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Area Physics for electrotechnology / Relativistic physics for electrotechnology

IEV ref 113-07-13

Symbol
L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36AD@

en
general Lorentz transformation
Lorentz transformation
transformation of four-vectors from one inertial frame S to another inertial frame S′ moving in any given direction

Note 1 to entry: General Lorentz transformations form a group. Denoting Ω L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaeyyQdC1aaSbaaSqaai aakYeaaeqaaaaa@359C@ the set of all general Lorentz transformations L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36AD@ , following rules are fulfilled:

  1. the identity transformation I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36AD@ belongs to Ω L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaeyyQdC1aaSbaaSqaai aakYeaaeqaaaaa@359C@ ;
  2. a composition of general Lorentz transformations is associative, i.e. L ( L L )=( L L ) L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaacmGaaiGacmabaaGcbaGabCitayaafaWaaeWaae aaceWHmbGbayaaceWHmbGbaibaaiaawIcacaGLPaaacaGI9aWaaeWa aeaaceWHmbGbauaaceWHmbGbayaaaiaawIcacaGLPaaaceWHmbGbai baaaa@3C48@ ;
  3. to any L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36AD@ exists an inverse one L 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaacmGaaiGacmabaaGcbaGaaCitamaaCaaaleqaba GaeyOeI0IaaGymaaaaaaa@35B5@ such that L L 1 =I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaacmGaaiGacmabaaGcbaGaaCitaiaahYeadaahaa WcbeqaaiabgkHiTiaaigdaaaGccqGH9aqpcaWHjbaaaa@386C@ .

Note 2 to entry: A general Lorentz transformation is a linear, rotational transformation in space-time.

Note 3 to entry: A general Lorentz transformation for synchronized S, S' can be expressed by x _ _ =L x _ _ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaWaaWqaaeaace WG4bGbauaaaaGaeyypa0JaaCitamaameaabaGaamiEaaaaaaa@39DB@ where L=( γ γ β x γ β y γ β z γ β x 1+ ( γ1 ) β x 2 β 2 ( γ1 ) β x β y β 2 ( γ1 ) β x β z β 2 γ β y ( γ1 ) β x β y β 2 1+ ( γ1 ) β y 2 β 2 ( γ1 ) β y β z β 2 γ β z ( γ1 ) β x β z β 2 ( γ1 ) β y β z β 2 1+ ( γ1 ) β z 2 β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCitaiabg2 da9maabmaabaqbaeqabqabaaaaaeaacqaHZoWzaeaacqGHsislcqaH ZoWzcqaHYoGydaWgaaWcbaGaamiEaaqabaaakeaacqGHsislcqaHZo WzcqaHYoGydaWgaaWcbaGaamyEaaqabaaakeaacqGHsislcqaHZoWz cqaHYoGydaWgaaWcbaGaamOEaaqabaaakeaacqGHsislcqaHZoWzcq aHYoGydaWgaaWcbaGaamiEaaqabaaakeaacaaIXaGaey4kaSYaaSaa aeaadaqadaqaaiabeo7aNjabgkHiTiaaigdaaiaawIcacaGLPaaacq aHYoGydaqhaaWcbaGaamiEaaqaaiaaikdaaaaakeaacqaHYoGydaah aaWcbeqaaiaaikdaaaaaaaGcbaWaaSaaaeaadaqadaqaaiabeo7aNj abgkHiTiaaigdaaiaawIcacaGLPaaacqaHYoGydaqhaaWcbaGaamiE aaqaaaaakiabek7aInaaBaaaleaacaWG5baabeaaaOqaaiabek7aIn aaCaaaleqabaGaaGOmaaaaaaaakeaadaWcaaqaamaabmaabaGaeq4S dCMaeyOeI0IaaGymaaGaayjkaiaawMcaaiabek7aInaaDaaaleaaca WG4baabaaaaOGaeqOSdi2aaSbaaSqaaiaadQhaaeqaaaGcbaGaeqOS di2aaWbaaSqabeaacaaIYaaaaaaaaOqaaiabgkHiTiabeo7aNjabek 7aInaaBaaaleaacaWG5baabeaaaOqaamaalaaabaWaaeWaaeaacqaH ZoWzcqGHsislcaaIXaaacaGLOaGaayzkaaGaeqOSdi2aa0baaSqaai aadIhaaeaaaaGccqaHYoGydaWgaaWcbaGaamyEaaqabaaakeaacqaH YoGydaahaaWcbeqaaiaaikdaaaaaaaGcbaGaaGymaiabgUcaRmaala aabaWaaeWaaeaacqaHZoWzcqGHsislcaaIXaaacaGLOaGaayzkaaGa eqOSdi2aa0baaSqaaiaadMhaaeaacaaIYaaaaaGcbaGaeqOSdi2aaW baaSqabeaacaaIYaaaaaaaaOqaamaalaaabaWaaeWaaeaacqaHZoWz cqGHsislcaaIXaaacaGLOaGaayzkaaGaeqOSdi2aa0baaSqaaiaadM haaeaaaaGccqaHYoGydaWgaaWcbaGaamOEaaqabaaakeaacqaHYoGy daahaaWcbeqaaiaaikdaaaaaaaGcbaGaeyOeI0Iaeq4SdCMaeqOSdi 2aaSbaaSqaaiaadQhaaeqaaaGcbaWaaSaaaeaadaqadaqaaiabeo7a NjabgkHiTiaaigdaaiaawIcacaGLPaaacqaHYoGydaqhaaWcbaGaam iEaaqaaaaakiabek7aInaaBaaaleaacaWG6baabeaaaOqaaiabek7a InaaCaaaleqabaGaaGOmaaaaaaaakeaadaWcaaqaamaabmaabaGaeq 4SdCMaeyOeI0IaaGymaaGaayjkaiaawMcaaiabek7aInaaDaaaleaa caWG5baabaaaaOGaeqOSdi2aaSbaaSqaaiaadQhaaeqaaaGcbaGaeq OSdi2aaWbaaSqabeaacaaIYaaaaaaaaOqaaiaaigdacqGHRaWkdaWc aaqaamaabmaabaGaeq4SdCMaeyOeI0IaaGymaaGaayjkaiaawMcaai abek7aInaaDaaaleaacaWG6baabaGaaGOmaaaaaOqaaiabek7aInaa CaaaleqabaGaaGOmaaaaaaaaaaGccaGLOaGaayzkaaaaaa@D064@

In the case where the representation of four-vectors is given by x _ _ =( x 0 ; x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaWaaWqaaeaaca WG4baaaiabg2da9iaacIcacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGa ai4oaiqadIhagaWcaiaacMcaaaa@3D00@ and their transposition by x _ _ T := ( x 0 ; x ) T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaWaaWqaaeaaca WG4baaamaaCaaaleqabaGaamivaaaakiacakSG6aGaeyypa0ZaaeWa aeaacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGaai4oaiqadIhagaWcaa GaayjkaiaawMcaamaaCaaaleqabaGaamivaaaaaaa@414E@ , then L=( γ γ β T γ β I+Γ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCitaiabg2 da9maabmaabaqbaeqabiGaaaqaaiabeo7aNbqaaiabgkHiTiabeo7a Njqbek7aIzaalaWaaWbaaSqabeaacaWGubaaaaGcbaGaeyOeI0Iaeq 4SdCMafqOSdiMbaSaaaeaacaWHjbGaey4kaSIaaC4KdaaaaiaawIca caGLPaaaaaa@4764@ , where I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCysaaaa@36AA@ is the 3×3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgE na0kaaiodaaaa@3969@ identity matrix and Γ= ( γ1 ) β 2 β β T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaC4Kdiabg2 da9maalaaabaWaaeWaaeaacqaHZoWzcqGHsislcaaIXaaacaGLOaGa ayzkaaaabaGaeqOSdi2aaWbaaSqabeaacaaIYaaaaaaakiqbek7aIz aalaGafqOSdiMbaSaadaahaaWcbeqaaiaadsfaaaaaaa@43E5@ is a three-dimensional matrix built from the dyadic product of the normalized velocity β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGafqOSdiMbaS aaaaa@378B@ .

Note 4 to entry: The coherent SI unit of the matrix L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36AD@ describing the general Lorentz transformation is one, symbol 1.


fr
transformation de Lorentz, f
transformation des quadrivecteurs d’un référentiel inertiel à un autre référentiel inertiel S′ qui se déplace dans toute direction donnée

Note 1 à l’article: Les transformations de Lorentz forment un groupe. En notant Ω L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaeyyQdC1aaSbaaSqaai aakYeaaeqaaaaa@359C@ l’ensemble des transformations de Lorentz L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36AD@ , on a les règles suivantes:

  1. la transformation identité I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36AD@ appartient à Ω L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaeyyQdC1aaSbaaSqaai aakYeaaeqaaaaa@359C@ ;
  2. une composition de transformations de Lorentz est associative, c’est-à-dire L ( L L )=( L L ) L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaacmGaaiGacmabaaGcbaGabCitayaafaWaaeWaae aaceWHmbGbayaaceWHmbGbaibaaiaawIcacaGLPaaacaGI9aWaaeWa aeaaceWHmbGbauaaceWHmbGbayaaaiaawIcacaGLPaaaceWHmbGbai baaaa@3C48@ ;
  3. pour toute transformation L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36AD@ , il existe une transformation inverse L 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaacmGaaiGacmabaaGcbaGaaCitamaaCaaaleqaba GaeyOeI0IaaGymaaaaaaa@35B5@ de telle sorte que L L 1 =I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaacmGaaiGacmabaaGcbaGaaCitaiaahYeadaahaa WcbeqaaiabgkHiTiaaigdaaaGccqGH9aqpcaWHjbaaaa@386C@ .

Note 2 à l’article: Une transformation de Lorentz est une transformation rotationnelle linéaire dans l’espace-temps.

Note 3 à l’article: Une transformation de Lorentz des référentiels inertiels S, S′ synchronisés peut être exprimée par x _ _ =L x _ _ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaWaaWqaaeaace WG4bGbauaaaaGaeyypa0JaaCitamaameaabaGaamiEaaaaaaa@39DB@ L=( γ γ β x γ β y γ β z γ β x 1+ ( γ1 ) β x 2 β 2 ( γ1 ) β x β y β 2 ( γ1 ) β x β z β 2 γ β y ( γ1 ) β x β y β 2 1+ ( γ1 ) β y 2 β 2 ( γ1 ) β y β z β 2 γ β z ( γ1 ) β x β z β 2 ( γ1 ) β y β z β 2 1+ ( γ1 ) β z 2 β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCitaiabg2 da9maabmaabaqbaeqabqabaaaaaeaacqaHZoWzaeaacqGHsislcqaH ZoWzcqaHYoGydaWgaaWcbaGaamiEaaqabaaakeaacqGHsislcqaHZo WzcqaHYoGydaWgaaWcbaGaamyEaaqabaaakeaacqGHsislcqaHZoWz cqaHYoGydaWgaaWcbaGaamOEaaqabaaakeaacqGHsislcqaHZoWzcq aHYoGydaWgaaWcbaGaamiEaaqabaaakeaacaaIXaGaey4kaSYaaSaa aeaadaqadaqaaiabeo7aNjabgkHiTiaaigdaaiaawIcacaGLPaaacq aHYoGydaqhaaWcbaGaamiEaaqaaiaaikdaaaaakeaacqaHYoGydaah aaWcbeqaaiaaikdaaaaaaaGcbaWaaSaaaeaadaqadaqaaiabeo7aNj abgkHiTiaaigdaaiaawIcacaGLPaaacqaHYoGydaqhaaWcbaGaamiE aaqaaaaakiabek7aInaaBaaaleaacaWG5baabeaaaOqaaiabek7aIn aaCaaaleqabaGaaGOmaaaaaaaakeaadaWcaaqaamaabmaabaGaeq4S dCMaeyOeI0IaaGymaaGaayjkaiaawMcaaiabek7aInaaDaaaleaaca WG4baabaaaaOGaeqOSdi2aaSbaaSqaaiaadQhaaeqaaaGcbaGaeqOS di2aaWbaaSqabeaacaaIYaaaaaaaaOqaaiabgkHiTiabeo7aNjabek 7aInaaBaaaleaacaWG5baabeaaaOqaamaalaaabaWaaeWaaeaacqaH ZoWzcqGHsislcaaIXaaacaGLOaGaayzkaaGaeqOSdi2aa0baaSqaai aadIhaaeaaaaGccqaHYoGydaWgaaWcbaGaamyEaaqabaaakeaacqaH YoGydaahaaWcbeqaaiaaikdaaaaaaaGcbaGaaGymaiabgUcaRmaala aabaWaaeWaaeaacqaHZoWzcqGHsislcaaIXaaacaGLOaGaayzkaaGa eqOSdi2aa0baaSqaaiaadMhaaeaacaaIYaaaaaGcbaGaeqOSdi2aaW baaSqabeaacaaIYaaaaaaaaOqaamaalaaabaWaaeWaaeaacqaHZoWz cqGHsislcaaIXaaacaGLOaGaayzkaaGaeqOSdi2aa0baaSqaaiaadM haaeaaaaGccqaHYoGydaWgaaWcbaGaamOEaaqabaaakeaacqaHYoGy daahaaWcbeqaaiaaikdaaaaaaaGcbaGaeyOeI0Iaeq4SdCMaeqOSdi 2aaSbaaSqaaiaadQhaaeqaaaGcbaWaaSaaaeaadaqadaqaaiabeo7a NjabgkHiTiaaigdaaiaawIcacaGLPaaacqaHYoGydaqhaaWcbaGaam iEaaqaaaaakiabek7aInaaBaaaleaacaWG6baabeaaaOqaaiabek7a InaaCaaaleqabaGaaGOmaaaaaaaakeaadaWcaaqaamaabmaabaGaeq 4SdCMaeyOeI0IaaGymaaGaayjkaiaawMcaaiabek7aInaaDaaaleaa caWG5baabaaaaOGaeqOSdi2aaSbaaSqaaiaadQhaaeqaaaGcbaGaeq OSdi2aaWbaaSqabeaacaaIYaaaaaaaaOqaaiaaigdacqGHRaWkdaWc aaqaamaabmaabaGaeq4SdCMaeyOeI0IaaGymaaGaayjkaiaawMcaai abek7aInaaDaaaleaacaWG6baabaGaaGOmaaaaaOqaaiabek7aInaa CaaaleqabaGaaGOmaaaaaaaaaaGccaGLOaGaayzkaaaaaa@D064@

Dans le cas où la représentation des quadrivecteurs est donnée par x _ _ =( x 0 ; x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaWaaWqaaeaaca WG4baaaiabg2da9iaacIcacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGa ai4oaiqadIhagaWcaiaacMcaaaa@3D00@ et transposée par x _ _ T := ( x 0 ; x ) T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaWaaWqaaeaaca WG4baaamaaCaaaleqabaGaamivaaaakiacakSG6aGaeyypa0ZaaeWa aeaacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGaai4oaiqadIhagaWcaa GaayjkaiaawMcaamaaCaaaleqabaGaamivaaaaaaa@414E@ , alors L=( γ γ β T γ β I+Γ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCitaiabg2 da9maabmaabaqbaeqabiGaaaqaaiabeo7aNbqaaiabgkHiTiabeo7a Njqbek7aIzaalaWaaWbaaSqabeaacaWGubaaaaGcbaGaeyOeI0Iaeq 4SdCMafqOSdiMbaSaaaeaacaWHjbGaey4kaSIaaC4KdaaaaiaawIca caGLPaaaaaa@4764@ , où I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCysaaaa@36AA@ est la matrice d’identité 3×3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgE na0kaaiodaaaa@3969@ et Γ= ( γ1 ) β 2 β β T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaC4Kdiabg2 da9maalaaabaWaaeWaaeaacqaHZoWzcqGHsislcaaIXaaacaGLOaGa ayzkaaaabaGaeqOSdi2aaWbaaSqabeaacaaIYaaaaaaakiqbek7aIz aalaGafqOSdiMbaSaadaahaaWcbeqaaiaadsfaaaaaaa@43E5@ est une matrice tridimensionnelle constituée à partir du produit tensoriel de la vitesse normalisée β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGafqOSdiMbaS aaaaa@378B@ .

Note 4 à l’article: L’unité SI cohérente de la matrice L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeWaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36AD@ qui décrit la transformation de Lorentz est un, symbole 1.


Publication date: 2022-06
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