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.

The foregoing sections treat calculations made
possible, and analogies with
classical forms which result, if one introduces the proper time/velocity
variables in the context of a single map frame. What happens when multiple
map-frames are required? In particular, are the Lorentz transform and other
multi-map relations similarly simplified or extended? The answer is yes,
although our insights in this area are limited since the focus of this paper
is introductory physics, and not special relativity.

This seems to be an improvement over the asymmetric equations normally used,
but of course requires a bit of matrix and 4-vector notation that your
students may not be ready to exploit.

The expression for length contraction, namely L =
L_{o}/gamma, is not
changed at all. The developments above do suggest that the concept of proper
length L_{o}, as the length of a yardstick
in the frame in which it is at
rest, may have broader use as well. The relativistic Doppler effect
expression, given as f = f_{o}
Sqrt[{1+(v/c)}/{1-(v/c)}] in terms of
coordinate velocity, also simplifies to f =
f_{o}/{gamma -(w/c)}. The
classical expression for the Doppler effected frequency of a wave of
velocity v_{wave} from a moving source of
frequency f_{o} is, for comparison, f =
f_{o}/{1-(v/v_{wave})}.

The most noticeable effect of proper velocity, on the multi-map
relationships considered here, involves simplification and symmetrization of
the velocity addition rule. The rule for adding coordinate velocities
v' and v to get relative coordinate velocity v'', namely
v_{||}'' =
(v' + v_{||})/(1+v_{||}v'/c^2)
and v_{perp}'' =/= v_{perp} with
subscripts referring to component orientation with respect to the direction
of v', is inherently complicated. Moreover, for
high speed calculations, the answer is usually uninteresting since large
coordinate velocities always add up to something very near to c. By
comparison, if one adds proper velocities w' = gamma' v'
and w = gammav
to get relative proper velocity w'',
one finds simply that the coordinate velocity factors
add while the gamma-factors multiply, i.e.

Note that the components transverse to the direction of v'
are unchanged. These equations are summarized for the
unidirectional motion case in Table III.

B. Classroom applications involving more than one map-frame.

Physically more interesting questions can be answered with Eqn. 13
than with the coordinate velocity addition rule commonly given to
students. For example, one might ask what the speed record is for
relative proper velocity between two objects accelerated by man. For the
world record in this particle-based demolition derby, consider colliding two
beams from an accelerator able to produce particles of known energy for
impact onto a stationary target. From Table I
for colliding 50[GeV] electrons in the LEP2 accelerator at CERN,
gamma and gamma'
are E/mc^2 ~ 50[GeV]/511[keV] ~ 10^5, v and v'
are essentially c, and w and w' are hence 10^5[ly/tyr]. Upon
collision, Eqn. 13 tells us that the relative proper speed
w'' is (10^5)^2(1+1) = 2 10^{10}[ly/tyr]. Investment in a
collider thus buys a factor of 2gamma =2 10^5 increase in the
momentum (and energy) of collision. Compared to the cost of building a
10[PeV] accelerator for the equivalent effect on a stationary target, the
collider is a bargain indeed!

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