Rapidity Explained

In special relativity, the classical concept of velocity is converted to rapidity to accommodate the limit determined by speed of light. Velocities must be combined by Einstein's velocity-addition formula. For low speeds, rapidity and velocity are almost exactly proportional but, for higher velocities, rapidity takes a larger value, with the rapidity of light being infinite.

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.

Using the inverse hyperbolic function, the rapidity corresponding to velocity is where is the velocity of light. For low speeds, is approximately . Since in relativity any velocity is constrained to the interval the ratio satisfies . The inverse hyperbolic tangent has the unit interval for its domain and the whole real line for its image; that is, the interval maps onto .

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 parameter replacing velocity was introduced in 1910 by Vladimir Varićak^[3] and by E. T. Whittaker.^[4] The parameter was named rapidity by Alfred Robb (1911)^[5] and this term was adopted by many subsequent authors, such as Ludwik Silberstein (1914), Frank Morley (1936) and Wolfgang Rindler (2001).

Area of a hyperbolic sector

(e^w, e^-w)

where is the rapidity, and is equal to the area of the hyperbolic sector from to these coordinates. Many authors refer instead to the unit hyperbola

x²-y²

, using rapidity for a parameter, as in the standard spacetime diagram. There the axes are measured by clock and meter-stick, more familiar benchmarks, and the basis of spacetime theory. So the delineation of rapidity as a hyperbolic parameter of beam-space is a reference to the seventeenth-century origin of our precious transcendental functions, and a supplement to spacetime diagramming.

Lorentz boost

The rapidity arises in the linear representation of a Lorentz boost as a vector-matrix product $\begin c t' \\ x' \end = \begin \cosh w & -\sinh w \\ -\sinh w & \cosh w \end \begin ct \\ x \end = \mathbf \Lambda (w) \begin ct \\ x \end.$

The matrix is of the type

\begin{pmatrix}p&q\ q&p\end{pmatrix}

with and satisfying, so that 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, can be expressed as

Λ(w)=e^Zw

, where is the negative of the anti-diagonal unit matrix

\mathbf Z = \begin 0 & -1 \\ -1 & 0 \end .

A key property of the matrix exponential is

e^Xs+Xt=e^Xse^Xt

from which immediately follows that

\mathbf(w_1 + w_2) = \mathbf(w_1)\mathbf(w_2).

This establishes the useful additive property of rapidity: if, and are frames of reference, then

w_= w_ + w_,

where denotes the rapidity of a frame of reference relative to a frame of reference . 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 $\gamma = \frac \equiv \cosh w,$ so the rapidity is implicitly used as a hyperbolic angle in the Lorentz transformation expressions using and . We relate rapidities to the velocity-addition formula $u = \frac$ by recognizing $\beta_i = \frac = \tanh$ and so $\tanh w = \frac= \tanh(w_1+ w_2)$

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 and appears frequently, and is from the above arguments $\beta \gamma = \tanh w \cosh w = \sinh w$

Exponential and logarithmic relations

From the above expressions we have $e^ = \gamma(1 + \beta) = \gamma \left(1 + \frac \right) = \sqrt \frac,$ and thus $e^ = \gamma(1 - \beta) = \gamma \left(1 - \frac \right) = \sqrt \frac.$ or explicitly $w = \ln \left[\gamma(1 + \beta)\right] = -\ln \left[\gamma(1 - \beta)\right] \, .$

The Doppler-shift factor associated with rapidity is

k=e^w

In experimental particle physics

The energy and scalar momentum of a particle of non-zero (rest) mass are given by: $\beginE &= \gamma mc^2 \\\left| \mathbf p \right| &= \gamma mv.\end$ With the definition of $w = \operatorname \frac,$ and thus with $\cosh w = \cosh \left(\operatorname \frac \right) = \frac = \gamma$ $\sinh w = \sinh \left(\operatorname \frac \right) = \frac = \beta \gamma,$ the energy and scalar momentum can be written as: $\beginE &= m c^2 \cosh w \\\left| \mathbf p \right| &= m c \, \sinh w.\end$

So, rapidity can be calculated from measured energy and momentum by $w = \operatorname \frac$

= \frac \ln \frac = \ln \frac ~.

However, experimental particle physicists often use a modified definition of rapidity relative to a beam axis $y = \frac \ln \frac,$ where is the component of momentum along the beam axis.^[6] 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.

Rapidity relative to a beam axis can also be expressed as $y = \ln \frac ~.$

Notes and references

- Vladimir Varićak (1910, 1912, 1924), see Vladimir Varićak#Publications Book: Whittaker . Edmund Taylor . E. T. Whittaker . . 1910 . 441.
Book: Robb, Alfred . Alfred Robb . 1911 . Optical geometry of motion, a new view of the theory of relativity . Cambridge . Heffner & Sons .
Émile Borel (1913), La théorie de la relativité et la cinématique (in French), Comptes rendus de l'Académie des Sciences, Paris: volume 156, pages 215-218; volume 157, pages 703-705
Book: Silberstein, Ludwik . Ludwik Silberstein . 1914 . The Theory of Relativity . London . Macmillan & Co. .
Vladimir Karapetoff (1936), "Restricted relativity in terms of hyperbolic functions of rapidities", American Mathematical Monthly, volume 43, page 70.
Frank Morley (1936), "When and Where", The Criterion, edited by Thomas Stearns Eliot, volume 15, pages 200-209.
Wolfgang Rindler (2001) Relativity: Special, General, and Cosmological, page 53, Oxford University Press.
Shaw, Ronald (1982) Linear Algebra and Group Representations, volume 1, page 229, Academic Press .
Book: Walter, Scott . 1999 . The non-Euclidean style of Minkowskian relativity . Jeremy John Gray . The Symbolic Universe: Geometry and Physics . 91–127 . Oxford University Press . http://www.univ-nancy2.fr/DepPhilo/walter/papers/nes.pdf . 2009-01-08 . 2013-10-16 . https://web.archive.org/web/20131016142709/http://www.univ-nancy2.fr/DepPhilo/walter/papers/nes.pdf . dead . (see page 17 of e-link)
Rhodes . John A. . Semon . Mark D. . 2004 . Relativistic velocity space, Wigner rotation, and Thomas precession . American Journal of Physics . 72 . 7 . 90–93 . 10.1119/1.1652040 . gr-qc/0501070 . 2004AmJPh..72..943R . 14764378.
Book: Jackson, John David . John David Jackson (physicist) . Classical Electrodynamics . registration . 3rd . 1999 . 1962 . 0-471-30932-X . . Chapter 11.

Notes and References

[Hermann Minkowski]
Sommerfeld, Phys. Z 1909
[Vladimir Varicak]
[E. T. Whittaker]
[Alfred Robb]
Amsler, C. et al., "The Review of Particle Physics", Physics Letters B 667 (2008) 1, Section 38.5.2