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

St. Louis MO 63121-4499, Phone: (314)516-5044, Fax:(314)516-6152

The most fundamental example of a non-coordinate kinematic centers about
time elapsed on the clocks of the traveling object. The time used in
describing map motions is thus proper time *tau* elapsed along the world
line of the traveler, while velocity in the traveler-kinematic, *u* = d*x*/d*tau*,
is the spatial component of 4-velocity with respect to the map frame.
Unlike coordinate-velocity, traveler-kinematic velocity *u* is thus
proportional to momentum, and obeys elegant addition/transformation rules.
In spite of its intuitive simplicity and usefulness for solving problems in
the context of a single inertial frame, discussion of this quantity (also
called ``proper speed'') has been with two exceptions{SearsBrehme,Shurcliff}
almost non-existent in basic texts{Shurcliff}.
The equations for constant proper acceleration look quite different in the
traveler kinematic, although they too have been rarely discussed (except for
example in the first edition of the classic text by Taylor and Wheeler{TaylorWheeler}).
Even in that case, they were not discussed as part of a
time-velocity pair defined in the context of one inertial frame.

Another interesting kinematic for describing motion, with respect to a
map-frame, centers about time elapsed on the clocks of a ``Galilean chase
plane''. For unidirectional motion, Galileo's equations for constant
acceleration *exactly* describe relativistic motions for our traveler in
such a kinematic. Therefore, with appropriate inter-kinematic conversions in
hand, Galileo's equations may offer the simplest path to some relativistic
answers! Galilean time and Galilean-kinematic velocity remain well defined
in (3+1)D special relativity, with kinetic energy held equal to half mass
times velocity squared. For non-unidirectional motion, however, the constant
acceleration equations containing Galilean-observer time look more like
elliptic integrals than like Galileo's original equations.

Since all kinematics share common distance measures in the context of one
inertial reference frame, they allow one to construct x-tv plots showing all
variables in the context of that frame, including universal constant proper
acceleration surfaces on which *all flat space-time problems* of this
sort plot. The single frame character of these non-coordinate kinematics
facilitates their graphical and quantitative use in pre-transform relativity
as well, i.e. by students not yet ready for multiple inertial frames{PreTransform}.

This paper is basically about three points of view: that of the map, the
traveler, and the observer. All un-primed variables refer to traveler
motion, and all distances are measured in the reference inertial coordinate
(*x*,*y*) frame. In short, everyone is talking about the traveler, and using
the same map! However, *traveler motion on the map* is described
variously: (i) from the map point of view with coordinate time *b* and
velocity *w*, (ii) from the traveler point of view with traveler time *tau*
and traveler kinematic velocity *u*, and (iii) from the observer or ``chase
plane'' point of view with observer-time *T* and observer-kinematic velocity
*V* = d*x*/d*T*. The velocity definitions used here are summarized
in Table 1.

The most direct way to specify a general observer kinematic might be via the
rate *h* at which coordinate-time *b* in the inertial map frame passes per
unit time *T* in the kinematic of interest. If we associate this observer
time with physical clock time elapsed along the world line of a chase plane
(primed frame) carrying the clocks of that kinematic, then this rate for a
given kinematic can be written

One can also relate this quantity *h* to velocity *V* in the kinematic of
interest. Since* V* = d*x*/d*T* and *w* = d*x*/d*b*, it
follows that the observer-kinematic velocity *V* of our traveling object,
with respect to the map frame, is related to coordinate-velocity *w* by
*V* = *wh*. Since *gamma* >= 1 with equality iff *V* equals *w*, any non-coordinate
velocity *V*, which describes the traveler motion with respect to a map
frame using times from a physical (albeit non-coordinate) clock, *will
be higher than the coordinate-velocity w* describing that same motion.

Time/velocity pairs (kinematics) like the ones examined here may be defined
by specifying *h* as a function of the speed of the traveler (e.g. as a
function of coordinate-velocity *w*). In this way the coordinate-kinematic
itself is specified by setting *h* to 1. If we set *h* equal to *gamma* =
d*b*/d*tau*=1/Sqrt[1-(*w*/*c*)^2], where *c* is the
coordinate-speed of light, then the resulting kinematic is that associated
with the time *tau* read by clocks moving with the traveler.

In order to determine *h* for a Galilean observer, we explore further the
relationship between *h* and energy factor *gamma* = E/*mc*^2 =1+
K/*mc*^2 = d*b*/d*tau*, where E is total relativistic energy for
our traveler, K is kinetic energy, and *m* is traveler rest mass. In
particular, given the function *gamma*[*V*], one can obtain *h*[*V*] from the
relation

If we define the Galilean kinematic as one for which kinetic energy is K = (1/2)*mV*^2,
then E = *mc*^2+ (1/2)*mV*^2, and hence *gamma*[*V*] = E/*mc*^2
= 1+(1/2)(*V*/*c*)^2. Then *V*[*gamma*] = *c*\Sqrt[2/(*gamma* -1)], which
substituted into (2) above gives *h* = *gamma* Sqrt[2/(*gamma*+1)].
Hence a ``chase plane'', which holds its own *gamma*'
value at *gamma* Sqrt[2/(*gamma*+1)] in terms of *gamma* for the
traveler, will observe a velocity *V* which is related to kinetic energy of
K = (1/2)*mV*^2, for the traveler in the reference inertial (map) frame.
Note that *gamma*' <= *gamma*, which means that the Galilean
chase plane always moves a bit more slowly than does the traveler. One can
also translate these ``chase plane instructions'' into coordinate-velocity
terms, by replacing *gamma* values above with the corresponding 1/Sqrt[1-(*w*/*c*)^2]
expression, and then solving for *w*'.

The generic observer-kinematic relationships discussed here, along with
derived values for coordinate, traveler, and Galilean kinematics, are also
listed in Table 1. What follows is a look at the equations
provided by each of these kinematics for describing map trajectories for our
traveler under constant proper acceleration *alpha*.

With no loss of generality, let's define *x* as the ''direction of proper
acceleration''. In other words, choose *x* as the direction in which
coordinate-velocity is changing. If the motion is not unidirectional, then
there will also be a component of coordinate-velocity which is not changing.
Choose the *y*-direction as the direction of this non-changing component of
coordinate-velocity, namely *w*_{y}. Holding *alpha* constant yields a
second order differential equation whose solution, for times and distances
measured from the ``turnabout point'' in the trajectory at which *w*_{x} -> 0, is

A first integral of the motion, which is independent of kinematic, is
helpful for exploring *intra-kinematic* solutions of the constant
acceleration equation. Since one thing in common to all kinematics are map
frame distances and expressions for *gamma*, we seek a relation between
initial/final *gamma* and initial/final position. This is nothing more
than the work-energy theorem for constant proper acceleration, which takes
the form *gamma*_{y} *m* *alpha* Delta[*x*] = Delta[E] = *mc*^2 Delta[*gamma*]. For the
special case when times and distances are measured from the turnabout point,
where *gamma* = *gamma*_{y} and *x*=0, this says that *gamma*/*gamma*_{y} = 1 + *alpha* *x*/*c*^2.

If motion is not unidirectional, the strategy above is complicated by the
fact that *V* is not d*x*/d*T*, but instead is connected to d*x*/d*T* through the relation *V*^2 = Sqrt[(d*x*/d*T*)^2 + (*h* *w*_{y})^2], where *h*
may also be a function of *V*. The strategy above still yields a first order
equation in terms of *x* and *T*, with only minor resulting complication in
the coordinate and traveler kinematic cases. The results of this for (3+1)D
motion, in the three example kinematics, are summarized in Table3.

As you can see from the Table on (3+1)D acceleration, all 14 variables (*x*,
*y*, *gamma*, *b*, *w*, *w*_{x}, *tau*, *u*, *u*_{x}, *u*_{y}, *t*, *v*, *v*_{x}, *v*_{y}) and 2 constants (*alpha*, *w*_{y}) are simply related, except for
Galilean time. We here take advantage of the fact that Galilean time *t* is
directly expressable in terms of traveler time *tau*, through the
dimensionless expression

As a matter of practice when dealing with multiple time and velocity types,
all defined within context of a single inertial frame, it is helpful to
append the kinematic name to the time units, to avoid confusion. Thus, time
in the traveler-kinematic may be measured in [traveler years], while
velocities in the traveler-kinematic are measured in [light-years per
traveler year]. Thus, from Table 4, a traveler-kinematic velocity of *u* = 1 [light-year per traveler year] corresponds to a coordinate-kinematic
velocity of *w* = 1/Sqrt[2] [*c*], where of course [*c*] is shorthand
for a [light-year per coordinate year]. This unitary value of
traveler-kinematic velocity may serve as a landmark in the transition
between non-relativistic and relativistic behaviors. For these reasons, we
expect that a shorthand units notation for traveler-kinematic velocity, in
particular for a [light-year per traveler year], may prove helpful in days
ahead.

Before leaving the acceleration equations, it is worthwhile summarizing the findings, in mixed kinematic form as integrated expressions for constant proper acceleration. For unidirectional motion, or (1+1)D special relativity, we have shown that

If we further divide distances in the *x*,*y* coordinate field by *c*^2/*alpha*, and restrict trajectories to those involving a single constant
acceleration trajectory, or to a family of such trajectories in (3+1)D, a
single universal acceleration plot in the velocity range of interest can be
assembled, on which all constant acceleration problems plot! A plot of this
sort, of the three example kinematic velocities and gamma, over the linear
velocity range from 0 to 3 [light-years per traveler year], is provided in
Figure 2.
Here the various velocity surfaces have been
parameterized using equal time increments in the kinematic of the velocity
being plotted, as well as with equally-spaced values of the constant
transverse coordinate-velocity *w*_{y}. It is thus easy to see that coordinate
time during constant acceleration passes more quickly, as a function of
position, than do either of the two kinematic times, as predicted more
generally by equation (1).

All of the foregoing has been done in the context of a single inertial
frame. The transformation properties of these various times and velocities
between frames also deserve mention. As is well known, and apparent from the
section on 4-vector acceleration above, *coordinate-time* is the time
component of the coordinate 4-vector, while *traveler-kinematic velocity*
provides spatial components for the velocity 4-vector. *Coordinate-velocity* enjoys no such role, and its comparatively messy
behaviors when transforming between frames need no further introduction.

A similar inelegance is found in the well-known rule of velocity addition
for coordinate-velocities. A comprehensive look at addition rules for the
velocities used here, and for *gamma* as well, shows that the simplest of
all (to this author at least) is the rule for the relative
traveler-kinematic velocity associated with particles A and B on a collision
course, namely *u*_{rel} = *gamma*_{A} *gamma*_{B}(*w*_{A} + *w*_{B}). Recall that within a
single frame, *u* = *gamma* *w*. Hence on addition, the *gamma* factors of
traveler-kinematic velocity multiply, while the coordinate-velocity factors
add.

For example, we lose the second time derivative of coordinate position as a
useful acceleration, and thus the coordinate kinematic loses *Galileo's
equations for constant acceleration*. These equations remain intact for
unidirectional motion if we put our clocks into a Galilean chase plane. The
role of this kinematic, in allowing proper acceleration to be written as a
second time derivative of position (i.e. for which *V* = *alpha* *T*), is
illuminated by noting that unidirectional motion from rest is described by
*alpha* *x*/*c*^2 = Cosh[*alpha* *tau*/*c*] - 1.
Hence *u*/*c* = *alpha* *b*/*c* = Sinh[*alpha* *tau*/*c*],
but *u* and *b* are in different
kinematics. If we want *V*/*c* = *alpha* *T*/*c* for __a__ time/velocity __pair__ in one
kinematic, a *V*[*tau*] obeying *alpha* d*x* = *alpha* *V*d*T* = *V*d*V* = *alpha* *c* Sinh[*alpha* *tau*/*c*] d*tau* is required. One can rewrite the right side of
this equation as 2*c* Sinh[*alpha* *tau*/2*c*] times *alpha* Cosh[*alpha* *tau*/2*c*] d*tau* = d[2*c* Sinh[*alpha* *tau*/2*c*], or as *V*d*V* if and only if *V*/*c* = 2 Sinh[*alpha* *tau*/2*c*] = Sqrt[2 *alpha* *x*/*c*^2]. Thus Galilean velocity is the only way to make proper
acceleration a second time derivative in (1+1)D. From similarities between *v*/*c* = *alpha* *t*/*c* = 2 Sinh[*alpha* *tau*/2*c*] and the expression above for *u*/*c* = *alpha* *b*/*c*, it is easy to see why the Galilean kinematic
provides an excellent low-velocity (small-argument) approximation to
traveler velocity and coordinate time.

On a deeper note, *the inertial nature of velocity* in Galilean
space-time (i.e. its proportionality to momentum), as well as its relatively
*elegant transformation properties*, do not belong to coordinate
velocity in Minkowski space-time. Velocity in the traveler kinematic
inherits them instead. The traveler-kinematic also has relatively
*simple equations for constant acceleration*. However, constant acceleration
is most simply expressed in mixed-kinematic terms, using the coordinate-time
derivative of a traveler-kinematic velocity component.

Thus we have a choice, to cast expressions which involve velocity in terms
of *either* the coordinate-time derivative of position (*w*), *or*
in terms of the dynamically and transformationally more robust traveler-time
derivative of position (*u*). Some of the connections discussed here might
be more clearly reflected than they are at present in the language that we
use, had historical strategies taken the alternate path.

Please share your thoughts via 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.

` (Thanks. /philf :)`