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 artanh, the rapidity φ corresponding to velocity v is φ = artanh(v / c). For low speeds, φ is approximately v / c. The speed of light c being finite, any velocity v is constrained to the interval −c < v < c and the ratio v / c satisfies −1 < v / c < 1. Since the inverse hyperbolic tangent has the unit interval (−1, 1) for its domain and the whole real line for its range, the interval −c < v < c maps onto −∞ < φ < ∞.
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. The rapidity was used by Whittaker 1910  and by Varićak 1910. The name "rapidity" is due to 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 φ arises in the linear representation of a Lorentz boost as a vector-matrix product
The matrix Λ(φ) 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 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(φ) 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 φPQ 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 φ:
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 Doppler-shift factor associated with rapidity φ is .
In more than one spatial dimension
From mathematical point of view, possible values of relativistic velocity form a manifold, where the metric tensor corresponds to the proper acceleration (see above). This is a non-flat space (namely, a hyperbolic space), and rapidity is the distance from the given velocity to the zero velocity in the given frame of reference. Although it is possible, as noted above, to add and subtract rapidities where corresponding relative velocities are parallel, in the general case the rapidity-addition formula is more complex because of negative curvature. For example, the result of "addition" of two perpendicular motions with rapidities φ1 and φ2 will be greater than expected by the Pythagorean theorem. Rapidity in two dimensions can be usefully visualized using the Poincaré disk. Points at the edge of the disk correspond to infinite rapidity. Geodesics correspond to steady accelerations. The Thomas precession is equal to minus the angular deficit of a triangle, or to minus the area of the triangle.
In experimental particle physics
The energy E and scalar momentum | p | of a particle of non-zero (rest) mass m are given by
and 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" 1910. In a later edition of this book in 1953 the rapidity is used consistently for the theory
- "Optical Geometry of Motion" p.9
- See his papers, available in translation in Wikisource
- John A. Rhodes & Mark D. Semon (2003) "Relativistic velocity space, Wigner rotation, and Thomas precession", American Journal of Physics 72(7):943–961
- Amsler, C. et al., "The Review of Particle Physics", Physics Letters B 667 (2008) 1, Section 38.5.2
- Whittaker, E.T. (1910). "1. Edition: A History of the theories of aether and electricity". Dublin: Longman, Green and Co. page 441
- 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)
- Varićak V (1910), (1912), (1924) See Vladimir Varićak#Publications