Proper acceleration

Gravitation therefore does not cause proper acceleration, because the same gravity acts equally on the inertial observer.

Thus the concept is useful: (i) with accelerated coordinate systems, (ii) at relativistic speeds, and (iii) in curved spacetime.

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).

Unaccelerated observers, of course, in their frame simply see their equal proper and coordinate accelerations vanish when they let go.

This cancels the downward geometric acceleration due to the choice of coordinate system (a so-called shell-frame[4]).

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.

It then accelerates downward (first slowing and then speeding up) over twice that period, followed by a 2*c/α upward deceleration to return to the original height.

The following alternate analyses of motion before the stone is released consider only forces acting in the radial direction.

Those components of coordinate acceleration not caused by physical forces (like direct contact or electrostatic attraction) are often attributed (as in the Newtonian example above) to forces that: (i) act on every gram of the object, (ii) cause mass-independent accelerations, and (iii) don't exist from all points of view.

Proper-acceleration's relationships to coordinate acceleration in a specified slice of flat spacetime follow[6] from Minkowski's flat-space metric equation (c dτ)2 = (c dt)2 − (dx)2.

Here a single reference frame of yardsticks and synchronized clocks define map position x and map time t respectively, the traveling object's clocks define proper time τ, and the "d" preceding a coordinate means infinitesimal change.

These relationships allow one to tackle various problems of "anyspeed engineering", albeit only from the vantage point of an observer whose extended map frame defines simultaneity.

Hence the change in proper-velocity w=dx/dτ is the integral of proper acceleration over map-time t i.e. Δw = αΔt for constant α.

For constant unidirectional proper-acceleration, similar relationships exist between rapidity η and elapsed proper time Δτ, as well as between Lorentz factor γ and distance traveled Δx.

For example, imagine a spaceship that can accelerate its passengers at "1 gee" (10 m/s2 or about 1.0 light year per year squared) halfway to their destination, and then decelerate them at "1 gee" for the remaining half so as to provide earth-like artificial gravity from point A to point B over the shortest possible time.

[8][9] For a map-distance of ΔxAB, the first equation above predicts a midpoint Lorentz factor (up from its unit rest value) of γmid = 1 + α(ΔxAB/2)/c2.

Unfortunately, sustaining 1-gee acceleration for years is easier said than done, as illustrated by the maximum payload to launch mass ratios shown in the figure at right.

After each round trip ship-pilots on this shuttle-run will have aged only half as much as colleagues stationed on earth.

Here U is the object's four-velocity, and Γ represents the coordinate system's 64 connection coefficients or Christoffel symbols.

The left hand side of this set of four equations (one each for the time-like and three spacelike values of index λ) is the object's proper-acceleration 3-vector combined with a null time component as seen from the vantage point of a reference or book-keeper coordinate system in which the object is at rest.

The first term on the right hand side lists the rate at which the time-like (energy/mc) and space-like (momentum/m) components of the object's four-velocity U change, per unit time τ on traveler clocks.

Let's solve for that first term on the right since at low speeds its spacelike components represent the coordinate acceleration.

This in turn can be broken down into parts due to proper and geometric components of acceleration and force.

If we further multiply the time-like component by lightspeed c, and define coordinate velocity as v = dx/dt, we get an expression for rate of energy change as well: Here ao is an acceleration due to proper forces and ag is, by default, a geometric acceleration that we see applied to the object because of our coordinate system choice.

Thus the equivalence principle extends the local usefulness of Newton's laws to accelerated coordinate systems and beyond.

On the other hand, for r ≫ rs, an upward proper force of only GMm/r2 is needed to prevent one from accelerating downward.

The spacetime equations of this section allow one to address all deviations between proper and coordinate acceleration in a single calculation.