Rapidity

In physics, rapidity is a parameter of the unit hyperbola used to relate frames of reference in special relativity. The term "rapidity" in English is usually a synonym of speed, but in special relativity rapidity is a continuous group parameter of the Lorentz group. Essentially, rapidity is a hyperbolic angle that differentiates a moving frame of reference from a fixed frame of reference associated with coordinates consisting of distance and time. Because the speed of light c is finite, subluminal velocity v is constrained to the interval (−c, c), and the ratio v/c is found in the unit interval (−1, 1). The hyperbolic tangent function has the whole real line for its domain and the unit interval for its range. The rapidity u associated with v is the hyperbolic angle for which the hyperbolic tangent is v/c. Using the inverse hyperbolic function artanh, the rapidity is given by u = artanh(v/c).

For low speeds, u is approximately v/c.

History

In 1910 E.T. Whittaker used this parameter in his description of the development of the Lorentz transformation in the first edition of his History of the Theories of Aether and Electricity. [1]

The rapidity parameter was named in 1911 by Alfred Robb; his term was acknowledged by Varićak (1912), Silberstein (1914), Eddington (1924), Morley (1936) and Rindler (2001).

The distillation of the rapidity concept occurred as linear algebra was developed. In the classical universe of absolute time and space, the Galilean transformations were used to relate frames of reference in relative motion. In the Lorentz transformation scenario, where Minkowski diagrams describe frames of reference, hyperbolic rotations move one frame to another. In 1848, William Rowan Hamilton and James Cockle developed the algebra of such transformations with their systems of biquaternions and tessarines respectively. A half-century of effort, including that by William Kingdon Clifford, Sophus Lie, and Alexander Macfarlane, was necessary before speed was displaced by rapidity in the modern view of kinematics.
In one spatial dimension

The rapidity φ arises in the linear representation of a Lorentz boost as a vector-matrix product

\( \begin{pmatrix} ct'\\x' \end{pmatrix} = \begin{pmatrix} \ \ \cosh\varphi & -\sinh\varphi\\-\sinh\varphi & \ \ \cosh\varphi \end{pmatrix} \begin{pmatrix} ct\\x \end{pmatrix} = \mathbf{\Lambda}(\varphi)\mathbf{v} . \)

The matrix Λ(φ) is of the type \begin{pmatrix} p & q \\ q & p \end{pmatrix} with p and q satisfying p^{2} - q^{2} {{=}} 1 , so that (p,q) lies on the unit hyperbola. Such matrices form a multiplicative group.

It is not hard to prove that

\( \mathbf{\Lambda}(\varphi_1 + \varphi_2) = \mathbf{\Lambda}(\varphi_1)\mathbf{\Lambda}(\varphi_2). \)

This establishes the useful additive property of rapidity: if \( \varphi_{PQ} \) denotes the rapidity of Q relative to P, then

\( \varphi_{AC} = \varphi_{AB} + \varphi_{BC} \,, \)

provided A, B and C all lie on the same straight line. The simplicity of this formula contrasts with the complexity of the corresponding velocity-addition formula.

The exponential function, logarithm, sinh, cosh, and tanh are all transcendental functions, requiring methods beyond algebraic expression. Conservatism in physical science explains the reluctance to rely on these functions in some presentations of relativity physics (see Scott Walter (1999)). Nevertheless, the Lorentz factor

\( \gamma {{=}} \frac {1} {\sqrt{ 1 - v^2 / c^2}} \)

identifies with \cosh\varphi where φ is rapidity. So the hyperbolic angle φ is implicit in the Lorentz transformation expressions using γ and β.

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.
In more than one spatial dimension
See also: Hyperboloid model

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 just the distance from the given velocity to the zero velocity in given frame of reference. Although it is possible, as noted above, to add and subtract rapidities where corresponding relative velocities are parallel, in 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 \( \varphi_1 \) and \( \varphi_2 \) will be greater than \( \sqrt{\varphi_1^2 + \varphi_2^2} \) expected by Pythagorean theorem. Rapidity in two dimensions can be usefully visualized using the Poincare disk.[2] 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

\( E = mc^2 \cosh \varphi\,
|\textbf{p}| = mc \, \sinh \varphi \)

and so rapidity can be calculated from measured energy and momentum by

\( \varphi = \tanh^{-1}{\frac{|\textbf{p}|c}{E}} = \frac{1}{2}\ln \frac{E + |\textbf{p}|c}{E - |\textbf{p}|c} \)

However, experimental particle physicists often use a modified definition of rapidity relative to a beam axis

\( y = \frac{1}{2}\ln \frac{E + p_zc}{E - p_zc} \)

where \( p_z \) is the component of momentum along the beam axis.[3] 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.

RelativityNotes 01 Angle and Rapidity (Hyperbolic Trigonometry)

References

^ Whittaker (1910) A History of the theories of aether and electricity (1. edition), page 441, Dublin: Longman, Green and Co. In 1953 when he published the second volume of the second edition of the History, the use of the rapidity parameter is found on page 32.
^ http://www.bates.edu/%7Emsemon/RhodesSemonFinal.pdf
^ Amsler, C. et al., "The Review of Particle Physics", Physics Letters B 667 (2008) 1, Section 38.5.2

Arthur Stanley Eddington (1924) The Mathematical Theory of Relativity, 2nd edition, Cambridge University Press, p.22.
Frank Morley (1936) "When and Where", The Criterion, edited by T.S. Eliot, 15:200-2009.
Vladimir Karapetoff (1936) "Restricted relativity in terms of hyperbolic functions of rapidities", American Mathematical Monthly 43:70.
Wolfgang Rindler (2001) Relativity: Special, General, and Cosmological, page 53, Oxford University Press.
Robb, Alfred (1911). Optical geometry of motion, a new view of the theory of relativity. Cambridge: Heffner & Sons.
Silberstein, Ludwik (1914). The Theory of Relativity. London: Macmillan & Co..
Walter, Scott (1999). "The non-Euclidean style of Minkowskian relativity". In J. Gray. The Symbolic Universe: Geometry and Physics. Oxford University Press. pp. 91–127.(see page 17 of e-link)
Whittaker, E.T. (1910). 1. Edition: A History of the theories of aether and electricity. Dublin: Longman, Green and Co..