Map-based relativity (i.e. relativity from the vantage point of a single space-time slice) allows one to solve constant acceleration problems at any speed, almost as easily as one might solve the Galilean constant acceleration problems common in today's introductory physics classes. It also makes for a nice transition from Newtonian physics, since Galileo's equations evolve naturally into the relativistic regime by simply asking the metric equation how moving clocks behave in context of a single reference map-frame. This list of mini-w{ebp}uzzlers, with equations, hints, and solutions where available, is meant to cover the full range of uni-directional constant proper-acceleration problem types. For some of these problems, closed-form solutions remain to be found. Maybe YOU can help!

map-distance

traveler-time

plus

Using

while map-defined simultaneity yields time-increments such that Δt > ΔT > Δτ.

Here is a comprehensive look at the familiar Newtonian equations for unidirectional constant acceleration at low (i.e. non-relativistic) speed:

The flat space metric equation,
(*c*dτ)^{2} = (*c*dt)^{2} - (dx)^{2}, requires
that at high speeds we keep track of proper time τ as well as
map-position x and time t. In addition to good old coordinate velocity
v = dx/dt, we therefore must also consider proper velocity w = dx/dτ and
"speed of map-time" γ = dt/dτ.

That velocity angle (or rapidity) η in particular comes in handy when considering the motion of accelerated objects, following the strategy used for the low-speed equations above. In this case however, the acceleration of interest is the proper acceleration α experienced by the traveler, rather than the coordinate acceleration a = dv/dt.

yields a family of kinetic energy versus momentum dispersion curves

that makes contact with much of everyday and modern physics.

A more comprehensive list of velocity inter-conversions is given below.

with gamma becoming a version of E/mc^2 that uses Newton's kinetic energy expression,

and all other velocity parameters become either V or V/c.

In other words, gamma = 1 + (1/2)(V/c)

The whole velocity-parameter conversion table thus simplifies at low speeds to...

at low speeds also reduce to their Newtonian form: α ~ (1/2) Δ(v

Here are a couple of recent plots about accelerated-twin round trips, seen from the perspective of a "stationary" spacetime slice. The first is a plot of velocity parameters and times on the horizontal axis, versus position on the vertical axis, for an accelerated roundtrip to a destination Δx a distance of 10c

*Note:* The Figure at right illustrates (using
a four-segment constant *proper-acceleration* roundtrip or
"twin-adventure") how *rapidity* is
the natural sister to *proper-time*, *proper-velocity* is the
natural sister to *map-time*, and *Galilean kinematic
velocity* is the natural sister to *chase-plane time*.
*Coordinate-velocity*, without a sister time, offers
only its parochial perspective on accelerated motion
at high speed.

Here are some MathCAD worksheets (example snapshot) on constant acceleration in two-clock and three-clock relativity, readable with MathSoft's free browser. These allow one to ``play with'' equations which might prove useful here in both uni-directional and 3D applications.

Many of the puzzles here are listed, with solving-algorithms, in four earlier-assembled MathCAD worksheets, one for Galilean-Kinematic (Chase-Plane Observer) Inputs, one for Coordinate-Kinematic (Non-Accelerated Observer) Inputs, one for Traveler-Kinematic (Proper-Time/Velocity) Inputs, and the last for Mixed-Kinematic (Multiple Elapsed-Time) Inputs. Beware of differences in variable notation used with earlier pages, like these, on this site.

Our browser-interactive acceleration solver only takes Galilean-kinematic inputs, but is available here. Lastly, you might want to try *graphical* solution of these problems. This link points to one of several pages under development toward this goal.

More coming soon on...

Send comments, possible solutions for problems posed, and/or complaints, to philf@newton.umsl.edu

` (Thanks. /philf :)`