# 电磁四维势

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

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

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

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

$\Box A_{\alpha} =\mu_0 \eta_{\alpha \beta} J^{\beta} \qquad \left( \Box A_{\alpha} =\frac{4 \pi}{c} \eta_{\alpha \beta} J^{\beta} \right)$

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

$\Box \phi = \frac{\rho}{\epsilon_0} \qquad \left(\Box \phi =4 \pi \rho \right)$
$\Box \vec{A} =\mu_0 \vec{j} \qquad \left( \Box \vec{A} = \frac{4 \pi}{c} \vec{j} \right)$

$\phi (\vec{x}, t) = \frac{1}{4 \pi \epsilon_0} \int \mathrm{d}^3 x^\prime \frac{\rho( \vec{x}^\prime, \tau)}{ \left| \vec{x} - \vec{x}^\prime \right|}$
$\vec A (\vec{x}, t) = \frac{\mu_0}{4 \pi} \int \mathrm{d}^3 x^\prime \frac{\vec{j}( \vec{x}^\prime, \tau)}{ \left| \vec{x} - \vec{x}^\prime \right|}$,

## 参考文献

• Rindler, Wolfgang. Introduction to Special Relativity (2nd). Oxford: Oxford University Press. 1991. ISBN 0-19-853952-5.
• Jackson, J D. Classical Electrodynamics (3rd). New York: Wiley. 1999. ISBN 0-471-30932-X.