Rapidity

In relativity, rapidity is commonly used as a measure for relativistic velocity. 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.

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 $-\infty < w < \infty$ .

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] It was used by Varićak (1910) and by Whittaker (1910).^[3] It was named "rapidity" by 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]

Area of a hyperbolic sector

The quadrature of the hyperbola xy = 1 by Gregoire de Saint-Vincent established the natural logarithm as the area of a hyperbolic sector, or an equivalent area against an asymptote. In spacetime theory, the connection of events by light divides the universe into Past, Future, or Elsewhere based on a Here and Now. On any line in space, a light beam may be directed left or right. Take the x-axis as the events passed by the right beam and the y-axis as the events of the left beam. Then a resting frame has time along the diagonal x = y. The rectangular hyperbola xy = 1 can be used to gauge velocities (in the first quadrant). Zero velocity corresponds to (1,1). Any point on the hyperbola has coordinates $(e^{w},\ e^{-w})$ where w is the rapidity, and is equal to the area of the hyperbolic sector from (1,1) to these coordinates. Many authors refer instead to the unit hyperbola $x^{2}-y^{2},$ using rapidity for 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 hyperbolic parameter of beam-space is a reference to the seventeenth century origin of our precious transcendental functions, and a supplement to spacetime diagramming.

In one spatial dimension

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

{\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 $p 2 - q 2 = 1$ , 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 $\mathbf {\Lambda } (w)=e^{\mathbf {Z} w}$ , 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

\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

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 $cosh w$

\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

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

and so

{\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

\beta \gamma =\sinh w\,.

Exponential and logarithmic relations

From the above expressions we have

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

and thus

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

or explicitly

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

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

In more than one spatial dimension

The relativistic velocity ${\boldsymbol {\beta }}$ is associated to the rapidity $\mathbf {w}$ of an object via^[6]

{\mathfrak {so}}(3,1)\supset \mathrm {span} \{K_{1},K_{2},K_{3}\}\approx \mathbb {R} ^{3}\ni \mathbf {w} ={\boldsymbol {\hat {\beta }}}\tanh ^{-1}\beta ,\quad {\boldsymbol {\beta }}\in \mathbb {B} ^{3},

where the vector $\mathbf {w}$ is thought of as Cartesian coordinates on the 3-dimensional subspace of the Lie algebra ${\mathfrak {o}}(3,1)\approx {\mathfrak {so}}(3,1)$ of the Lorentz group spanned by the boost generators $K_{1},K_{2},K_{3}$ – in complete analogy with the one-dimensional case ${\mathfrak {o}}(1,1)$ discussed above – and velocity space is represented by the open ball $\mathbb {B} ^{3}$ with radius $1$ since $|\beta |<1$ . The latter follows from that $c$ is a limiting velocity in relativity (with units in which $c=1$ ).

The general formula for composition of rapidities is^[7]^{[nb 1]}

\mathbf {w} ={\boldsymbol {\hat {\beta }}}\tanh ^{-1}\beta ,\quad {\boldsymbol {\beta }}={\boldsymbol {\beta }}_{1}\oplus {\boldsymbol {\beta }}_{2},

where ${\boldsymbol {\beta }}_{1}\oplus {\boldsymbol {\beta }}_{2}$ refers to relativistic velocity addition and ${\boldsymbol {\hat {\beta }}}$ is a unit vector in the direction of ${\boldsymbol {\beta }}$ . This operation is not commutative nor associative. Rapidities $\mathbf {w} _{1},\mathbf {w} _{2}$ with directions inclined at an angle $\theta$ have a resultant norm $w\equiv |\mathbf {w} |$ (ordinary Euclidean length) given by the hyperbolic law of cosines,^[8]

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

The geometry on rapidity space is inherited from the hyperbolic geometry on velocity space via the map stated. This geometry, in turn, can be inferred from the addition law of relativistic velocities.^[9] Rapidity in two dimensions can thus be usefully visualized using the Poincaré disk.^[10] Geodesics correspond to steady accelerations. Rapidity space in three dimensions can in the same way be put in isometry with the hyperboloid model (isometric to the $3$ -dimensional Poincaré disk (or ball)). This is detailed in geometry of Minkowski space.

The addition of two rapidities results not only in a new rapidity; the resultant total transformation is the composition of the transformation corresponding to the rapidity given above and a rotation parametrized by the vector ${\boldsymbol {\theta }}$ ,

\Lambda =e^{-i{\boldsymbol {\theta }}\cdot \mathbf {J} }e^{-i\mathbf {w} \cdot \mathbf {K} },

where the physicist convention for the exponential mapping is employed. This is a consequence of the commutation rule

[K_{i},K_{j}]=-i\epsilon _{ijk}J_{k},

where $J_{k},k=1,2,3,$ are the generators of rotation. This is related to the phenomenon of Thomas precession. For the computation of the parameter ${\boldsymbol {\theta }}$ , the linked article is referred to.

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}

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

With the definition of $w$

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

and thus with

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

\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:

E=mc^{2}\cosh w

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

So rapidity can be calculated from measured energy and momentum by

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.^[11] 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.

Remarks

↑ This is to be understood in the sense that given two velocities, the resulting rapidity is the rapidity corresponding to the two velocities relativistically added. Rapidities also have the ordinary addition inherited from $\mathbb {R} ^{3}$ , and context decides which operation to use.

Notes and references

↑ Minkowski, H., Fundamental Equations for Electromagnetic Processes in Moving Bodies" 1908
↑ Sommerfeld, Phys. Z 1909
↑ "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
↑ "Optical Geometry of Motion" p.9
↑ See his papers, available in translation in Wikisource
↑ Jackson 1999, p. 547
↑ Rhodes & Semon 2003
↑ Robb 1910, Varićak 1910, Borel 1913
↑ Landau & Lifshitz 2002, Problem p. 38
↑ Rhodes & Semon 2003
↑ Amsler, C. et al., "The Review of Particle Physics", Physics Letters B 667 (2008) 1, Section 38.5.2

Varićak V (1910), (1912), (1924) See Vladimir Varićak#Publications
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)
Rhodes, J. A.; Semon, M. D. (2004). "Relativistic velocity space, Wigner rotation, and Thomas precession". Am. J. Phys. 72: 93–90. arXiv:gr-qc/0501070. Bibcode:2004AmJPh..72..943R. doi:10.1119/1.1652040.
Jackson, J. D. (1999) [1962]. "Chapter 11". Classical Electrodynamics (3d ed.). John Wiley & Sons. ISBN 0-471-30932-X.

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[8] This is to be understood in the sense that given two velocities, the resulting rapidity is the rapidity corresponding to the two velocities relativistically added. Rapidities also have the ordinary addition inherited from $\mathbb {R} ^{3}$ , and context decides which operation to use.

[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" In a later 1953 edition of this book it was used consistently for the theory

[4] "Optical Geometry of Motion" p.9

[5] See his papers, available in translation in Wikisource

[6] Jackson 1999, p. 547

[7] Rhodes & Semon 2003

[9] Robb 1910, Varićak 1910, Borel 1913

[10] Landau & Lifshitz 2002, Problem p. 38

[11] Rhodes & Semon 2003

[12] Amsler, C. et al., "The Review of Particle Physics", Physics Letters B 667 (2008) 1, Section 38.5.2