Lorentz transformation

The Lorentz transformation is the linear coordinate transformation between two inertial frames S and S' in special relativity, where S' moves at velocity v relative to S along the +x direction. The transformation reads t' = γ(t − vx/c²), x' = γ(x − vt), y' = y, z' = z, where γ = 1/(1 − β²)^(1/2) is the Lorentz factor and β = v/c. Time and the spatial coordinate along the boost direction mix linearly; perpendicular coordinates are unchanged. The transformation preserves the spacetime interval ds² = c²dt² − dx² − dy² − dz² (the Minkowski-metric scalar) and reduces to the Galilean transformation t' = t, x' = x − vt in the limit β → 0. From it follow time dilation (set Δx = 0 in S, get Δt' = γ Δt), length contraction (require simultaneity in S', solve for Δx in S), and the relativistic velocity-addition formula u' = (u − v)/(1 − uv/c²).

Hendrik Antoon Lorentz, Dutch, 1853–1928 — distinct from Ludvig Lorenz (no T), Danish, 1829–1891, the radiation-gauge physicist of EM §11.5. The two are routinely confused because their names are nearly identical and both worked on classical electromagnetism. Hendrik Lorentz derived the transformation in 1899–1904 as a mathematical device that left Maxwell's equations covariant, treating the local time t' as an auxiliary parameter rather than a physical coordinate. Henri Poincaré in 1905 named the transformation after Lorentz and recognised that it formed a group. Einstein's 1905 paper On the Electrodynamics of Moving Bodies derived the same transformation directly from the two postulates, without an aether, and elevated the local time to the actual time measured by clocks at rest in the moving frame. Hermann Minkowski's 1908 reformulation showed the transformation is a hyperbolic rotation in 4D pseudo-Euclidean spacetime — making the geometric content explicit and the special-relativistic kinematics manifestly invariant.