Proper acceleration is the physical acceleration experienced by an object relative to a locally co-moving free-float frame. This may be quite different than the coordinate-acceleration seen in a separate reference frame (inertial or not) of co-moving yardsticks and synchronized clocks.
The proper-acceleration 3-vector, combined with a null time-component, yields the object's four-acceleration (as measured by the object itself) which makes proper-acceleration's magnitude Lorentz-invariant. Thus proper-acceleration comes in handy: (i) with accelerated coordinate systems, (ii) at relativistic speeds, and (iii) in curved spacetime.
Proper-acceleration reduces to coordinate-acceleration in an inertial coordinate system in flat spacetime (i.e. in the absence of gravity), provided the magnitude of the object's proper-velocity (momentum per unit mass) is much less than the speed of light c. Here we focus on situations where proper-acceleration and coordinate-acceleration are not always the same.
