In relativity theory, proper acceleration 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.
Discussion In an accelerating rocket after launch, or even in a rocket standing on the launch pad, the proper acceleration is the acceleration felt by the occupants, and which is described as g-force (which is not a force but rather an acceleration; see that article for more discussion) delivered by the vehicle only. The "acceleration of gravity" (involved in the "force of gravity") never contributes to proper acceleration in any circumstances, and thus the proper acceleration felt by observers standing on the ground is due to the mechanical force from the ground, not due to the "force" or "acceleration" of gravity. If the ground is removed and the observer allowed to free-fall, the observer will experience coordinate acceleration, but no proper acceleration, and thus no g-force. Generally, objects in a state of inertial motion, also called free-fall or a ballistic path (including objects in orbit) experience no proper acceleration (neglecting small tidal accelerations for inertial paths in gravitational fields). This state is also known as "zero gravity" ("zero-g") or "free-fall," and it produces a sensation of weightlessness. 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. Only in such situations is coordinate acceleration entirely felt as a g-force (i.e. a proper acceleration, also defined as one that produces measurable weight). In situations in which gravitation is absent but the chosen coordinate system is not inertial, but is accelerated with the observer (such as the accelerated reference frame of an accelerating rocket, or a frame fixed upon objects in a centrifuge), then g-forces and corresponding proper accelerations felt by observers in these coordinate systems are caused by the mechanical forces which resist their weight in such systems. This weight, in turn, is produced by fictitious forces or "inertial forces" which appear in all such accelerated coordinate systems, in a manner somewhat like the weight produced by the "force of gravity" in systems where objects are fixed in space with regard to the gravitating body (as on the surface of the Earth). The total (mechanical) force that is calculated to induce the proper acceleration on a mass at rest in a coordinate system that has a proper acceleration, via Newton's law F = ma, is called the proper force. As seen above, the proper force is equal to the opposing reaction force that is measured as an object's "operational weight" (i.e. its weight as measured by a device like a spring scale, in vacuum, in the object's coordinate system). Thus, the proper force on an object is always equal and opposite to its measured weight.
Examples When holding onto a carousel that turns at constant angular velocity an observer experiences a radially inward (centripetal) proper-acceleration due to the interaction between the handhold and the observer's hand. This cancels the radially outward geometric acceleration associated with their spinning coordinate frame. This outward acceleration (from the spinning frame's perspective) will become the coordinate acceleration when they let go, causing them to fly off along a zero proper-acceleration (geodesic) path. Unaccelerated observers, of course, in their frame simply see their equal proper and coordinate accelerations vanish when they let go.
Similarly, standing on a non-rotating planet (and on earth for practical purposes) observers experience an upward proper-acceleration due to the normal force exerted by the earth on the bottom of their shoes. This cancels the downward geometric acceleration due to the choice of coordinate system (a so-called shell-frame). That downward acceleration becomes coordinate if they inadvertently step off a cliff into a zero proper-acceleration (geodesic or rain-frame) trajectory.
Geometric accelerations (due to the connection term in the coordinate system's covariant derivative below) act on every gram of our being, while proper-accelerations are usually caused by an external force. Introductory physics courses often treat gravity's downward (geometric) acceleration as due to a mass-proportional force. This, along with diligent avoidance of unaccelerated frames, allows them to treat proper and coordinate acceleration as the same thing. Even then if an object maintains a constant proper-acceleration from rest over an extended period in flat spacetime, observers in the rest frame will see the object's coordinate acceleration decrease as its coordinate velocity approaches lightspeed. The rate at which the object's proper-velocity goes up, nevertheless, remains constant.
Thus the distinction between proper-acceleration and coordinate acceleration allows one to track the experience of accelerated travelers from various non-Newtonian perspectives. These perspectives include those of accelerated coordinate systems (like a carousel), of high speeds (where proper and coordinate times differ), and of curved spacetime (like that associated with gravity on Earth).
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