Rapidity

In relativity rapidity is an alternative to speed as a method of measuring motion. 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 $\phi$ of an object relative to a frame of reference is the hyperbolic angle defined as

$\phi =\operatorname {artanh} {\frac {v}{c}}\,$

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 Silberstein (1914), Eddington (1924) and Morley (1936).

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

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

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

$\mathbf {\Lambda } (\phi _{1}+\phi _{2})=\mathbf {\Lambda } (\phi _{1})\mathbf {\Lambda } (\phi _{2})$.

This establishes the useful additive property of rapidity: if $\phi _{PQ}$ denotes the rapidity of Q relative to P, then

$\phi _{AC}=\phi _{AB}+\phi _{BC}\,$,

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 φ 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

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.

References

• Arthur Stanley Eddington (1924) The Mathematical Theory of Relativity, 2nd edition, Cambridge University Press, p.22.
• 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. {{cite book}}: Cite has empty unknown parameter: |1= (help)
• Frank Morley (1936) “When and Where”, The Criterion, edited by T.S. Eliot, 15:200-2009.
• 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, Dublin: Longman, Green and Co. {{citation}}: External link in |title= (help)