For motion along the $x$-axis, the Lorentz transformation is $t' = \gamma\left(t - \frac{vx}{c^2}\right)$ and $x' = \gamma(x - vt)$, with $\gamma = 1/\sqrt{1-v^2/c^2}$. These relations preserve the spacetime interval $$s^2 = c^2 t^2 - x^2 - y^2 - z^2.$$

Events are points in a four-dimensional spacetime. Light rays satisfy $ds^2 = 0$, defining the null cone. Timelike, spacelike, and null separations classify causal structure and determine what can influence what.

Because $t'$ depends on both $t$ and $x$, two events simultaneous in one inertial frame need not be simultaneous in another. This is not a paradox but a geometric consequence of spacetime structure.