Flat-Space Motion in 3D

Motion may be described in terms of displacement vector x, time t, and simultaneity referenced to a map frame, with traveling clock time (τ) inferred from the metric
Useful rates for a traveling object include its: coordinate velocity (v=dx/dt), proper velocity (w=dx/dτ), speed of map time (γ=dt/dτ), "gamma perp" (defined below), parallel rapidity (η||=sinh[w||/c]), coordinate acceleration (a=dv/dt), and proper acceleration α. These rates are related by the conversions:
and by the constant proper acceleration integrals:
The equations on the right apply when v<<c: γ~1, t~τ, v~w, and a~α. This also simplifies displacement-time equations from the integrals above.
Note: "||" and "perpendicular" denote directions wrt/acceleration, which for sideways relativistic objects is not the direction of proper-velocity (hence momentum) change.
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