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Caution: This single-frame approach provides students with quantitative tools for dealing with time dilation, relativistic acceleration, and even curved space-time. However, only a multi-frame approach can fully protect students against everyday-language confusion regarding other practical matters manifest at high speeds, like frame-dependent simultaneity and length contraction. For students with only single-frame perspectives on space-time, one bit of advice therefore is to stick to only distances measured using map-frame yardsticks.
Begin with a "map frame", across which is distributed a set of co-moving yardsticks and synchronized clocks. Simultaneity throughout this page will be defined in the context of this single frame of motion. Nonetheless, clocks carried by observers moving with respect to the map frame (i.e. travelers) will keep their own time. Metric equations are taylor-made* for this type of asymmetric situation...
Minkowski's space-time version of Pythagoras' Theorem (the flat-space metric equation) for frame-invariant proper (here "traveler") time is:
(c*Map_Time_Elapsed)^2 - Map_Distance_Traveled^2
Useful velocity or rate measures that arise because time's passage is frame-variant (i.e. everyone's clocks tell a different story) include...
ArcSinh[Proper_Velocity/c] = ArcTanh[Coordinate_Velocity/c] =
From the metric equation, one can obtain the last two equivalences above, as well as the Velocity Conversions below:
Sqrt[1+(Proper_Velocity/c)^2] = Cosh[Rapidity]
Concerning rates of velocity change, the metric equation (along with the metric relation between second proper-time derivatives of map-coordinates) allows us to define frame-invariant proper (felt) acceleration, coordinate (frame-variant) acceleration a, and the relation between them as well:
Lastly, the metric equation yields the Motion Integrals of Constant Proper Acceleration:
(Final_Proper_Velocity - Initial_Proper_Velocity)/Map_Time_Elapsed
Proper_Acceleration = c*(Final_Rapidity -
These integrals can be combined to predict changes in any of these variables, in terms of changes in another. For example, a displacement-time equation for constant proper acceleration may be obtained by combining the Rapidity and Speed_of_Map_Time Integrals above:
On the right of the two equations above, you'll note that these integrals reduce to the simpler Galilean integrals of constant acceleration, when Coordinate_Velocity is much less than Light_Speed c.
Concerning insight into the causes of motion, experiment further provides evidence for the below dynamically conserved quantities which likewise depend on one's frame of reference:
Mass*Proper_Velocity; Mass_Energy =
* no pun intended, even though the metric equation plays a central role in much of Edwin Taylor's recent work.
Reconcile yourself to simultaneity as manifest to map-frame observers, specify alternate values for the three green variables below, and then click:
Equations for each of these quantities, in a given segment of the trip, are quite easy to obtain from the integrals of constant Proper_Acceleration given above. Thus with the right travel arrangements, you can find out how your younger sibling behaves when they find themselves senior to you by ten years! There is risk, of course, because if you don't like the way they behave, it may not be possible to go back to being the older of the two, unless you can persuade them to take a similar trip.