Rapidity

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In relativity, rapidity is an alternative to speed as a measure of rate 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.

Using the inverse hyperbolic function artanh, the rapidity φ corresponding to velocity v is φ = artanh(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 maps 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.[1] This angle therefore represents (in one spatial dimension) a simple additive measure of the velocity between frames.[2] The rapidity was used by Whittaker 1910 [3] and by Varićak 1910. The name "rapidity" is due to Alfred Robb 1911 [4] and this term was adopted by many subsequent authors, such as Varićak 1912, Silberstein (1914), Morley (1936) and Rindler (2001). The development of the theory of rapidity is mainly due to Varićak in writings from 1910 to 1924.[5]

In one spatial dimension

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

$\begin{pmatrix} c t' \\ x' \end{pmatrix} = \begin{pmatrix} \cosh \varphi & \sinh \varphi \\ \sinh \varphi & \cosh \varphi \end{pmatrix} \begin{pmatrix} ct \\ x \end{pmatrix} = \mathbf \Lambda (\varphi) \begin{pmatrix} ct \\ x \end{pmatrix}$.

The matrix Λ(φ) is of the type $\begin{pmatrix} p & q \\ q & p \end{pmatrix}$ with p and q satisfying p2q2 = 1, so that (p, q) lies on the unit hyperbola. Such matrices form the indefinite orthogonal group O(1,1) with one-dimensional Lie algebra spanned by the anti-diagonal unit matrix, showing that the rapidity is the coordinate on this Lie algebra. This action may be depicted in a spacetime diagram. In matrix exponential notation, Λ(φ) can be expressed as $\mathbf \Lambda (\varphi) = e^{\mathbf Z \varphi}$, where Z is the anti-diagonal unit matrix

$\mathbf Z = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} .$

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 A, B and C are frames of reference, then

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

where φPQ denotes the rapidity of a frame of reference Q relative to a frame of reference P. The simplicity of this formula contrasts with the complexity of the corresponding velocity-addition formula.

As we can see from the Lorentz transformation above, the Lorentz factor identifies with cosh φ

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

so the rapidity φ is implicitly used as a hyperbolic angle in the Lorentz transformation expressions using γ and β. We relate rapidities to the velocity-addition formula

$u=(u_1+u_2)/(1+u_1u_2/c^2)$

by recognizing

$\beta_i = \frac{u_i}{c} = \tanh{\varphi_i}$

and so

\begin{align} \tanh\varphi &= \frac{\tanh\varphi_1 +\tanh\varphi_2}{1+\tanh\varphi_1\tanh\varphi_2} \\ &= \tanh(\varphi_1+\varphi_2) \end{align}

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.

The product of β and γ appears frequently, and is from the above arguments

$\beta \gamma = \sinh\varphi \,.$

Exponential and logarithmic relations

From the above expressions we have

$e^{\varphi} = \gamma(1+\beta) = \gamma \left( 1 + \frac{v}{c} \right) = \sqrt \frac{1 + \tfrac{v}{c}}{1 - \tfrac{v}{c}},$

and thus

$e^{-\varphi} = \gamma(1-\beta) = \gamma \left( 1 - \frac{v}{c} \right) = \sqrt \frac{1 - \tfrac{v}{c}}{1 + \tfrac{v}{c}}.$

or explicitly

$\varphi = \ln \left[\gamma(1+\beta)\right] = -\ln \left[\gamma(1-\beta)\right] \, .$

The Doppler-shift factor associated with rapidity φ is $k = e^\varphi$.

In more than one spatial dimension

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 the distance from the given velocity to the zero velocity in the 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 the Pythagorean theorem. Rapidity in two dimensions can be usefully visualized using the Poincaré disk.[6] 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:

$E = m \gamma c^2$
$| \mathbf p | = m \gamma v.$

With the definition of φ

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

and thus with

$\cosh \varphi = \cosh \left( \operatorname{artanh} \frac{v}{c} \right) = \frac {1}{ \sqrt { 1- \frac{v^2}{c^2} }} = \gamma$
$\sinh \varphi = \sinh \left( \operatorname{artanh} \frac{v}{c} \right) = \frac {\frac{v}{c}}{ \sqrt { 1- \frac{v^2}{c^2} }} = \gamma \frac{v}{c},$

the energy and scalar momentum can be written as:

$E = m c^2 \cosh \varphi$
$| \mathbf p | = m c \, \sinh \varphi.$

So rapidity can be calculated from measured energy and momentum by

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

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

$y = \frac{1}{2} \ln \frac{E + p_z c}{E - p_z c}$

where pz is the component of momentum along the beam axis.[7] 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.

Notes and references

1. Minkowski, H., Fundamental Equations for Electromagnetic Processes in Moving Bodies" 1908
2. Sommerfeld, Phys. Z 1909
3. "A History of the Theories of the Aether and Electricity" 1910. In a later edition of this book in 1953 the rapidity is used consistently for the theory
4. "Optical Geometry of Motion" p.9
5. See his papers, available in translation in Wikisource
6. John A. Rhodes & Mark D. Semon (2003) "Relativistic velocity space, Wigner rotation, and Thomas precession", American Journal of Physics 72(7):943–961
7. Amsler, C. et al., "The Review of Particle Physics", Physics Letters B 667 (2008) 1, Section 38.5.2