Traveler representation of constant proper-acceleration: For constant proper-acceleration α in the x-direction, relative to a free-float Minkowski-metric map-frame with respect to which our traveler is moving at proper-velocity w in an arbitrary direction, the traveler representation^[1] of key physical quantities like momentum, force, energy and power might take the following form.

Imagine that you "feel" the strength and direction of the proper-force/acceleration to which you are being subject e.g. as a result of the thrusters on your spaceship, and that you can also sense the rate (or proper-velocity) at which you are passing "map-frame landmarks" in your vicinity. In short, you are given your proper-acceleration α, your mass m, and your proper-velocity w = dr/dτ where τ measures the proper-time elapsed on your clocks.

Lorentz factor γ, constants v_⊥ and γ_⊥, as well as your parallel proper-velocity w_|| and associated rapidity η = asinh[w_||/c] then follow from this information, as shown above. If proper-acceleration is constant, of course, then time-dependences throughout also follow from the simple fact that rapidity η[τ] = ατ/c.

In that presumed-constant context we are defining both distance and time values to have an origin at the point of (acceleration-parallel) x-velocity match between our traveler and the map-frame. Only the bottom row of inferred quantities (absolute map distances {x,y} and map time {t} elapsed from the origin point) actually depend on this assumption.

In this traveler-perspective context we also only look at power and force values in terms of proper-time τ. However, coordinate versions of both power and force (i.e. in terms of map-frame time t) are obtained by dividing through by Lorentz-factor γ[τ] ≡ dt/dτ.