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| {{For|the derivations|Derivations of the Lorentz transformations}}
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| {{spacetime}}
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| In [[physics]], the '''Lorentz transformation''' (or '''transformations''') is named after the Dutch [[physicist]] [[Hendrik Lorentz]]. It was the result of attempts by Lorentz and others to explain how the speed of [[light]] was observed to be independent of the [[frame of reference|reference frame]], and to understand the symmetries of the laws of [[electromagnetism]]. The Lorentz transformation is in accordance with [[special relativity]], but was derived well before special relativity.
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| The transformations describe how measurements of space and time by two observers are related. They reflect the fact that observers moving at different [[velocity|velocities]] may measure different [[Length contraction|distances]], [[time dilation|elapsed times]], and even different [[Relativity of simultaneity|orderings of events]]. They supersede the [[Galilean transformation]] of [[Newtonian physics]], which assumes an absolute space and time (see [[Galilean relativity]]). The Galilean transformation is a good approximation only at relative speeds much smaller than the speed of light.
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| The Lorentz transformation is a [[linear transformation]]. It may include a rotation of space; a rotation-free Lorentz transformation is called a '''Lorentz boost'''.
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| In the [[Minkowski space]], the Lorentz transformations preserve the [[spacetime interval]] between any two events. They describe only the transformations in which the spacetime event at the origin is left fixed, so they can be considered as a [[hyperbolic rotation]] of Minkowski space. The more general set of transformations that also includes translations is known as the [[Poincaré group]].
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| ==History==
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| {{main|History of Lorentz transformations}}
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| Many physicists, including [[Woldemar Voigt]], [[George FitzGerald]], [[Joseph Larmor]], and [[Hendrik Lorentz]] himself had been discussing the physics implied by these equations since 1887.<ref>
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| {{citation |first1 = John J. |last1 = O'Connor |first2 = Edmund F. |last2 = Robertson
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| |title = A History of Special Relativity
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| |url = http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Special_relativity.html}}
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| </ref> | |
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| Early in 1889, [[Oliver Heaviside]] had shown from [[Maxwell's equations]] that the [[electric field]] surrounding a spherical distribution of charge should cease to have [[spherical symmetry]] once the charge is in motion relative to the ether. FitzGerald then conjectured that Heaviside’s distortion result might be applied to a theory of intermolecular forces. Some months later, FitzGerald published the conjecture that bodies in motion are being contracted, in order to explain the baffling outcome of the 1887 ether-wind experiment of [[Michelson–Morley experiment|Michelson and Morley]]. In 1892, Lorentz independently presented the same idea in a more detailed manner, which was subsequently called [[Length contraction|FitzGerald–Lorentz contraction hypothesis]].<ref>{{citation |first = Harvey R. |last = Brown
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| |title = Michelson, FitzGerald and Lorentz: the Origins of Relativity Revisited
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| |url = http://philsci-archive.pitt.edu/id/eprint/987}}
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| </ref>
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| Their explanation was widely known before 1905.<ref>
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| {{citation |first = Tony |last = Rothman
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| |title = Lost in Einstein's Shadow
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| |url = http://www.americanscientist.org/libraries/documents/200622102452_866.pdf
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| |journal = American Scientist |volume = 94 |issue = 2 |pages = 112f. |year = 2006}}
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| </ref>
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| Lorentz (1892–1904) and Larmor (1897–1900), who believed the [[luminiferous ether]] hypothesis, were also seeking the transformation under which [[Maxwell's equations]] are invariant when transformed from the ether to a moving frame. They extended the FitzGerald–Lorentz contraction hypothesis and found out that the time coordinate has to be modified as well ("[[relativity of simultaneity|local time]]"). [[Henri Poincaré]] gave a physical interpretation to local time (to first order in v/c) as the consequence of clock synchronization, under the assumption that the speed of light is constant in moving frames.<ref>{{Citation
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| |author=Darrigol, Olivier
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| |title=The Genesis of the theory of relativity
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| |year=2005
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| |journal=Séminaire Poincaré
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| |volume=1
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| |pages=1–22
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| |url=http://www.bourbaphy.fr/darrigol2.pdf}}</ref> Larmor is credited to have been the first to understand the crucial [[time dilation]] property inherent in his equations.<ref>
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| {{citation |first = Michael N. |last = Macrossan
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| |title = A Note on Relativity Before Einstein
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| |url = http://espace.library.uq.edu.au/view.php?pid=UQ:9560
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| |journal=Brit. Journal Philos. Science |volume = 37 |year=1986 |pages= 232–34}}
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| </ref>
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| In 1905, Poincaré was the first to recognize that the transformation has the properties of a [[group (mathematics)|mathematical group]],
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| and named it after Lorentz.<ref>
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| The reference is within the following paper:
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| {{citation |first = Henri |last = Poincaré |author-link = Henri Poincaré
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| |title = [[s:Translation:On the Dynamics of the Electron (June)|On the Dynamics of the Electron]]
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| |journal = Comptes rendus hebdomadaires des séances de l'Académie des sciences
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| |volume = 140 |pages = 1504–1508 |year = 1905}}
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| </ref>
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| Later in the same year [[Albert Einstein]] published what is now called [[special relativity]], by deriving the Lorentz transformation under the assumptions of the [[principle of relativity]] and the constancy of the speed of light in any [[inertial reference frame]], and by abandoning the mechanical aether.<ref>{{Citation
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| |author=Einstein, Albert
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| |year=1905
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| |title=Zur Elektrodynamik bewegter Körper
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| |journal=Annalen der Physik
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| |volume=322
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| |issue=10
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| |pages=891–921
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| |url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_891-921.pdf
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| |doi=10.1002/andp.19053221004|bibcode = 1905AnP...322..891E }}. See also: [http://www.fourmilab.ch/etexts/einstein/specrel/ English translation].</ref>
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| ==Lorentz transformation for frames in standard configuration==
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| Consider two observers ''O'' and ''O''′, each using their own [[Cartesian coordinate system]] to measure space and time intervals. ''O'' uses (''t'', ''x'', ''y'', ''z'') and ''O''′ uses (''t''′, ''x''′, ''y''′, ''z''′). Assume further that the coordinate systems are oriented so that, in 3 dimensions, the ''x''-axis and the ''x''′-axis are [[collinearity|collinear]], the ''y''-axis is parallel to the ''y''′-axis, and the ''z''-axis parallel to the ''z''′-axis. The relative velocity between the two observers is ''v'' along the common ''x''-axis; ''O'' measures ''O′'' to move at velocity ''v'' along the coincident ''xx′'' axes, while ''O′'' measures ''O'' to move at velocity −''v'' along the coincident ''xx′'' axes. Also assume that the origins of both coordinate systems are the same, that is, coincident times and positions. If all these hold, then the coordinate systems are said to be in '''standard configuration'''.
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| The [[inverse function|inverse]] of a Lorentz transformation relates the coordinates the other way round; from the coordinates ''O''′ measures (''t''′, ''x''′, ''y''′, ''z''′) to the coordinates ''O'' measures (''t'', ''x'', ''y'', ''z''), so ''t'', ''x'', ''y'', ''z'' are in terms of ''t''′, ''x''′, ''y''′, ''z''′. The mathematical form is nearly identical to the original transformation; the only difference is the negation of the uniform relative velocity (from ''v'' to −''v''), and exchange of primed and unprimed quantities, because ''O''′ moves at velocity ''v'' relative to ''O'', and equivalently, ''O'' moves at velocity −''v'' relative to ''O''′. This symmetry makes it effortless to find the inverse transformation (carrying out the exchange and negation saves a lot of rote algebra), although more fundamentally; it highlights that all physical laws should remain unchanged under a Lorentz transformation.<ref>{{cite book|title=3000 Solved Problems in Physics|series=Schaum Series|author=A. Halpern|publisher=Mc Graw Hill|year=1988|isbn=978-0-07-025734-4|page=688}}</ref>
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| {{anchor|boost}}Below, the Lorentz transformations are called "boosts" in the stated directions.
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| ===Boost in the ''x''-direction===
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| [[File:Lorentz transforms 2.svg|thumb|right|482px|"480px"|The spacetime coordinates of an event, as measured by each observer in their inertial reference frame (in standard configuration) are shown in the speech bubbles.<br />Top: frame '''F'''′ moves at velocity ''v'' along the ''x''-axis of frame '''F'''.<br />Bottom: frame '''F''' moves at velocity −''v'' along the ''x''′-axis of frame '''F'''′.<ref>University Physics – With Modern Physics (12th Edition), H.D. Young, R.A. Freedman (Original edition), Addison-Wesley (Pearson International), 1st Edition: 1949, 12th Edition: 2008, ISBN (10-) 0-321-50130-6, ISBN (13-) 978-0-321-50130-1</ref>]]
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| These are the simplest forms. The Lorentz transformation for frames in standard configuration can be shown to be (see for example <ref>Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Manchester Physics Series, John Wiley & Sons Ltd, ISBN 978-0-470-01460-8</ref> and <ref>http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html. Hyperphysics, web-based physics material hosted by Georgia State University, USA.</ref>):
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| :<math>\begin{align}
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| t' &= \gamma \left( t - \frac{vx}{c^2} \right) \\
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| x' &= \gamma \left( x - v t \right)\\
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| y' &= y \\
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| z' &= z
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| \end{align}</math>
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| where:
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| * ''v'' is the relative velocity between frames in the ''x''-direction,
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| * ''c'' is the [[speed of light]],
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| * <math>\ \gamma = \frac{1}{ \sqrt{1 - { \beta^2}}}</math> is the [[Lorentz factor]] ([[Greek alphabet|Greek]] lowercase [[gamma]]),
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| * <math>\ \beta = \frac{v}{c}</math> is the velocity coefficient ([[Greek alphabet|Greek]] lowercase [[beta]]), again for the ''x''-direction.
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| The use of ''β'' and ''γ'' is standard throughout the literature.<ref name="Relativity DeMystified 2006">Relativity DeMystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145545-0</ref> For the remainder of the article – they will be also used throughout unless otherwise stated. Since the above is a linear system of equations (more technically a [[linear transformation]]), they can be written in [[matrix (mathematics)|matrix]] form:
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| :<math>
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| \begin{bmatrix}
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| c t' \\ x' \\ y' \\ z'
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| \gamma&-\beta \gamma&0&0\\
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| -\beta \gamma&\gamma&0&0\\
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| 0&0&1&0\\
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| 0&0&0&1\\
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| \end{bmatrix}
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| \begin{bmatrix}
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| c\,t \\ x \\ y \\ z
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| \end{bmatrix} ,
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| </math>
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| According to the principle of relativity, there is no privileged frame of reference, so the inverse transformations frame ''F''′ to frame ''F'' must be given by simply negating ''v'':
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| :<math>\begin{align}
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| t &= \gamma \left( t' + \frac{vx'}{c^2} \right) \\
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| x &= \gamma \left( x' + v t' \right)\\
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| y &= y' \\
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| z &= z',
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| \end{align}</math>
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| where the value of ''γ'' remains unchanged.
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| ===Boost in the ''y'' or ''z'' directions===
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| The above collection of equations apply only for a boost in the ''x''-direction. The standard configuration works equally well in the ''y'' or ''z'' directions instead of ''x'', and so the results are similar.
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| For the ''y''-direction:
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| :<math>\begin{align}
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| t' &= \gamma \left( t - \frac{vy}{c^2} \right) \\
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| x' &= x \\
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| y' &= \gamma \left( y - vt \right)\\
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| z' &= z
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| \end{align}</math>
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| summarized by
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| :<math>
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| \begin{bmatrix}
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| c t' \\ x' \\ y' \\ z'
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| \gamma&0&-\beta \gamma&0\\
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| 0&1&0&0\\
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| -\beta \gamma&0&\gamma&0\\
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| 0&0&0&1\\
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| \end{bmatrix}
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| \begin{bmatrix}
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| c\,t \\ x \\ y \\ z
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| \end{bmatrix} ,
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| </math>
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| where ''v'' and so ''β'' are now in the ''y''-direction.
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| For the ''z''-direction:
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| :<math>\begin{align}
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| t' &= \gamma \left( t - \frac{vz}{c^2} \right) \\
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| x' &= x \\
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| y' &= y \\
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| z' &= \gamma \left( z - v t \right)\\
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| \end{align}</math>
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| summarized by
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| :<math>
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| \begin{bmatrix}
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| c t' \\ x' \\ y' \\ z'
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| \gamma&0&0&-\beta \gamma\\
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| 0&1&0&0\\
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| 0&0&1&0\\
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| -\beta \gamma&0&0&\gamma\\
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| \end{bmatrix}
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| \begin{bmatrix}
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| c\,t \\ x \\ y \\ z
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| \end{bmatrix} ,
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| </math>
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| where ''v'' and so ''β'' are now in the ''z''-direction.
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| The Lorentz or boost matrix is usually denoted by '''Λ''' ([[Greek alphabet|Greek]] capital [[lambda]]). Above the transformations have been applied to the [[four-vector|four-position]] '''X''',
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| :<math>
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| \mathbf{X} = \begin{bmatrix}
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| c\,t \\ x \\ y \\ z
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| \end{bmatrix}\ , \quad \mathbf{X}' = \begin{bmatrix}
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| c\,t' \\ x' \\ y' \\ z'
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| \end{bmatrix},
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| </math>
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| The Lorentz transform for a boost in one of the above directions can be compactly written as a single matrix equation:
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| :<math>\mathbf{X}' = \boldsymbol{\Lambda}(v)\mathbf{X} .</math>
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| ===Boost in any direction===
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| [[File:Lorentz transforms 3.svg|402px|thumb|Boost in an arbitrary direction.]]
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| ====Vector form====
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| {{further|Euclidean vector|vector projection}}
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| For a boost in an arbitrary direction with velocity '''v''', that is, ''O'' observes ''O′'' to move in direction '''v''' in the ''F'' coordinate frame, while ''O′'' observes ''O'' to move in direction −'''v''' in the ''F′'' coordinate frame, it is convenient to decompose the spatial vector '''r''' into components perpendicular and parallel to '''v''':
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| :<math>\mathbf{r}=\mathbf{r}_\perp+\mathbf{r}_\|</math>
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| so that
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| :<math>\mathbf{r} \cdot \mathbf{v} = \mathbf{r}_\bot \cdot \mathbf{v} + \mathbf{r}_\parallel \cdot \mathbf{v} = r_\parallel v </math>
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| where '''•''' denotes the [[dot product]] (see also [[orthogonality]] for more information). Then, only time and the component '''r'''<sub>‖</sub> in the direction of '''v''';
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| :<math>\begin{align}
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| t' & = \gamma \left(t - \frac{\mathbf{r} \cdot \mathbf{v}}{c^{2}} \right) \\
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| \mathbf{r'} & = \mathbf{r}_\perp + \gamma (\mathbf{r}_\| - \mathbf{v} t)
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| \end{align}</math>
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| are "warped" by the Lorentz factor:
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| :<math>\gamma(\mathbf{v}) = \frac{1}{\sqrt{1 - \tfrac{\mathbf{v} \cdot \mathbf{v}}{c^{2}}}} = \frac{1}{\sqrt{1 - \tfrac{v^2}{c^2}}}</math>.
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| The parallel and perpendicular components can be eliminated, by substituting <math>\mathbf{r}_\bot = \mathbf{r} - \mathbf{r}_\parallel</math> into '''r′''':
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| :<math>\mathbf{r}' = \mathbf{r} + \left(\gamma - 1 \right)\mathbf{r}_\parallel - \gamma\mathbf{v}t \,. </math>
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| Since '''r'''<sub>‖</sub> and '''v''' are parallel we have
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| :<math>\mathbf{r}_\parallel = r_\parallel \dfrac{\mathbf{v}}{v} = \left(\dfrac{\mathbf{r}\cdot\mathbf{v}}{v}\right) \frac{\mathbf{v}}{v}</math>
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| where geometrically and algebraically:
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| * '''v'''/''v'' is a dimensionless [[unit vector]] pointing in the same direction as '''r'''<sub>‖</sub>,
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| * ''r''<sub>‖</sub> = ('''r''' • '''v''')/''v'' is the [[Scalar projection|projection]] of '''r''' into the direction of '''v''',
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| substituting for '''r'''<sub>‖</sub> and factoring '''v''' gives
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| :<math>\mathbf{r}' = \mathbf{r} + \left(\frac{\gamma-1}{v^2}\mathbf{r}\cdot\mathbf{v} - \gamma t \right)\mathbf{v}\,. </math>
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| This method, of eliminating parallel and perpendicular components, can be applied to any Lorentz transformation written in parallel-perpendicular form.
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| ====Matrix forms====
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| These equations can be expressed in [[block matrix]] form as
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| :<math>
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| \begin{bmatrix}
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| c t' \\
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| \mathbf{r'}
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| \gamma & - \gamma \boldsymbol{\beta}^\mathrm{T} \\
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| -\gamma\boldsymbol{\beta} & \mathbf{I} + (\gamma-1) \boldsymbol{\beta}\boldsymbol{\beta}^\mathrm{T}/\beta^2 \\
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| \end{bmatrix}
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| \begin{bmatrix}
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| c t \\
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| \mathbf{r}
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| \end{bmatrix}\,,
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| </math>
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| where '''I''' is the 3×3 [[identity matrix]] and '''β''' = '''v'''/c is the relative velocity vector (in units of ''c'') as a ''[[column vector]]'' – in [[cartesian coordinates|cartesian]] and [[tensor index notation]] it is:
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| :<math>\boldsymbol{\beta} = \frac{\bold{v}}{c}
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| \equiv \begin{bmatrix}
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| \beta_x \\ \beta_y \\ \beta_z
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| \end{bmatrix}
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| = \frac{1}{c}\begin{bmatrix}
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| v_x \\ v_y \\ v_z
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| \end{bmatrix}
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| \equiv \begin{bmatrix}
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| \beta_1 \\ \beta_2 \\ \beta_3
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| \end{bmatrix}
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| = \frac{1}{c}\begin{bmatrix}
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| v_1 \\ v_2 \\ v_3
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| \end{bmatrix}</math>
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| '''β'''<sup>T</sup> = '''v'''<sup>T</sup>/c is the [[transpose]] – a [[row vector]]:
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| :<math>\boldsymbol{\beta}^\mathrm{T} = \frac{\bold{v}^\mathrm{T}}{c}
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| \equiv \begin{bmatrix}
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| \beta_x & \beta_y & \beta_z
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| \end{bmatrix}
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| = \frac{1}{c}\begin{bmatrix}
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| v_x & v_y & v_z
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| \end{bmatrix}
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| \equiv \begin{bmatrix}
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| \beta_1 & \beta_2 & \beta_3
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| \end{bmatrix}
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| = \frac{1}{c}\begin{bmatrix}
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| v_1 & v_2 & v_3 \\
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| \end{bmatrix}</math>
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| and ''β'' is the [[magnitude (vector)|magnitude]] of '''β''':
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| :<math>\beta = |\boldsymbol{\beta}| = \sqrt{\beta_x^2 + \beta_y^2 + \beta_z^2}\,.</math>
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| More explicitly stated:
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| :<math>
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| \begin{bmatrix}
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| c\,t' \\ x' \\ y' \\ z'
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| \gamma&-\gamma\,\beta_x&-\gamma\,\beta_y&-\gamma\,\beta_z\\
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| -\gamma\,\beta_x&1+(\gamma-1)\dfrac{\beta_x^2}{\beta^2}&(\gamma-1)\dfrac{\beta_x \beta_y}{\beta^2}&(\gamma-1)\dfrac{\beta_x \beta_z}{\beta^2}\\
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| -\gamma\,\beta_y&(\gamma-1)\dfrac{\beta_y \beta_x}{\beta^2}&1+(\gamma-1)\dfrac{\beta_y^2}{\beta^2}&(\gamma-1)\dfrac{\beta_y \beta_z}{\beta^2}\\
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| -\gamma\,\beta_z&(\gamma-1)\dfrac{\beta_z \beta_x}{\beta^2}&(\gamma-1)\dfrac{\beta_z \beta_y}{\beta^2}&1+(\gamma-1)\dfrac{\beta_z^2}{\beta^2}\\
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| \end{bmatrix}
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| \begin{bmatrix}
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| c\,t \\ x \\ y \\ z
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| \end{bmatrix}\,.
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| </math>
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| The transformation '''Λ''' can be written in the same form as before,
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| :<math>\mathbf{X}' = \boldsymbol{\Lambda}(\mathbf{v})\mathbf{X}.</math>
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| which has the structure:<ref name="Gravitation, J.A. Wheeler 1973"/>
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| | |
| :<math>\begin{bmatrix}
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| c\,t' \\ x' \\ y' \\ z'
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| \end{bmatrix} =
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| | |
| \begin{bmatrix}
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| \Lambda_{00} & \Lambda_{01} & \Lambda_{02} & \Lambda_{03} \\
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| \Lambda_{10} & \Lambda_{11} & \Lambda_{12} & \Lambda_{13} \\
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| \Lambda_{20} & \Lambda_{21} & \Lambda_{22} & \Lambda_{23} \\
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| \Lambda_{30} & \Lambda_{31} & \Lambda_{32} & \Lambda_{33} \\
| |
| \end{bmatrix}
| |
| | |
| \begin{bmatrix}
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| c\,t \\ x \\ y \\ z
| |
| \end{bmatrix}.
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| </math>
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| | |
| and the components deduced from above are:
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| :<math> \begin{align} \Lambda_{00} & = \gamma, \\
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| \Lambda_{0i} & = \Lambda_{i0} = - \gamma \beta_{i}, \\
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| \Lambda_{ij} & = \Lambda_{ji} = ( \gamma - 1 )\dfrac{\beta_{i}\beta_{j}}{\beta^{2}} + \delta_{ij}= ( \gamma - 1 )\dfrac{v_i v_j}{v^2} + \delta_{ij}, \\
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| \end{align}
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| \,\!</math>
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| | |
| where δ<sub>''ij''</sub> is the [[Kronecker delta]], and by convention: [[Latin alphabet|Latin letters]] for indices take the values 1, 2, 3, for spatial components of a 4-vector ([[Greek alphabet|Greek]] indices take values 0, 1, 2, 3 for time and space components).
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| | |
| Note that this transformation is only the "boost," i.e., a transformation between two frames whose ''x'', ''y'', and ''z'' axis are parallel and whose spacetime origins coincide. The most general proper Lorentz transformation also contains a rotation of the three axes, because the composition of two boosts is not a pure boost but is a boost followed by a rotation. The rotation gives rise to [[Thomas precession]]. The boost is given by a [[symmetric matrix]], but the general Lorentz transformation matrix need not be symmetric.
| |
| | |
| ===Composition of two boosts===
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| | |
| The composition of two Lorentz boosts B('''u''') and B('''v''') of velocities '''u''' and '''v''' is given by:<ref name="relcompara">{{cite journal | first = A. A. | last = Ungar | url = http://www.springerlink.com/content/g157304vh4434413/ | title = The relativistic velocity composition paradox and the Thomas rotation| journal = [[Foundations of Physics]] | volume = 19 | pages = 1385–1396 | year = 1989 |bibcode = 1989FoPh...19.1385U |doi = 10.1007/BF00732759 }}</ref><ref>{{cite journal | first = A. A. | last = Ungar | title = The relativistic composite-velocity reciprocity principle | id = {{citeseerx|10.1.1.35.1131}} | journal = [[Foundations of Physics]] | year = 2000 | publisher = Springer | volume = 30 | issue = 2 | pages = 331–342 }}</ref>
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| | |
| :<math>B(\mathbf{u})B(\mathbf{v})=B\left ( \mathbf{u}\oplus\mathbf{v} \right )\mathrm{Gyr}\left [ \mathbf{u},\mathbf{v}\right ]=\mathrm{Gyr}\left [\mathbf{u},\mathbf{v} \right ]B \left ( \mathbf{v}\oplus\mathbf{u} \right )</math>,
| |
| | |
| where
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| *B('''v''') is the 4 × 4 matrix that uses the components of '''v''', i.e. v<sub>1</sub>, v<sub>2</sub>, v<sub>3</sub> in the entries of the matrix, or rather the components of '''v'''/c in the representation that is used above,
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| *<math>\mathbf{u}\oplus\mathbf{v}</math> is the [[velocity-addition formula|velocity-addition]],
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| *Gyr['''u''','''v'''] (capital G) is the rotation arising from the composition. If the 3 × 3 matrix form of the rotation applied to spatial coordinates is given by gyr['''u''','''v'''], then the 4 × 4 matrix rotation applied to 4-coordinates is given by:<ref name="relcompara"/>
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| ::<math>
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| \mathrm{Gyr}[\mathbf{u},\mathbf{v}]=
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| \begin{pmatrix}
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| 1 & 0 \\
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| 0 & \mathrm{gyr}[\mathbf{u},\mathbf{v}]
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| \end{pmatrix}\,,
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| </math>
| |
| *gyr (lower case g) is the [[gyrovector space]] abstraction of the ''gyroscopic Thomas precession'', defined as an operator on a velocity '''w''' in terms of velocity addition:
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| ::<math>\text{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u} \oplus \mathbf{v}) \oplus (\mathbf{u} \oplus (\mathbf{v} \oplus \mathbf{w}))</math>
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| :for all '''w'''.
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| | |
| The composition of two Lorentz transformations ''L''('''u''', ''U'') and ''L''('''v''', ''V'') which include rotations ''U'' and ''V'' is given by:<ref>eq. (55), Thomas rotation and the parameterization of the Lorentz transformation group, AA Ungar – Foundations of Physics Letters, 1988</ref>
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| | |
| :<math>L(\mathbf{u},U)L(\mathbf{v},V)=L(\mathbf{u}\oplus U\mathbf{v}, \mathrm{gyr}[\mathbf{u},U\mathbf{v}]UV)</math>
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| | |
| ==Visualizing the transformations in Minkowski space==
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| {{main|Minkowski space}}
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| | |
| Lorentz transformations can be depicted on the [[Hermann Minkowski|Minkowski]] [[light cone]] [[spacetime diagram]].
| |
| | |
| [[File:Lorentz transform of world line.gif|right|framed|The momentarily co-moving inertial frames along the [[world line]] of a rapidly accelerating observer (center).
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| The vertical direction indicates time, while the horizontal indicates distance, the dashed line is the [[spacetime]] trajectory ("[[world line]]") of the observer. The small dots are specific events in spacetime. If one imagines these events to be the flashing of a light, then the events that pass the two diagonal lines in the bottom half of the image (the past [[light cone]] of the observer in the origin) are the events visible to the observer.
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| The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the momentarily co-moving inertial frame changes when the observer accelerates.]]
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| | |
| {{multiple image
| |
| | align = center
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| | position
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| | direction = horizontal
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| | footer = Lorentz transformations on the [[Hermann Minkowski|Minkowski]] [[light cone]] [[spacetime diagram]], for one space and one time dimension.
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| | image1 = Minkowski lightcone lorentztransform inertial.svg
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| | caption1 = Particle travelling at constant velocity (straight [[worldline]] coincident with time ''t''′ axis).
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| | width1 = 300
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| | image2 = Minkowski lightcone lorentztransform.svg
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| | caption2 = [[Acceleration|Accelerating]] particle (curved [[worldline]]).
| |
| | width2 = 300
| |
| }}
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| The yellow axes are the rest frame of an observer, the blue axes correspond to the frame of a moving observer
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| | |
| The red lines are [[world line]]s, a continuous sequence of events: straight for an object travelling at constant velocity, curved for an object accelerating. Worldlines of [[light]] form the boundary of the light cone.
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| | |
| The purple [[hyperbolae]] indicate this is a [[hyperbolic rotation]], the hyperbolic angle ϕ is called [[rapidity]] (see below). The greater the relative speed between the reference frames, the more "warped" the axes become. The relative velocity cannot exceed ''c''.
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| The black arrow is a [[Displacement (vector)|displacement]] [[four-vector]] between two events (not necessarily on the same world line), showing that in a Lorentz boost; [[time dilation]] (fewer time intervals in moving frame) and length contraction (shorter lengths in moving frame) occur. The axes in the moving frame are orthogonal (even though they do not look so).
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| | |
| ===Rapidity===
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| The Lorentz transformation can be cast into another useful form by defining a parameter ''ϕ'' called the [[rapidity]] (an instance of [[hyperbolic angle]]) such that
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| :<math>e^{\phi} = \gamma(1+\beta) = \gamma \left( 1 + \frac{v}{c} \right) = \sqrt \frac{1 + \tfrac{v}{c}}{1 - \tfrac{v}{c}},</math>
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| | |
| and thus
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| :<math>e^{-\phi} = \gamma(1-\beta) = \gamma \left( 1 - \frac{v}{c} \right) = \sqrt \frac{1 - \tfrac{v}{c}}{1 + \tfrac{v}{c}}.</math>
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| | |
| Equivalently:
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| | |
| :<math>\phi = \ln \left[\gamma(1+\beta)\right] = -\ln \left[\gamma(1-\beta)\right] \, </math>
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| | |
| Then the Lorentz transformation in standard configuration is:
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| | |
| :<math>\begin{align}
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| & c t-x = e^{- \phi}(c t' - x') \\
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| & c t+x = e^{\phi}(c t' + x') \\
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| & y = y' \\
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| & z = z'.
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| \end{align}</math>
| |
| | |
| ====Hyperbolic expressions====
| |
| From the above expressions for e<sup>''φ''</sup> and e<sup>''−φ''</sup>
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| | |
| :<math> \gamma = \cosh\phi = { e^{\phi} + e^{-\phi} \over 2 }, </math>
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| :<math> \beta \gamma = \sinh\phi = { e^{\phi} - e^{-\phi} \over 2 },</math>
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| | |
| and therefore,
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| | |
| :<math> \beta = \tanh\phi = { e^{\phi} - e^{-\phi} \over e^{\phi} + e^{-\phi} } .</math>
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| | |
| ====Hyperbolic rotation of coordinates====
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| | |
| Substituting these expressions into the matrix form of the transformation, it is evident that
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| :<math>
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| \begin{bmatrix}
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| c t' \\ x' \\ y' \\ z'
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| \cosh\phi &-\sinh\phi & 0 & 0 \\
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| -\sinh\phi & \cosh\phi & 0 & 0 \\
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| 0 & 0 & 1 & 0 \\
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| 0 & 0 & 0 & 1 \\
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| \end{bmatrix}
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| \begin{bmatrix}
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| c t \\ x \\ y \\ z
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| \end{bmatrix}\ .
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| </math>
| |
| | |
| Thus, the Lorentz transformation can be seen as a [[hyperbolic rotation]] of coordinates in [[Minkowski space]], where the parameter {{mvar|ϕ}} represents the hyperbolic angle of rotation, often referred to as [[rapidity]]. This transformation is sometimes illustrated with a [[Minkowski diagram]], as displayed above.
| |
| | |
| This 4-by-4 boost matrix can thus be written compactly as a [[Matrix exponential]],
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| ::<math>\begin{bmatrix}
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| \cosh\phi &-\sinh\phi & 0 & 0 \\
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| -\sinh\phi & \cosh\phi & 0 & 0 \\
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| 0 & 0 & 1 & 0 \\
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| 0 & 0 & 0 & 1 \\
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| \end{bmatrix}= \exp \left( - \phi \begin{bmatrix}
| |
| 0 &1 & 0 & 0 \\
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| 1 & 0 & 0 & 0 \\
| |
| 0 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & 0 \\
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| \end{bmatrix}\right)\equiv \exp (-\phi K_x),</math>
| |
| where the simpler [[Lie algebra|Lie-algebraic]] hyperbolic rotation generator {{math|''K<sub>x</sub>''}} is called a '''boost generator'''.
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| | |
| == Transformation of other physical quantities ==
| |
| | |
| {{For|the notation used|Ricci calculus}}
| |
| | |
| The transformation matrix is universal for all [[four-vector]]s, not just 4-dimensional spacetime coordinates. If '''Z''' is any four-vector, then:<ref name="Gravitation, J.A. Wheeler 1973">Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0</ref>
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| | |
| :<math> \mathbf{Z}' = \boldsymbol{\Lambda}(\mathbf{v})\mathbf{Z}.</math>
| |
| | |
| or in [[tensor index notation]]:
| |
| | |
| :<math> Z^{\alpha'} = \Lambda^{\alpha'}{}_\alpha Z^\alpha \,.</math>
| |
| | |
| in which the primed indices denote indices of '''Z''' in the primed frame. | |
| | |
| More generally, the transformation of any [[tensor]] quantity '''''T''''' is given by:<ref>{{cite book
| |
| |title=Spacetime and Geometry: An Introduction to General Relativity |edition=illustrated |first1=Sean |last1=M. Carroll |publisher=Addison Wesley |year=2004 |isbn=0-8053-8732-3 |page=22 |url=http://books.google.com/books?id=1SKFQgAACAAJ}}</ref>
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| | |
| :<math>T^{\alpha' \beta' \cdots \zeta'}_{\theta' \iota' \cdots \kappa'} =
| |
| \Lambda^{\alpha'}{}_{\mu} \Lambda^{\beta'}{}_{\nu} \cdots \Lambda^{\zeta'}{}_{\rho}
| |
| \Lambda_{\theta'}{}^{\sigma} \Lambda_{\iota'}{}^{\upsilon} \cdots \Lambda_{\kappa'}{}^{\phi}
| |
| T^{\mu \nu \cdots \rho}_{\sigma \upsilon \cdots \phi}</math>
| |
| | |
| where <math>\Lambda_{\chi'}{}^{\psi} \,</math> is the [[inverse matrix]] of <math>\Lambda^{\chi'}{}_{\psi} \,.</math>
| |
| | |
| ==Special relativity==
| |
| | |
| The crucial insight of Einstein's clock-setting method is the idea that ''time is relative''. In essence, each observer's frame of reference is associated with a unique set of clocks, the result being that time as measured for a location passes at different rates for different observers.<ref name="lire1">{{Cite web|last=Einstein |first=Albert |year=1916 |title=Relativity: The Special and General Theory |format=PDF |pages= |url=http://www.archive.org/stream/cu31924011804774#page/n35/mode/2up |accessdate=2012-01-23}}</ref> This was a direct result of the Lorentz transformations and is called [[time dilation]]. We can also clearly see from the Lorentz "local time" transformation that the concept of the relativity of simultaneity and of the relativity of length contraction are also consequences of that clock-setting hypothesis.<ref>Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978 0 470 01460 8</ref>
| |
| | |
| === Transformation of the electromagnetic field ===
| |
| | |
| {{For|the transformation rules|classical electromagnetism and special relativity}}
| |
| | |
| Lorentz transformations can also be used to prove that [[magnetic fields|magnetic]] and [[electric fields|electric]] fields are simply different aspects of the same force — the [[electromagnetic force]], as a consequence of relative motion between [[electric charge]]s and observers.<ref>Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 9-780471-927129</ref> The fact that the electromagnetic field shows relativistic effects becomes clear by carrying out a simple thought experiment:<ref>Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3</ref>
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| * Consider an observer measuring a charge at rest in a reference frame F. The observer will detect a static electric field. As the charge is stationary in this frame, there is no electric current, so the observer will not observe any magnetic field.
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| * Consider another observer in frame F′ moving at relative velocity '''v''' (relative to F and the charge). ''This'' observer will see a different electric field because the charge is moving at velocity −'''v''' in their rest frame. Further, in frame F′ the moving charge constitutes an electric current, and thus the observer in frame F′ will also see a magnetic field.
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| | |
| This shows that the Lorentz transformation also applies to electromagnetic field quantities when changing the frame of reference, given below in vector form.
| |
| | |
| ===The correspondence principle===
| |
| | |
| For relative speeds much less than the speed of light, the Lorentz transformations reduce to the [[Galilean transformation]] in accordance with the [[correspondence principle]].
| |
| | |
| The correspondence limit is usually stated mathematically as: as ''v'' → 0, ''c'' → ∞. In words: as velocity approaches 0, the speed of light (seems to) approach infinity. Hence, it is sometimes said that nonrelativistic physics is a physics of "instantaneous action at a distance".<ref name="lire1" />
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| | |
| == Spacetime interval ==
| |
| | |
| In a given coordinate system ''x''<sup>μ</sup>, if two [[Spacetime#Basic concepts|events]] ''A'' and ''B'' are separated by
| |
| | |
| :<math>(\Delta t, \Delta x, \Delta y, \Delta z) = (t_B-t_A, x_B-x_A, y_B-y_A, z_B-z_A)\ ,</math>
| |
| | |
| the [[Spacetime#Spacetime intervals|spacetime interval]] between them is given by
| |
| | |
| :<math>s^2 = - c^2(\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2\ .</math>
| |
| | |
| This can be written in another form using the [[Minkowski metric]]. In this coordinate system,
| |
| | |
| :<math>\eta_{\mu\nu} =
| |
| \begin{bmatrix} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix}\ .
| |
| </math>
| |
| | |
| Then, we can write
| |
| | |
| :<math>
| |
| s^2 = \begin{bmatrix}c \Delta t & \Delta x & \Delta y & \Delta z \end{bmatrix}
| |
| \begin{bmatrix} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix}
| |
| \begin{bmatrix} c \Delta t \\ \Delta x \\ \Delta y \\ \Delta z \end{bmatrix}
| |
| </math>
| |
| or, using the [[Einstein summation convention]],
| |
| | |
| : <math>s^2= \eta_{\mu\nu} x^\mu x^\nu\ .</math>
| |
| | |
| Now suppose that we make a coordinate transformation ''x''<sup>μ</sup> → ''x''′ <sup>μ</sup>. Then, the interval in this coordinate system is given by
| |
| | |
| :<math>
| |
| s'^2 = \begin{bmatrix}c \Delta t' & \Delta x' & \Delta y' & \Delta z' \end{bmatrix}
| |
| \begin{bmatrix} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix}
| |
| \begin{bmatrix} c \Delta t' \\ \Delta x' \\ \Delta y' \\ \Delta z' \end{bmatrix}
| |
| </math>
| |
| | |
| or
| |
| | |
| :<math>s'^2= \eta_{\mu\nu} x'^\mu x'^\nu\ .</math>
| |
| | |
| It is a result of [[special relativity]] that the interval is an [[Invariant (physics)|invariant]]. That is, ''s''<sup>2</sup> = ''s''′ <sup>2</sup>. For this to hold, it can be shown<ref>{{citation |first = Steven |last = Weinberg |title = Gravitation and Cosmology |author-link = Steven Weinberg |place = New York, [NY.] |publisher = Wiley |year = 1972 |isbn = 0-471-92567-5}}: (Section 2:1)</ref> that it is necessary (but not sufficient) for the coordinate transformation to be of the form
| |
| | |
| : <math>x'^\mu = x^\nu \Lambda^\mu_\nu + C^\mu\ .</math>
| |
| | |
| Here, ''C''<sup>μ</sup> is a constant vector and Λ<sup>μ</sup><sub>ν</sub> a constant matrix, where we require that
| |
| | |
| : <math>\eta_{\mu\nu}\Lambda^\mu_\alpha \Lambda^\nu_\beta = \eta_{\alpha\beta}\ .</math>
| |
| | |
| Such a transformation is called a ''[[Poincaré group|Poincaré transformation]]'' or an ''inhomogeneous Lorentz transformation''.<ref>{{citation |first = Steven |last = Weinberg |author-link = Steven Weinberg |title = The quantum theory of fields (3 vol.) |place = Cambridge, [England] ; New York, [NY.] |publisher = Cambridge University Press |year = 1995 |isbn = 0-521-55001-7}} : volume 1.</ref> The ''C<sup>a</sup>'' represents a spacetime translation. When ''C<sup>a</sup>'' = 0, the transformation is called an ''homogeneous Lorentz transformation'', or simply a ''Lorentz transformation''.
| |
| | |
| Taking the [[determinant]] of
| |
| | |
| :<math>\eta_{\mu\nu}{\Lambda^\mu}_\alpha{\Lambda^\nu}_\beta = \eta_{\alpha\beta}</math>
| |
| | |
| gives us
| |
| | |
| :<math>\det (\Lambda^a_b) = \pm 1\ .</math>
| |
| | |
| The cases are:
| |
| *'''Proper Lorentz transformations''' have det(Λ<sup>μ</sup><sub>ν</sub>) = +1, and form a [[subgroup]] called the [[special orthogonal group]] SO(1,3).
| |
| *'''Improper Lorentz transformations''' are det(Λ<sup>μ</sup><sub>ν</sub>) = −1, which do not form a subgroup, as the product of any two improper Lorentz transformations will be a proper Lorentz transformation.
| |
| | |
| From the above definition of Λ it can be shown that (Λ<sup>0</sup><sub>0</sub>)<sup>2</sup> ≥ 1, so either Λ<sup>0</sup><sub>0</sub> ≥ 1 or Λ<sup>0</sup><sub>0</sub> ≤ −1, called [[orthochronous]] and non-orthochronous respectively. An important subgroup of the proper Lorentz transformations are the '''proper orthochronous Lorentz transformations''' which consist purely of boosts and rotations. Any Lorentz transform can be written as a proper orthochronous, together with one or both of the two discrete transformations; [[Parity (physics)|space inversion]] ''P'' <!--surely space inversion refers to parity inversion? correct if wrong, thanks...--> and [[T-symmetry|time reversal]] ''T'', whose non-zero elements are:
| |
| | |
| : <math>P^0_0=1, P^1_1=P^2_2=P^3_3=-1</math>
| |
| : <math>T^0_0=-1, T^1_1=T^2_2=T^3_3=1</math>
| |
| | |
| The set of Poincaré transformations satisfies the properties of a group and is called the [[Poincaré group]]. Under the [[Erlangen program]], [[Minkowski space]] can be viewed as the [[geometry]] defined by the Poincaré group, which combines Lorentz transformations with translations. In a similar way, the set of all Lorentz transformations forms a group, called the [[Lorentz group]].
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| | |
| A quantity invariant under Lorentz transformations is known as a [[Lorentz scalar]].
| |
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| ==See also==
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| *[[Ricci calculus]]
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| *[[Electromagnetic field]]
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| *[[Galilean transformation]]
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| *[[split-complex number#Geometry|Hyperbolic rotation]]
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| *[[Invariance mechanics]]
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| *[[Lorentz group]]
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| *[[Principle of relativity]]
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| *[[Velocity-addition formula]]
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| *[[Algebra of physical space]]
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| *[[Relativistic aberration]]
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| *[[Prandtl–Glauert transformation]]
| |
| | |
| ==References==
| |
| {{Reflist|colwidth=30em}}
| |
| | |
| ==Further reading==
| |
| *{{Citation |first = Albert |last = Einstein |author-link = Albert Einstein |title = Relativity: The Special and the General Theory |place = New York|url = http://www.marxists.org/reference/archive/einstein/works/1910s/relative/ | publisher = Three Rivers Press|year = 1961|publication-date = 1995|isbn = 0-517-88441-0}}
| |
| *{{Citation |first1 = A. |last1 = Ernst |first2 = J.-P. |last2 = Hsu |title = First proposal of the universal speed of light by Voigt 1887 |journal = Chinese Journal of Physics |volume = 39 |issue = 3|url = http://psroc.phys.ntu.edu.tw/cjp/v39/211.pdf
| |
| |pages = 211–230 |year = 2001|bibcode = 2001ChJPh..39..211E }}
| |
| *{{Citation |first1 = Stephen T. |last1 = Thornton |first2 = Jerry B. |last2 = Marion |title = Classical dynamics of particles and systems |edition = 5th |place = Belmont, [CA.] |publisher = Brooks/Cole |year = 2004 |pages = 546–579 |isbn = 0-534-40896-6}}
| |
| *{{Citation |first = Woldemar |last = Voigt |author-link = Woldemar Voigt |title = Über das Doppler'sche princip |journal = Nachrichten von der Königlicher Gesellschaft den Wissenschaft zu Göttingen |volume = 2 |pages = 41–51 |year = 1887}}
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| | |
| ==External links==
| |
| {{Wikisource portal|Relativity}}
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| {{wikibooks|special relativity}}
| |
| *[http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf Derivation of the Lorentz transformations]. This web page contains a more detailed derivation of the Lorentz transformation with special emphasis on group properties.
| |
| *[http://casa.colorado.edu/~ajsh/sr/paradox.html The Paradox of Special Relativity]. This webpage poses a problem, the solution of which is the Lorentz transformation, which is presented graphically in its next page.
| |
| *[http://www.lightandmatter.com/html_books/0sn/ch07/ch07.html Relativity] – a chapter from an online textbook
| |
| *[http://physnet.org/home/modules/pdf_modules/m12.pdf ''Special Relativity: The Lorentz Transformation, The Velocity Addition Law''] on [http://www.physnet.org Project PHYSNET]
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| *[http://www.adamauton.com/warp/ Warp Special Relativity Simulator]. A computer program demonstrating the Lorentz transformations on everyday objects.
| |
| *[http://www.youtube.com/watch?v=C2VMO7pcWhg Animation clip] visualizing the Lorentz transformation.
| |
| *[http://math.ucr.edu/~jdp/Relativity/Lorentz_Frames.html Lorentz Frames Animated] ''from John de Pillis.'' Online Flash animations of Galilean and Lorentz frames, various paradoxes, EM wave phenomena, ''etc''.
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| {{Relativity}}
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| {{DEFAULTSORT:Lorentz Transformation}}
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| [[Category:Equations]]
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| [[Category:Minkowski spacetime]]
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| [[Category:Special relativity]]
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| [[Category:Concepts in physics]]
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| [[Category:Functions and mappings]]
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| [[Category:Time]]
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