Metric-first motion
Hence it is no surprise that in recent decades, senior undergraduate and introductory textbooks increasingly recognize the special nature of proper-time i.e. the frame-invariant time-elapsed on a traveler's clock. This move away from always considering two equivalent frames, with independent arrays of synchronized clocks, to the use of one primary bookkeeper frame (as well as to engineering problems that measure time in seconds and distances in meters cf. Cook2004) is moving forward in a variety of other ways as well (Fraundorf2011). Some of these are listed on this now ancient website.
There will likely be downsides as well as advantages to use of frame-invariant and/or synchrony-free measures for describing motion from the beginning (Fraundorf2012), and to the use of a single bookkeeper-frame as primary. Explanations in terms of length-contraction may, for example, be difficult with this approach. The elucidation of these pros and cons awaits more detailed work on kinematics content-modernization by physics education researchers everywhere.
Stay tuned...
Some related references:
Edwin F. Taylor and John Archibald Wheeler (2000) Exploring Black Holes (original draft title" "Scouting Black Holes with Calculus") Addison-Wesley-Longman, NY; free 2017 2nd edition.
R. J. Cook (2004) Physical time and physical space in general relativity, Am. J. Phys. 72:214-219.
P. Fraundorf (2011) "Metric-first & entropy-first surprises", arXiv:1106.4698 [physics.gen-ph].
P. Fraundorf (2012) "A traveler-centered intro to kinematics", arxiv:1206.2877 [physics.pop-ph].
Hermann Minkowski's metric-equation has been central to relativistic problem-solving since Albert Einstein recognized its importance to general relativity in the 1910-1920 time-frame. Derivatives with respect to proper-time continue to play a central role in most modern studies of spacetime.
The metric-equation contains within it all of special relativity, as well as many of the consequences of general relativity, even though it is little more than a space-time version of Pythagoras' theorem. Of special importance is the fact that it relates frame-invariant traveler-time to local space and time in a single bookkeeper or map frame, as illustrated in the top half of the figure at right.
Hence it does not require use of multiple frames of synchronized clocks, which are generally difficult to conceptualize and in curved-spacetimes may be difficult to create as well. Does that mean that metric-first kinematics can provide a more robust foundation to beginners as well as to experts?
