{"history":[],"article":{"id":"3c0a0ce9-b27f-46af-8020-729c673dd551","title":"Rapidity","slug":"rapidity","content":":::summary\n:::\n\n# \n[[File:Inverse Hyperbolic Tangent.svg|right|thumb|250px|Rapidity is the value of\nartanh(v / c)\nfor velocity v and speed of light c]]\nIn [special relativity](https://bliptext.com/articles/special-relativity), the classical concept of [velocity](https://bliptext.com/articles/velocity) is converted to **rapidity** to accommodate the limit determined by the [speed of light](https://bliptext.com/articles/speed-of-light).\nVelocities must be combined by Einstein's [velocity-addition formula](https://bliptext.com/articles/velocity-addition-formula).\nFor low speeds, rapidity and velocity are almost exactly proportional but, for higher velocities, rapidity takes a larger value, with the rapidity of light being infinite.\nMathematically, rapidity can be defined as the [hyperbolic angle](https://bliptext.com/articles/hyperbolic-angle) that differentiates two [frames of reference](https://bliptext.com/articles/frames-of-reference) in relative motion, each frame being associated with [distance](https://bliptext.com/articles/distance) and [time](https://bliptext.com/articles/time) coordinates.\nUsing the [inverse hyperbolic function](https://bliptext.com/articles/inverse-hyperbolic-function)\nartanh\n, the rapidity w corresponding to velocity v is\nwhere c is the speed of light.\nFor low speeds, by the [small-angle approximation](https://bliptext.com/articles/small-angle-approximation), w is approximately\nv / c\n. Since in relativity any velocity v is constrained to the interval\n−c < v < c\nthe ratio\nv / c\nsatisfies\n−1 < v / c < 1\n. The inverse hyperbolic tangent has the unit interval\n(−1, 1)\nfor its [domain](https://bliptext.com/articles/domain-of-a-function) and the whole [real line](https://bliptext.com/articles/real-line) for its [image](https://bliptext.com/articles/image-mathematics); that is, the interval\n−c < v < c\nmaps onto\n−∞ < w < ∞\n.\n\n# History\nIn 1908 [Hermann Minkowski](https://bliptext.com/articles/hermann-minkowski) explained how the [Lorentz transformation](https://bliptext.com/articles/lorentz-transformation) could be seen as simply a [hyperbolic rotation](https://bliptext.com/articles/hyperbolic-rotation) of the [spacetime coordinates](https://bliptext.com/articles/coordinate-time), i.e., a rotation through an imaginary angle.\nThis angle therefore represents (in one spatial dimension) a simple additive measure of the velocity between frames.\nThe rapidity parameter replacing velocity was introduced in 1910 by [Vladimir Varićak](https://bliptext.com/articles/vladimir-vari-ak) and by [E. T. Whittaker](https://bliptext.com/articles/e-t-whittaker).\nThe parameter was named rapidity by [Alfred Robb](https://bliptext.com/articles/alfred-robb) (1911) and this term was adopted by many subsequent authors, such as [Ludwik Silberstein](https://bliptext.com/articles/ludwik-silberstein) (1914), [Frank Morley](https://bliptext.com/articles/frank-morley) (1936) and [Wolfgang Rindler](https://bliptext.com/articles/wolfgang-rindler) (2001).\n# [200px | upload.wikimedia.org/wikipedia/commons/c/cd/Hyperbolic///sector.svg]\n\n## Area of a hyperbolic sector\nThe [quadrature](https://bliptext.com/articles/quadrature-mathematics) of the hyperbola\nby [Grégoire de Saint-Vincent](https://bliptext.com/articles/gr-goire-de-saint-vincent) established the natural logarithm as the area of a hyperbolic sector or an equivalent area against an asymptote.\nIn spacetime theory, the connection of events by light divides the universe into Past, Future, or Elsewhere based on a Here and Now.\nOn any line in space, a light beam may be directed left or right.\nTake the x-axis as the events passed by the right beam and the y-axis as the events of the left beam.\nThen a resting frame has [time](https://bliptext.com/articles/time) along the diagonal\n. The rectangular hyperbola\ncan be used to gauge velocities (in the first quadrant).\nZero velocity corresponds to\n(1, 1)\n. Any point on the hyperbola has [light-cone coordinates](https://bliptext.com/articles/light-cone-coordinates) where w is the rapidity, and is equal to the area of the [hyperbolic sector](https://bliptext.com/articles/hyperbolic-sector) from\n(1, 1)\nto these coordinates.\nMany authors refer instead to the [unit hyperbola](https://bliptext.com/articles/unit-hyperbola) x^2 - y^2, using rapidity for a parameter, as in the standard [spacetime diagram](https://bliptext.com/articles/spacetime-diagram).\nThere the axes are measured by clock and meter-stick, more familiar benchmarks, and the basis of spacetime theory.\nSo the delineation of rapidity as a hyperbolic parameter of beam-space is a reference to the seventeenth-century origin of our precious [transcendental functions](https://bliptext.com/articles/transcendental-function), and a supplement to spacetime diagramming.\n\n# Lorentz boost\nThe rapidity w arises in the linear representation of a [Lorentz boost](https://bliptext.com/articles/lorentz-boost) as a vector-matrix product\nThe matrix\nΛ(w)\nis of the type with p and q satisfying\np2 – q2 = 1\n, so that\n(p, q)\nlies on the [unit hyperbola](https://bliptext.com/articles/unit-hyperbola).\nSuch matrices form the [indefinite orthogonal group O(1,1)](https://bliptext.com/articles/indefinite-orthogonal-group) with one-dimensional Lie algebra spanned by the anti-diagonal unit matrix, showing that the rapidity is the coordinate on this Lie algebra.\nThis action may be depicted in a [spacetime diagram](https://bliptext.com/articles/spacetime-diagram).\nIn [matrix exponential](https://bliptext.com/articles/matrix-exponential) notation,\nΛ(w)\ncan be expressed as, where\nZ\nis the negative of the anti-diagonal unit matrix\nA key property of the matrix exponential is from which immediately follows that\nThis establishes the useful additive property of rapidity: if\nA\n,\nB\nand\nC\nare [frames of reference](https://bliptext.com/articles/frames-of-reference), then\nwhere\nwPQ\ndenotes the rapidity of a frame of reference\nQ\nrelative to a frame of reference\nP\n. The simplicity of this formula contrasts with the complexity of the corresponding [velocity-addition formula](https://bliptext.com/articles/velocity-addition-formula).\nAs we can see from the Lorentz transformation above, the [Lorentz factor](https://bliptext.com/articles/lorentz-factor) identifies with\ncosh w\nso the rapidity w is implicitly used as a hyperbolic angle in the Lorentz transformation expressions using γ and β.\nWe relate rapidities to the [velocity-addition formula](https://bliptext.com/articles/velocity-addition-formula)\nby recognizing\nand so\n[Proper acceleration](https://bliptext.com/articles/proper-acceleration) (the acceleration 'felt' by the object being accelerated) is the rate of change of rapidity with respect to [proper time](https://bliptext.com/articles/proper-time) (time as measured by the object undergoing acceleration itself).\nTherefore, 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.\nThe product of β and γ appears frequently, and is from the above arguments\n\n## Exponential and logarithmic relations\nFrom the above expressions we have\nand thus\nor explicitly\nThe [Doppler-shift](https://bliptext.com/articles/relativistic-doppler-effect) factor associated with rapidity w is k = e^w.\n\n# In experimental particle physics\nThe energy E and scalar momentum\nof a particle of non-zero (rest) mass m are given by:\nWith the definition of w\nand thus with\nthe energy and scalar momentum can be written as:\nSo, rapidity can be calculated from measured energy and momentum by\nHowever, experimental particle physicists often use a modified definition of rapidity relative to a beam axis\nwhere\npz\nis the component of momentum along the beam axis.\nThis 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.\nRelated to this is the concept of [pseudorapidity](https://bliptext.com/articles/pseudorapidity).\nRapidity relative to a beam axis can also be expressed as\n\n# Notes and references","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T07:23:52.229Z","updated_at":"2025-01-26T07:23:52.229Z","search_vector":"'rapid':1","search_content":" \n \n\n# \n[[File:Inverse Hyperbolic Tangent.svg|right|thumb|250px|Rapidity is the value of\nartanh(v / c)\nfor velocity v and speed of light c]]\nIn [special relativity](https://bliptext.com/articles/special-relativity), the classical concept of [velocity](https://bliptext.com/articles/velocity) is converted to **rapidity** to accommodate the limit determined by the [speed of light](https://bliptext.com/articles/speed-of-light).\nVelocities must be combined by Einstein's [velocity-addition formula](https://bliptext.com/articles/velocity-addition-formula).\nFor low speeds, rapidity and velocity are almost exactly proportional but, for higher velocities, rapidity takes a larger value, with the rapidity of light being infinite.\nMathematically, rapidity can be defined as the [hyperbolic angle](https://bliptext.com/articles/hyperbolic-angle) that differentiates two [frames of reference](https://bliptext.com/articles/frames-of-reference) in relative motion, each frame being associated with [distance](https://bliptext.com/articles/distance) and [time](https://bliptext.com/articles/time) coordinates.\nUsing the [inverse hyperbolic function](https://bliptext.com/articles/inverse-hyperbolic-function)\nartanh\n, the rapidity w corresponding to velocity v is\nwhere c is the speed of light.\nFor low speeds, by the [small-angle approximation](https://bliptext.com/articles/small-angle-approximation), w is approximately\nv / c\n. Since in relativity any velocity v is constrained to the interval\n−c < v < c\nthe ratio\nv / c\nsatisfies\n−1 < v / c < 1\n. The inverse hyperbolic tangent has the unit interval\n(−1, 1)\nfor its [domain](https://bliptext.com/articles/domain-of-a-function) and the whole [real line](https://bliptext.com/articles/real-line) for its [image](https://bliptext.com/articles/image-mathematics); that is, the interval\n−c < v < c\nmaps onto\n−∞ < w < ∞\n.\n\n# History\nIn 1908 [Hermann Minkowski](https://bliptext.com/articles/hermann-minkowski) explained how the [Lorentz transformation](https://bliptext.com/articles/lorentz-transformation) could be seen as simply a [hyperbolic rotation](https://bliptext.com/articles/hyperbolic-rotation) of the [spacetime coordinates](https://bliptext.com/articles/coordinate-time), i.e., a rotation through an imaginary angle.\nThis angle therefore represents (in one spatial dimension) a simple additive measure of the velocity between frames.\nThe rapidity parameter replacing velocity was introduced in 1910 by [Vladimir Varićak](https://bliptext.com/articles/vladimir-vari-ak) and by [E. T. Whittaker](https://bliptext.com/articles/e-t-whittaker).\nThe parameter was named rapidity by [Alfred Robb](https://bliptext.com/articles/alfred-robb) (1911) and this term was adopted by many subsequent authors, such as [Ludwik Silberstein](https://bliptext.com/articles/ludwik-silberstein) (1914), [Frank Morley](https://bliptext.com/articles/frank-morley) (1936) and [Wolfgang Rindler](https://bliptext.com/articles/wolfgang-rindler) (2001).\n# [200px | upload.wikimedia.org/wikipedia/commons/c/cd/Hyperbolic///sector.svg]\n\n## Area of a hyperbolic sector\nThe [quadrature](https://bliptext.com/articles/quadrature-mathematics) of the hyperbola\nby [Grégoire de Saint-Vincent](https://bliptext.com/articles/gr-goire-de-saint-vincent) established the natural logarithm as the area of a hyperbolic sector or an equivalent area against an asymptote.\nIn spacetime theory, the connection of events by light divides the universe into Past, Future, or Elsewhere based on a Here and Now.\nOn any line in space, a light beam may be directed left or right.\nTake the x-axis as the events passed by the right beam and the y-axis as the events of the left beam.\nThen a resting frame has [time](https://bliptext.com/articles/time) along the diagonal\n. The rectangular hyperbola\ncan be used to gauge velocities (in the first quadrant).\nZero velocity corresponds to\n(1, 1)\n. Any point on the hyperbola has [light-cone coordinates](https://bliptext.com/articles/light-cone-coordinates) where w is the rapidity, and is equal to the area of the [hyperbolic sector](https://bliptext.com/articles/hyperbolic-sector) from\n(1, 1)\nto these coordinates.\nMany authors refer instead to the [unit hyperbola](https://bliptext.com/articles/unit-hyperbola) x^2 - y^2, using rapidity for a parameter, as in the standard [spacetime diagram](https://bliptext.com/articles/spacetime-diagram).\nThere the axes are measured by clock and meter-stick, more familiar benchmarks, and the basis of spacetime theory.\nSo the delineation of rapidity as a hyperbolic parameter of beam-space is a reference to the seventeenth-century origin of our precious [transcendental functions](https://bliptext.com/articles/transcendental-function), and a supplement to spacetime diagramming.\n\n# Lorentz boost\nThe rapidity w arises in the linear representation of a [Lorentz boost](https://bliptext.com/articles/lorentz-boost) as a vector-matrix product\nThe matrix\nΛ(w)\nis of the type with p and q satisfying\np2 – q2 = 1\n, so that\n(p, q)\nlies on the [unit hyperbola](https://bliptext.com/articles/unit-hyperbola).\nSuch matrices form the [indefinite orthogonal group O(1,1)](https://bliptext.com/articles/indefinite-orthogonal-group) with one-dimensional Lie algebra spanned by the anti-diagonal unit matrix, showing that the rapidity is the coordinate on this Lie algebra.\nThis action may be depicted in a [spacetime diagram](https://bliptext.com/articles/spacetime-diagram).\nIn [matrix exponential](https://bliptext.com/articles/matrix-exponential) notation,\nΛ(w)\ncan be expressed as, where\nZ\nis the negative of the anti-diagonal unit matrix\nA key property of the matrix exponential is from which immediately follows that\nThis establishes the useful additive property of rapidity: if\nA\n,\nB\nand\nC\nare [frames of reference](https://bliptext.com/articles/frames-of-reference), then\nwhere\nwPQ\ndenotes the rapidity of a frame of reference\nQ\nrelative to a frame of reference\nP\n. The simplicity of this formula contrasts with the complexity of the corresponding [velocity-addition formula](https://bliptext.com/articles/velocity-addition-formula).\nAs we can see from the Lorentz transformation above, the [Lorentz factor](https://bliptext.com/articles/lorentz-factor) identifies with\ncosh w\nso the rapidity w is implicitly used as a hyperbolic angle in the Lorentz transformation expressions using γ and β.\nWe relate rapidities to the [velocity-addition formula](https://bliptext.com/articles/velocity-addition-formula)\nby recognizing\nand so\n[Proper acceleration](https://bliptext.com/articles/proper-acceleration) (the acceleration 'felt' by the object being accelerated) is the rate of change of rapidity with respect to [proper time](https://bliptext.com/articles/proper-time) (time as measured by the object undergoing acceleration itself).\nTherefore, 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.\nThe product of β and γ appears frequently, and is from the above arguments\n\n## Exponential and logarithmic relations\nFrom the above expressions we have\nand thus\nor explicitly\nThe [Doppler-shift](https://bliptext.com/articles/relativistic-doppler-effect) factor associated with rapidity w is k = e^w.\n\n# In experimental particle physics\nThe energy E and scalar momentum\nof a particle of non-zero (rest) mass m are given by:\nWith the definition of w\nand thus with\nthe energy and scalar momentum can be written as:\nSo, rapidity can be calculated from measured energy and momentum by\nHowever, experimental particle physicists often use a modified definition of rapidity relative to a beam axis\nwhere\npz\nis the component of momentum along the beam axis.\nThis 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.\nRelated to this is the concept of [pseudorapidity](https://bliptext.com/articles/pseudorapidity).\nRapidity relative to a beam axis can also be expressed as\n\n# Notes and references","revisionCount":0},"hasMore":false,"nextCursor":null}