Rapidity

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).

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 ${\displaystyle {\begin{pmatrix}p&q\\q&p\end{pmatrix}}}$ 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 negative of the anti-diagonal unit matrix

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

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

$${\displaystyle u={\frac {u_{1}+u_{2}}{1+{\frac {u_{1}u_{2}}{c^{2}}}}}}$$

by recognizing

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

and so

$${\displaystyle \tanh w={\frac {\tanh w_{1}+\tanh w_{2}}{1+\tanh w_{1}\tanh w_{2}}}=\tanh(w_{1}+w_{2})}$$

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 β and γ appears frequently, and is from the above arguments

$${\displaystyle \beta \gamma =\tanh w\cosh w=\sinh w}$$

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 ${\displaystyle k=e^{w}}$.

The energy E and scalar momentum |p| of a particle of non-zero (rest) mass m are given by:

$${\displaystyle {\begin{aligned}E&=\gamma mc^{2}\\\left|\mathbf {p} \right|&=\gamma mv.\end{aligned}}}$$

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:

$${\displaystyle {\begin{aligned}E&=mc^{2}\cosh w\\\left|\mathbf {p} \right|&=mc\,\sinh w.\end{aligned}}}$$

So, rapidity can be calculated from measured energy and momentum by

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

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

$${\displaystyle y={\frac {1}{2}}\ln {\frac {E+p_{z}c}{E-p_{z}c}},}$$

where p_z is the component of momentum along the beam axis.[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

$${\displaystyle y=\ln {\frac {E+p_{z}c}{\sqrt {m^{2}c^{4}+p_{T}^{2}c^{2}}}}~.}$$

• Bondi k-calculus
• Lorentz transformation
• Pseudorapidity
• Proper velocity
• Theory of relativity

• Vladimir Varićak (1910, 1912, 1924), see Vladimir Varićak#Publications
• Whittaker, Edmund Taylor (1910). "A History of the Theories of Aether and Electricity": 441. {{cite journal}}: Cite journal requires |journal= (help)
• Robb, Alfred (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
• 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, 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 ISBN 0-12-639201-3.
• Walter, Scott (1999). "The non-Euclidean style of Minkowskian relativity" (PDF). In Jeremy John Gray (ed.). The Symbolic Universe: Geometry and Physics. Oxford University Press. pp. 91–127. Archived from the original (PDF) on 2013-10-16. Retrieved 2009-01-08.(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. arXiv:gr-qc/0501070. Bibcode:2004AmJPh..72..943R. doi:10.1119/1.1652040. S2CID 14764378.
• Jackson, John David (1999) [1962]. "Chapter 11". Classical Electrodynamics (3rd ed.). John Wiley & Sons. ISBN 0-471-30932-X.