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Rapidity
In relativity, rapidity is an alternative to speed as a measure of rate of motion. On parallel velocities (say, in one-dimensional space) rapidities are simply additive, unlike speeds at relativistic velocities. For low speeds, rapidity and speed are proportional, but for high speeds, rapidity takes a larger value. The rapidity of light is infinite.
Using the inverse hyperbolic function , the rapidity corresponding to velocity is . For low speeds, is approximately . The speed of light being finite, any velocity is constrained to the interval and the ratio satisfies . Since the inverse hyperbolic tangent has the unit interval for its domain and the whole real line for its range, the interval maps onto .
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.
==History==
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.〔Minkowski, H., Fundamental Equations for Electromagnetic Processes in Moving Bodies" 1908〕 This angle therefore represents (in one spatial dimension) a simple additive measure of the velocity between frames.〔Sommerfeld, Phys. Z 1909〕 The rapidity was used by Whittaker 1910 〔"A History of the Theories of the Aether and Electricity" 1910. In a later edition of this book in 1953 the rapidity is used consistently for the theory〕 and by Varićak 1910. The name "rapidity" is due to Alfred Robb 1911 〔"Optical Geometry of Motion" p.9〕 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.〔See his papers, available in translation in Wikisource〕
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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