Relativity of simultaneity ($\Delta t^{\prime }=-\gamma v\Delta x/c^2(\Delta t=0)$)

Spatially separated simultaneous events in one inertial frame are generally not simultaneous in a relatively moving frame. The Lorentz transformation explicitly mixes position into time. This is essential when comparing distances between moving light pulses: one must compare their positions at the same time in the measuring frame.

For two photons emitted simultaneously a distance $d$ apart in one frame and moving in the same positive direction, the world-line intercept $x-ct$ transforms to $\gamma (1+v/c)(x-ct)$. Their equal-time separation in the boosted frame is therefore the displayed constant $d^{\prime }$. It is different from length contraction of the emitters, because the emission events are not simultaneous after the boost.