# Map-Based Relativity: Synopsis of the Science

By Phil Fraundorf
Dept. of Physics & Astronomy, University of Missouri - StL, St. Louis MO.

If you consider motion unfolding on a map of landmarks fixed with respect to a single inertial reference frame, the map-frame choice defines both distances and simultaneity! The two times in the metric equation (map and traveler) then give rise to two useful velocities (coordinate and proper), two accelerations (coordinate and invariant -- only the latter connects to the physics directly), and two time-integrals of the equations of motion (a vector momentum equation associated with action-reaction, and an ``oriented-scalar'' impulse equation associated with the force "felt" by the traveler).

As a result, invariance in relativity as it differs from Newtonian physics can be described clearly, and tons of problems including constant acceleration can be worked, before inertial frames in relative motion (with length-contraction, velocity-addition, and frame-dependent simultaneity) need be considered quantitatively. The study of relative motion also simplifies when the variables defined above are put to use.

While at it, should you consider accelerated motion of a traveler on the map using clocks in a chase-plane not quite keeping up with the traveler, one can also find a context in which Galileo's 1-D constant acceleration equations provide relativistically correct predictions as well. In fact, Galileo discovered the simplest set of equations for describing relativistic constant acceleration in 1-D, although at high speed it deals only with the behavior of a rather special subset of clocks.

This page contains original work, so if you choose to echo on paper, in application, or on the web, a citation would be appreciated. ( Thanks. /philf :)

For source, cite URL at http://www.umsl.edu/~fraundor/mbrsynop.html
Version release date: 12 Apr 2005.

Information on applications, and on related publications, are listed in our Map-Based Relativity Table-of-Contents. Send your thoughts and suggestions via e-mail to philf@newton.umsl.edu.

• Copyright (1970-97) by Phil Fraundorf
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