"For Newton's Laws, relativity brought the **bad news**
that coordinate-velocity has an upper limit and neither elapsed-time nor force
are frame-invariant at high speed.
The **good news** was that: (i) a 'one-gee round trip' takes travelers
*further than predicted* for a given elapsed-time on their clocks,
(ii) high speed *colliders are a bargain* because gamma-factors for
relative proper-velocity
multiply
rather than add, (iii) acceleration
*properly*-defined remains frame-invariant, and (iv) Newton's laws
apply *locally* even in accelerated (non-rain) frames using
geometric (connection-coefficient) forces like
gravity, centrifugal, and
Coriolis
that act on every ounce of an object as if they might be removed by
choice of a different frame.",
Anonymous 2007

*Proper-acceleration*
and *proper-force* are nothing more than
the 3-vector acceleration and net force experienced in the
frame of an accelerated object. Like *proper-time* and
*proper-length*, they describe quantities
from the vantage point of a special frame, in this case the frame
of the object to which the acceleration and forces are
applied. Their magnitude also happens to be the Lorentz-invariant magnitude
of the acceleration and net force four-vectors *seen from all
frames*, since in the rest frame of the accelerated object the
time-like component of these four-vectors is zero. Because
forces might be caused by objects moving with respect to one
another, proper force **F**_{o} doesn't obey
an action-reaction principle
like 4-vector and locally-defined frame-variant forces do.
However, it's an excellent
bridge between locally-measured coordinate-acceleration **a**
and force **F**=d**p**/dt (two things that Isaac Newton was
fond of relating), since **F**_{o} is precisely
m**a**γ^{3}/γ_{⊥}
while **F** differs from **F**_{o} only when
velocity is not-parallel to acceleration. In that case **F**'s
component parallel to **F**_{o} is divided by
γ_{⊥} as **F** rotates toward the velocity line.

Using these observations, the figure at right illustrates
relativistic adjustments to Newton's 2nd law (d**p**/dt~m**a**)
over the domain of allowed values for coordinate-velocity d**x**/dt.
For a fixed
*rightward* proper-force **F**_{o}=m**α** and
proper-acceleration **α**,
the contours show how the magnitude of coordinate-acceleration
**a**=d^{2}**x**/dt^{2} (always
parallel to proper-force) falls off as velocities
approach lightspeed in any direction. In this way,
*coordinate-acceleration literally fades away* as a useful concept at high
speeds. The arrows, on the other hand,
show how frame-variant force d**p**/dt falls off in magnitude
as the component of coordinate-velocity perpendicular to
proper-force
increases, and how the
direction of apparent force on an accelerated object aligns with its velocity
direction, rather than with the proper-force
that it experiences, as that velocity approaches c.
Thus the white part of the bull's eye is the only region
where relativistic adjustments to Newton's 2nd Law (d**p**/dt=m**a**)
are small.

The equivalence principle of general relativity, combined with the concept of local affine-connection (or geometric) forces, makes proper-force useful even in non-inertial settings (like gravitationally-curved spacetime). The utility of Newton's Laws for describing gravity on earth as "just one more inter-object force" is living proof of that, since gravity (like magnetism) is an example of everyday relativity.

Below are some related plots. In the first plot, a trajectory for "1-gee" constant downward proper-acceleration is illustrated. This is the trajectory followed by a fly ball (neglecting air resistance) as well as by the Boeing 707 "vomit comet" used to give astronauts a taste of freefall, here implemented with help from a rocket engine on the accelerated object so that constancy can be maintained over much longer distances than possible with gravitational acceleration near the surface of a planet. Because the rocket trajectory is an accelerated trajectory in an unaccelerated frame (rather than a freefall trajectory in the presence of gravitational acceleration), the passengers on board will sense an earth-like gravity pulling down if their up is the direction of proper acceleration. As the field of view widens, you'll go from seeing only part of earth's orbit to seeing the sun and other planetary orbits, and eventually to seeing the Oort cloud and the nearby Centauri star-cluster shrink until the path looks like that of a Vogon demolition ship making a hyper-relativistic (127

Minimally-variant quantities like proper-time,
relativistic energy and momentum (proper-velocity),
and proper-acceleration are illustrated *on the right*-hand side of the plot
while some highly frame-dependent but locally-useful counterparts (e.g.
map-time, coordinate-velocity, kinetic-energy and rates of
momentum/energy change) are shown *on the left*.
The total duration of the "burn" on fixed-observer (map) clocks is
eight times the listed interval between large dots on the trajectory
profile (time ticks), and
runs from 3.65 days to 100 years of earth time. This amounts to between
3.65 days and about 9 years on the clocks of those on board the spaceship.
Imagine a drag racer capable of holding the pedal
to the metal for days or years. We probably should have done
experiments on this when gasoline was only 32¢ per gallon!

The transverse velocity (vperp) is adjusted to keep the trajectory
footprint square, the colored tick-marks have decimal separations of
10^{-6} through 10^{0} lightyears, and the back to back
arrows mark spatial separations increasing by powers of ten from
2×10^{-6} lightyears (yellow) to 2 lightyears (red).
Notice the changes in shape from parabolic to hyperbolic as burn
time increases. The transition occurs primarily over trajectories
whose range is between 0.02 and 20 lightyears, as average velocity makes the
transition from sub to hyper relativistic. Eventually
the *color* of the arrows in this animation will convey information on the
magnitude of the first (scalar) quantity labeling that arrow.
The *length* of the arrow will convey information on the magnitude of
the second (vector) quantity labeling that arrow. For the
moment, only vector directions are conveyed. This changes most
noticeably for the dp/dt vector on the left hand side, as
relativistic Wigner rotation aligns the force seen in the map frame with
the velocity line when the direction of proper acceleration remains fixed.

In the two plots on velocity space below, are those arrows perpendicular to those contours? If so, why?

For those with a more technical bent, a set
of fairly general equations for constant proper-acceleration
α=F_{o}/m in
the x-direction are listed by *Mathematica*
below. Four-vectors are listed with the time-like component first. Using
a Minkowski-signature of (+---), as expected u4·u4=c^{2},
α4·α4=-α^{2}, and u4·α4=0.
One might generalize these further to arbitrary initial
v_{x0}/c = β_{x0}, not by a Lorentz-boost
in the acceleration direction (which with nonzero v_{⊥}
Wigner-rotates the proper acceleration out of the observer's x-direction) but
by replacing proper-time τ with τ - τ_{o}, where
τ_{o} = -(c/α)ArcTanh[γ_{⊥}β_{x0}].
A bit of algebraic gymnastics with these equations might allow you,
for example, to test the assertions here and/or to plot
trajectories of your favorite projectile on the figures above.

- An application to the relativistic roots of magnetism.
- In this context you might also enjoy
this
older note on
*Anyspeed Acceleration - The Movie*. - More notes on proper quantities and map-based motion at anyspeed.