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.
In one spatial dimension
The rapidity w arises in the linear representation of a Lorentz boost as a vector-matrix product
The matrix Λ(w) is of the type with p and q satisfying p2 − q2 = 1, so that (p, q) lies on the unit hyperbola. Such matrices form the indefinite orthogonal group O(1,1) with one-dimensional Lie algebra spanned by the anti-diagonal unit matrix, showing that the rapidity is the coordinate on this Lie algebra. This action may be depicted in a spacetime diagram. In matrix exponential notation, Λ(w) can be expressed as , where Z is the anti-diagonal unit matrix
It is not hard to prove that
This establishes the useful additive property of rapidity: if A, B and C are frames of reference, then
where wPQ denotes the rapidity of a frame of reference Q relative to a frame of reference P. The simplicity of this formula contrasts with the complexity of the corresponding velocity-addition formula.
As we can see from the Lorentz transformation above, the Lorentz factor identifies with cosh w
Proper acceleration (the acceleration 'felt' by the object being accelerated) is the rate of change of rapidity with respect to proper time (time as measured by the object undergoing acceleration itself). Therefore, the rapidity of an object in a given frame can be viewed simply as the velocity of that object as would be calculated non-relativistically by an inertial guidance system on board the object itself if it accelerated from rest in that frame to its given speed.
The product of β and γ appears frequently, and is from the above arguments
Exponential and logarithmic relations
From the above expressions we have
The Doppler-shift factor associated with rapidity w is .
In experimental particle physics
The energy E and scalar momentum |p| of a particle of non-zero (rest) mass m are given by:
With the definition of w
and thus with
the energy and scalar momentum can be written as:
So rapidity can be calculated from measured energy and momentum by
However, experimental particle physicists often use a modified definition of rapidity relative to a beam axis
where pz is the component of momentum along the beam axis. This is the rapidity of the boost along the beam axis which takes an observer from the lab frame to a frame in which the particle moves only perpendicular to the beam. Related to this is the concept of pseudorapidity.
Notes and references
- Minkowski, H., Fundamental Equations for Electromagnetic Processes in Moving Bodies" 1908
- Sommerfeld, Phys. Z 1909
- "A History of the Theories of the Aether and Electricity" In a later 1953 edition of this book it was used consistently for the theory
- "Optical Geometry of Motion" p.9
- See his papers, available in translation in Wikisource
- Amsler, C. et al., "The Review of Particle Physics", Physics Letters B 667 (2008) 1, Section 38.5.2
- Varićak V (1910), (1912), (1924) See Vladimir Varićak#Publications
- Whittaker, E. T. (1910). "A history of the theories of aether and electricity": 441. Retrieved 22 January 2016.
- Robb, Alfred (1911). Optical geometry of motion, a new view of the theory of relativity. Cambridge: Heffner & Sons.
- Borel E (1913) La théorie de la relativité et la cinématique, Comptes Rendus Acad Sci Paris 156 215-218; 157 703-705
- Silberstein, Ludwik (1914). The Theory of Relativity. London: Macmillan & Co.
- Vladimir Karapetoff (1936)"Restricted relativity in terms of hyperbolic functions of rapidities",American Mathematical Monthly 43:70.
- Frank Morley (1936) "When and Where", The Criterion, edited by T.S. Eliot, 15:200-2009.
- Wolfgang Rindler (2001) Relativity: Special, General, and Cosmological, page 53, Oxford University Press.
- Shaw, Ronald (1982) Linear Algebra and Group Representations, v. 1, page 229, Academic Press ISBN 0-12-639201-3.
- Walter, Scott (1999). "The non-Euclidean style of Minkowskian relativity" (PDF). In J. Gray. The Symbolic Universe: Geometry and Physics. Oxford University Press. pp. 91–127.(see page 17 of e-link)
- Rhodes J A.& Semon M D (2004) Relativistic velocity space, Wigner rotation, and Thomas precession, Amer. J. Phys. 72 93-90; DOI: 10.1119/1.1652040