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Area Physics for electrotechnology / Relativistic physics for electrotechnology

IEV ref 113-07-12

en
special Lorentz transformation
transformation of four-vectors from one inertial frame S to another inertial frame S′ with parallel coordinate axes x k || x k ,k=1,2,3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacaabaaGcbaGaamiEamaaBaaaleaaca WGRbaabeaakiaacYhacaGG8bGabmiEayaafaWaaSbaaSqaaiaadUga aeqaaOGaaiilaiaaywW7caWGRbGaeyypa0JaaGymaiaacYcacaaIYa Gaaiilaiaaiodaaaa@411F@ , while moving along one of these axes usually denoted by k=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacaabaaGcbaGaam4Aaiabg2da9iaaig daaaa@35B6@

Note 1 to entry: The term “special” in "special Lorentz transformation" is used with a different meaning than that in the term “special theory of relativity”.

Note 2 to entry: For a position vector ( x 0 = c 0 t,x,y,z ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaeWaaeaacaWG4bWaaS baaSqaaiaaicdaaeqaaOGaeyypa0Jaam4yamaaBaaaleaacaaIWaaa beaakiaadshacaGGSaGaamiEaiaacYcacaWG5bGaaiilaiaadQhaai aawIcacaGLPaaaaaa@3F60@ and β =( β,0,0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaa8HaaeaacaaMi8Uaiu gGjk7aaiaawEniaiabg2da9maabmaabaGaaKOSdiaacYcacaaIWaGa aiilaiaaicdaaiaawIcacaGLPaaaaaa@3F1E@ as the velocity of S′ regarding to S, the special Lorentz transformation reads

c 0 t =γ( c 0 tβx ) x =γ( xβ c 0 t ) y =y z =z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGceaqabeaacaWGJbWaaSbaaS qaaiaaicdaaeqaaOGabmiDayaafaGaeyypa0JaaK4SdmaabmaabaGa am4yamaaBaaaleaacaaIWaaabeaakiaadshacqGHsislcGaDaMOSdi aayIW7caWG4baacaGLOaGaayzkaaaabaGabmiEayaafaGaeyypa0Ja aK4SdmaabmaabaGaamiEaiabgkHiTiaajk7acaaMc8Uaam4yamaaBa aaleaacaaIWaaabeaakiaajshaaiaawIcacaGLPaaaaeaaceWG5bGb auaacqGH9aqpcaWG5baabaGabmOEayaafaGaeyypa0JaamOEaaaaaa@54C9@

For a position vector ( x 0 =j c 0 t, x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaeWaaeaacaWG4bWaaS baaSqaaiaaicdaaeqaaOGaeyypa0JaciOAaiaadogadaWgaaWcbaGa aGimaaqabaGccaWG0bGaaiilaiaadIhadaWgaaWcbaGaaGymaaqaba GccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaacYcacaWG4bWa aSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaaaaa@4323@ in a complex form with pseudo-Euclidian metric, and β =( β,0,0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8Haaeaaca aMi8UamugGek7aIbGaay51GaGaeyypa0ZaaeWaaeaacqaHYoGycaGG SaGaaGimaiaacYcacaaIWaaacaGLOaGaayzkaaaaaa@42AE@ as the velocity of S′ regarding to S, the special Lorentz transformation reads

x 0 =γ x 0 jβγ x 1 x 1 =jβγ x 0 +γ x 1 x 2 = x 2 x 3 = x 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGceaqabeaaceWG4bGbauaada WgaaWcbaGaaGimaaqabaGccqGH9aqpcaqIZoGaamiEamaaBaaaleaa caaIWaaabeaakiabgkHiTiGacQgacGaDaMOSdiaajo7acaaMi8Uaam iEamaaBaaaleaacaaIXaaabeaaaOqaaiqadIhagaqbamaaBaaaleaa caaIXaaabeaakiabg2da9iGacQgacaqIYoGaaK4SdiaayIW7caWG4b WaaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaaK4SdiaadIhadaWgaaWc baGaaGymaaqabaaakeaaceWG4bGbauaadaWgaaWcbaGaaGOmaaqaba GccqGH9aqpcaWG4bWaaSbaaSqaaiaaikdaaeqaaaGcbaGabmiEayaa faWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaamiEamaaBaaaleaaca aIZaaabeaaaaaa@59EF@

showing that the special Lorentz transformation is a rotation in a complex plane ( x 0 ; x 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaeWaaeaacaWG4bWaaS baaSqaaiaaicdaaeqaaOGaai4oaiaadIhadaWgaaWcbaGaaGymaaqa baaakiaawIcacaGLPaaaaaa@392C@ with a complex angle φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaaKOXdaaa@345D@ where tanφ=β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaciiDaiaacggacaGGUb GaaKOXdiabg2da9iaajk7aaaa@3974@ .

Note 3 to entry: Two special Lorentz transformations along the same axis result in a special Lorentz transformation along the same axis. Two special Lorentz transformations along different axes usually result in a general Lorentz transformation.


fr
transformation de Lorentz spéciale, f
transformation des quadrivecteurs d’un référentiel inertiel S à un autre référentiel inertiel S′ avec des axes de coordonnées parallèles x k || x k ,k=1,2,3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacaabaaGcbaGaamiEamaaBaaaleaaca WGRbaabeaakiaacYhacaGG8bGabmiEayaafaWaaSbaaSqaaiaadUga aeqaaOGaaiilaiaaywW7caWGRbGaeyypa0JaaGymaiaacYcacaaIYa Gaaiilaiaaiodaaaa@411F@ , conjointement à un déplacement le long de l’un de ces axes généralement désigné par k=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacaabaaGcbaGaam4Aaiabg2da9iaaig daaaa@35B6@

Note 1 à l’article: En anglais, le terme “special” dans "special Lorentz transformation" est utilisé avec une signification différente de "special theory of relativity".

Note 2 à l’article: Pour un quadrivecteur position ( x 0 = c 0 t,x,y,z ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaeWaaeaacaWG4bWaaS baaSqaaiaaicdaaeqaaOGaeyypa0Jaam4yamaaBaaaleaacaaIWaaa beaakiaadshacaGGSaGaamiEaiaacYcacaWG5bGaaiilaiaadQhaai aawIcacaGLPaaaaaa@3F60@ et une vitesse β =( β,0,0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaa8HaaeaacaaMi8Uaiu gGjk7aaiaawEniaiabg2da9maabmaabaGaaKOSdiaacYcacaaIWaGa aiilaiaaicdaaiaawIcacaGLPaaaaaa@3F1E@ de S′ par rapport à S, la transformation de Lorentz spéciale s’écrit comme suit

c 0 t =γ( c 0 tβx ) x =γ( xβ c 0 t ) y =y z =z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGceaqabeaacaWGJbWaaSbaaS qaaiaaicdaaeqaaOGabmiDayaafaGaeyypa0JaaK4SdmaabmaabaGa am4yamaaBaaaleaacaaIWaaabeaakiaadshacqGHsislcGaDaMOSdi aayIW7caWG4baacaGLOaGaayzkaaaabaGabmiEayaafaGaeyypa0Ja aK4SdmaabmaabaGaamiEaiabgkHiTiaajk7acaaMc8Uaam4yamaaBa aaleaacaaIWaaabeaakiaajshaaiaawIcacaGLPaaaaeaaceWG5bGb auaacqGH9aqpcaWG5baabaGabmOEayaafaGaeyypa0JaamOEaaaaaa@54C9@

Pour un quadrivecteur position complexe ( x 0 =j c 0 t, x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaeWaaeaacaWG4bWaaS baaSqaaiaaicdaaeqaaOGaeyypa0JaciOAaiaadogadaWgaaWcbaGa aGimaaqabaGccaWG0bGaaiilaiaadIhadaWgaaWcbaGaaGymaaqaba GccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaacYcacaWG4bWa aSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaaaaa@4323@ avec métrique pseudo-euclidienne et une vitesse β =( β,0,0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8Haaeaaca aMi8UamugGek7aIbGaay51GaGaeyypa0ZaaeWaaeaacqaHYoGycaGG SaGaaGimaiaacYcacaaIWaaacaGLOaGaayzkaaaaaa@42AE@ de S′ par rapport à S, la transformation de Lorentz spéciale s’écrit comme suit

x 0 =γ x 0 jβγ x 1 x 1 =jβγ x 0 +γ x 1 x 2 = x 2 x 3 = x 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGceaqabeaaceWG4bGbauaada WgaaWcbaGaaGimaaqabaGccqGH9aqpcaqIZoGaamiEamaaBaaaleaa caaIWaaabeaakiabgkHiTiGacQgacGaDaMOSdiaajo7acaaMi8Uaam iEamaaBaaaleaacaaIXaaabeaaaOqaaiqadIhagaqbamaaBaaaleaa caaIXaaabeaakiabg2da9iGacQgacaqIYoGaaK4SdiaayIW7caWG4b WaaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaaK4SdiaadIhadaWgaaWc baGaaGymaaqabaaakeaaceWG4bGbauaadaWgaaWcbaGaaGOmaaqaba GccqGH9aqpcaWG4bWaaSbaaSqaaiaaikdaaeqaaaGcbaGabmiEayaa faWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaamiEamaaBaaaleaaca aIZaaabeaaaaaa@59EF@

ce qui indique que la transformation de Lorentz spéciale est une rotation dans un plan complexe ( x 0 ; x 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaeWaaeaacaWG4bWaaS baaSqaaiaaicdaaeqaaOGaai4oaiaadIhadaWgaaWcbaGaaGymaaqa baaakiaawIcacaGLPaaaaaa@392C@ avec un angle complexe φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaaKOXdaaa@345D@ tanφ=β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaciiDaiaacggacaGGUb GaaKOXdiabg2da9iaajk7aaaa@3974@ .

Note 3 à l’article: Deux transformations de Lorentz spéciales le long du même axe produisent une transformation de Lorentz spéciale le long du même axe. Deux transformations de Lorentz spéciales le long d’axes différents produisent habituellement une transformation de Lorentz.


Publication date: 2022-06
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