What is Four-vectors?

A four-vector is a quadruple $a^\mu = (a^0, \vect a)$, $\mu = 0, 1, 2, 3$, whose components mix under a boost exactly as $(ct, \vect r)$ do:

for a boost at $\beta c$ along $x$. Any two four-vectors give the invariant scalar product $a\cdot b = a^0b^0 - \vect a\cdot\vect b$, the same number in every frame — the pattern behind $c^2t^2 - x^2$ and $E^2 - p^2c^2$. Position $(ct, \vect r)$ and four-momentum $(E/c, \vect p)$ are the two we have; electromagnetism now supplies three more: current, potential, and wave-vector.