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 of an object relative to a frame of reference is the hyperbolic angle φ such that tanh φ = v/c where v is the velocity of the object relative to the same frame of reference, c is the velocity of light, and tanh is the 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 p2 − q2 = 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 φPQ denotes the rapidity of Q relative to P, then

φAC = φAB + φBC,

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.

Mathematically, the rapidity can be viewed as a re-linearization of the velocity, since the naively linear v becomes absurd as v approaches c. This re-linearization constrains the future to a quadrant of a spacetime plane.

References

Wikipedia content modification information:

  • This page was last modified on 12 November 2008, at 13:34.

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