電磁場テンソル

${\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}$

${\displaystyle A^{\mu }=\left({\frac {\phi }{c}},{\boldsymbol {A}}\right),~A_{\mu }=\eta _{\mu \nu }A^{\nu }=\left(-{\frac {\phi }{c}},{\boldsymbol {A}}\right)}$

${\displaystyle \partial _{\mu }={\frac {\partial }{\partial x^{\mu }}}=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)}$

${\displaystyle {\boldsymbol {E}}=-\nabla \phi -{\frac {\partial {\boldsymbol {A}}}{\partial t}}}$

${\displaystyle {\boldsymbol {B}}=\nabla \times {\boldsymbol {A}}}$

${\displaystyle E_{j}/c=\partial _{j}A_{0}-\partial _{0}A_{j}=-F_{0j}}$

${\displaystyle B_{i}\epsilon _{ijk}=\partial _{j}A_{k}-\partial _{k}A_{j}=F_{jk}}$

${\displaystyle (F_{01},F_{02},F_{03})=(-E_{x}/c,-E_{y}/c,-E_{z}/c)}$

${\displaystyle (F_{23},F_{31},F_{12})=(B_{x},B_{y},B_{z})}$

${\displaystyle (F^{01},F^{02},F^{03})=(E_{x}/c,E_{y}/c,E_{z}/c)}$

${\displaystyle (F^{23},F^{31},F^{12})=(B_{x},B_{y},B_{z})}$

${\displaystyle (F_{\mu \nu })={\begin{bmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&B_{z}&-B_{y}\\E_{y}/c&-B_{z}&0&B_{x}\\E_{z}/c&B_{y}&-B_{x}&0\\\end{bmatrix}},}$

${\displaystyle (F^{\mu \nu })={\begin{bmatrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&B_{z}&-B_{y}\\-E_{y}/c&-B_{z}&0&B_{x}\\-E_{z}/c&B_{y}&-B_{x}&0\\\end{bmatrix}}}$

${\displaystyle {\tilde {F}}^{\mu \nu }={\frac {1}{2}}\epsilon ^{\mu \nu \rho \sigma }F_{\rho \sigma }}$

${\displaystyle ({\tilde {F}}^{01},{\tilde {F}}^{02},{\tilde {F}}^{03})=(F_{23},F_{31},F_{12})=(B_{x},B_{y},B_{z})}$

${\displaystyle ({\tilde {F}}^{23},{\tilde {F}}^{31},{\tilde {F}}^{12})=(F_{01},F_{02},F_{03})=(-E_{x}/c,-E_{y}/c,-E_{z}/c)}$

${\displaystyle ({\tilde {F}}^{\mu \nu })={\begin{bmatrix}0&B_{x}&B_{y}&B_{z}\\-B_{x}&0&-E_{z}/c&E_{y}/c\\-B_{y}&E_{z}/c&0&-E_{x}/c\\-B_{z}&-E_{y}/c&E_{x}/c&0\\\end{bmatrix}}}$

${\displaystyle (G^{01},G^{02},G^{03})=(D_{x},D_{y},D_{z})}$

${\displaystyle (G^{23},G^{31},G^{12})=(H_{x}/c,H_{y}/c,H_{z}/c)}$

${\displaystyle G^{\mu \nu }={\frac {1}{Z_{0}}}F^{\mu \nu }+P^{\mu \nu }={\frac {1}{c\mu _{0}}}F^{\mu \nu }+P^{\mu \nu }}$

${\displaystyle (P^{01},P^{02},P^{03})=(P_{x},P_{y},P_{z})}$

${\displaystyle (P^{23},P^{31},P^{12})=(-M_{x}/c,-M_{y}/c,-M_{z}/c)}$

${\displaystyle (G^{\mu \nu })={\begin{bmatrix}0&D_{x}&D_{y}&D_{z}\\-D_{x}&0&H_{z}/c&-H_{y}/c\\-D_{y}&-H_{z}/c&0&H_{x}/c\\-D_{z}&H_{y}/c&-H_{x}/c&0\\\end{bmatrix}}}$

${\displaystyle (P^{\mu \nu })={\begin{bmatrix}0&P_{x}&P_{y}&P_{z}\\-P_{x}&0&-M_{z}/c&M_{y}/c\\-P_{y}&M_{z}/c&0&-M_{x}/c\\-P_{z}&-M_{y}/c&M_{x}/c&0\\\end{bmatrix}}}$

${\displaystyle A_{\mu }=\left(-{\frac {\phi }{c}},A_{r},rA_{\theta },rA_{\varphi }\sin \theta \right)}$

${\displaystyle (F_{\mu \nu })={\begin{bmatrix}0&-E_{r}/c&-rE_{\theta }/c&-rE_{\varphi }\sin \theta /c\\E_{r}/c&0&rB_{\varphi }&-rB_{\theta }\sin \theta \\rE_{\theta }/c&-rB_{\varphi }&0&r^{2}B_{r}\sin \theta \\rE_{\varphi }\sin \theta /c&rB_{\theta }\sin \theta &-r^{2}B_{r}\sin \theta &0\\\end{bmatrix}}}$

${\displaystyle \eta _{\mu \nu }=\operatorname {diag} (-1,1,r^{2},r^{2}\sin ^{2}\theta )}$

${\displaystyle \eta ^{\mu \nu }=\operatorname {diag} (-1,1,{\tfrac {1}{r^{2}}},{\tfrac {1}{r^{2}\sin ^{2}\theta }})}$

${\displaystyle (F^{\mu \nu })={\begin{bmatrix}0&E_{r}/c&E_{\theta }/cr&E_{\varphi }/cr\sin \theta \\-E_{r}/c&0&B_{\varphi }/r&-B_{\theta }/r\sin \theta \\-E_{\theta }/cr&-B_{\varphi }/r&0&B_{r}/r^{2}\sin \theta \\-E_{\varphi }/cr\sin \theta &B_{\theta }/r\sin \theta &-B_{r}/r^{2}\sin \theta &0\\\end{bmatrix}}}$

${\displaystyle (F^{\mu \nu })={\begin{bmatrix}0&{\frac {1}{4\pi \epsilon _{0}c}}{\frac {q}{r^{2}}}&0&0\\-{\frac {1}{4\pi \epsilon _{0}c}}{\frac {q}{r^{2}}}&0&0&0\\0&0&0&0\\0&0&0&0\\\end{bmatrix}}}$

${\displaystyle \partial _{\rho }F_{\mu \nu }+\partial _{\mu }F_{\nu \rho }+\partial _{\nu }F_{\rho \mu }=0}$

${\displaystyle \partial _{\mu }{\tilde {F}}^{\mu \nu }={\frac {1}{2}}\epsilon ^{\mu \nu \rho \sigma }\partial _{\mu }F_{\rho \sigma }=0}$

${\displaystyle \nabla \cdot {\boldsymbol {B}}=0}$

${\displaystyle \nabla \times {\boldsymbol {E}}+{\frac {\partial {\boldsymbol {B}}}{\partial t}}=\mathbf {0} }$

${\displaystyle \partial _{\mu }F^{\mu \nu }=-{\frac {Z_{0}}{c}}\,j^{\nu }=-\mu _{0}j^{\nu }}$

${\displaystyle \nabla \cdot {\boldsymbol {E}}={\frac {\rho }{\epsilon _{0}}}}$

${\displaystyle \nabla \times {\boldsymbol {B}}-{\frac {1}{c^{2}}}{\frac {\partial {\boldsymbol {E}}}{\partial t}}=\mu _{0}{\boldsymbol {j}}}$

${\displaystyle \partial _{\mu }G^{\mu \nu }=-{\frac {1}{c}}\,j^{\nu }}$

${\displaystyle \nabla \cdot {\boldsymbol {D}}=\rho }$

${\displaystyle \nabla \times {\boldsymbol {H}}-{\frac {\partial {\boldsymbol {D}}}{\partial t}}={\boldsymbol {j}}}$

${\displaystyle {\dot {p}}_{\mu }=-q{\dot {X}}^{\nu }F_{\nu \mu }(X)}$

${\displaystyle F_{\mu \nu }={\mathcal {D}}_{\mu }A_{\nu }-{\mathcal {D}}_{\nu }A_{\mu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}$

${\displaystyle {\mathcal {D}}_{\mu }F^{\mu \nu }=-{\frac {Z_{0}}{c}}\,j^{\nu }}$