Proper Velocity

Proper Velocity

In relativity, proper-velocity, also known as celerity, is an alternative to velocity for measuring motion. Whereas velocity relative to an observer is distance per unit time where both distance and time are measured by the observer, proper velocity relative to an observer divides observer-measured distance by the time elapsed on the clocks of the traveling object. Proper velocity equals velocity at low speeds. Proper velocity at high speeds, moreover, retains many of the properties that velocity loses in relativity compared with Newtonian theory.

For example proper-velocity equals momentum per unit mass at any speed, and therefore has no upper limit. At high speeds, as shown in the figure at right, it is proportional to an object's energy as well.

Proper-velocity is one of three related derivatives in special relativity (coordinate velocity v = dx/dt, proper-velocity w = dx/dτ, and Lorentz factor γ = dt/dτ) that describe an object's rate of travel. For unidirectional motion, each of these is also simply related to a traveling object's hyperbolic velocity angle or rapidity η by

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