The concept of a tachyon does not inherently break the mathematics of special relativity. Rather, it represents an unexplored mathematical domain within the Lorentz transformations. To understand tachyons, we must look closely at the equations that govern energy, momentum, and spacetime when velocity ($v$) strictly exceeds the speed of light ($c$).
1. The Energy-Momentum Inversion
In special relativity, the total energy ($E$) and momentum ($p$) of a particle with rest mass ($m₀$) and velocity ($v$) are given by the Lorentz equations:
E = m₀c² / √(1 - v²/c²)
p = m₀v / √(1 - v²/c²)
For ordinary matter (bradyons), $v < c$, so the term under the square root, $(1 - v²/c²)$, is positive. The denominator is real, the rest mass is real, and therefore energy and momentum are real observable quantities. As $v$ approaches $c$, the denominator approaches zero, driving the energy toward infinity. This is why ordinary matter cannot reach the speed of light—it would require infinite energy.
For a tachyon, $v > c$. This makes the term $(1 - v²/c²)$ negative. Taking the square root of a negative number yields an imaginary denominator. For the energy $E$ and momentum $p$ to remain real, observable quantities, the numerator must also be imaginary. This leads to the requirement that the tachyon's rest mass $m₀$ must be imaginary:
Where $i = √(-1)$ and $μ$ is a real number representing the magnitude of the tachyon's mass. The energy equation then becomes:
The Speed Limit from Below
Look closely at the modified energy equation above. If a tachyon's velocity $v$ decreases and approaches $c$ (from above), the denominator $(v²/c² - 1)$ approaches zero, and the tachyon's energy approaches infinity. Conversely, as $v$ approaches infinity, the denominator grows infinitely large, and the energy $E$ approaches zero. A tachyon with zero energy is traveling at infinite speed. It cannot slow down to $c$ any more than a regular particle can speed up to $c$.
2. The Invariant Mass Equation
The relationship between energy, momentum, and rest mass can also be expressed through the relativistic invariant equation:
For a tachyon with imaginary mass $m₀ = iμ$, the square of the mass $(m₀)² = (iμ)² = -μ²$. The equation becomes:
This indicates that for a tachyon, the squared momentum $(pc)²$ is always strictly greater than its squared energy $E²$. In four-dimensional Minkowski spacetime, the energy-momentum four-vector of a tachyon is spacelike, whereas for ordinary matter it is timelike, and for light it is lightlike (or null).
3. Feinberg's Reinterpretation Principle
The most severe physical complication of tachyons is causality. Because the tachyon's four-momentum is spacelike, different observers in different inertial reference frames will disagree on the temporal order of events.
If Observer A sees a tachyon emitted at event $E_1$ with positive energy and absorbed at event $E_2$ later in time ($t_2 > t_1$), there exists another valid frame of reference, Observer B, traveling at a relative velocity $v < c$, who will see the events in reverse order ($t_1' > t_2'$). Furthermore, in Observer B's frame, the energy of the tachyon will mathematically appear negative.
To resolve this, Gerald Feinberg introduced the Reinterpretation Principle (also independently formulated by Sudarshan). The principle states that a negative-energy tachyon traveling backward in time is physically indistinguishable from a positive-energy anti-tachyon traveling forward in time.
If Observer B sees a particle moving from $E_2$ to $E_1$ backward in time with negative energy, they must reinterpret this as an anti-particle moving from $E_1$ to $E_2$ forward in time with positive energy. The act of "emission" and "absorption" simply swaps depending on the observer's frame of reference. This restores localized thermodynamic stability, though it does not fully solve the macro-scale causality paradoxes like the Tachyonic Antitelephone.
4. Quantum Spin and Helicity
If tachyons exist as quantum particles, they must possess quantum numbers like spin. However, the Wigner classification of the representations of the Poincaré group shows that tachyonic states are highly anomalous. For a spacelike momentum vector, the "little group" (the subgroup of Lorentz transformations that leaves the momentum vector unchanged) is $SO(2,1)$, which is non-compact.
This implies that a quantum tachyon would have an infinite number of polarization states (a continuous spin), unless the state is constrained to zero helicity (a scalar tachyon). Because we do not observe elementary particles with infinite continuous spin states, theoretical physicists generally model tachyons as spin-0 scalar fields.
Conclusion
The physics of tachyons requires us to invert our usual intuition about energy and velocity. As mathematically elegant as the imaginary mass and the reinterpretation principle are, they describe a universe where causality is deeply observer-dependent. While physical tachyons remain unverified, this exact mathematical framework—specifically the spacelike momentum and imaginary mass—forms the basis of understanding field instabilities (tachyon condensation) in modern string theory and the Higgs mechanism.