properacceleration

Proper acceleration is the acceleration that an observer would measure with an accelerometer carried along by the object. It is invariant and independent of the chosen coordinate system. In special relativity it is defined as the magnitude of the four-acceleration a^μ, which is the derivative of the four-velocity u^μ with respect to the object's proper time τ: a^μ = du^μ/dτ. The proper acceleration α is the invariant quantity α = sqrt(-a^μ a_μ) (the sign depends on the metric convention). Physically, α represents the physical “felt” acceleration or g‑force an object experiences.

In Minkowski spacetime, a body undergoing constant proper acceleration traces a hyperbolic worldline. If α is constant, the coordinate acceleration as seen in a given frame diminishes at high speed even though the onboard accelerometer reads a constant value. Thus proper acceleration is not the same as coordinate acceleration.

Relation to gravity and motion is central in the equivalence principle. An object at rest in a gravitational field experiences nonzero proper acceleration, since it must be supported against gravity. An object in free fall follows a geodesic and has zero proper acceleration. This distinction underlies the general relativistic view that gravity is not a force in the locally inertial frame, but a manifestation of curved spacetime; proper acceleration is what remains as a measurable, non-geodesic acceleration in any local frame.

In summary, proper acceleration is the invariant, frame-independent acceleration sensed by an object’s own accelerometer, equal to the magnitude of its four-acceleration, and fundamental to distinguishing inertial motion, thrust, and gravity.