Four-acceleration

Four-acceleration is the second proper-time derivative of a particle's spacetime position, a^μ = d²x^μ/dτ² = du^μ/dτ. It is the Lorentz-covariant generalisation of ordinary three-acceleration, expressed in a way that transforms cleanly between inertial frames. In the instantaneous rest frame of the particle (where γ = 1 and v = 0), the spatial components of a^μ reduce exactly to the ordinary three-acceleration dv/dt; the time component vanishes in that frame. In any other frame the components mix, and the four-acceleration carries both the change in spatial velocity and a contribution from the changing γ.

Because the four-velocity has a fixed Minkowski norm u^μ u_μ = c², differentiating both sides with respect to τ yields 2 a^μ u_μ = 0 — the four-acceleration is always orthogonal to the four-velocity in the Minkowski metric. Geometrically, this is the same statement as a unit-tangent vector on a curve being orthogonal to its second derivative on a sphere: the world-line is constrained to move at "unit Minkowski speed," so the only freedom is to change direction, never magnitude. The four-acceleration vanishes identically on inertial (geodesic, straight-line) world-lines and is non-zero everywhere on accelerated ones — it is the precise relativistic measure of "how non-inertial" a trajectory is. The twin paradox's asymmetry resides exactly here: the travelling twin's world-line carries non-vanishing four-acceleration during turnaround; the stay-at-home twin's does not.