Four-velocity
The Lorentz four-vector u^μ = dx^μ/dτ = γ(c, v_x, v_y, v_z), the tangent to a particle's timelike world-line parametrized by proper time. Its norm u^μ u_μ = c² is constant on every timelike world-line; differentiating it gives four-acceleration, and m·u^μ is the four-momentum.
Definition
Four-velocity is the tangent vector to a particle's timelike world-line, parametrized by the proper time τ rather than by any frame's coordinate time: u^μ = dx^μ/dτ. Expanding the chain rule dτ = dt/γ gives the explicit form u^μ = γ(c, v_x, v_y, v_z), where v is the particle's three-velocity in the chosen inertial frame and γ = 1/√(1 − β²) is the Lorentz factor. Because τ is a Lorentz scalar, differentiating x^μ with respect to it yields a manifestly covariant four-vector — the first major payoff of the proper-time parametrization.
The Minkowski norm of four-velocity is fixed: u^μ u_μ = γ²(c² − |v|²) = c². This is a constant of motion along every timelike world-line, in every frame, regardless of how the particle accelerates. The invariance has a geometric meaning — the four-velocity is a unit timelike vector when c = 1, and the world-line traces a curve in spacetime at "unit speed" in proper-time parametrization. Differentiating u^μ once more by τ produces the four-acceleration a^μ = du^μ/dτ, which because u·u is constant must satisfy a·u = 0 — four-acceleration is always orthogonal to four-velocity in the Minkowski sense. Multiplying by rest mass gives the four-momentum p^μ = m·u^μ, recovering the dynamical quantity from the kinematic one. Four-velocity is the bridge from the geometry of world-lines to the dynamics of forces and momenta.