Rapidity

In relativity, rapidity is an alternative to speed as a framework for measuring 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. Rapidity can also link geometric techniques to Minkowski diagrams. The rapidity concept is not widely (nor fashionably) used; speed has the advantage of more direct measurability.

Detail

The rapidity φ of an object relative to a frame of reference is the hyperbolic angle defined as

$\varphi =\operatorname {arctanh} {\frac {v}{c}}\,$

where

v is the speed of the object relative to the same frame of reference,

c is the speed of light, and

artanh is the inverse hyperbolic tangent function.

For low speeds, φ is approximately v/c.

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 (page 441). 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. 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).

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 $p^{2}-q^{2}{=}1$. The study of all matrices ${\begin{pmatrix}p&q\\q&p\end{pmatrix}}$ with p, q ∈ R is taken up in the article split-complex number. 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 simple formula (as well as the matrix above) is analogous to a formula for adding angles of two ordinary rotations in the Euclidean space (and for a rotation matrix): indeed, rapidity may be understood as a "hyperbolic angle" or a Minkowskian counterpart of an angle.

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.

Mathematically, the rapidity can be viewed as a re-linearization of the speed, since the naively linear v becomes absurd as v approaches c.

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 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 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

$\varphi =\operatorname {artanh} {\frac {|{\textbf {p}}|c}{E}}={\frac {1}{2}}\ln {\frac {E+|{\textbf {p}}|c}{E-|{\textbf {p}}|c}}$

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

$\varphi _{z}={\frac {1}{2}}\ln {\frac {E+p_{z}c}{E-p_{z}c}}$

where $p_{z}$ is the component of momentum along the beam axis. Related to this is the concept of pseudorapidity.

References

• 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.
• 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" (PDF). In J. Gray (ed.). 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" (Document). Dublin: Longman, Green and Co. {{cite document}}: External link in |title= (help)CS1 maint: postscript (link)