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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 −∞ < w < ∞.

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 Silberstein (1914), Morley (1936) and Rindler (2001).

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 (ew, e−w)displaystyle (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 x2−y2,displaystyle 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

(ct′x′)=(cosh⁡w−sinh⁡w−sinh⁡wcosh⁡w)(ctx)=Λ(w)(ctx)displaystyle beginpmatrixct'\x'endpmatrix=beginpmatrixcosh w&-sinh w\-sinh w&cosh wendpmatrixbeginpmatrixct\xendpmatrix=mathbf Lambda (w)beginpmatrixct\xendpmatrix.

The matrix Λ(w) is of the type (pqqp)displaystyle beginpmatrixp&q\q&pendpmatrix 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 Λ(w)=eZwdisplaystyle mathbf Lambda (w)=e^mathbf Z w, where Z is the anti-diagonal unit matrix

Z=(0110).displaystyle mathbf Z =beginpmatrix0&1\1&0endpmatrix.

It is not hard to prove that

Λ(w1+w2)=Λ(w1)Λ(w2)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

wAC=wAB+wBCdisplaystyle w_textAC=w_textAB+w_textBC

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

γ=11−v2/c2≡cosh⁡wdisplaystyle gamma =frac 1sqrt 1-v^2/c^2equiv 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=(u1+u2)/(1+u1u2/c2)displaystyle u=(u_1+u_2)/(1+u_1u_2/c^2)

by recognizing

βi=uic=tanh⁡widisplaystyle beta _i=frac u_ic=tanh w_i

and so

tanh⁡w=tanh⁡w1+tanh⁡w21+tanh⁡w1tanh⁡w2=tanh⁡(w1+w2)displaystyle beginalignedtanh w&=frac tanh w_1+tanh w_21+tanh w_1tanh w_2\&=tanh(w_1+w_2)endaligned

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

βγ=sinh⁡w.displaystyle beta gamma =sinh w,.

Exponential and logarithmic relations

From the above expressions we have

ew=γ(1+β)=γ(1+vc)=1+vc1−vc,displaystyle e^w=gamma (1+beta )=gamma left(1+frac vcright)=sqrt frac 1+tfrac vc1-tfrac vc,

and thus

e−w=γ(1−β)=γ(1−vc)=1−vc1+vc.displaystyle e^-w=gamma (1-beta )=gamma left(1-frac vcright)=sqrt frac 1-tfrac vc1+tfrac vc.

or explicitly

w=ln⁡[γ(1+β)]=−ln⁡[γ(1−β)].displaystyle w=ln left[gamma (1+beta )right]=-ln left[gamma (1-beta )right],.

The Doppler-shift factor associated with rapidity w is k=ewdisplaystyle k=e^w.

In more than one spatial dimension

The relativistic velocity βdisplaystyle boldsymbol beta is associated to the rapidity wdisplaystyle mathbf w of an object via^[6]

so(3,1)⊃spanK1,K2,K3≈R3∋w=β^tanh−1⁡β,β∈B3,displaystyle mathfrak so(3,1)supset mathrm span K_1,K_2,K_3approx mathbb R ^3ni mathbf w =boldsymbol hat beta tanh ^-1beta ,quad boldsymbol beta in mathbb B ^3,

where the vector wdisplaystyle mathbf w is thought of as Cartesian coordinates on the 3-dimensional subspace of the Lie algebra o(3,1)≈so(3,1)displaystyle mathfrak o(3,1)approx mathfrak so(3,1) of the Lorentz group spanned by the boost generators K1,K2,K3displaystyle K_1,K_2,K_3 – in complete analogy with the one-dimensional case o(1,1)displaystyle mathfrak o(1,1) discussed above – and velocity space is represented by the open ball B3displaystyle mathbb B ^3 with radius 1displaystyle 1 since |β|<1displaystyle . The latter follows from that cdisplaystyle c is a limiting velocity in relativity (with units in which c=1displaystyle c=1).

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

w=β^tanh−1⁡β,β=β1⊕β2,displaystyle mathbf w =boldsymbol hat beta tanh ^-1beta ,quad boldsymbol beta =boldsymbol beta _1oplus boldsymbol beta _2,

where β1⊕β2displaystyle boldsymbol beta _1oplus boldsymbol beta _2 refers to relativistic velocity addition and β^displaystyle boldsymbol hat beta is a unit vector in the direction of βdisplaystyle boldsymbol beta . This operation is not commutative nor associative. Rapidities w1,w2displaystyle mathbf w _1,mathbf w _2 with directions inclined at an angle θdisplaystyle theta have a resultant norm w≡|w| (ordinary Euclidean length) given by the hyperbolic law of cosines,^[8]

cosh⁡w=cosh⁡w1cosh⁡w2+sinh⁡w1sinh⁡w2cos⁡θ.displaystyle cosh w=cosh w_1cosh w_2+sinh w_1sinh w_2cos 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 θdisplaystyle boldsymbol theta ,

Λ=e−iθ⋅Je−iw⋅K,displaystyle Lambda =e^-iboldsymbol theta cdot mathbf J e^-imathbf w cdot mathbf K ,

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

[Ki,Kj]=−iϵijkJk,displaystyle [K_i,K_j]=-iepsilon _ijkJ_k,

where Jk,k=1,2,3,displaystyle 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 θdisplaystyle 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=γmc2displaystyle E=gamma mc^2

|p|=γmv.displaystyle

With the definition of w

w=artanh⁡vc,displaystyle w=operatorname artanh frac vc,

and thus with

cosh⁡w=cosh⁡(artanh⁡vc)=11−v2c2=γdisplaystyle cosh w=cosh left(operatorname artanh frac vcright)=frac 1sqrt 1-frac v^2c^2=gamma

sinh⁡w=sinh⁡(artanh⁡vc)=vc1−v2c2=βγ,displaystyle sinh w=sinh left(operatorname artanh frac vcright)=frac frac vcsqrt 1-frac v^2c^2=beta gamma ,

the energy and scalar momentum can be written as:

E=mc2cosh⁡wdisplaystyle E=mc^2cosh w

|p|=mcsinh⁡w.displaystyle

So rapidity can be calculated from measured energy and momentum by

w=artanh⁡|p|cE=12ln⁡E+|p|cE−|p|cdisplaystyle w=operatorname artanh frac E=frac 12ln frac E+E-

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

y=12ln⁡E+pzcE−pzcdisplaystyle y=frac 12ln frac E+p_zcE-p_zc

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.

See also

• Bondi k-calculus


• Lorentz transformation


• Pseudorapidity


• Proper velocity


• Theory of relativity

Remarks

Notes and references

• 
  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..mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
  


• 
  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.