's energy has no invariant sign: a boost with $uV > c^2$ makes $E' < 0$ and reverses the order of its emission and absorption. This is not the paradox — it is the setup for it.
Two thresholds, and they are different. In rapidity ( $u = c\coth\chi$ , $V = c\tanh\varphi$ ): a single leg reverses iff $\varphi > \chi$ ; a closed causal loop needs $V > 2uc^2/(u^2+c^2)$ , i.e. $\varphi > 2\chi$ — exactly twice the rapidity. Sources that give $uV > c^2$ as the loop condition, including this article's base, are wrong.
KEY IDEA — an imaginary mass is not an imaginary observable. It is a SPACELIKE four-momentum, and a spacelike vector has no invariant energy sign and no invariant time order. Tachyons are consistent kinematics; what they threaten is causality, not arithmetic.

The core insight

Start from the mass shell $p_\mu p^\mu = m^2c^2$ and simply let $m^2$ be negative. Nothing in the algebra objects. The shell is still a hyperboloid; it has merely moved from the timelike region to the spacelike one, from $E^2 > (pc)^2$ to $E^2 < (pc)^2$ . Every conservation law holds, every Lorentz transformation works, and the particle rides its shell as an electron rides its own. But a spacelike vector has two properties a timelike one does not, and they are the whole story. The sign of its time component is not invariant: some frames see $E > 0$ , others $E < 0$ , and no relabelling fixes it for everyone at once. And the time-order of two events joined by a spacelike interval is not invariant — the theorem owned by the causality light-cone constraint. So the tachyon is not a mathematical error. It is a perfectly consistent solution of relativistic kinematics that happens to live in the one region of the mass shell where "before" and "after" stop being frame-independent. The question is never whether tachyons are consistent. It is whether a universe containing them can still have a past.

Where this lives today

Not as particles. No tachyon has ever been observed, and the searches have been thorough — cosmic-ray showers arriving before their primaries, and OPERA's cable. The sharpest is the neutrino: beta-decay experiments fit for $m^2$ , not $m$ , precisely because $m^2$ is what the kinematics contains, and a genuinely negative best-fit value would be a tachyon's fingerprint. KATRIN's current bound is $m(\nu_e) < 0.45\ \mathrm{eV}$ at $90\ \%$ confidence (2025). The two-sided barrier of Derivation 2 explains why such searches must look for pre-existing tachyons rather than made ones — and why the negative results are less decisive than they look.

As instabilities — where the word actually lives. In modern field theory "tachyonic" almost always means $m^2 < 0$ in a Lagrangian, a statement about a vacuum, not a velocity: you have expanded around a maximum of the potential, and the field will roll. This is the engine of the Higgs mechanism, of cosmological tachyonic preheating, and of the instability that condemned the original bosonic string — whose tachyon forced the invention of superstrings. The most useful thing to know about tachyons is that when the word appears in a modern paper, it almost certainly refers to none of the kinematics in this article.

As a boundary marker. The tachyon's real service is negative and clarifying: it shows the light barrier is not a consequence of the kinematics alone — the kinematics is perfectly happy with $v > c$ — but of the kinematics plus the demand that cause precede effect. It isolates which principle does the forbidding, and it is not the one most people would name.

The physical picture

Draw the mass shell in the $(E, pc)$ plane. For an ordinary particle it is a hyperbola opening upward, with its vertex at $E = mc^2$ on the energy axis: no matter how you boost, $E$ stays above $mc^2$ , and the branch you are on ( $E > 0$ or $E < 0$ ) is an invariant — you cannot boost from the upper branch to the lower one, because to do so you would have to cross the region $|E| < mc^2$ , which the shell does not visit. That is why an electron's energy is positive in every frame. It is a topological fact about a timelike hyperboloid, not a convention. Now let $m^2 < 0$ . The hyperbola rotates by a quarter turn: it now opens sideways, with its vertices on the momentum axis at $pc = \pm\mu c^2$ , and it crosses the line $E = 0$ . The two branches are no longer separated. A boost slides you continuously along the curve, through $E = 0$ , and out the other side into $E < 0$ . Nothing forbids it, because there is no gap to jump. The tachyon's energy changes sign under boosts for the same reason the electron's cannot: the geometry of its own mass shell. Everything else falls out of that picture. The speed $v = pc^2/E$ — the same Hamilton relation an ordinary particle obeys — now exceeds $c$ , because on this hyperbola $pc > E$ . As $E \to 0$ the speed diverges; as $E \to \infty$ the curve becomes asymptotic to $E = pc$ , so $v \to c$ from above. Energy pushes a tachyon down towards the light cone, never through it. The light cone is the pair of asymptotes shared by both hyperbolas. Neither curve reaches it. They are two families of solutions to one equation, separated by a boundary that costs infinite energy to touch from either side — the two-sided barrier, which is so far only a picture. The proof is below.

Figure

The two-sided barrier, drawn — the article's owned result. Energy in units of Mc² (M = m for a bradyon, μ for a tachyon) plotted against β = v/c. LEFT of the light barrier, in BLUE, is the ordinary world: E = mc²/√(1−β²), which starts at the rest energy mc² when β = 0 and RISES without bound as β → 1⁻. This is the familiar wall. RIGHT of the barrier, in RED, is the tachyon: E = μc²/√(β²−1), which is ALSO infinite at β = 1 — and then FALLS as the particle goes faster, decaying towards ZERO as β → ∞. Read that again: the red curve slopes DOWNWARD. Take energy away from a tachyon and it SPEEDS UP. A tachyon of zero energy travels at infinite speed (carrying momentum μc, so it is not nothing). PUNCHLINE, and it is what this article owns: THE WALL AT β = 1 IS INFINITELY HIGH FROM BOTH SIDES, AND IT RISES SYMMETRICALLY. Near the barrier both branches behave as E ≃ Mc²/√(2|1−β|) — the SAME law, mirrored. The dashed grey curve is that common asymptotic form, and it hugs both. At |1−β| = 0.01 it gives 7.0711; the true bradyon value is 7.0888 and the true tachyon value is 7.0535 — one 0.25 % above, the other 0.25 % below, symmetric to first order. No finite-energy trajectory crosses from blue to red. The two sectors are dynamically disjoint, and c is not a speed limit so much as a partition.

The core equations

Signature $\eta_{\mu\nu} = \mathrm{diag}(+1,-1,-1,-1)$ , re-derived, never transcribed. A timelike vector has positive norm; a spacelike vector has negative norm. Units are SI.

The mass shell, for anything. For a bradyon this is the familiar

p_\mu p^\mu \;=\; \frac{E^2}{c^2} - \mathbf{p}^2 \;=\; +\,m^2c^2 \;>\; 0 \qquad (\text{timelike}) .

The tachyon. Set $m = i\mu$ with $\mu > 0$ real, so $m^2 = -\mu^2$ :

\boxed{\;p_\mu p^\mu \;=\; \frac{E^2}{c^2} - \mathbf{p}^2 \;=\; -\,\mu^2c^2 \;<\; 0 \qquad (\text{spacelike})\;}

Equivalently $E^2 < (pc)^2$ , equivalently $v > c$ — three ways of saying one thing.

Energy and momentum, real for every $\beta = v/c > 1$ :

\boxed{\;E \;=\; \frac{\mu c^2}{\sqrt{\beta^2 - 1}}, \qquad p \;=\; \frac{\mu v}{\sqrt{\beta^2 - 1}}\;}

Note the order under the root: $\beta^2 - 1$ , not $1 - \beta^2$ . That reversal is the whole substitution, and it is what keeps $E$ and $p$ real.

Velocity from the Hamilton relation — unchanged: $v = pc^2/E$ , so $\beta = pc/E > 1$ , because $pc > E$ on a spacelike shell.

Energy falls with speed:

\boxed{\;\frac{\mathrm{d}E}{\mathrm{d}v} \;=\; -\,\frac{\mu\,\beta}{(\beta^2-1)^{3/2}}\;\cdot c \;<\; 0\;}

Energy sign under a boost. Under a boost of speed $V$ along the tachyon's line of flight, $E' = \gamma_V\big(E - V p\big)$ , so, using $E/p = c^2/u$ for a tachyon of speed $u$ ,

\boxed{\;E' < 0 \quad\Longleftrightarrow\quad uV > c^2\;}

Field theory. The Klein-Gordon equation with $m^2 = -\mu^2$ becomes $\big(\partial_\mu\partial^\mu - \mu^2c^2/\hbar^2\big)\phi = 0$ , whose potential $V(\phi) \propto -\mu^2\phi^2$ has a maximum at $\phi = 0$ . That is the modern reading, and it is not about superluminal motion at all.

Working it out

Derivation AImaginary mass, real physics — and the spacelike shell it lands you on

The goal is to put $m = i\mu$ into relativistic kinematics and follow it honestly, checking at each step that nothing observable turns imaginary. It does not. What we get instead is a four-momentum on the other side of the light cone, and every subsequent oddity — falling energy, sign-changing energy, the two-sided wall — is a consequence of that single relocation.

Step 1. Substitute. The standard results are $E = mc^2/\sqrt{1-\beta^2}$ and $p = mv/\sqrt{1-\beta^2}$ . Put $m = i\mu$ and take $\beta > 1$ , so $1 - \beta^2 < 0$ and $\sqrt{1-\beta^2} = \sqrt{-(\beta^2-1)} = i\sqrt{\beta^2-1}$ .

Step 2. Watch the imaginaries cancel.

E \;=\; \frac{i\mu c^2}{i\sqrt{\beta^2-1}} \;=\; \frac{\mu c^2}{\sqrt{\beta^2-1}}, \qquad p \;=\; \frac{i\mu v}{i\sqrt{\beta^2-1}} \;=\; \frac{\mu v}{\sqrt{\beta^2-1}} .

Both are real and positive for every $\beta > 1$ . Why this is the crux: the imaginary mass never appears alone. It appears only in the combination $m/\sqrt{1-\beta^2}$ , and for $\beta > 1$ the denominator supplies a second factor of $i$ that kills the first. An imaginary parameter has produced an entirely real observable. Nothing here is more mysterious than $i \times i = -1$ .

Step 3. Compute the invariant — factorised, never as a difference of two large squares.

\frac{E^2}{c^2} - p^2 \;=\; \frac{\mu^2c^2}{\beta^2-1} - \frac{\mu^2v^2}{\beta^2-1} \;=\; \frac{\mu^2c^2\big(1 - \beta^2\big)}{\beta^2-1} \;=\; -\,\mu^2c^2 .

\boxed{\;p_\mu p^\mu \;=\; -\,\mu^2 c^2 \;<\; 0\;}

Why the sign is the whole article: in $(+,-,-,-)$ a negative norm means spacelike — the tachyon's four-momentum points outside the light cone. It is not that the tachyon has a strange energy; it is that its energy-momentum vector lives where relativity's guarantees about time order and energy sign do not apply. Those guarantees are theorems about timelike vectors, nothing more.

Step 4. Check consistency with the Hamilton relation. For an ordinary particle, $v = pc^2/E$ . Does a tachyon obey it?

\frac{pc^2}{E} \;=\; \frac{\mu v}{\sqrt{\beta^2-1}}\cdot\frac{\sqrt{\beta^2-1}}{\mu c^2}\cdot c^2 \;=\; v \qquad\checkmark

Why bother: it shows the tachyon is not a new dynamics bolted on, but the same dynamics on a different branch of the same shell — and it gives $\beta = pc/E > 1$ immediately, since a spacelike shell has $pc > E$ by definition. The superluminal speed is not a postulate. It is forced.

Step 5. Differentiate. From $E = \mu c^2(\beta^2-1)^{-1/2}$ , with $\beta = v/c$ ,

\frac{\mathrm{d}E}{\mathrm{d}v} \;=\; \mu c^2 \cdot \left(-\tfrac12\right)(\beta^2-1)^{-3/2}\cdot 2\beta \cdot \frac1c \;=\; -\,\frac{\mu c\,\beta}{(\beta^2-1)^{3/2}} \;<\; 0 .

Energy decreases as the tachyon goes faster. Why this is not a paradox: on a sideways hyperbola, moving up in energy means moving inward in momentum and therefore down in speed. The relation $v = pc^2/E$ never said "more energy means more speed" — that was true only because, for a bradyon, $p$ grows faster than $E$ . On this branch it does not.

Step 6. The limits. As $\beta \to 1^+$ , $E \to \infty$ . As $\beta \to \infty$ , $E \to 0$ while $p = \mu c\beta/\sqrt{\beta^2-1} \to \mu c$ . A tachyon of zero energy carries momentum $\mu c$ and moves at infinite speed — the vertex of the spacelike hyperbola, and the exact mirror of a bradyon of zero momentum still carrying $mc^2$ . Verified symbolically and numerically in `verify_aq-sr-tachyons-superluminal-particles.py`.

Go deeper · Derivation B · 2 minTHE OWNED RESULT: the barrier is two-sided — and it rises symmetrically

Derivation BTHE OWNED RESULT: the barrier is two-sided — and it rises symmetrically

Everyone knows the light barrier is infinitely high from below. This derivation establishes the half nobody states: it is equally infinitely high from above — and the two divergences obey the same law. That upgrades "tachyons cannot decelerate through $c$ " from an assertion to a theorem, and it forbids any process turning an ordinary particle into a tachyon.

Step 1. The familiar side. For a bradyon, factorise the root rather than leaving it as $1 - \beta^2$ :

E \;=\; \frac{mc^2}{\sqrt{1-\beta^2}} \;=\; \frac{mc^2}{\sqrt{(1-\beta)(1+\beta)}} \;\xrightarrow{\;\beta\to1^-\;}\; \infty .

Step 2. The unfamiliar side. For a tachyon, factorise identically:

E \;=\; \frac{\mu c^2}{\sqrt{\beta^2-1}} \;=\; \frac{\mu c^2}{\sqrt{(\beta-1)(\beta+1)}} \;\xrightarrow{\;\beta\to1^+\;}\; \infty .

Why the factorisation matters and is not cosmetic: written this way, the two expressions differ only in which of $(1-\beta)$ and $(\beta-1)$ appears — that is, only in the sign of $1 - \beta$ . Both are $\;\propto |1-\beta|^{-1/2}$ . The divergence is the same divergence.

Step 3. Extract the common asymptotic law. Near the barrier, $\beta \to 1$ makes $(1+\beta) \to 2$ on both sides. Hence, with $M$ standing for $m$ or $\mu$ as appropriate,

\boxed{\;E \;\simeq\; \frac{Mc^2}{\sqrt{2\,|1-\beta|}} \qquad\text{on BOTH sides of } \beta = 1\;}

This is the owned result. The wall is not merely infinite in both directions; it is the same wall, approached with the same square-root law, from outside and from inside. Stand $1\ \%$ away on each side and compare. Bradyon at $\beta = 0.99$ : $E/mc^2 = 1/\sqrt{(1-0.99)(1+0.99)} = 1/\sqrt{0.0199} = \mathbf{7.088812}$ . Tachyon at $\beta = 1.01$ : $E/\mu c^2 = 1/\sqrt{(1.01-1)(1.01+1)} = 1/\sqrt{0.0201} = \mathbf{7.053456}$ . The common law predicts $1/\sqrt{2\times0.01} = \mathbf{7.071068}$ for both — the bradyon $\mathbf{0.251\ \%}$ above it, the tachyon $\mathbf{0.249\ \%}$ below. Symmetric to three significant figures, and the asymmetry vanishes as the barrier is approached. The wall does not merely have two faces; it has the same face twice.

Step 4. Draw the conclusion. Suppose some interaction were to take a bradyon of finite energy and turn it into a tachyon. The particle's speed must pass through $c$ , either continuously or in a jump. Continuously is impossible: $E \to \infty$ on approach from either side, and no finite reservoir supplies it. And a jump is not available either, because energy-momentum conservation is local — the four-momentum is continuous along the worldline, so a transition from the timelike branch of the shell to the spacelike branch would have to pass through the light cone, where $p_\mu p^\mu = 0$ , and the shells $\pm M^2c^2$ do not meet it at any finite $E$ .

\boxed{\;\text{No finite-energy trajectory connects the bradyon and tachyon sectors.}\;}

Why this is the strongest statement in the article: tachyons, if they exist, are a separate species, not a state ordinary matter can be excited into. They could not be produced by accelerating anything — which is why every search has had to look for pre-existing tachyons, and why "we did not make one in a collider" was never evidence against them. Verified symbolically and numerically in `verify_aq-sr-tachyons-superluminal-particles.py`.

Go deeper · Derivation C · 2 minTwo thresholds, not one — the energy flip, and the correction this article owes its base

Derivation CTwo thresholds, not one — the energy flip, and the correction this article owes its base

A tachyon's energy has no invariant sign, and its emission and absorption can be seen in either order. Both follow from the spacelike shell in two lines. The third claim — that these let you build a causal loop — is separate and stronger, requires a strictly larger boost, and this article's own source gets it wrong. We state both thresholds exactly, then hand the causal argument to its owners.

Step 1. Boost the energy. Under a boost of speed $V$ along the tachyon's direction of flight,

E' \;=\; \gamma_V\big(E - V p\big) \qquad\Longrightarrow\qquad E' < 0 \;\Longleftrightarrow\; V\,p > E \;\Longleftrightarrow\; V > \frac{E}{p} .

Step 2. Eliminate $E/p$ . From Step 4 of Derivation 1, $v = pc^2/E$ , so for a tachyon of speed $u$ , $E/p = c^2/u$ exactly. Therefore

\boxed{\;E' < 0 \quad\Longleftrightarrow\quad uV > c^2\;}

Why this is not yet a paradox: the same condition reverses the time order of the tachyon's emission and absorption — one statement, since a negative-energy tachyon going forwards is a positive-energy tachyon going backwards. That is the reinterpretation move, and whether it rescues anything belongs to the reinterpretation principle. One leg is not a loop.

Step 3. Say it in rapidity, where the geometry is visible. Parametrise the tachyon's superluminal speed by $u = c\coth\chi$ (legitimate: $\coth\chi > 1$ for $\chi > 0$ , so this covers every $u > c$ ) and the observer's subluminal boost by $V = c\tanh\varphi$ . Then

uV > c^2 \;\Longleftrightarrow\; \coth\chi\,\tanh\varphi > 1 \;\Longleftrightarrow\; \tanh\varphi > \tanh\chi \;\Longleftrightarrow\; \boxed{\;\varphi > \chi\;}

Step 4. The loop is a different, larger condition. A causal loop needs two legs — a tachyon out and a tachyon back — and its threshold is not the single-leg one. The correct condition is $V > 2uc^2/(u^2+c^2)$ , and the reason to trust it is that it collapses to something exact in rapidity. Substituting $u = c\coth\chi$ and $\coth\chi = \cosh\chi/\sinh\chi$ :

\frac{2uc^2}{u^2+c^2} \;=\; \frac{2c\coth\chi}{\coth^2\chi + 1} \;=\; \frac{2c\,\dfrac{\cosh\chi}{\sinh\chi}}{\dfrac{\cosh^2\chi + \sinh^2\chi}{\sinh^2\chi}} \;=\; \frac{2c\sinh\chi\cosh\chi}{\cosh 2\chi} \;=\; c\,\frac{\sinh 2\chi}{\cosh 2\chi} \;=\; c\tanh 2\chi .

Hence the loop condition is $c\tanh\varphi > c\tanh 2\chi$ , that is

\boxed{\;\varphi \;>\; 2\chi \qquad\text{— the loop needs EXACTLY TWICE the rapidity of a single reversal.}\;}

Step 5. Refuse the base, and quantify the error. The source document behind this article states, in its antitelephone derivation, that $uv > c^2$ implies the reply precedes the original message — that is, it offers the single-leg condition $\varphi > \chi$ as the loop condition. It does not follow, and the gap is not small. Take $u = 1.5\,c$ , so $\chi = \operatorname{artanh}(2/3) = 0.80472$ . Then

\text{single-leg reversal: } V > c\tanh\chi = \mathbf{0.66667\,c}, \qquad \text{closed loop: } V > c\tanh 2\chi = \mathbf{0.92308\,c} ,

and the second agrees with $2uc^2/(u^2+c^2) = 3/3.25 = \mathbf{0.92308}$ exactly. For every boost between $0.667c$ and $0.923c$ the base's criterion announces a causal loop where none exists — a quarter of the available range of $V$ , misclassified. We do not repeat the claim. We correct it.

Step 6. Hand it over. The construction itself belongs to the reinterpretation principle and causal paradoxes; the theorem that order reversal requires spacelike separation belongs to the causality light-cone constraint and causal structure. This article owns the kinematics that make the question askable — not the answer. Verified symbolically and numerically in `verify_aq-sr-tachyons-superluminal-particles.py`.

Figure

The two thresholds, in rapidity — and exactly where the base is wrong. The tachyon's superluminal speed u is written u = c·coth χ, and the observer's boost V = c·tanh φ; both are exact parametrisations, and in these coordinates the whole problem becomes a comparison of two numbers. Horizontal axis: the tachyon rapidity χ. Vertical axis: the boost rapidity φ needed. The LOWER (blue) line is φ = χ — cross it and the tachyon's energy goes NEGATIVE and its emission and absorption swap order. That is the single-leg condition, and it is equivalent to uV > c². The UPPER (red) line is φ = 2χ — cross THAT and a closed causal loop becomes possible, because a loop has TWO legs. It is equivalent to V > 2uc²/(u²+c²), and the identity 2uc²/(u²+c²) ≡ c·tanh(2χ) is what makes the two statements the same statement. THE AMBER BAND BETWEEN THE LINES IS THE ERROR. In it, a single leg reverses but no loop closes. The base document for this article puts the loop condition at uv > c² — the LOWER line — and so claims a paradox throughout the amber band. It is not there. WORKED POINT (black dot): u = 1.5c gives χ = 0.80472; the single leg reverses for V > 0.66667c but the loop needs V > 0.92308c. PUNCHLINE: THE LOOP COSTS EXACTLY TWICE THE RAPIDITY, AND HALF THE LITERATURE CHARGES ONLY ONCE.

Limiting cases

regime	what happens	the reading
$\beta \to 1^+$	$E \to \infty$	the barrier from above — a tachyon can never slow to $c$
$\beta \to \infty$	$E \to 0$ , $p \to \mu c$	the transcendent tachyon: no energy, finite momentum, infinite speed
$uV \to c^2$	$E' \to 0$	the boost at which the energy changes sign
$\varphi > \chi$	one leg reverses	necessary, not sufficient
$\varphi > 2\chi$	the loop can close	the real threshold; see the reinterpretation principle

Worked examples

1 · A 1.00 MeV tachyon, at three speeds

Take $\mu c^2 = 1.00\ \mathrm{MeV}$ — the tachyon analogue of a rest energy, though nothing is ever at rest. Then $E = \mu c^2/\sqrt{\beta^2-1}$ and $pc = \mu c^2\beta/\sqrt{\beta^2-1}$ :

$\beta = v/c$	$E$ (MeV)	$pc$ (MeV)	$E^2 - (pc)^2$ (MeV²)
$1.5$	$\mathbf{0.894427}$	$\mathbf{1.341641}$	$\mathbf{-1.000000}$
$2.0$	$\mathbf{0.577350}$	$\mathbf{1.154701}$	$\mathbf{-1.000000}$
$10.0$	$\mathbf{0.100504}$	$\mathbf{1.005038}$	$\mathbf{-1.000000}$

Three things to read off. The energy falls as the speed rises — $0.894$ MeV at $1.5c$ , $0.101$ MeV at $10c$ . The momentum falls towards a floor: $pc \to \mu c^2 = 1.00$ MeV as $\beta \to \infty$ , never below. And the invariant is $\mathbf{-1.000000\ MeV^2} = -(\mu c^2)^2$ at every speed, to fifty digits — negative, as a spacelike vector must be, and constant, as an invariant must be. Compare the bradyon of the same $|m|$ : at $\beta = 0$ it has $E = 1.00$ MeV and $p = 0$ ; the tachyon's mirror is $\beta = \infty$ , where $E = 0$ and $pc = 1.00$ MeV. The two sectors are reflections across the light cone, energy and momentum trading places.

2 · OPERA, and what a real superluminal claim looks like

In September 2011 the OPERA collaboration reported muon neutrinos arriving at Gran Sasso from CERN $60.7 \pm 6.9\ \text{(stat)} \pm 7.4\ \text{(sys)}\ \mathrm{ns}$ earlier than light would have, over a baseline of $730\ \mathrm{km}$ . Turn that into a fractional excess: light takes

t \;=\; \frac{730\,000\ \mathrm{m}}{299\,792\,458\ \mathrm{m/s}} \;=\; \mathbf{2.4350\ ms} ,

so the claimed excess speed was

\frac{v - c}{c} \;=\; \frac{60.7\ \mathrm{ns}}{2.4350\ \mathrm{ms}} \;=\; \mathbf{2.4928\times10^{-5}} ,

in agreement with the $2.48\times10^{-5}$ the collaboration quoted. It was wrong — traced in 2012 to a loose fibre-optic connector in the timing chain, CERN announcing consistency with $c$ that June. But note what the claim would have implied: at $\beta - 1 = 2.4928\times10^{-5}$ the factor $1/\sqrt{(\beta-1)(\beta+1)} = \mathbf{141.6}$ , so the neutrino would have been a tachyon of energy $141.6\,\mu c^2$ — perfectly finite, perfectly consistent kinematics. Which is exactly why the result had to be killed by metrology rather than by theory. Relativity does not forbid what OPERA reported. A cable did.

Common trapThe single-leg condition is NOT the loop condition — and this article's base gets it wrong

The claim " $uV > c^2$ , therefore the reply arrives before the question" is false as stated, and it appears in the source document for this very article. What $uV > c^2$ buys you is that one leg of the exchange runs backwards in one frame: the tachyon's energy goes negative and its emission and absorption swap order. That is a necessary condition for a paradox. It is not a sufficient one. A causal loop needs both legs to come home, and the threshold is strictly larger:

V \;>\; \frac{2uc^2}{u^2+c^2} , \qquad\text{equivalently}\qquad \varphi > 2\chi \quad (u = c\coth\chi,\; V = c\tanh\varphi) ,

because $2uc^2/(u^2+c^2) \equiv c\tanh 2\chi$ identically. The loop costs exactly twice the rapidity. For $u = 1.5c$ a single leg reverses at $V = 0.667c$ but the loop does not close until $V = 0.923c$ ; every boost in between is misclassified by the naive criterion. Do not quote $uV > c^2$ as an antitelephone condition. The construction is owned, correctly, by the reinterpretation principle and causal paradoxes.

Physics of the portrait

Origins · 3 minTachyon and how this was discovered

Tachyon

1904–2012 · Sommerfeld studies superluminal electrons BEFORE relativity exists (1904-05) · Tolman poses the antitelephone (1917) · Bilaniuk, Deshpande and Sudarshan resurrect the idea and offer the reinterpretation principle (1962) · Feinberg names the tachyon (1967) · Benford, Book and Newcomb sharpen the paradox (1970) · OPERA, and a loose cable (2011-12)

The idea is older than the theory it appears to violate. Arnold Sommerfeld, in 1904 — a year before Einstein's relativity paper — worked out the electrodynamics of an electron moving faster than light, and for a few months believed Paschen's gamma-ray data might have found one, writing to Lorentz that December that he must assume "motion faster than light really exists". By 1905 he had reversed himself: superluminal motion "is in no way a physically real phenomenon". Then relativity arrived and the question looked closed. It was not closed, because relativity forbids crossing the barrier and says nothing about being born beyond it. Richard Tolman saw the danger early, laying out in The Theory of the Relativity of Motion (1917) the argument that a superluminal signal permits a message to one's own past — the antitelephone, before it had the name. The subject lay quiet four decades, until Bilaniuk, Deshpande and Sudarshan published "'Meta' Relativity" (American Journal of Physics 30, 718, 1962), taking the imaginary-mass solutions seriously and offering the reinterpretation principle: a negative-energy particle running backwards is a positive-energy particle running forwards, so relabel and the energetics are saved. Gerald Feinberg named them in "Possibility of Faster-Than-Light Particles" (Physical Review 159, 1089, 1967) — tachyon, from the Greek for swift. The rescue did not hold. Benford, Book and Newcomb ("The Tachyonic Antitelephone", Physical Review D 2, 263, 1970) showed that reinterpretation saves the energetics of an uncontrolled tachyon while leaving signalling untouched: a paradox living in a correlation between two experimenters' choices is not repaired by a change of dictionary, because the two ends do not agree on which dictionary to use. That argument is owned by the reinterpretation principle and causal paradoxes — and it is where the loop condition $\varphi > 2\chi$ must be got right, because it is very often got wrong. The last act was farce: in 2011 OPERA measured neutrinos beating light to Gran Sasso by 60.7 ns, and in 2012 traced it to an imperfectly seated fibre-optic connector. Sommerfeld's question survived a hundred and eight years and was answered by a cable. The word, meanwhile, escaped into field theory and now means something else entirely — an unstable vacuum — the usual fate of a beautiful wrong idea: too useful to discard, so quietly repurposed.

History

1904

— Sommerfeld, a year before special relativity, computes the electrodynamics of superluminal electrons and briefly believes Paschen's gamma-ray data support them.

1905

— Sommerfeld reverses himself, concluding that motion faster than light "is in no way a physically real phenomenon"; Einstein's relativity paper follows within months.

1917

— Tolman, in "The Theory of the Relativity of Motion", sets out the argument that a superluminal signal permits a message to one's own past — the antitelephone, before it had the name.

1934

— Cherenkov observes the radiation of a charge outrunning light in a medium; Frank and Tamm explain it in 1937, and the three share the 1958 Nobel Prize. It is superluminal only with respect to c/n, and it is the closest nature comes.

1962

— Bilaniuk, Deshpande and Sudarshan publish "'Meta' Relativity" (American Journal of Physics 30, 718), taking imaginary mass seriously and proposing the reinterpretation principle.

1967

— Feinberg, "Possibility of Faster-Than-Light Particles" (Physical Review 159, 1089), coins the word tachyon and attempts a field theory.

1970

— Benford, Book and Newcomb, "The Tachyonic Antitelephone" (Physical Review D 2, 263), show that reinterpretation rescues the energetics but not the signalling.

2011

— OPERA reports muon neutrinos arriving 60.7 ns early over 730 km, a fractional excess of 2.48e-5.

2012

— The anomaly is traced to a loose fibre-optic connector; CERN announces that the neutrino speed is consistent with c.

Going further

The causality question this article leaves open is answered by its owners: the causality light-cone constraint for the theorem that order reversal requires spacelike separation, causal structure for the geometry, and the reinterpretation principle and causal paradoxes for the antitelephone and for exactly where Feinberg's rescue fails. For the interval classification underlying all of it, see timelike, spacelike and lightlike separations. And for the machinery this article inverted, see the relativistic Hamiltonian formalism, whose Klein-Gordon equation with $m^2 < 0$ is the tachyonic field modern physics actually uses.

If tachyons have imaginary mass, isn't the whole thing obviously nonsense?

No. The imaginary mass never appears alone in anything measurable: it appears only inside $m/\sqrt{1-\beta^2}$ , and for $\beta > 1$ the square root is itself imaginary, so the two factors of $i$ cancel and $E$ and $p$ come out real and positive. What the kinematics contains is $m^2$ , not $m$ — and $m^2 = -\mu^2$ is simply a negative real number. Writing $m = i\mu$ is bookkeeping. The correct objection is not that tachyons are imaginary (they are not) but that their four-momentum is spacelike, and spacelike vectors have no invariant energy sign and no invariant time order.

Doesn't relativity forbid faster-than-light travel outright?

It forbids accelerating through the light barrier, which is a different and weaker statement. The energy required diverges as $v \to c^-$ , so no finite process pushes a massive particle up to $c$ , let alone past it. But that argument says nothing whatever about an object that was never slower than light. This article's owned result closes the loophole from the other direction: energy also diverges as $v \to c^+$ , so a tachyon can never decelerate to $c$ either. The barrier is two-sided, and the two sectors are dynamically disjoint. Relativity does not forbid tachyons. It forbids the transition — which means, if they exist, they are a separate species that has always been superluminal, and cannot be manufactured.

Is Cherenkov radiation an example of faster-than-light motion?

No, and the distinction is exact. Cherenkov radiation occurs when a charge moves faster than the phase speed of light in a medium, $c/n$ , which is less than $c$ . The particle is entirely subluminal with respect to $c$ ; it merely outruns the light in that material, shedding the excess as a cone with $\cos\theta_c = c/(nv)$ . Nothing in relativity is troubled. It appears here as an analogy and a warning: a tachyon in vacuum would be expected to radiate the same way — and since a tachyon gains speed as it loses energy, the radiation would accelerate it, making it radiate harder. A runaway with no equilibrium. That instability is one of the better arguments against tachyons, and it is entirely separate from the causality argument.

What does it mean that a tachyon's energy can be negative?

It means the sign of $E$ is not a Lorentz invariant for a spacelike four-vector — a theorem, not a defect of the model. For a timelike four-momentum the branches $E > 0$ and $E < 0$ are disconnected and no boost carries you between them; that is exactly why an electron's energy is positive in every frame. A tachyon's hyperboloid opens sideways and crosses $E = 0$ , so a boost slides smoothly through. Concretely, $uV > c^2$ makes $E' < 0$ . The reinterpretation principle reads such a particle as a positive-energy tachyon going the other way — which fixes the energetics and, as the reinterpretation principle shows, does not fix the signalling.

Is a tachyonic mass term in a Lagrangian the same as a tachyon?

Almost never, and this is the single most useful thing to know about the word. "Tachyonic mass" means $m^2 < 0$ in the Lagrangian, and its physical content is that the potential $V(\phi) \propto -\mu^2\phi^2$ has a maximum at $\phi = 0$ : the field is balanced on a hilltop and will roll off. It signals an unstable vacuum, not superluminal propagation. Expand about the true minimum instead and the excitations have $m^2 > 0$ and propagate strictly inside the light cone. This is precisely the Higgs mechanism — the Higgs potential's $-\mu^2|\phi|^2$ term is a tachyonic mass — and it is why the Standard Model contains a tachyonic mass term and no tachyons at all. When you meet the word in a modern paper, it almost certainly means an instability.

Did OPERA find tachyons in 2011?

No. OPERA reported muon neutrinos arriving at Gran Sasso $60.7 \pm 6.9\ \text{(stat)} \pm 7.4\ \text{(sys)}$ ns earlier than light over 730 km — a fractional excess of $2.48\times10^{-5}$ . In 2012 it was traced to a loose fibre-optic connector in the GPS timing chain, and CERN announced consistency with $c$ . The episode illustrates this article's point: the claim, had it held, would not have contradicted relativistic kinematics — a neutrino at $\beta - 1 = 2.4928\times10^{-5}$ is a perfectly consistent tachyon of energy $141.6\,\mu c^2$ . It would have contradicted the theory's causal structure, and would have had to be reconciled with the two-sided barrier, since those neutrinos were made from ordinary pions. It was refuted by metrology, not theory — exactly as it should have been.