# 电磁四维势

${\displaystyle A_{\alpha }=\left({\frac {\phi }{c}},-{\vec {A}}\right)\qquad \left(A_{\alpha }=(\phi ,-{\vec {A}})\right)}$

${\displaystyle {\vec {E}}=-{\vec {\nabla }}\phi -{\frac {\partial {\vec {A}}}{\partial t}}\qquad \left(-{\vec {\nabla }}\phi -{\frac {1}{c}}{\frac {\partial {\vec {A}}}{\partial t}}\right)}$
${\displaystyle {\vec {B}}={\vec {\nabla }}\times {\vec {A}}}$

${\displaystyle A_{\alpha }g^{\alpha \beta }A_{\beta }={\frac {\phi ^{2}}{c^{2}}}-|{\vec {A}}|^{2}\qquad \left(A_{\alpha }g^{\alpha \beta }A_{\beta }\,=\phi ^{2}-|{\vec {A}}|^{2}\right)}$

${\displaystyle \phi \qquad \rightarrow \qquad \phi +{\frac {\partial \lambda }{\partial t}}\,}$
${\displaystyle {\vec {A}}\qquad \rightarrow \qquad {\vec {A}}-\nabla \lambda \,}$

${\displaystyle \Box A_{\alpha }=\mu _{0}\eta _{\alpha \beta }J^{\beta }\qquad \left(\Box A_{\alpha }={\frac {4\pi }{c}}\eta _{\alpha \beta }J^{\beta }\right)}$

${\displaystyle \Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}}$达朗贝尔算符

${\displaystyle \Box \phi ={\frac {\rho }{\epsilon _{0}}}\qquad \left(\Box \phi =4\pi \rho \right)}$
${\displaystyle \Box {\vec {A}}=\mu _{0}{\vec {j}}\qquad \left(\Box {\vec {A}}={\frac {4\pi }{c}}{\vec {j}}\right)}$

${\displaystyle \phi ({\vec {x}},t)={\frac {1}{4\pi \epsilon _{0}}}\int \mathrm {d} ^{3}x^{\prime }{\frac {\rho ({\vec {x}}^{\prime },\tau )}{\left|{\vec {x}}-{\vec {x}}^{\prime }\right|}}}$
${\displaystyle {\vec {A}}({\vec {x}},t)={\frac {\mu _{0}}{4\pi }}\int \mathrm {d} ^{3}x^{\prime }{\frac {{\vec {j}}({\vec {x}}^{\prime },\tau )}{\left|{\vec {x}}-{\vec {x}}^{\prime }\right|}}}$,