Relativity and the Speed Limit

Albert Einstein’s Special Theory of Relativity is often cited as the ultimate speed limit of the universe: nothing can travel faster than light. But the actual physics is more nuanced than this popular summary suggests. To understand where tachyons fit, we need to return to the foundations of relativity itself.

Einstein’s 1905 Paper

On June 30, 1905, Albert Einstein submitted his paper “Zur Elektrodynamik bewegter Körper” (“On the Electrodynamics of Moving Bodies”) to the journal Annalen der Physik. This paper did not mention “relativity” by name, and it contained no complex mathematics beyond algebra. What it did contain were two deceptively simple postulates that would reshape physics forever.

Postulate 1 (The Principle of Relativity): The laws of physics are the same in all inertial reference frames. No experiment performed inside a sealed, uniformly moving laboratory can determine its absolute velocity.

Postulate 2 (The Invariance of the Speed of Light): The speed of light in a vacuum, $c$, is the same for all observers regardless of the motion of the light source or the observer. Its measured value is approximately 299,792,458 meters per second.

From these two statements alone, Einstein derived the Lorentz transformations, time dilation, length contraction, and the relativistic addition of velocities. The consequences were radical: simultaneity is relative, moving clocks run slow, and moving rulers shrink in their direction of motion.

The Lorentz Factor and the Light Barrier

The mathematical heart of special relativity is the Lorentz factor, denoted by the Greek letter gamma:

$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$

This factor appears in nearly every relativistic equation. At everyday speeds, $\gamma$ is indistinguishable from 1. At 87% of light speed, $\gamma$ equals 2, meaning time dilation doubles. At 99.5% of light speed, $\gamma$ reaches 10.

As $v$ approaches $c$, the denominator approaches zero and $\gamma$ diverges toward infinity. This has direct physical consequences:

Relativistic mass (or more precisely, relativistic momentum) grows without bound, requiring ever-increasing force to produce the same acceleration
Time dilation becomes extreme; a particle’s internal clock nearly stops relative to the laboratory
The energy required to push a massive particle to $c$ becomes literally infinite, as the total relativistic energy is $E = \gamma m c^2$

This is why no particle with positive real mass has ever been observed at or above the speed of light. Every particle accelerator ever built confirms this: protons at the Large Hadron Collider reach 99.999999% of $c$, but never $c$ itself.

The Popular Misunderstanding

The popular version of the speed limit, “nothing can travel faster than light,” is an oversimplification. What special relativity actually forbids is more specific: no object that is currently traveling slower than light can be accelerated to or beyond $c$. The equations show that crossing the light barrier from below requires infinite energy. But the equations are silent about what might already exist on the other side.

This distinction is critical. The light speed barrier is not a one-sided ceiling. It is a two-sided wall. Particles below $c$ cannot reach $c$. But if particles above $c$ exist, they equally cannot decelerate to $c$. The speed of light is an impenetrable dividing line, not an absolute maximum.

Arnold Sommerfeld, the great German theoretical physicist, recognized this as early as 1904. Before Einstein had even published his relativity paper, Sommerfeld was studying the theoretical properties of electrons moving faster than light, analyzing the electromagnetic radiation such particles would produce. His work, though largely forgotten, anticipated key features of what we now call tachyon physics by over six decades.

The Three-Class Particle Taxonomy

This understanding leads to a natural classification of all particles into three families, based on their relationship to the speed of light:

Tardyons (Bradyons)

Speed: Always less than $c$
Rest mass: Positive and real
Examples: Electrons, protons, neutrons, quarks, neutrinos
Energy behavior: Gaining energy increases speed, approaching but never reaching $c$
The name “tardyon” comes from the Latin tardus (slow); “bradyon” comes from the Greek bradys (slow)

Luxons

Speed: Always exactly $c$
Rest mass: Zero
Examples: Photons, gluons, and (if massless) gravitons
Energy behavior: Always travel at $c$ regardless of energy; gaining energy increases frequency, not speed
The name comes from the Latin lux (light)

Tachyons

Speed: Always greater than $c$
Rest mass: Imaginary (a real number multiplied by $i = \sqrt{-1}$)
Examples: None confirmed; hypothetical
Energy behavior: Gaining energy decreases speed toward $c$; losing energy increases speed toward infinity
The name comes from the Greek tachys (swift), coined by Gerald Feinberg in 1967

This taxonomy was formalized by physicists George Sudarshan and Ole-Peder Bilaniuk in their landmark 1962 paper, which argued that the mathematics of special relativity permits all three classes equally. The equations do not prefer one class over the others.

The Energy-Momentum Relation

The full energy-momentum relation in special relativity is:

$$E^2 = (pc)^2 + (mc^2)^2$$

where $E$ is total energy, $p$ is momentum, $m$ is rest mass, and $c$ is the speed of light. For tardyons, both $E$ and $m$ are real, and this equation works straightforwardly. For tachyons, $m$ is imaginary, but $m^2$ is negative, which means the equation still yields real values for energy and momentum as long as $p > mc$ in magnitude.

This connects to the geometry of spacetime. In the four-dimensional spacetime of relativity, events can be separated by timelike, lightlike, or spacelike intervals:

Timelike intervals correspond to separations that tardyons can traverse. The interval squared is positive (in the $+---$ sign convention), and a causal ordering of events is preserved across all reference frames.
Lightlike (null) intervals correspond to the paths of luxons. The interval squared is zero.
Spacelike intervals correspond to separations that only tachyons could traverse. The interval squared is negative, and crucially, different observers disagree about which event happened first.

It is this last point that creates the profound causality problems associated with tachyons. If a tachyon connects two spacelike-separated events, some observers see it traveling forward in time while others see it traveling backward in time. This is not a quirk of measurement; it is a fundamental feature of the Lorentz transformations applied to superluminal motion.

Why This Matters for Tachyon Physics

The framework of special relativity does not forbid tachyons outright. It forbids crossing the light barrier, which is a different statement entirely. The mathematics accommodates three classes of particles separated by an impassable wall at $v = c$.

However, relativity does impose severe constraints on any tachyon theory. The causality violations implied by spacelike propagation must be addressed. The imaginary mass must be given a coherent physical interpretation. And any detection scheme must reckon with the bizarre fact that a tachyon with zero energy would travel at infinite speed.

These constraints have led theoretical physicists down two very different paths. One path, pursued by Feinberg and others, attempts to construct a consistent quantum field theory of tachyon particles. The other path, dominant in modern string theory and cosmology, reinterprets the tachyon not as a particle but as a sign that the vacuum state of a field is unstable, leading to the phenomenon of tachyon condensation.

Both paths begin with Einstein’s equations. Both paths take the mathematics seriously. And both paths remind us that special relativity, far from closing the door on faster-than-light physics, left it tantalizingly ajar.

Einstein's Relativity and Faster-Than-Light Particles