Two events that are simultaneous in one inertial frame need not be simultaneous in another moving frame. This relativity of simultaneity follows from the invariant speed of light and is the key reason space and time coordinates must mix.

For frames whose axes are aligned and whose relative speed is $v$ along $x$, the Lorentz transformation is

$$x'=\gamma(x-vt),$$

$$t'=\gamma\left(t-\frac{vx}{c^2}\right),$$

where

$$\gamma=\frac{1}{\sqrt{1-v^2/c^2}}.$$

Coordinates perpendicular to the motion are unchanged.

Low-speed limit

When $v\ll c$,

$$\gamma\approx1$$

and the transformation approaches the familiar Galilean result $x'\approx x-vt$, $t'\approx t$.

Space and time mix

The term $vx/c^2$ means that assigning a time coordinate to a distant event depends on the observer's inertial frame. Time dilation and length contraction are consequences of this same transformation, not independent corrections added afterward.