Relative speed

Difficulty level:   ★ ★

Suppose two observers at the same place and time (that is, “event”) move with 4-velocities u and v respectively, then they measure their relative speed as follows. The Lorentz factor is simply

    \[\gamma=-\fvec u\cdot\fvec v\]

(The dot is not the Euclidean dot product, but uses the metric: g_{\alpha\beta}u^\alpha v^\beta where the indices \alpha and \beta are summed over by the Einstein summation convention.) The proof is based on the axiom that some local inertial frame exists, although interestingly one does not need to explicitly construct it.

The relative 3-speed V, may then be recovered via:

    \[V=\sqrt{1-\gamma^{-2}}\]

See for instance Carroll (end of §2.5) who terms it “ordinary three-velocity”. Other sources express the first formula more indirectly, in terms of the energy and momentum measured by an observer \fvec u: E=-\fvec u\cdot\fvec p where \fvec p=m\fvec v is the 4-momentum of another observer/object, and combine this with E=m\gamma (MTW Exercise 2.5 in §2.8 term it “ordinary velocity”, or Hartle §5.6, and Example 9.1 in §9.3).

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