the four-acceleration
Just as four-velocity fixes the frame-dependence of ordinary velocity by differentiating with respect to proper time, the four-acceleration does the same for acceleration. It measures how a particle's four-velocity — its direction through spacetime — bends along the worldline, and it is what an accelerometer carried by the particle actually responds to.
The four-acceleration is A^mu = dU^mu/dtau = d^2 x^mu/dtau^2, the proper-time derivative of the four-velocity. Because U·U = c^2 is constant, differentiating gives 2 U·A = 0, so the four-acceleration is always Minkowski-orthogonal to the four-velocity: U_mu A^mu = 0. For an object whose rest-frame (proper) acceleration has magnitude alpha, the invariant square is A·A = -alpha^2 (signature +,-,-,-) — space-like, since a genuine four-acceleration can never be time-like. In terms of lab quantities it is messier than U, involving gamma and its time derivative, but its Minkowski magnitude is simply the proper acceleration.
The four-acceleration distinguishes true acceleration (nonzero A^mu, felt as a force, radiated by a charge) from mere coordinate motion. For 'hyperbolic motion' — constant proper acceleration alpha — a particle traces a hyperbola in spacetime, x^2 - (ct)^2 = (c^2/alpha)^2, the relativistic analog of uniformly accelerated motion; this is the worldline of Rindler observers underlying much of accelerated-frame physics. Caveat: constant proper acceleration does not mean constant lab acceleration — as v approaches c the coordinate acceleration falls to zero even though the accelerometer reading stays fixed, because you can never reach light speed.
A rocket burning to give its crew a steady 1 g (alpha = 9.8 m/s^2) as felt onboard has |A^mu| = alpha throughout, yet an outside observer sees its coordinate acceleration dv/dt shrink toward zero as v approaches c.
Proper acceleration stays constant while lab-frame acceleration must vanish near light speed.
Zero four-acceleration (A^mu = 0) is the covariant definition of inertial, force-free motion — a straight worldline. In general relativity the same condition A^mu = 0 (with a proper-time covariant derivative) defines a geodesic, i.e. free fall.