As in the other pre-transform relativity stuff described here, if one measures all distances with respect to one inertial frame but defines several velocities, then relativistic acceleration and time dilation effects can be dealt with both graphically and quantitatively before multiple frames and its complications need be considered. These several velocities are the usual

If we plot displacement *x* in a direction parallel to the
coordinate acceleration, and displacement *y* in whatever
direction our accelerated object is moving perpendicular to *x*,
then by putting everything in suitably dimensionless units*, all
times and velocities (plus energy) can be plotted as a function of
position in this frame, with a single surface per variable on which
*all* constant acceleration problems in (3+1)D special
relativity plot! The graph below introduces you to some of
those surfaces.

To enhance viewability, instead of plotting coordinate, Galilean and
proper times *in addition to* *E**/mc*^2 and all three
velocities, we've provided information on the passage of time along each
trajectory with the help of "tick marks" with which the various
trajectory surfaces are parameterized. We've attempted to space the
tick marks by 0.25 in the appropriate dimensionless units. However, a
comparitive view of time surfaces as well is constructive
(*cf.* the 1+1D examples elsewhere on these pages), as is a
comparable view of Galilean constant acceleration set in a Galilean
(rather than a Lorentz or Minkowski) space time. After all,
*most of the accelerations* humans have encountered in
history *plot in the stem* of this flower, where the
Galilean surface is indistinguishable from the other two! These will
be linked hereto, when we get the chance.

Since we have plotted velocity magnitudes which are always greater
than zero, but wanted to provide surfaces both for incoming and
outgoing motion (relative to the rest event), the upward sequence of
velocity surfaces is repeated in the downward direction as well. This
means that when an incoming traveler hits the *y*=0 plane (this
is also the 1+1D universal plot), they switch instantly by definition
to the companion outgoing surface. If anyone wants to put together
some animations in this regard, for example in conjuction with the
solution of specific types of constant acceleration problems, by all
means let me know!

*To make the units "dimensionless", plot *distances* in units of
*c*^2 over the proper acceleration *a*_{o},
*times* in units of *c*/*a*_{o}, and
*velocities* in units of lightspeed *c*.

For those who might like a more interactive look at this naturally occurring "trajectory flower", we've put one such flower into a 380kB VRML file linked to the image below. If you have a viewer for such files, like Netscape Navigator 2.0 with Live3D, then you can spin, and zoom around through, the structure as you wish. Here is a link to a javaview translation of the same model.

The section of the universal plot provided here covers trajectories for
*coordinate velocity* (green) during *map times* from -2 to +2,
and *proper velocity* (blue) during *proper times* from -2 to +2,
with all times measured relative to the rest event in units of
*c*/*a*_{o} (or years at 1 gee). Values of
*transverse coordinate velocity* (a constant for a given
trajectory) range from -0.9c to 0.9c (i.e. around 2 lightyears/traveler
year). There is also a red surface representing *E*/*mc*^2 for
the *x-distance traveled* relative to rest of 0 to +2, in units of
*c*^2/*a*_{o} (or lightyears at 1 gee). See if you can
determine, with help from your browser, how deep into the flower the
*E*/*mc*^2 "cap" extends.

Send comments, your 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 :)`