Relativity of simultaneity

The relativity of simultaneity is the result that two events that occur at different spatial locations and are judged simultaneous in one inertial frame are not simultaneous in any other inertial frame moving relative to the first. It is a direct consequence of the constancy of c: if two events at positions x_A and x_B emit light flashes that reach the midpoint x_M = (x_A + x_B)/2 at the same instant in frame S, then in frame S' moving at velocity v relative to S, the midpoint is itself moving, and the light from A and B reaches it at different instants. The two events therefore cannot be simultaneous in S' — there is no frame-independent answer to "did A happen at the same time as B?" once the events are spatially separated.

Quantitatively, if events A and B are simultaneous in S with spatial separation Δx along the boost direction, then in S' moving at velocity v their time difference is Δt' = −γ v Δx / c². The sign of Δt' depends on the sign of v: an observer moving in the +x direction sees the rear event (smaller x) precede the front event; an observer moving in the −x direction sees the reverse. Hermann Minkowski's 1908 reformulation made the geometric content explicit: simultaneity slices in spacetime are space-like hyperplanes orthogonal to a particular timelike direction (the time axis of a particular frame), and rotating that timelike direction (i.e. boosting to another frame) tilts the simultaneity slice. There is no privileged absolute "now" — there are only frame-specific simultaneity slices, each as valid as the others. This is the deepest break special relativity makes with Newtonian intuition, and the geometric content of the Lorentz transformation's time-mixing.