Proper acceleration

In relativity theory, proper acceleration^[1] is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured. Gravitation therefore does not cause proper acceleration, because the same gravity acts equally on the inertial observer. As a consequence, all inertial observers always have a proper acceleration of zero.

Proper acceleration contrasts with coordinate acceleration, which is dependent on choice of coordinate systems and thus upon choice of observers (see three-acceleration in special relativity).

In the standard inertial coordinates of special relativity, for unidirectional motion, proper acceleration is the rate of change of proper velocity with respect to coordinate time.

In an inertial frame in which the object is momentarily at rest, the proper acceleration 3-vector, combined with a zero time-component, yields the object's four-acceleration, which makes proper-acceleration's magnitude Lorentz-invariant. Thus the concept is useful: (i) with accelerated coordinate systems, (ii) at relativistic speeds, and (iii) in curved spacetime.