PACS: 03.30.+p, 01.40.Gm, 01.55.+b

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Unfortunately for these relativistic masses, no deep sense in which mass either changes or has directional dependence has emerged. They put familiar relationships to use in keeping track of non-classical behaviors (item A above), but do not (item B above) provide frame-invariant insights or make other relationships simpler as well. As a result, majority acceptance of their use seems further away now than it did several decades ago {Goldstein}.

A more subtle trend in the pedagogical literature has been toward the
definition of a quantity called proper velocity {cf. SearsBrehme,Shurcliff},
which can be written as **w** = *gamma* **v**.
We use the symbol *w* here because it is not in common use elsewhere in
relativity texts, and because *w* looks like *gamma* *v* from a distance.
This quantity also lets us cast the momentum expression above in classical
form as a mass times a velocity, ie. as **p** = *m***w**.
Hence it serves at least one of the ``type A'' goals served by *m*' above.

We show here that, when introduced as part of a non-coordinate time/velocity
pair in pre-transform special relativity, proper velocity allows us to
introduce relativistic momentum, time-dilation, *and* frame-invariant
relativistic acceleration in context of a single inertial frame. Moreover,
through use of proper velocity many relationships (including post-transform
relationships like velocity addition which require consideration of multiple
frames) are made simpler and sometimes more useful. Hence it appears to
serve goals of ``type B'' mentioned above as well.

It follows from above that we might also consider two velocities, namely
coordinate velocity **v** = d**x**/d*t*, and
proper (or traveler-kinematic) velocity **w** = d**x**/d*t*_{o}.
It is helpful to distinguish the units used to
measure these velocities, by saying that the first measures distance
traveled per unit coordinate time, while the latter measures distance
traveled per unit traveler time. Convenient units are [lightyears per
coordinate year] and [lightyears per traveler year], respectively. Each of
these velocities can be calculated from the other by knowing the
velocity-dependence of the ``traveler's speed of **time**'' *gamma*=d*t*/d*t*_{o}, via
the equation

A number of useful relationships for this ``traveler's speed of time'',
including the familiar relationship to coordinate velocity, follow simply
from the (3+1)D nature of the flat spacetime metric (specifically from the
frame-invariant dot-product rule applied to the displacement and velocity
4-vectors). Their derivation therefrom is outlined in the
Appendix. For
introductory students, we can simply quote Einstein's prediction that
spacetime is tied together so that instead of *gamma* =1, one has *gamma*
=1/Sqrt[1-(*v*/*c*)^2}=*E*/*mc*^2, where E is Einstein's ``relativistic energy''
and *c* is the speed of light. It then follows from above that:

Given these tools to describe the motion of an object with respect to single
inertial reference frame, perhaps the easiest type of relativistic problem
to solve is that of time dilation. From the very definition of *gamma* as a
``traveler's speed of time'', and the velocity relations which show that
*gamma* is greater than or equal to 1, it is easy to see that the traveler's clock will always run
slower than coordinate time. Hence if the traveler is going at a constant
speed, one has from equation (2) that traveler time is dilated
(spread out over a larger interval) relative to coordinate time, by the
relation

Equations (2) above also allow one to relate these velocities to energy. Hence an important part of relativistic dynamics is in hand as well. Another important part of relativistic dynamics, as mentioned in the introduction, takes on a familiar form since momentum at any speed is

One might already imagine that proper speed *w* is the important speed to a
relativistic traveler trying to get somewhere with minimum traveler time.
Equation (4) shows that it is also a more interesting speed from
the point of view of law enforcement officials wishing to minimize
fatalities on a highway in which relativistic speeds are an option. This a
``type B'' benefit of proper velocity. In this context, it is not surprising
that the press doesn't report ``land speed record'' for the fastest
accelerated particle. New progress only changes the value of *v* in the 7th
or 8th decimal place. The story of increasing proper velocity, in the
meantime, goes untold to a public whose imagination might be captured
thereby. After all, the educated lay public (comprised of those who have had
only one physics course) is by and large under the impression that the
lightspeed limit rules out major progress along these lines.

By again examining the frame-invariant scalar product of a 4-vector (this
time of the acceleration 4-vector), one can show (as we do in the Appendix)
that a ``proper acceleration'' for a given object, which is the same to all
inertial observers and thus ``frame-invariant'', can be written in terms of
components for the classical acceleration vector (i.e. the second
coordinate-time *t* derivative of displacement **x**) by:

Before considering integrals of the motion for constant proper acceleration
**alpha**, let's review the classical integrals of motion
for constant acceleration **a**. These can be written as
*a* = Delta *v*_{||}/Delta *t* = (1/2) Delta(*v*^2)/Delta *x*_{||}. The
first of these is associated with conservation of momentum in the
absence of acceleration, and the second with the work-energy theorem. These
may look more familiar in the form *v*_{||f} = *v*_{||i} + *a* Delta
*t*, and *v*_{||f}^2 = *v*_{||i}^2 + 2 *a* Delta *x*_{||}.
Given that coordinate velocity has an upper limit at the speed of light, it
is easy to imagine why holding coordinate acceleration constant in
relativistic situations requires forces which change even from the
traveler's point of view, and is not possible at all for Delta
*t* greater than (*c*-*v*_{||i})/*a*.

Provided that proper time *t*_{o}, proper velocity *w*, and
time-speed *gamma*
can be used as variables, three simple integrals of the proper
acceleration can be obtained by a procedure which works for integrating
other non-coordinate velocity/time expressions as well{Noncoord}. The
resulting integrals are summarized in compact form, like those above, as

In order to visualize the relationships defined by equation 6,
it is helpful to plot for the (1+1)D or *gamma*_{perp}=1 case all
velocities and times versus *x* in dimensionless form from a common origin
on a single graph (i.e. as *v*/*c*, *alpha* *t*_{o}/*c*,
*w*/*c*=*alpha t*/*c*, and
gamma versus *alpha x*/*c*^2). As shown in Fig. 1, *v*/*c* is asymptotic to
1, *alpha t*_{o}/*c* is exponential for large arguments,
*w*/*c*=*alpha t*/*c* are
hyperbolic, and also tangent to a linear *gamma* for large arguments. The
equations underlying this plot, from eqn 6 for *gamma*_{perp}=1
and coordinates sharing a common origin, can be written simply as:

In classical kinematics, the rate at which traveler energy *E* increases
with time is not frame-independent, but the rate at which momentum
*p* increases is invariant. In special relativity, these rates (when figured with
respect to proper time) relate to each other as time and space components,
respectively, of the acceleration 4-vector. Both are frame-*dependent*
at high speed. However, we can define proper force
separately as the force *felt* by an accelerated object. We show in the
Appendix that this is simply **F**_{o} = *m* **alpha**. That is, all accelerated objects *feel*
a frame-invariant 3-vector force **F**_{o} in the direction of their acceleration.
The magnitude of this force can be calculated from any inertial frame, by multiplying the rate
of momentum change *in the acceleration direction* times *gamma*_{perp},
or by multiplying mass times the proper acceleration *alpha*. The classical
relation *F* = d*p*/d*t* = *m*d*v*/d*t* = *m*d^2*x*/d*t*^2 = *ma* then becomes:

Although they depend on the observer's inertial frame, it is instructive
to write out the rates of momentum and energy change in terms of the proper
force magnitude *F*_{o}. The classical equation relating rates of momentum
change to force is d**p**/d*t* = **F** = *ma*
**i**_{||}, where **i**_{||} is
the unit vector in the direction of acceleration. This becomes

As mentioned above, the rate at which traveler energy increases with time
classically depends on traveler velocity through the relation
d*E*/d*t* = *Fv*_{||} = *m***a*****v**.
Relativistically, this becomes

Similarly, the classical relationship between work, force, and impulse can
be summarized with the relation
d*E*/d*x*_{||} = F = d*p*_{||}/d*t*.
Relativistically, this becomes

The Lorentz transform itself is simplified, in that it can be written using proper velocity in the symmetric matrix form:

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 frequency *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 most noticeable effect of proper velocity, in the post-transform
relativity we've considered so far, involves simplification and
symmetrization of the velocity addition rule. The rule for adding coordinate
velocities **v**' and **v** to get
relative coordinate velocity **v**'' in inherently complicated, namely
*v*_{||}'' =
(*v*'+*v*_{||})/(1+*v*_{||}*v*'/*c*^2)
with *v*_{perp}'' NOT equal to *v*_{perp}. Here
subscripts refer to component orientation with respect to the direction
of **v**'. Moreover, for
high speed calculations, the answer is usually un-interesting 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** = *gamma* **v** to get relative proper velocity **w**'',
one finds simply that the coordinate velocity factors
add while the *gamma*-factors multiply, i.e.

Physically more interesting questions can be answered with equation (12)
than with the coordinate velocity addition rule. For example, one
might ask what relative proper velocity (and hence momentum) is attainable
with colliding beams from an accelerator able to produce particles of proper
speed *w* for impact onto a stationary target. If one is using 50GeV
electrons in the LEP2 accelerator at CERN, *gamma* and *gamma*'
are *E*/*mc*^2 = 50GeV/511keV or 10^5, *v* and *v*' are essentially
*c*, and *w* and *w*' are hence 10^5*c*. Upon collision, equation
(12) tells us that in a collider the relative proper speed
*w*'' is (10^5)^2(*c*+*c*) = 2 x 10^10 *c*. Investment in a
collider thus buys a factor of 2 *gamma* = 2 x 10^5 increase in the
momentum (and energy) of collision. Compared to the cost of building a
10PeV accelerator for the equivalent effect on a stationary target, the
collider is a bargain indeed!

We show further that a *frame-invariant* proper acceleration 3-vector
has three simple integrals of the motion, in terms of these variables. Hence
students can speak of the proper acceleration and force 3-vectors for an
object in frame-independent terms, and solve relativistic constant
acceleration problems much as they now do for non-relativistic problems in
introductory courses.

We further show that the use of these variables does more than ``superficially preserve classical forms''. Not only are more than one classical equation preserved with minor change with these variables. In addition, more interesting physics is accessible to students more quickly with the equations that result. The relativistic addition rule for proper velocities is a special case of the latter in point. Hence we argue that the trend in the pedagogical literature, away from relativistic masses and toward use of proper time and velocity in combination, may be a robust one which provides: (B) deeper insight, as well as (A) more value from lessons first-taught.

For more on this subject, see our table of contents. Please share your thoughts using our review template, or send comments, answers to problems posed, and/or complaints, to philf@newton.umsl.edu. This page contains original material, so if you choose to echo in your work, in print, or on the web, a citation would be cool.

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