Relativistic velocity addition

Relativistic velocity addition is the rule for combining velocities consistent with the Lorentz transformation: if a particle has velocity u in frame S and S' moves at velocity v relative to S along the same direction, the particle's velocity in S' is u' = (u − v)/(1 − uv/c²) for collinear motion. The denominator is the new content versus the Galilean rule u' = u − v: it ensures that no combination of subluminal velocities (|u|, |v| < c) ever yields a superluminal result. Set u = c (a light pulse): u' = (c − v)/(1 − v/c) = c. Light's speed is c in every frame, exactly as the second postulate demands. Set u, v ≪ c: the denominator ≈ 1 and the formula reduces to the Galilean rule, recovering Newtonian intuition in the low-speed limit.

The 1851 Fizeau experiment provided the first quantitative test before special relativity existed. Hippolyte Fizeau measured the speed of light in flowing water and found a dragging coefficient (1 − 1/n²) — exactly what the relativistic velocity-addition formula predicts when applied to light at speed c/n in water moving at flow velocity v, expanded to first order in v/c. Fizeau's 1851 result was a direct verification of relativistic velocity addition fifty-four years before Einstein wrote it down — the kind of empirical hint that, in retrospect, made special relativity feel less like a leap and more like an organisation of pre-existing experimental anomalies. The formula generalises to non-collinear motion via the perpendicular-velocity correction u'_⊥ = u_⊥/(γ(1 − uv/c²)), which encodes the kinematic component of relativistic aberration.