5. Conclusions
To recap, we suggest here a few paragraphs of introduction about: (i) time as a local variable for describing motion regardless of speed using a traveler-kinematic involving only frame-invariant or synchrony-free variables, and (ii) the Galilean-kinematic approximation, which is the subject of most courses because it works so well for engineering problems on earth. We further provide example problems for applying the traveler-kinematic which are mathematically accessible to most introductory physics students, should they be tempted to explore them further outside the bounds of a traditional class. Appendices are provided as background on the metric equations, Lorentz-factor calculations, and constant acceleration integrals that underlie these well-known results.
Concerning wider applications for the traveler kinematic, four-vectors are of course written in the traveler-kinematic i.e. in terms of derivatives with respect to proper-time. Moreover the free-particle Lagrangian in curved space-time, when parameterized in terms of proper-time, is simply -mc^2. This yields the most elegant and comprehensive prediction of free-particle motion yet: In the absence of proper-acceleration, objects move so as to maximize aging (Taylor2003, Moore2004) i.e. elapsed proper-time.
More generally of course the metric-equation strategy used above for relating proper-forces to bookkeeper-acceleration works in any curved space-time or accelerated-frame. Of course the relation between the two involves a covariant-derivative with connection-coefficients, whose multi-component sums are beyond the interest of most undergraduates.
Nonetheless, a closer look at everyday curvatures (like the gravity around a planet) may be of interest to some. Questions to explore there, for instance, might include: (a) how satellite orbits can be predicted by choosing the path of maximal aging, (b) how the Schwarschild-metric yields within one expression the equations for both shell-frame gravity and accelerated-frame centrifugal-force, or (c) how gravitational effects might change if lightspeed really were only about 2.5 mph.
Acknowledgements: The author(s) would (hopefully) like to thank Stephen Gull, Roger Hill, Bill Shurcliff, and Edwin Taylor for their enthusiasm about new ways to look at old things.
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