Rapidity

In relativity, rapidity is an alternative to speed as a measure 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. The rapidity concept is not widely (nor fashionably) used; speed has the advantage of more direct measurability.

Using the inverse hyperbolic function tanh⁻¹, the rapidity φ corresponding to velocity v is φ = tanh⁻¹(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 becomes mapped onto -∞<φ<∞.

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, and this angle therefore represents (in one spatial dimension) a simple additive measure of the velocity between frames. Minkowski's angle of rotation was given the name "rapidity" in 1911 by Alfred Robb, and this term was adopted by many subsequent authors, such as Varićak (1912), Silberstein (1914), Eddington (1924), Morley (1936) and Rindler (2001).

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: using the notation $\varphi_{PQ}$ to denote the rapidity of a frame of reference Q relative to a frame of reference P,

$\varphi_{AC} = \varphi_{AB} + \varphi_{BC} \,$,

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 Lorentz factor identifies with $\cosh\varphi$:

$\gamma {{=}} \frac {1} {\sqrt{ 1 - v^2 / c^2}} \equiv \cosh\varphi$,

so the rapidity $\varphi$ is implicitly used as a hyperbolic angle 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 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 $\sqrt{\varphi_1^2 + \varphi_2^2}$ expected by Pythagorean theorem. Rapidity in two dimensions can be usefully visualized using the Poincare 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

$E = mc^2 \cosh \varphi\,$

$|\textbf{p}| = mc \, \sinh \varphi$

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. 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.

References

• Varićak V (1910), (1912), (1924) See Vladimir Varićak#Publications
• 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. 
• 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.
• 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)
• Wolfgang Rindler (2001) Relativity: Special, General, and Cosmological, page 53, Oxford University Press.