Rapidity

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In relativity rapidity is an alternative to velocity as a method of measuring motion. At low speeds, rapidity and velocity are proportional, but for high speeds, rapidity takes a larger value than velocity. The rapidity of light is infinite.

The rapidity $φ$ 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 velocity 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. For this reason, some authors define rapidity to be cφ, giving it the same units as velocity.

The rapidity concept was initially identified by Alfred Robb; his idea was acknowledged by Silberstein (1914) 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 $φ P Q$ 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. 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.