# 经典电磁理论的协变形式

## 协变量

### 电磁张量

$F_{\alpha \beta} = \left( \begin{matrix} 0 & \frac{E_x}{c} & \frac{E_y}{c} & \frac{E_z}{c} \\ \frac{-E_x}{c} & 0 & -B_z & B_y \\ \frac{-E_y}{c} & B_z & 0 & -B_x \\ \frac{-E_z}{c} & -B_y & B_x & 0 \end{matrix} \right)$

$F^{\mu \nu} \, \stackrel{\mathrm{def}}{=} \, \eta^{\mu \alpha} \, F_{\alpha \beta} \, \eta^{\beta \nu} = \left( \begin{matrix} 0 & \frac{-E_x}{c} & \frac{-E_y}{c} & \frac{-E_z}{c} \\ \frac{E_x}{c} & 0 & -B_z & B_y \\ \frac{E_y}{c} & B_z & 0 & -B_x \\ \frac{E_z}{c} & -B_y & B_x & 0 \end{matrix} \right).$

### 四维电流密度

$J^{\alpha} = \, (c \rho, \mathbf{J} )$

### 电磁四维势

$A_{\alpha} = \left( \phi/c,- \mathbf{A} \right)$

$F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha} \,$

$\partial_\alpha = \frac{\partial}{\partial x^{\alpha}} = \left( \frac{1}{c} \frac{\partial}{\partial t}, \mathbf{\nabla} \right) \,.$

### 电磁应力-能量张量

$T^{\alpha\beta} = \begin{bmatrix} \frac{1}{2}(\epsilon_{0}E^2 + \frac{1}{\mu_{0}}B^2) & S_x/c & S_y/c & S_z/c \\ S_x/c & -\sigma_{xx} & -\sigma_{xy} & -\sigma_{xz} \\ S_y/c & -\sigma_{yx} & -\sigma_{yy} & -\sigma_{yz} \\ S_z/c & -\sigma_{zx} & -\sigma_{zy} & -\sigma_{zz} \end{bmatrix}$

$\mathbf{S} = \frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B} \,$

$\sigma_{ij} = \epsilon_{0}E_{i}E_{j} + \frac{1}{\mu_{0}}B_{i}B_{j} - \tfrac{1}{2}(\epsilon_{0}E^2 + \frac{1}{\mu_{0}}B^2)\delta_{ij} \,.$

$T^{\alpha\beta} = \frac{-1}{\mu_{0}} ( F^{\alpha \gamma}\eta_{\gamma \nu}F^{\nu \beta} + \frac{1}{4}\eta^{\alpha\beta}F_{\gamma \nu}F^{\gamma \nu})$

$\epsilon_{0} \mu_{0} c^2 = 1\,.$

### 其他非电磁学协变量

$x^\alpha = (ct, x, y, z) \,.$
$u^\alpha = \gamma(c,\mathbf{u}) \,$

$p_\alpha = ( E/c, - \mathbf{p}) = m \, \eta_{\alpha \nu} \, u^{\nu} \,$

## 馬克士威方程組

$\eta^{\gamma \nu} \partial_{\gamma} \partial_{\nu} F^{\alpha\beta} \, \stackrel{\mathrm{def}}{=} \, \Box F^{\alpha\beta} \, \stackrel{\mathrm{def}}{=} \, \nabla^2 F^{\alpha\beta} - {1 \over c^2 } { \partial^2 F^{\alpha\beta} \over {\partial t }^2 } = 0 \,.$

### 其他符号记法

$\sum_{x^\alpha =ct,x,y,z}{\partial F^{\alpha \beta}\over\partial x^\alpha} = \mu_{0} J^{\beta} \qquad\hbox{and}\qquad 0 = {\partial F_{\alpha\beta}\over\partial x^\gamma} + {\partial F_{\beta\gamma}\over\partial x^\alpha} + {\partial F_{\gamma\alpha}\over\partial x^\beta}$

${\partial F^{\alpha\beta}\over \partial x^\gamma} \, \stackrel{\mathrm{def}}{=} \, \partial_\gamma F^{\alpha\beta} \, \stackrel{\mathrm{def}}{=} \, {F^{\alpha\beta}}_{,\gamma} \,.$

## 连续性方程

${J^{\alpha}}_{,\alpha} \, \stackrel{\mathrm{def}}{=} \, \partial_{\alpha} J^{\alpha} \, = \, 0 \,.$

## 洛伦兹力

$\frac{d p_{\alpha}}{d \tau} \, = q \, F_{\alpha \beta} \, u^\beta$

${ d p_{\alpha} \over { d t } } = q \, F_{\alpha \beta} \, \frac{d x^\beta}{d t} \,.$

$f_{\mu} = F_{\mu\nu}J^{\nu} .\!$

## 电磁应力-能量张量的微分方程

$\eta_{\alpha \nu} { T^{\nu \beta } }_{,\beta} + F_{\alpha \beta} J^{\beta} = 0 \,$

## 洛伦茨规范条件

$\eta^{\alpha \nu} \, \partial_{\alpha} A_{\nu} = 0 \,.$

### 洛伦茨规范下的麦克斯韦方程组

$\eta^{\sigma \nu} \, \Box A_{\nu} = - \mu_{0} \, J^{\sigma}$

## 介质中麦克斯韦方程组的协变形式

$J^{\alpha} = {J^{\alpha}}_{\text{free}} + {J^{\alpha}}_{\text{bound}} \,.$

$\mathcal{M}^{\mu \nu} = \begin{pmatrix} 0 & -P_xc & -P_yc & -P_zc \\ P_xc & 0 & M_z & -M_y \\ P_yc & -M_z & 0 & M_x \\ P_zc & M_y & -M_x & 0 \end{pmatrix}$

${J^{\mu}}_{\text{bound}} = \partial_{\nu} \mathcal{M}^{\mu \nu} \,.$

$\mathcal{D}^{\mu \nu} = \begin{pmatrix} 0 & D_xc & D_yc & D_zc \\ -D_xc & 0 & H_z & -H_y \\ -D_yc & -H_z & 0 & H_x \\ -D_zc & H_y & -H_x & 0 \end{pmatrix}.$

$\mathcal{D}^{\mu \nu} = \frac{1}{\mu_{0}} F^{\mu \nu} - \mathcal{M}^{\mu \nu} \,$

${J^{\mu}}_{\text{free}} = \partial_{\nu} \mathcal{D}^{\mu \nu} \,.$

$\partial_{\mu} {J^{\mu}}_{\text{bound}} = 0 \,$ :$\partial_{\mu} {J^{\mu}}_{\text{free}} = 0 \,.$

$\mathbf{J}_{\text{free}} = \sigma \mathbf{E} \,$
$\mathbf{P} = \epsilon_0 \chi_e \mathbf{E} \,$
$\mathbf{M} = \chi_m \mathbf{H} \,$

## 拉格朗日量

$\mathcal{L} \, = \, \mathcal{L}_{\mathrm{field}} + \mathcal{L}_{\mathrm{int}} = - \frac{1}{4 \mu_0} F^{\alpha \beta} F_{\alpha \beta} + A_{\alpha} J^{\alpha} \,.$

$\mathcal{L} \, = \, - \frac{1}{4 \mu_0} F^{\alpha \beta} F_{\alpha \beta} + A_{\alpha} J^{\alpha}_{\text{free}} + \frac12 F_{\alpha \beta} \mathcal{M}^{\alpha \beta} \,.$ 对应的非相对论形式为
$\mathcal{L} \, = \, \frac12 (\epsilon_{0} E^2 - \frac{1}{\mu_{0}} B^2) - \phi \, \rho_{\text{free}} + \mathbf{A} \cdot \mathbf{J}_{\text{free}} + \mathbf{E} \cdot \mathbf{P} + \mathbf{B} \cdot \mathbf{M} \,.$

## 广义相对论中的推广

$F_{\alpha \beta} \, = \, \partial_{\alpha} A_{\beta} \, - \, \partial_{\beta} A_{\alpha} \,$
$\mathcal{D}^{\mu\nu} \, = \, \frac{1}{\mu_{0}} \, g^{\mu\alpha} \, F_{\alpha\beta} \, g^{\beta\nu} \, \sqrt{-g} \,$
$J^{\mu} \, = \, \partial_\nu \mathcal{D}^{\mu \nu} \,$
$f_\mu \, = \, F_{\mu\nu} \, J^\nu \,$

## 参考文献

1. ^ 1.0 1.1 Jackson, John David, Classical Electrodynamic 3rd., USA: John Wiley & Sons, Inc., pp. 1–2, 1999, ISBN 978-0-471-30932-1
• Einstein, A. Relativity: The Special and General Theory. New York: Crown. 1961. ISBN 0-517-02961-8.
• Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald. Gravitation. San Francisco: W. H. Freeman. 1973. ISBN 0-7167-0344-0.
• Landau, L. D. and Lifshitz, E. M. Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. 1975. ISBN 0-08-018176-7.
• R. P. Feynman, F. B. Moringo, and W. G. Wagner. Feynman Lectures on Gravitation. Addison-Wesley. 1995. ISBN 0-201-62734-5.