Physics & Mathematics

Special Relativity Foundation

Tachyons emerge from the equations of Einstein's special theory of relativity, published in 1905. Special relativity unified space and time into a four-dimensional spacetime and revealed that the speed of light is a universal constant that nothing with mass can reach or exceed - or so it seemed.

The theory is built on two fundamental postulates:

• The laws of physics are the same in all inertial reference frames
• The speed of light in vacuum is constant for all observers, regardless of their motion

Energy-Momentum Relations

The fundamental equation relating energy (E), momentum (p), mass (m), and the speed of light (c) is:

For ordinary particles with positive mass m² > 0, this equation ensures they travel slower than light. For photons with m² = 0, they travel at exactly the speed of light. But what if m² < 0?

When m² is negative, we can write m = iμ, where i is the imaginary unit and μ is a real positive number. This gives tachyons an imaginary rest mass, allowing them to travel faster than light while still satisfying the energy-momentum relation.

Velocity and Energy Relationship

For a tachyon with imaginary mass m = iμ, the velocity v is related to energy E by:

This equation reveals several remarkable properties:

• When E → ∞, velocity approaches c from above (v → c⁺)
• When E = μc², velocity is infinite
• When E → 0, velocity approaches infinity
• The tachyon can never slow down to the speed of light or below

Lorentz Transformations

The Lorentz transformation equations describe how space and time coordinates change between reference frames. The Lorentz factor γ (gamma) is:

For ordinary particles (v < c), γ is real and greater than 1. For tachyons (v > c), the term under the square root becomes negative, making γ imaginary. This leads to the peculiar property that tachyons appear to travel backward in time in some reference frames.

The time dilation formula becomes especially interesting for tachyons, as observers in different reference frames may disagree not just on when events occur, but on their temporal ordering.

The Reinterpretation Principle

Gerald Feinberg proposed the "reinterpretation principle" to address causality concerns. According to this principle, a tachyon traveling backward in time in one reference frame can be reinterpreted as an anti-tachyon traveling forward in time.

This means:

• A tachyon moving from A to B (backward in time) equals an anti-tachyon moving from B to A (forward in time)
• Emission and absorption events are observer-dependent
• The apparent direction of causality depends on the reference frame

While this preserves some form of consistency, it doesn't fully resolve all causality paradoxes, especially in cases involving closed timelike curves.

Tachyons in Quantum Field Theory

In quantum field theory, a tachyonic field is one whose squared mass term is negative. The appearance of such fields often indicates an instability in the vacuum state rather than the existence of actual faster-than-light particles.

Important applications include:

• Spontaneous Symmetry Breaking: Tachyonic instabilities can drive systems to more stable configurations
• Higgs Mechanism: The Higgs field starts as a tachyonic field before symmetry breaking
• String Theory: Tachyons appear in some string theory configurations, often indicating an unstable state

In these contexts, tachyonic behavior doesn't imply faster-than-light particles, but rather represents mathematical properties of quantum fields.

Mathematical Constraints and Paradoxes

Several mathematical and physical constraints challenge the existence of tachyons: