4.2.7 the anyspeed 2nd-law
Let's try to find "the best joints for carving up" the phenomenon of accelerated motion i.e. with the greatest frame-independence, as well as the least need for extended-networks of synchronized-clocks (which are difficult to come by on accelerated platforms and in curved spacetime). Relativists for a long time have expressed unhappiness with coordinate acceleration and force (for good reason), but have also pointed out that general relativity makes a case for the local-validity of Newton's laws in all frames provided that we consider geometric (frame-dependent or connection-coefficient) as well as proper forces whenever we find ourselves in a non-"free-float" trajectory.
As shown in the figure at right, 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 of key physical quantities like momentum, force, energy and power 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.
The quantities in green are synchrony-free i.e. they do not require an extended frame of synchronized clocks, while quantities in blue are in addition frame-invariant i.e. they are seen to be the same from all frames of reference.
A draft paper on this topic may now be found here.