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...

Thus anyspeed acceleration solver's may be built on the fact that constant acceleration problems involving motion in one spatial direction can be described

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/d*t* 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-velocities^{Note01}
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 = (V_{f} - V_{o})/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 d*t* 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 = c^{2}dγ/dx =
c^{2}dγ/dw×dw/dx =
c^{2}(w/γc^{2})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 = γ^{3} and
dt/dτ = γ, these include for example (these notation
conversions here still need to
be double-checked/pf...)

where η_{o} = ArcSinh[w_{o}/c] and
γ_{o} = 1/Sqrt[1-(v_{o}/c)^{2}]
from above.
Distance traveled is then obtained in terms of time elapsed (and
initial velocity) in each of the kinematics using
Δx = (γ_{f} - γ_{o})c^{2}/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.

Send ideas/ comments/ questions/ complaints, and