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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