Lorentz Transformations

When two observers move at constant velocity relative to each other, they each measure different values for the position and time of the same event. In classical (Newtonian) physics, these differences were handled by simple addition and subtraction — a ball thrown forward on a moving train has a speed equal to the train's speed plus the throwing speed. This works perfectly at everyday speeds, but fails catastrophically when velocities approach the speed of light. The Lorentz transformations are the precise mathematical rules that correctly convert coordinates and time between two inertial (non-accelerating) reference frames for any speed, including those near c. Named after Dutch physicist Hendrik Lorentz, these equations were the mathematical foundation on which Einstein built Special Relativity in 1905. The key insight is that space and time are not separate, independent quantities — they mix together when you change reference frames. A measurement that is purely a time interval in one frame may appear as a combination of both time and distance in another. This mixing is captured elegantly by the Lorentz factor γ (gamma), a number that equals 1 at low speeds and grows rapidly as velocity approaches c. The transformations predict and quantify two of special relativity's most striking consequences: time dilation (moving clocks run slow) and length contraction (moving objects shrink along the direction of motion). These are not illusions or measurement errors — they are real physical effects confirmed by countless experiments, from the survival of cosmic-ray muons reaching Earth's surface to the precise corrections needed in GPS satellite clocks every single day.

Key takeaways

Setting up two reference frames
The Lorentz factor γ
The transformation equations
Recovering time dilation and length contraction
Relativistic velocity addition

Setting up two reference frames

Imagine frame S (the 'lab') and frame S' (a rocket), where S' moves at constant velocity v in the positive x-direction relative to S. At the moment the origins coincide, both clocks read zero. An event (like a firecracker popping) has coordinates (x, t) in S and (x', t') in S'. We want equations connecting these four quantities.

The Lorentz factor γ

The core quantity in all Lorentz transformations is the Lorentz factor: γ = 1 / √(1 − v²/c²). When v is small compared to c, the term v²/c² ≈ 0, so γ ≈ 1 and everything reduces to classical physics. As v → c, the denominator approaches zero, making γ → ∞. At v = 0.866c, γ = 2; at v = 0.99c, γ ≈ 7.1. This rapid growth is why relativistic effects become dramatic only at very high speeds.

The transformation equations

The full Lorentz transformation from S to S' is: x' = γ(x − vt), y' = y, z' = z, t' = γ(t − vx/c²). The y and z coordinates are unchanged because the motion is only along x. Notice that x' mixes x and t, and t' mixes t and x — space and time are genuinely intertwined. The inverse transformation (from S' back to S) is obtained by replacing v with −v: x = γ(x' + vt'), t = γ(t' + vx'/c²).

Recovering time dilation and length contraction

Time dilation emerges naturally: if a clock sits at rest in S' (x' = 0), then from the inverse transform t = γt', meaning the lab measures more time than the rocket clock ticks — the moving clock runs slow by factor γ. Length contraction follows from measuring both ends of a rod simultaneously (same t) in the lab frame: the rod's length in S is L = L₀/γ, where L₀ is the proper length measured in the rod's own rest frame. Both effects are automatic consequences of the same transformation equations.

Relativistic velocity addition

The Lorentz transformations also give the correct rule for combining velocities. If an object moves at speed u' in S', its speed in S is u = (u' + v) / (1 + u'v/c²). When u' and v are both small, the denominator ≈ 1 and we get classical addition u ≈ u' + v. Crucially, if u' = c (a photon), then u = (c + v)/(1 + v/c) = c(c + v)/(c + v) = c — light always travels at c in every inertial frame, exactly as Einstein's second postulate demands.

The Key Equation

x' = \gamma(x - vt), \quad t' = \gamma\!\left(t - \frac{vx}{c^2}\right), \quad \gamma = \frac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}

x' — Transformed position (m): Spatial coordinate of the event as measured in the moving frame S'
t' — Transformed time (s): Time of the event as measured in the moving frame S'
x — Position in lab frame (m): Spatial coordinate of the event as measured in the rest frame S
t — Time in lab frame (s): Time of the event as measured in the rest frame S
v — Relative velocity (m/s): Constant velocity of frame S' relative to frame S along the x-axis
c — Speed of light (m/s): Universal constant equal to 2.998 × 10⁸ m/s
\gamma — Lorentz factor (dimensionless): Factor that quantifies how much time and space transform; always ≥ 1

Worked Example

Problem. A spaceship travels at v = 0.60c relative to Earth. A firecracker explodes at position x = 9.0 × 10⁸ m and time t = 5.0 s in the Earth frame. What are the position x' and time t' of this explosion in the spaceship's frame?

x' = 0 \text{ m}, \quad t' = 4.0 \text{ s}

Answer. x' = 0 m (explosion is at the ship's location), t' = 4.0 s

Figure

Diagram

Origins · 1 minHendrik Antoon Lorentz and how this was discovered

Hendrik Antoon Lorentz

1853–1928 · Dutch

Lorentz derived his transformation equations in 1904 while trying to explain why the Michelson–Morley experiment found no evidence for the ether — a medium then believed to carry light waves. He showed mathematically that if objects contract in their direction of motion and clocks slow down, the ether would remain undetectable. Einstein then reinterpreted these same equations not as mechanical effects from moving through ether, but as fundamental properties of space and time itself. Lorentz won the Nobel Prize in Physics in 1902 (shared with Pieter Zeeman) for earlier work on the influence of magnetism on radiation. Albert Einstein deeply admired Lorentz and called him 'the greatest and most noble man' he had ever known. When Lorentz died in 1928, Einstein delivered the eulogy at his funeral.

Postulates of Special Relativity — The Lorentz transformations are the direct mathematical consequence of Einstein's two postulates: the laws of physics are the same in all inertial frames, and the speed of light is constant.
Time Dilation — Time dilation (Δt = γΔt₀) is derived directly from the Lorentz time-transformation equation by setting the spatial separation in the moving frame to zero.
Length Contraction — Length contraction (L = L₀/γ) follows from the Lorentz position-transformation equation when both ends of an object are measured simultaneously in the lab frame.
Spacetime Intervals — The spacetime interval s² = c²t² − x² is invariant under Lorentz transformations — it has the same value in every inertial frame, making it a fundamental tool in relativity.
Relativistic Momentum and Energy — The famous E = mc² and the relativistic momentum p = γmv are derived by applying Lorentz transformations to the laws of conservation of energy and momentum.