the four-velocity

Ordinary velocity, dx/dt, has a hidden flaw for relativity: it divides by t, a frame-dependent time. To get a genuine four-vector you must instead differentiate an object's position with respect to something everyone agrees on — the particle's own proper time. The four-velocity is the tangent to a worldline measured this way: which direction, and how fast, an object moves through spacetime itself.

The four-velocity is U^mu = dx^mu/dtau, where tau is proper time along the worldline. Writing gamma = 1/sqrt(1 - v^2/c^2) and using dt/dtau = gamma, its components are U^mu = gamma(c, v_x, v_y, v_z) = (gamma c, gamma v). Because it is dx^mu (a four-vector) divided by dtau (an invariant scalar), U^mu transforms correctly as a four-vector. Its magnitude is fixed: U·U = eta_munu U^mu U^nu = gamma^2(c^2 - v^2) = c^2 always (signature +,-,-,-). Every object's four-velocity has the same 'length' c — objects differ only in its direction in spacetime.

The constancy U·U = c^2 is deeply useful: differentiating it shows the four-acceleration is always Minkowski-orthogonal to the four-velocity, U·A = 0. Multiplying U^mu by rest mass gives the four-momentum p^mu = m U^mu. Caveat: the low-speed limit of U^mu is (c, v), not (0, v) — the time component does not vanish, which is the seed of the rest energy E = m c^2. Light has no four-velocity: a photon has no rest frame and dtau = 0 along a null worldline, so one parametrizes null paths by an affine parameter instead.