Rapidity

Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with distance and time coordinates.

In relativity, rapidity denoted by w is commonly used as a measure for relativistic velocity. For one-dimensional motion, rapidities are additive whereas velocities must be combined by Einstein's Velocity-addition formula. For low speeds, rapidity and velocity are proportional, but for higher velocities, rapidity takes a larger value, the rapidity of light being infinite.

Using the inverse hyperbolic function artanh, the rapidity w corresponding to velocity v is w = artanh(v / c) where c is the velocity of light. For low speeds, w is approximately v / c. Since in relativity any velocity v is constrained to the interval −c < v < c the ratio v / c satisfies −1 < v / c < 1. The inverse hyperbolic tangent has the unit interval (−1, 1) for its domain and the whole real line for its range, and so the interval −c < v < c maps onto −∞ < w < ∞.

In 1908 Hermann Minkowski explained how the Lorentz transformation could be seen as simply a hyperbolic rotation of the spacetime coordinates, i.e., a rotation through an imaginary angle. This angle therefore represents (in one spatial dimension) a simple additive measure of the velocity between frames. It was used by Varićak 1910 by Whittaker 1910 It was named "rapidity" by Alfred Robb 1911 and this term was adopted by many subsequent authors, such as Varićak 1912, Silberstein (1914), Morley (1936) and Rindler (2001). The development of the theory of rapidity is mainly due to Varićak in writings from 1910 to 1924.

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