6.2. proper-acceleration (α) from the Lorentz factor

In flat spacetime, the proper-velocity and the Lorentz-factor make up components of the 4-vector velocity whose proper-time derivative is the 4-vector proper-acceleration. Moreover the magnitude α of the proper-acceleration 3-vector in the traveler frame equals the frame-invariant magnitude of that 4-vector proper-acceleration, since to the traveler the proper-acceleration 4-vector has a zero time-component. Hence finding a relation between magnitude α and parameters like velocity and position, with help from the Lorentz-factor calculated from the metric, is relatively easy in the Minkowski (flat-space) metric. 

Although the direction of α in the traveler frame may be an interesting question, except for its sign in the unidirectional case we concentrate here on its magnitude. In flat spacetime the relation to use is simply:

.

The first of the terms in the middle expression is simply the bookkeeper-acceleration (dw/dτ ≡ d2x/dτ2) squared, while the second is simply the Lorentz-factor (γ ≡ dt/dτ) rate-of-change squared. 

The problem is more complicated in curved spacetime and in accelerated frames because of the need for connection (i.e. geometric acceleration) terms in the covariant derivative that describes the 4-vector proper-acceleration. Here we consider only two special cases: Low speed radial motion relative to the shell-frame of a gravitating sphere, and travelers whose radar-distance from an accelerating map-frame is fixed.