Lorentz transformations

Classical mechanics converts an event's position and time between two inertial frames moving at relative speed v using the Galilean transformations: x' = x − vt and t' = t, treating time as an absolute quantity that every observer agrees on. This works extremely well at everyday speeds, but it is inconsistent with a constant speed of light for all observers, since under a Galilean transformation velocities simply add, which would let a beam of light appear to travel faster or slower than c depending on the observer's own speed — directly contradicting the second postulate of special relativity.

The Lorentz transformations are the replacement that keeps the speed of light invariant. Written for a frame S' moving at speed v along the x-axis relative to frame S, they read x' = γ(x − vt) and t' = γ(t − vx/c²), where γ = 1/√(1 − v²/c²) is the Lorentz factor that also governs time dilation and length contraction. Crucially, time is no longer transformed independently of position — the t' equation depends on x as well as t, which is exactly the mathematical origin of the relativity of simultaneity: two events with the same t but different x in frame S generally end up with different t' in frame S'. Time dilation and length contraction are not separate, independently derived rules bolted onto this picture; both follow directly as consequences of applying the Lorentz transformation to a clock's tick or a rod's endpoints.

The Lorentz transformations do not discard classical mechanics; they contain it. Whenever v is much smaller than c, γ approaches 1, the vx/c² term in the time equation becomes negligible, and the Lorentz transformations reduce smoothly to the ordinary Galilean transformations — which is exactly why Newtonian mechanics works so well for cars, planes, and everyday objects despite being, strictly speaking, only an approximation valid at low relative speeds.