4.2.5. adding 3-vector velocities
The traveler-kinematic also extrapolates nicely into a description of (3+1)D accelerated motion, as illustrated in the figure at right. We’ve already discussed some advantages of unidirectional proper-velocity addition, in comparison e.g. to the addition of relative coordinate-velocities at high speed. We here explore the familiar “tail-to-nose" addition of three-vector velocities, and in particular some extreme-physics applications of introductory physics' dreaded “relative-velocity equation" vAC = vAB + vBC.Coordinate-velocity 3-vectors only add in the usual way at speeds small compared to that of light. However, regardless of speed, local momenta per unit mass or proper-velocities w ≡ dx/dτ do add as 3-vectors, provided that we re-scale/direct the "out-of-frame" component wAB and time-dilate wBC according to:
where the first term is the non-proper "shared clock but not metric" velocity:
while the second term is the non-proper "shared metric but not clock" velocity:
.
One consequence of this is that wCA (which uses A's rather than C's definition of simultaneity) will have the same magnitude as, but a different direction than, wAC. Because the gamma factors are always greater than one, it is easy to see from the equation that the proper-velocity sum direction will be "Thomas-precessed" from wAB+wBC toward the direction of the intermediate frame's proper-velocity wBC.
This makes possible problems like the following:
Example Problem 4.2.5a: A starfleet battle-cruiser drops out of hyperspace in the orbital plane of a ringworld, traveling at 1 ly/ty radially away from the ringworld's star. An enemy cruiser drops out of hyperspace nearby at the same time, traveling 1 ly/ty in the rotation-direction of the ringworld's orbit, and in a direction perpendicular to the starfleet cruiser's radial-trajectory. What is the proper-velocity (magnitude and direction) of the enemy cruiser relative to the starfleet ship?