Summary

DescriptionAddingUnitVelocities.png	English: Adding unit proper-velocities: In this case object A is traveling in various directions with respect to object B as shown by the red wAB vectors in the figure above, while object B is traveling east with respect to object C as shown by the green wBC vectors in the figure above. This illustrates how vector-addition can help us figure out the proper velocity of object A with respect to object C, as shown by the blue wAC vectors in the figure above. Note that when adding proper-velocities (i.e. map-distance r per unit traveler-time τ) at high speed e.g. to get wAC from wAB and wBC, the out-of-frame component (here A with respect to B) must be rescaled (orange extension) to become (wAB)C by a factor ≥ 1 before the vector-addition is performed. At high speeds, this magnitude-rescaling is needed in general (even in the uni-directional case).
Date	20 January 2012, 12:35:40
Source	Own work
Author	P. Fraundorf
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Additional information

As pointed out here^[1], local proper-velocities w ≡ dx/dτ add as 3-vectors, provided that we rescale the magnitude of the "out-of-frame" vector i.e. in this case (w_AB)_C:

${\displaystyle {\vec {w}}_{\text{AC}}=\left({\vec {w}}_{\text{AB}}\right)_{\text{C}}+{\vec {w}}_{\text{BC}}=\gamma _{\text{AC}}{\vec {v}}_{\text{AC}}}$.

Coordinate-velocity 3-vectors do not similarly add. Moreover, both proper-time and proper-acceleration are frame invariants. This makes 3-vector velocities and accelerations much more useful at high-speeds in the traveler-kinematic^[2] than in the Galilean-kinematic.

An application of relative proper-velocity vector-addition is illustrated in the figure at right.

Footnotes

1. ↑ P. Fraundorf (2011/2012) "Metric-first & entropy-first surprises", arXiv:1106.4698 [physics.gen-ph].
2. ↑ P. Fraundorf (2012) "A traveler-centered intro to kinematics", arxiv:1206.2877 [physics.pop-ph].

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