Rapidity

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.

In relativity, rapidity denoted by w is commonly used as a measure for relativistic velocity. For one-dimensional motion, rapidities are additive whereas velocities must be combined by Einstein's Velocity-addition formula. For low speeds, rapidity and velocity are proportional, but for higher velocities, rapidity takes a larger value, the rapidity of light being infinite.

Using the inverse hyperbolic function artanh, the rapidity w corresponding to velocity v is w = artanh(v / c) where c is the velocity of light. For low speeds, w is approximately v / c. Since in relativity any velocity v is constrained to the interval −c < v < c the ratio v / c satisfies −1 < v / c < 1. The inverse hyperbolic tangent has the unit interval (−1, 1) for its domain and the whole real line for its range, and so the interval −c < v < c maps onto −∞ < w < ∞.

Rapidity w does not have the dimensions of velocity. A more direct relation with ordinary velocity v comes from re-scaling it and defining the relativistic velocity V as cw.^[1] Then V reduces to ordinary velocity v when v ≪ c.

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.^[2] This angle therefore represents (in one spatial dimension) a simple additive measure of the velocity between frames.^[3] It was used by Varićak 1910 by Whittaker 1910 ^[4] It was named "rapidity" by Alfred Robb 1911 ^[5] 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.^[6]

In one spatial dimension

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

${\displaystyle {\begin{pmatrix}ct'\\x'\end{pmatrix}}={\begin{pmatrix}\cosh w&\sinh w\\\sinh w&\cosh w\end{pmatrix}}{\begin{pmatrix}ct\\x\end{pmatrix}}=\mathbf {\Lambda } (w){\begin{pmatrix}ct\\x\end{pmatrix}}}$.

The matrix Λ(w) is of the type ${\begin{pmatrix}p&q\\q&p\end{pmatrix}}$ with p and q satisfying p² − q² = 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(w) can be expressed as ${\displaystyle \mathbf {\Lambda } (w)=e^{\mathbf {Z} w}}$, 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

${\displaystyle \mathbf {\Lambda } (w_{1}+w_{2})=\mathbf {\Lambda } (w_{1})\mathbf {\Lambda } (w_{2})}$.

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

${\displaystyle w_{\text{AC}}=w_{\text{AB}}+w_{\text{BC}}}$

where w_PQ 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 w

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

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

$u=(u_{1}+u_{2})/(1+u_{1}u_{2}/c^{2})$

by recognizing

${\displaystyle \beta _{i}={\frac {u_{i}}{c}}=\tanh {w_{i}}}$

and so

${\displaystyle {\begin{aligned}\tanh w&={\frac {\tanh w_{1}+\tanh w_{2}}{1+\tanh w_{1}\tanh w_{2}}}\\&=\tanh(w_{1}+w_{2})\end{aligned}}}$

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

${\displaystyle \beta \gamma =\sinh w\,.}$

Exponential and logarithmic relations

From the above expressions we have

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

and thus

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

or explicitly

${\displaystyle w=\ln \left[\gamma (1+\beta )\right]=-\ln \left[\gamma (1-\beta )\right]\,.}$

The Doppler-shift factor associated with rapidity w is ${\displaystyle k=e^{w}}$.

In more than one spatial dimension

In more than one spatial dimension rapidities lie in a hyperbolic space having unit radius of negative curvature and they may be combined by the hyperbolic law of cosines. Rapidities ${\displaystyle w_{1},w_{2}}$ with directions inclined at an angle $\theta$ have a resultant rapidity $w$ given by

${\displaystyle \cosh w=\cosh w_{1}\cosh w_{2}+\sinh w_{1}\sinh w_{2}\cos \theta }$

This was one of the first results to be proved in the hyperbolic theory of special relativity.^[7] Note that the two rapidities must be added sequentially "head to tail"; it is not possible to combine rapidities starting from a common origin as can be done with velocity vectors in Euclidean space.^[8] This is the failure of equipollence in hyperbolic space.

Directed values of rapidity form a hyperbolic space of unit radius of negative curvature and can be represented by the hyperboloid model. The metric tensor corresponds to the proper acceleration (see above).

Rapidity in two dimensions can be usefully visualized using the Poincaré disk.^[9] 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:

$E=\gamma mc^{2}$

${\displaystyle |\mathbf {p} |=\gamma mv.}$

With the definition of w

${\displaystyle w=\operatorname {artanh} {\frac {v}{c}},}$

and thus with

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

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

the energy and scalar momentum can be written as:

${\displaystyle E=mc^{2}\cosh w}$

${\displaystyle |\mathbf {p} |=mc\,\sinh w.}$

So rapidity can be calculated from measured energy and momentum by

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

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 p_z is the component of momentum along the beam axis.^[10] 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.

See also

• Theory of relativity
• Lorentz transformation
• Pseudorapidity
• proper velocity

Notes and references

1. This was implicit in Varićak 1910, Borel 1913
2. Minkowski, H., Fundamental Equations for Electromagnetic Processes in Moving Bodies" 1908
3. Sommerfeld, Phys. Z 1909
4. "A History of the Theories of the Aether and Electricity" In a later 1953 edition of this book it was used consistently for the theory
5. "Optical Geometry of Motion" p.9
6. See his papers, available in translation in Wikisource
7. Robb 1910, Varićak 1910, Borel 1913
8. Silberstein L:The Theory of Relativity 1914 pp.178,179
9. John A. Rhodes & Mark D. Semon (2003) "Relativistic velocity space, Wigner rotation, and Thomas precession", American Journal of Physics 72(7):943–961
10. Amsler, C. et al., "The Review of Particle Physics", Physics Letters B 667 (2008) 1, Section 38.5.2

• Whittaker, E. T. (1910). "A history of the theories of aether and electricity": 441. Retrieved 22 January 2016.
• Robb, Alfred (1911). Optical geometry of motion, a new view of the theory of relativity. Cambridge: Heffner & Sons.
• Borel E (1913) La théorie de la relativité et la cinématique, Comptes Rendus Acad Sci Paris 156 215-218; 157 703-705
• Silberstein, Ludwik (1914). The Theory of Relativity. London: Macmillan & Co.
• Vladimir Karapetoff (1936) "Restricted relativity in terms of hyperbolic functions of rapidities", American Mathematical Monthly 43:70.
• Frank Morley (1936) "When and Where", The Criterion, edited by T.S. Eliot, 15:200-2009.
• Wolfgang Rindler (2001) Relativity: Special, General, and Cosmological, page 53, Oxford University Press.
• Shaw, Ronald (1982) Linear Algebra and Group Representations, v. 1, page 229, Academic Press ISBN 0-12-639201-3.
• Walter, Scott (1999). "The non-Euclidean style of Minkowskian relativity" (PDF). In J. Gray. The Symbolic Universe: Geometry and Physics. Oxford University Press. pp. 91–127.(see page 17 of e-link)
• Varićak V (1910), (1912), (1924) See Vladimir Varićak#Publications