If space and time were separate and universal, simple motion formulas would be enough. But because the speed of light is fixed for all observers, we need new equations to relate measurements in different inertial frames.

The key quantity is $$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$ where $v$ is the relative speed between frames.

For motion in one line, the coordinates transform as $$x' = \gamma(x - vt)$$ and $$t' = \gamma\left(t - \frac{vx}{c^2}\right)$$. These equations replace the old Newtonian way of translating coordinates.

The Lorentz transformation is the mathematical backbone of special relativity. It makes the postulates precise and lets us calculate real effects.