from ``A one-map two-clock approach to teaching relativity in introductory physics'', arXiv:physics/9611011 (xxx.lanl.gov archive, Los Alamos, NM, 1996).

Symbols: This is a paper has a v-w-u shift in notation from earlier non-coordinate kinematic papers, particularly those which involve the Galilean kinematic.

This appendix provides a more
elegant view of matters discussed in the body
of this paper by using space-time 4-vectors not used there, along with some
promised derivations. We postulate first that: (i) displacements
between events in space and time may be described by a displacement 4-vector
X for which the time--component may be put into distance-units by
multiplying by the speed of light c; (ii) subtracting the sum of
squares of space-related components of any 4-vector from the time component
squared yields a scalar ``dot-product'' which is frame-invariant, i.e.
which has a value which is the same for all inertial observers; and
(iii) translational momentum and energy, two physical quantities which are
conserved in the absence of external intervention, are components of the
momentum-energy 4-vector P = m dX/dtau, where m
is the object's rest mass and tau is the frame-invariant displacement in
time-units along its trajectory.

From above, the 4-vector displacement between two events in space-time
is described in terms of the position and time coordinate values for those
two events, and can be written as:

Here the usual Delta is used to represent the value of final
minus initial. The dot-product of the displacement 4-vector is defined as
the square of the frame-invariant proper-time interval between those two
events. In other words,

Since this dot-product can be positive or negative, proper time intervals
can be real (time-like) or imaginary (space-like). It is easy to rearrange
this equation for the case when the displacement is infinitesimal, to
confirm the first two equalities in equation (2) via:

Here we've also taken the liberty to use a velocity 4-vectorU
= dX/dtau. The equality in
Eqn. 2 between
gamma and E/mc^2 follows immediately.
The frame-invariant dot-product
of this 4-vector, times c squared, yields the familiar relativistic
relation between total energy E, momentum p, and frame-invariant rest
mass-energy mc^2:

If we define kinetic energy as the difference between rest mass-energy and
total energy using K = E-mc^2, then the last equality in
Eqn. 2
for gamma follows as well. Another useful relation which follows is the
relation between infinitesimal uncertainties, namely
dE/dp = dx/dt.

Lastly, the force-power 4-vector may be defined as the proper time
derivative of the momentum-energy 4-vector, i.e.:

Here we've taken the liberty to define acceleration 4-vectorA
= d^2X/dtau ^2 as well.

The dot-product of the force-power 4-vector is always negative. It may
therefore be used to define the frame-invariant proper acceleration alpha,
by writing:

We still must show that this frame-invariant proper acceleration has the
magnitude specified in the text (Eqn. 5). To relate
proper acceleration alpha to coordinate acceleration a =
dv/dt = d^2x/dt^2, note first that
c dgamma/dtau = gamma^4 a v_{||}/c,
that dw_{||}/dtau = a gamma^4/gamma_{perp}^2,
and that dw_{perp}/dtau =
a gamma^3 v_{perp}v_{||}/c^2.
Putting these results into the dot-product expression
for the fourth term in Eqn. A6
and simplifying yields
alpha^2 = a^2 gamma^6/gamma_{perp}^2
as required.

As mentioned in the text, power is classically frame-dependent, but
frame-dependence for the components of momentum change only asserts itself
at high speed. This is best illustrated by writing out the force 4-vector
components for a trajectory with constant proper acceleration, in terms of
frame-invariant proper time/acceleration variables tau and alpha.
If we consider separately the momentum-change components parallel and
perpendicular to the unchanging and frame-independent acceleration 3-vector
alpha, one gets

where eta_{o} is simply the initial value for eta_{||} =
ArcSinh[w/c].

The force responsible for motion, as distinct from the frame-dependent rates
of momentum change described above, is that seen by the accelerated object
itself. As Eqn. A8 shows
for tau, v_{perp},
and eta_{o} set to zero, this is nothing more than F
= malpha. Thus some utility for the rapidity/proper
time integral of the equations of constant proper acceleration (3rd term in
Eqn. 6) is illustrated as well.

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