Relativistic velocity addition

Classical mechanics adds velocities directly: if a ball is thrown forward at speed u from a train moving at speed v, an observer on the ground sees the ball moving at u + v. This works well at everyday speeds, but it cannot be exactly right in special relativity, because it would allow the observed speed of light to depend on the speed of its source — directly violating the postulate that light travels at speed c for every inertial observer, regardless of the source's motion.

Applying the Lorentz transformations to the definition of velocity (distance transformed divided by time transformed) yields the correct relativistic velocity-addition formula: if an object moves at speed u in a frame that is itself moving at speed v relative to a second frame (all velocities along the same line), the object's speed u' in the second frame is u' = (u + v)/(1 + uv/c²), not simply u + v. At low speeds, where u and v are both much smaller than c, the term uv/c² in the denominator becomes negligible, and the formula reduces to the familiar classical u' ≈ u + v — consistent with why classical velocity addition works perfectly well for cars, planes, and thrown balls.

At speeds approaching c, the denominator's correction becomes essential. The formula's structure guarantees that combining any two speeds each less than c always produces a result still less than c: the denominator grows in exactly the way needed to suppress the naive sum. Take the extreme case of adding the speed of light to any other speed — set u = c in the formula and it returns u' = c regardless of v, exactly consistent with light travelling at c for every observer. This built-in ceiling is not an approximation or a rule imposed separately; it emerges directly from the same Lorentz transformations that produce time dilation, length contraction, and the relativity of simultaneity, and it is the reason nothing observed to be moving slower than light can ever be combined with another sub-light speed to exceed it.