Queries, comments, suggestions? Please contact us.



Area Physics for electrotechnology / Relativistic physics for electrotechnology

IEV ref 113-07-11

Symbol
A _ _ <in special theory of relativity> <en relativité restreinte>

en
four-vector
4-vector
vector in space-time consisting of a one-dimensional time-related component and a spatial three-dimensional vector

Note 1 to entry: Four-vector symbols can be written using two different forms of presentation:

  1. a light face single letter in italics with a double underscore, which is that form mostly used in the special theory of relativity (STR) when the first component is imaginary, by analogy with the underscoring of symbols of complex quantities, e.g. x _ _ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacaabaaGcbaGaaGPaVpaameaabaGaam iEaaaaaaa@359F@ ;
  2. a light face single letter in italics with a subscript (denoting the covariant component) or a superscript (denoting the contravariant component), which can or cannot be enclosed in braces (curly brackets), and which is that form mostly used in theoretical physics in both special theory of relativity and general theory of relativity (GTR), e.g. { x μ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacaabaaGcbaWaaiWaaeaacaWLa8Uaam iEamaaBaaaleaacaqI8oaabeaaaOGaay5Eaiaaw2haaaaa@393C@ or { x μ } , x μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacqaH8oqBaeqaaaaa@38B6@ or x μ .

Note 2 to entry: In STR, the time-related component can be expressed as an imaginary quantity, using symbol j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaciOAaaaa@33F9@ as the imaginary unit. Then, pseudo-Euclidean metric can be used with rules of Euclidean metric but allowing negative magnitudes | x _ _ |<0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaqWaaeaadaadbaqaai aadIhaaaaacaGLhWUaayjcSdGaeyipaWJaaGimaaaa@38F7@ and zero magnitudes | x _ _ |=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaqWaaeaadaadbaqaai aadIhaaaaacaGLhWUaayjcSdGaeyypa0JaaGimaaaa@38F9@ even for x _ _ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaWqaaeaacaWG4baaai abgcMi5kaaicdaaaa@3698@ . See IEV 113-07-18.

In case time-related component is real, it is denoted as the fourth component x 4 =ct MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaamiEamaaBaaaleaaca aI0aaabeaakiabg2da9iaadogacaWG0baaaa@37E1@ and the space-related components are x 1 , x 2 , x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaamiEamaaBaaaleaaca aIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiil aiaadIhadaWgaaWcbaGaaG4maaqabaaaaa@3A2C@ . The corresponding components of the metric tensor yielding the four-scalar product and squared four-magnitude have opposite signs, e.g., for flat space-time in STR g 11 = g 22 = g 33 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaam4zamaaBaaaleaaca aIXaGaaGymaaqabaGccqGH9aqpcaWGNbWaaSbaaSqaaiaaikdacaaI Yaaabeaakiabg2da9iaadEgadaWgaaWcbaGaaG4maiaaiodaaeqaaO Gaeyypa0JaaGymaaaa@3EA4@ , g 44 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaam4zamaaBaaaleaaca aI0aGaaGinaaqabaGccqGH9aqpcqGHsislcaaIXaaaaa@3855@ or g 11 = g 22 = g 33 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaam4zamaaBaaaleaaca aIXaGaaGymaaqabaGccqGH9aqpcaWGNbWaaSbaaSqaaiaaikdacaaI Yaaabeaakiabg2da9iaadEgadaWgaaWcbaGaaG4maiaaiodaaeqaaO Gaeyypa0JaeyOeI0IaaGymaaaa@3F91@ , g 44 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaam4zamaaBaaaleaaca aI0aGaaGinaaqabaGccqGH9aqpcaaIXaaaaa@3768@ . In GTR, the non-diagonal metric tensor is used.

Note 3 to entry: The representations used in this part of IEC 60050 are x _ _ =( x 0 , x 1 , x 2 , x 3 )={ x μ }=( x 0 ,{ x m } ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWqaaeaaca WG4baaaiabg2da9iaacIcacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGa aiilaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamiEamaaBa aaleaacaaIYaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaiodaaeqa aOGaaiykaiabg2da9maacmaabaGaaCjaVlaadIhadaWgaaWcbaGaeq iVd0gabeaaaOGaay5Eaiaaw2haaiabg2da9maabmaabaGaamiEamaa BaaaleaacaaIWaaabeaakiaacYcadaGadaqaaiaadIhadaWgaaWcba GaamyBaaqabaaakiaawUhacaGL9baaaiaawIcacaGLPaaaaaa@5438@ , where x 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacaabaaGcbaGaamiEamaaBaaaleaaca aIWaaabeaaaaa@34E9@ is the time-related component and x m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGTbaabeaaaaa@37F2@ are the space-related components. In three-dimensional space, components of three-dimensional vectors are denoted using lowercase Latin letters for indices (i,j,k,l,m,) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaaiikaiaadMgacaGGSa GaamOAaiaacYcacaWGRbGaaiilaiaadYgacaGGSaGaamyBaiaacYca caGGUaGaaiOlaiaac6cacaGGPaaaaa@3E98@ .

In four-dimensional space, components of four-dimensional vectors are denoted using lowercase Greek letters for indices, ( ι,κ,λ,μ,ν, ) . In STR, indices range usually from 0 to 3, where 0 is used for the imaginary time-related component, and in GTR, indices range usually from 1 to 4 where 4 is used for the real time-related component.

Examples in STR are the position four-vector x _ _ :=( x 0 , x 1 , x 2 , x 3 )=(j c 0 t,x,y,z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaWqaaeaacaWG4baaai aaysW7cGaGGkOoaiabg2da9iaacIcacaWG4bWaaSbaaSqaaiaaicda aeqaaOGaaiilaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaam iEamaaBaaaleaacaaIYaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaa iodaaeqaaOGaaiykaiabg2da9iaacIcaciGGQbGaam4yamaaBaaale aacaaIWaaabeaakiaadshacaGGSaGaamiEaiaacYcacaWG5bGaaiil aiaadQhacaGGPaaaaa@4EDD@ and the electromagnetic four-potential A _ _ =( jV/ c 0 ; A x , A y , A z )=( jV/ c 0 ; A ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaWqaaeaacaWGbbaaai abg2da9maabmaabaGaciOAaiaadAfacaGGVaGaam4yamaaBaaaleaa caaIWaaabeaakiaabUdacaWGbbWaaSbaaSqaaiaadIhaaeqaaOGaai ilaiaadgeadaWgaaWcbaGaamyEaaqabaGccaGGSaGaamyqamaaBaaa leaacaWG6baabeaaaOGaayjkaiaawMcaaiabg2da9maabmaabaGaci OAaiaadAfacaGGVaGaam4yamaaBaaaleaacaaIWaaabeaakiaabUda daWhcaqaaiaadgeaaiaawEniaaGaayjkaiaawMcaaaaa@4CEE@ .

Note 4 to entry: If there is no risk of misunderstanding, “free index symbolic” is used, e.g. a component x μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacqaH8oqBaeqaaaaa@38B6@ instead of full vector { x μ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacaabaaGcbaWaaiWaaeaacaWLa8Uaam iEamaaBaaaleaacaqI8oaabeaaaOGaay5Eaiaaw2haaaaa@393C@ . Index μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaaKiVdaaa@3453@ is then called “free index”.


fr
quadrivecteur, m
vecteur dans l’espace-temps qui comprend une composante temporelle unidimensionnelle et un vecteur spatial tridimensionnel

Note 1 à l’article: Les symboles des quadrivecteurs peuvent s’écrire sous deux formes différentes:

  1. une lettre unique en italique ordinaire avec double soulignement, qui est la forme utilisée principalement dans la relativité restreinte lorsque la première composante est imaginaire, par analogie au soulignement des symboles de grandeurs complexes, par exemple x _ _ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacaabaaGcbaGaaGPaVpaameaabaGaam iEaaaaaaa@359F@ ;
  2. une lettre unique en italique ordinaire avec un indice (indiquant une composante covariante) ou un exposant (indiquant une composante contravariante), délimitée ou non par des accolades, qui est la forme principalement utilisée en physique théorique dans la relativité restreinte et la relativité générale, par exemple { x μ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacaabaaGcbaWaaiWaaeaacaWLa8Uaam iEamaaBaaaleaacaqI8oaabeaaaOGaay5Eaiaaw2haaaaa@393C@ ou { x μ } , x μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacqaH8oqBaeqaaaaa@38B6@ ou x μ .

Note 2 à l’article: En relativité restreinte, la composante temporelle peut être exprimée en tant que grandeur imaginaire, avec le symbole j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaciOAaaaa@33F9@ comme unité imaginaire. Puis, la métrique pseudo-euclidienne peut être utilisée avec les règles de la métrique euclidienne, mais en admettant des amplitudes négatives | x _ _ |<0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaqWaaeaadaadbaqaai aadIhaaaaacaGLhWUaayjcSdGaeyipaWJaaGimaaaa@38F7@ et des amplitudes nulles | x _ _ |=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaqWaaeaadaadbaqaai aadIhaaaaacaGLhWUaayjcSdGaeyypa0JaaGimaaaa@38F9@ même pour x _ _ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaWqaaeaacaWG4baaai abgcMi5kaaicdaaaa@3698@ . Voir IEV 113-07-18.

Lorsque la composante temporelle est réelle, elle est désignée comme la quatrième composante x 4 =ct MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaamiEamaaBaaaleaaca aI0aaabeaakiabg2da9iaadogacaWG0baaaa@37E1@ et les composantes spatiales sont x 1 , x 2 , x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaamiEamaaBaaaleaaca aIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiil aiaadIhadaWgaaWcbaGaaG4maaqabaaaaa@3A2C@ . Les composantes correspondantes du tenseur métrique qui génèrent le produit quadriscalaire et la norme au carré, ont des signes opposés, par exemple, pour un espace-temps plat en relativité restreinte g 11 = g 22 = g 33 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaam4zamaaBaaaleaaca aIXaGaaGymaaqabaGccqGH9aqpcaWGNbWaaSbaaSqaaiaaikdacaaI Yaaabeaakiabg2da9iaadEgadaWgaaWcbaGaaG4maiaaiodaaeqaaO Gaeyypa0JaaGymaaaa@3EA4@ , g 44 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaam4zamaaBaaaleaaca aI0aGaaGinaaqabaGccqGH9aqpcqGHsislcaaIXaaaaa@3855@ ou g 11 = g 22 = g 33 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaam4zamaaBaaaleaaca aIXaGaaGymaaqabaGccqGH9aqpcaWGNbWaaSbaaSqaaiaaikdacaaI Yaaabeaakiabg2da9iaadEgadaWgaaWcbaGaaG4maiaaiodaaeqaaO Gaeyypa0JaeyOeI0IaaGymaaaa@3F91@ , g 44 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaam4zamaaBaaaleaaca aI0aGaaGinaaqabaGccqGH9aqpcaaIXaaaaa@3768@ . En relativité générale, le tenseur métrique non diagonal est utilisé.

Note 3 à l’article: Les représentations utilisées dans la présente partie de l’IEC 60050 sont x _ _ =( x 0 , x 1 , x 2 , x 3 )={ x μ }=( x 0 ,{ x m } ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWqaaeaaca WG4baaaiabg2da9iaacIcacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGa aiilaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamiEamaaBa aaleaacaaIYaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaiodaaeqa aOGaaiykaiabg2da9maacmaabaGaaCjaVlaadIhadaWgaaWcbaGaeq iVd0gabeaaaOGaay5Eaiaaw2haaiabg2da9maabmaabaGaamiEamaa BaaaleaacaaIWaaabeaakiaacYcadaGadaqaaiaadIhadaWgaaWcba GaamyBaaqabaaakiaawUhacaGL9baaaiaawIcacaGLPaaaaaa@5438@ , où x 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacaabaaGcbaGaamiEamaaBaaaleaaca aIWaaabeaaaaa@34E9@ est la composante temporelle et les x m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGTbaabeaaaaa@37F2@ représentent les composantes spatiales.

Dans l’espace tridimensionnel, les composantes des vecteurs tridimensionnels sont désignées au moyen de lettres latines minuscules pour les indices (i,j,k,l,m,) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaaiikaiaadMgacaGGSa GaamOAaiaacYcacaWGRbGaaiilaiaadYgacaGGSaGaamyBaiaacYca caGGUaGaaiOlaiaac6cacaGGPaaaaa@3E98@ .

Dans l’espace quadridimensionnel, les composantes des vecteurs quadridimensionnels sont désignées au moyen de lettres grecques minuscules pour les indices, ( ι,κ,λ,μ,ν, ) . En relativité restreinte, les indices sont généralement compris entre 0 et 3, où 0 est utilisé pour la composante temporelle imaginaire, et en relativité générale, les indices sont généralement compris entre 1 et 4 où 4 est utilisé pour la composante temporelle réelle.

Exemples en relativité restreinte: quadrivecteur position x _ _ :=( x 0 , x 1 , x 2 , x 3 )=(j c 0 t,x,y,z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaWqaaeaacaWG4baaai aaysW7cGaGGkOoaiabg2da9iaacIcacaWG4bWaaSbaaSqaaiaaicda aeqaaOGaaiilaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaam iEamaaBaaaleaacaaIYaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaa iodaaeqaaOGaaiykaiabg2da9iaacIcaciGGQbGaam4yamaaBaaale aacaaIWaaabeaakiaadshacaGGSaGaamiEaiaacYcacaWG5bGaaiil aiaadQhacaGGPaaaaa@4EDD@ et quadrivecteur potentiel électromagnétique A _ _ =( jV/ c 0 ; A x , A y , A z )=( jV/ c 0 ; A ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaWaaWqaaeaacaWGbbaaai abg2da9maabmaabaGaciOAaiaadAfacaGGVaGaam4yamaaBaaaleaa caaIWaaabeaakiaabUdacaWGbbWaaSbaaSqaaiaadIhaaeqaaOGaai ilaiaadgeadaWgaaWcbaGaamyEaaqabaGccaGGSaGaamyqamaaBaaa leaacaWG6baabeaaaOGaayjkaiaawMcaaiabg2da9maabmaabaGaci OAaiaadAfacaGGVaGaam4yamaaBaaaleaacaaIWaaabeaakiaabUda daWhcaqaaiaadgeaaiaawEniaaGaayjkaiaawMcaaaaa@4CEE@ .

Note 4 à l’article: En l’absence de tout risque d’incompréhension, une “symbolique à indice libre” est utilisée, par exemple, une composante x μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacqaH8oqBaeqaaaaa@38B6@ au lieu du vecteur complet { x μ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacaabaaGcbaWaaiWaaeaacaWLa8Uaam iEamaaBaaaleaacaqI8oaabeaaaOGaay5Eaiaaw2haaaaa@393C@ . L’indice μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm 0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9ad baGaaiGabaqaamaaceGaaiGacmabaaGcbaGaaKiVdaaa@3453@ est alors appelé "indice libre".


Publication date: 2022-06
Copyright © IEC 2022. All Rights Reserved.