Flat-Space Motion in 1D

Motion may be described using displacement x, time t, and simultaneity referenced to a map frame, with time on traveling clocks (τ) inferred from the metric relation
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), rapidity (η=sinh[w/c]), coordinate acceleration (a=dv/dt), and proper acceleration α. For unidirectional motion, these rates are related by the conversions:
and by the constant proper acceleration integrals:
Note the simplifications when v<<c: γ~1, t~τ, v~w~cη, a~α. The displacement-time equation (from γ and η integrals above) also simplifies:
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