Four-vector

A four-vector is a quantity X^μ with four components (X⁰, X¹, X², X³) — one "time-like" and three "space-like" — that transforms under Lorentz boosts the same way the spacetime coordinates (ct, x, y, z) do. The transformation rule is X'^μ = Λ^μ_ν X^ν, where Λ^μ_ν is the Lorentz boost matrix. The four-vector is the basic building block of special-relativistic physics: every physical quantity that pairs a "scalar" with a "three-vector" — energy with momentum, charge density with current density, scalar potential with vector potential — turns out to be a four-vector when properly examined.

The Lorentz-invariant inner product of two four-vectors uses the Minkowski metric η_{μν} = diag(+1, −1, −1, −1) (in the mostly-minus convention used by Griffiths and Jackson) or its sign-flipped sibling. For a single four-vector X^μ, the Minkowski "length-squared" X^μ X_μ = (X⁰)² − |X|² is a Lorentz scalar — the same in every frame. For X^μ = (E/c, p) this is m²c², the rest-mass invariant; for X^μ = (cρ, J) it is c²ρ² − |J|², an invariant of the four-current. The transformation properties also dictate which combinations of more elementary quantities can themselves be four-vectors: ∂_μ = (∂/c∂t, ∇) is a covariant four-vector, and the four-velocity U^μ = γ(c, v) of a particle is a contravariant four-vector. These structural rules make the construction of Lorentz-covariant physical theories almost mechanical — choose your dynamical variables to transform as four-vectors (or higher tensors), build invariants by index contraction, and any equation that equates two tensors of the same rank holds in every frame.