Lorentz Transformation Equations
The Lorentz transformation equations convert spacetime coordinates between two inertial reference frames moving at constant velocity relative to each other.
The Formulas
t' = γ(t − vx/c²)
y' = y
z' = z
γ = 1 / √(1 − v²/c²)
The Lorentz transformations tell you how spacetime coordinates (x, y, z, t) in one inertial reference frame S relate to coordinates (x', y', z', t') in another frame S' moving at velocity v along the x-axis. They are the mathematical foundation of special relativity and replace the classical Galilean transformations, which fail at high speeds.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| x, y, z | Spatial coordinates in frame S (the stationary frame) | meters |
| t | Time in frame S | seconds |
| x', y', z' | Spatial coordinates in frame S' (the moving frame) | meters |
| t' | Time in frame S' — different from t! | seconds |
| v | Relative velocity between the two frames (along x-axis) | m/s |
| γ | Lorentz factor = 1/√(1 − v²/c²) | dimensionless |
| c | Speed of light = 2.998 × 10⁸ m/s | m/s |
The inverse transformation (from S' back to S) is identical but with v replaced by −v:
x = γ(x' + vt'), t = γ(t' + vx'/c²)
Example 1 — Coordinate Transformation
Frame S' moves at v = 0.6c relative to S. An event occurs at x = 10 m, t = 0 in frame S. Find x' and t' in S'.
γ = 1/√(1 − 0.36) = 1/√0.64 = 1/0.8 = 1.25
x' = 1.25 × (10 − 0.6c × 0) = 1.25 × 10 = 12.5 m
t' = 1.25 × (0 − 0.6c × 10 / c²) = 1.25 × (−6/c)
t' = −25 ns — the event occurred 25 nanoseconds "before" the origins coincided in S'
Example 2 — Relativity of Simultaneity
Two events are simultaneous in S (t₁ = t₂ = 0) but occur at different positions x₁ = 0 and x₂ = 1000 m. Frame S' moves at v = 0.5c. Are they simultaneous in S'?
γ = 1/√(1 − 0.25) = 1/√0.75 = 1.155
t₁' = 1.155 × (0 − 0.5c × 0/c²) = 0
t₂' = 1.155 × (0 − 0.5c × 1000/c²) = 1.155 × (−1.667 × 10⁻⁶) = −1.93 × 10⁻⁶ s
The events are NOT simultaneous in S' — they differ by 1.93 microseconds. Simultaneity is relative!
When to Use Them
Use the Lorentz transformations when:
- Converting spacetime coordinates between two reference frames moving at relativistic speeds
- Deriving length contraction and time dilation (both follow directly from these equations)
- Proving that the speed of light is the same in all inertial frames
- Analyzing the relativity of simultaneity — events simultaneous in one frame may not be in another
- Working in particle physics where particles travel at near-light speeds
At low speeds (v ≪ c), γ → 1 and the Lorentz transformations reduce to the classical Galilean transformations: x' = x − vt, t' = t. This is why Newtonian mechanics works perfectly well at everyday speeds.