FlatSpace 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 displacementtime
equation (from γ and η integrals above) also simplifies:

Some
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active.
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