We are here to bring good news. A dynamical quantity called proper acceleration has wonderful and elegant properties, if we instead begin with the "one-map two-clock" metric equation (Minkowski's spacetime version of Pythagoras' theorem) that underlies Einstein's extension of relativity to the treatment of gravity with curved spacetime. In other words, as shown below you can access spacetime either by the long trip over "Multiple-Inertial-Frame (MIF) Mountain", or by the much shorter and familiar path through "Minkowski Meadow". Moreover, Minkowski's path leads to that side of "The Lake of SpaceTime" from which general relativity and string theory can be most easily accessed.
In relativistic research circles, it is hence common knowledge that constant acceleration problems can be neatly posed and solved. We recently noticed that even Galileo's familiar constant acceleration equations can be put to work to describe proper acceleration at any speed, at least for unidirectional motion, if one recognizes that inertial frame and traveler clocks follow a cadence different from each other (and from Galilean clocks) at high speed...
For conceptual clarity, we use T and V≡dx/dT to denote Galilean "chase-plane" time and the non-coordinate "Galilean-kinematic" velocity of our accelerated traveler (map-distance traveled per unit chase-plane time), t and v ≡ dx/dt for map-time and coordinate-velocity (map distance traveled per unit map-time), and τ (or tau) and w ≡ dx/dτ for the traveler's "proper-time" and "proper-velocity" (map-distance traveled per unit traveler time), respectively. Notice that all of these velocities are defined in terms of distances (dx) measured with respect to our map-frame. Even though many of us seem to think routinely of distances in this way, the simplification introduced here has likely been missed in the teaching of special relativity because of the laudable enthusiasm of its inventors (Albert Einstein, in particular) for a frame-independent perspective. To further maintain clarity, we often quote map frame times in [years], traveler times in [tyears], coordinate-velocities in units of [c] (lightspeed in lightyears/year), and proper-velocitiesNote01 in [ly/ty] or rodden-berries [rb]. For every 1D constant acceleration problem, these variables (11 in all) coexist and have values with predictive power therein.
Using the given relationships and the definitions of v and w above, a bit of algebra will give you the "velocity interconversions":
This equation provides relationships between the various velocities as well as to gamma. For example, some rearrangement of the terms with w and V in them will yield the useful result that w = V Sqrt[1+¼(V/c)2]. We can also infer that v = w/γ = w / Sqrt[1+(w/c)2]. In this way, given a velocity in any of the three kinematics, it is a simple matter (as shown in the figure) to convert it to the corresponding velocity in another.
Velocity-time relationships are only a bit harder to get. For Galilean-kinematic variables T and V, note that
Hey -- this looks familiar! Hence ΔT = (Vf - Vo)/α, as we might have expected. This same trick works with the other kinematics.
For the map-kinematic,
Note that proper acceleration is NOT a simple time derivative of velocity in the inertial kinematic! If we put dt on the left hand side of this equation, and write γ as a function of v, we can integrate this equation to get
For the traveler kinematic,
α = c2dγ/dx = c2dγ/dw×dw/dx = c2(w/γc2)dw/dx = (1/γ)w dw/dx = (1/γ)dx/dτ×dw/dx = (1/γ)dw/dτ.
Once again, in the traveler kinematic acceleration is not a simple time derivative of velocity. If we put dτ on the left side of this equation, and write γ as a function of w, we can integrate this to get
where η=ArcSinh(w/c) is called the "hyperbolic velocity angle" or rapidity. Thus in terms of traveler time, acceleration IS a simple time derivative of this velocity angle η (times the speed of light). For more on this see SpaceTime Physics by Taylor and Wheeler (W.H. Freeman, San Francisco, 1963 & 1992).
All of these velocity-distance/time relations can be summarized by one big series of equations for the proper acceleration:
These expressions are effectively integrated, so that the lower case d's have been replaced by delta's. A large number of differential relations (i.e. not integrated) could also be added to this list. If we note first that dw/dv = γ3 and dt/dτ = γ, these include for example (these notation conversions here still need to be double-checked/pf...)
where ηo = ArcSinh[wo/c] and γo = 1/Sqrt[1-(vo/c)2] from above. Distance traveled is then obtained in terms of time elapsed (and initial velocity) in each of the kinematics using Δx = (γf - γo)c2/α. As a result, the distance is parabolic in chase-plane time (as well as in Galilean-kinematic velocity), hyperbolic in map-time (as well as in proper-velocity), and exponential in proper-time, as shown in the graph and in our equation group photo. The graph also shows that inertial velocity is asymptotic to lightspeed, and that gamma is linear in distance traveled from rest.