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| In [[Theory of relativity|relativity]], '''rapidity''' is an alternative to [[speed]] as a measure of motion. On [[parallel (geometry)|parallel]] velocities (say, in one-dimensional space) rapidities are simply additive, unlike [[Velocity-addition_formula#Special_theory_of_relativity|speeds at relativistic velocities]]. For low speeds, rapidity and speed are proportional, but for high speeds, rapidity takes a larger value. The rapidity of light is infinite.
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| Using the [[inverse hyperbolic function]] {{math|tanh<sup>−1</sup>}}, the rapidity {{math|<var>φ</var>}} corresponding to velocity {{math|<var>v</var>}} is {{math|<var>φ</var> {{=}} tanh<sup>−1</sup>(<var>v</var> / <var>c</var>)}}. For low speeds, {{math|<var>φ</var>}} is approximately {{math|<var>v</var> / <var>c</var>}}. The speed of light c being finite, any velocity v is constrained to the interval {{math|−<var>c</var> < <var>v</var> < <var>c</var>}} and the ratio v/c satisfies {{math|−1 < <var>v</var> / <var>c</var> < 1}}. Since the inverse hyperbolic tangent has the unit interval {{math|(−1, 1)}} for its [[domain (mathematics)|domain]] and the whole [[real line]] for its [[range (function)|range]], the interval {{math|−<var>c</var> < <var>v</var> < <var>c</var>}} becomes mapped onto {{math|−∞ < <var>φ</var> < ∞}}.
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| 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.
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| ==History==
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| In 1908 [[Hermann Minkowski]] explained how the [[Lorentz transformation]] could be seen as simply a [[hyperbolic rotation]] of the [[Coordinate time|spacetime coordinates]], i.e., a rotation through an imaginary angle, and this angle therefore represents (in one spatial dimension) a simple additive measure of the velocity between frames.<ref>Minkowski, H., Fundamental Equations for Electromagnetic Processes in Moving Bodies", 1908</ref> Minkowski's angle of rotation was given the name "rapidity" in 1911 by [[Alfred Robb]], and this term was adopted by many subsequent authors, such as [[Vladimir Varićak|Varićak]] (1912), [[Ludwik Silberstein|Silberstein]] (1914), [[Arthur Eddington|Eddington]] (1924), [[Edward Morley|Morley]] (1936) and [[Wolfgang Rindler|Rindler]] (2001).
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| ==In one spatial dimension==
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| The rapidity {{math|<var>φ</var>}} arises in the linear representation of a [[Lorentz boost]] as a vector-matrix product
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| :<math>
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| \begin{pmatrix}
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| c t' \\
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| x'
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| \end{pmatrix}
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| =
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| \begin{pmatrix}
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| \cosh \varphi & - \sinh \varphi \\
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| - \sinh \varphi & \cosh \varphi
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| \end{pmatrix}
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| \begin{pmatrix}
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| ct \\
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| x
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| \end{pmatrix}
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| = \mathbf \Lambda (\varphi) \mathbf v</math>.
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| The matrix {{math|'''Λ'''(<var>φ</var>)}} is of the type <math>\begin{pmatrix} p & q \\ q & p \end{pmatrix} </math> with {{math|<var>p</var>}} and {{math|<var>q</var>}} satisfying {{math|<var>p</var><sup>2</sup> − <var>q</var><sup>2</sup> {{=}} 1}}, so that {{math|(<var>p</var>, <var>q</var>)}} lies on the [[unit hyperbola]]. Such matrices form a multiplicative [[group (mathematics)|group]].
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| It is not hard to prove that
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| :<math>\mathbf{\Lambda}(\varphi_1 + \varphi_2) = \mathbf{\Lambda}(\varphi_1)\mathbf{\Lambda}(\varphi_2)</math>.
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| This establishes the useful additive property of rapidity: if <var>A</var>, <var>B</var> and <var>C</var> are frames of reference and they all lie on the same straight line, then
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| :<math>\varphi_{AC}= \varphi_{AB} + \varphi_{BC}</math>
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| where {{math|<var>φ</var><sub><var>P</var><var>Q</var></sub>}} denotes the rapidity of a frame of reference <var>Q</var> relative to a frame of reference <var>P</var>. The simplicity of this formula contrasts with the complexity of the corresponding [[velocity-addition formula#Special theory of relativity|velocity-addition formula]].
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| The [[Lorentz factor]] identifies with {{math|cosh <var>φ</var>}}:
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| :<math>\gamma = \frac{1}{\sqrt{1 - v^2 / c^2}} \equiv \cosh \varphi</math>,
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| so the rapidity {{math|<var>φ</var>}} is implicitly used as a hyperbolic angle in the [[Lorentz transformation]] expressions using {{math|<var>γ</var>}} and <var>β</var>.
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| [[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.
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| ==In more than one spatial dimension==
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| {{see also|Hyperboloid model}}
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| From mathematical point of view, possible values of relativistic velocity form a manifold, where the [[metric tensor]] corresponds to the proper acceleration (see above). This is a non-flat space (namely, a [[hyperbolic space]]), and rapidity is just the [[metric space|distance]] from the given velocity to the zero velocity in given frame of reference. Although it is possible, as noted above, to add and subtract rapidities where corresponding relative velocities are parallel, in the general case the rapidity-addition formula is more complex because of negative [[scalar curvature|curvature]].<!-- BTW could somebody to write this explicitly? --> For example, the result of "addition" of two perpendicular motions with rapidities {{math|<var>φ<var><sub>1</sub>}} and {{math|<var>φ<var><sub>2</sub>}} will be greater than <math>\sqrt{\varphi_1^2 + \varphi_2^2}</math> expected by [[Pythagorean theorem]]. Rapidity in two dimensions can be usefully visualized using the [[Poincaré disk model|Poincaré disk]].<ref>John A. Rhodes & Mark D. Semon (2003) [http://www.bates.edu/%7Emsemon/RhodesSemonFinal.pdf "Relativistic velocity space, Wigner roation, and Thomas precession"], [[American Journal of Physics]] 72(7):943–61</ref> 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.
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| ==In experimental particle physics==
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| The energy {{math|<var>E</var>}} and scalar momentum {{math|{{!}} '''p''' {{!}}}} of a particle of non-zero (rest) mass {{math|<var>m</var>}} are given by
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| :<math>E = m c^2 \cosh \varphi </math>
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| :<math>| \mathbf p | = m c \, \sinh \varphi </math>
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| and so rapidity can be calculated from measured energy and momentum by
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| :<math>\varphi = \tanh^{-1} \frac{| \mathbf p | c}{E}= \frac{1}{2} \ln \frac{E + | \mathbf p | c}{E - | \mathbf p | c} </math>
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| However, experimental particle physicists often use a modified definition of rapidity relative to a beam axis
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| :<math>y = \frac{1}{2} \ln \frac{E + p_z c}{E - p_z c} </math>
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| where {{math|<var>p</var><sub>''z''</sub>}} is the component of momentum along the beam axis.<ref>Amsler, C. ''et al.'', [http://pdg.lbl.gov/2009/reviews/rpp2009-rev-kinematics.pdf "The Review of Particle Physics"], ''Physics Letters B'' '''667''' (2008) 1, Section 38.5.2</ref> 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]].
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| ==See also==
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| * [[Theory of relativity]]
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| * [[Lorentz transformation]]
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| * [[Pseudorapidity]]
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| ==Notes and references==
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| {{Reflist}}
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| * {{Cite document | author=Whittaker, E.T. | year=1910 | title= 1. Edition: [http://www.archive.org/details/historyoftheorie00whitrich A History of the theories of aether and electricity] | place =Dublin |publisher=Longman, Green and Co. | postscript=.}} page 441
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| *{{Cite book|last=Robb|first=Alfred|authorlink=Alfred Robb|year=1911|title=Optical geometry of motion, a new view of the theory of relativity|location=Cambridge|publisher=Heffner & Sons|url=http://www.archive.org/details/opticalgeometryo00robbrich}}
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| * [[Émile Borel|Borel E]] (1913) La théorie de la relativité et la cinématique, Comptes Rendus Acad Sci Paris 156 215-218; 157 703-705
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| *{{Cite book|last=Silberstein|first=Ludwik|authorlink=Ludwik Silberstein|year=1914|title=The Theory of Relativity|location=London|publisher=Macmillan & Co.|url=http://www.archive.org/details/theoryofrelativi00silbrich}}
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| * [[Arthur Stanley Eddington]] (1924) [[List of publications in physics#The Mathematical Theory of Relativity|The Mathematical Theory of Relativity]], 2nd edition, [[Cambridge University Press]], p. 22.
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| * [[Vladimir Karapetoff]] (1936) "Restricted relativity in terms of hyperbolic functions of rapidities", [[American Mathematical Monthly]] 43:70.
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| * [[Frank Morley]] (1936) "When and Where", ''The Criterion'', edited by [[T.S. Eliot]], 15:200-2009.
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| * [[Wolfgang Rindler]] (2001) ''Relativity: Special, General, and Cosmological'', page 53, [[Oxford University Press]].
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| * Shaw, Ronald (1982) ''Linear Algebra and Group Representations'', v. 1, page 229, [[Academic Press]] ISBN 0-12-639201-3.
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| *{{Cite book|author=Walter, Scott|year=1999|contribution=The non-Euclidean style of Minkowskian relativity|editor=J. Gray|title=The Symbolic Universe: Geometry and Physics|pages=91–127|publisher=Oxford University Press|contribution-url=http://www.univ-nancy2.fr/DepPhilo/walter/papers/nes.pdf}}(see page 17 of e-link)
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| * [[Vladimir Varićak|Varićak V]] (1910), (1912), (1924) See [[Vladimir Varićak#Publications]]
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| [[Category:Special relativity]]
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| [[pl:Pospieszność]]
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She is known by the title of Myrtle Shryock. Since she was eighteen she's been operating as a receptionist but her promotion by no means arrives. To collect badges is what her family and her enjoy. Minnesota is exactly where he's been living for many years.
Take a look at my blog post ... at home std test