# 经典电磁理论的协变形式

## 协变量

### 电磁张量

${\displaystyle F_{\alpha \beta }=\left({\begin{matrix}0&{\frac {E_{x}}{c}}&{\frac {E_{y}}{c}}&{\frac {E_{z}}{c}}\\{\frac {-E_{x}}{c}}&0&-B_{z}&B_{y}\\{\frac {-E_{y}}{c}}&B_{z}&0&-B_{x}\\{\frac {-E_{z}}{c}}&-B_{y}&B_{x}&0\end{matrix}}\right)}$

${\displaystyle F^{\mu \nu }\,{\stackrel {\mathrm {def} }{=}}\,\eta ^{\mu \alpha }\,F_{\alpha \beta }\,\eta ^{\beta \nu }=\left({\begin{matrix}0&{\frac {-E_{x}}{c}}&{\frac {-E_{y}}{c}}&{\frac {-E_{z}}{c}}\\{\frac {E_{x}}{c}}&0&-B_{z}&B_{y}\\{\frac {E_{y}}{c}}&B_{z}&0&-B_{x}\\{\frac {E_{z}}{c}}&-B_{y}&B_{x}&0\end{matrix}}\right).}$

### 四维电流密度

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

### 电磁四维势

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

${\displaystyle F_{\alpha \beta }=\partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha }\,}$

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

### 电磁应力-能量张量

${\displaystyle T^{\alpha \beta }={\begin{bmatrix}{\frac {1}{2}}(\epsilon _{0}E^{2}+{\frac {1}{\mu _{0}}}B^{2})&S_{x}/c&S_{y}/c&S_{z}/c\\S_{x}/c&-\sigma _{xx}&-\sigma _{xy}&-\sigma _{xz}\\S_{y}/c&-\sigma _{yx}&-\sigma _{yy}&-\sigma _{yz}\\S_{z}/c&-\sigma _{zx}&-\sigma _{zy}&-\sigma _{zz}\end{bmatrix}}}$

${\displaystyle \mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} \,}$

${\displaystyle \sigma _{ij}=\epsilon _{0}E_{i}E_{j}+{\frac {1}{\mu _{0}}}B_{i}B_{j}-{\tfrac {1}{2}}(\epsilon _{0}E^{2}+{\frac {1}{\mu _{0}}}B^{2})\delta _{ij}\,.}$

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

${\displaystyle \epsilon _{0}\mu _{0}c^{2}=1\,.}$

### 其他非电磁学协变量

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

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

## 馬克士威方程組

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

### 其他符号记法

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

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

## 连续性方程

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

## 洛伦兹力

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

${\displaystyle {dp_{\alpha } \over {dt}}=q\,F_{\alpha \beta }\,{\frac {dx^{\beta }}{dt}}\,.}$

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

## 电磁应力-能量张量的微分方程

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

## 洛伦茨规范条件

${\displaystyle \eta ^{\alpha \nu }\,\partial _{\alpha }A_{\nu }=0\,.}$

### 洛伦茨规范下的麦克斯韦方程组

${\displaystyle \eta ^{\sigma \nu }\,\Box A_{\nu }=-\mu _{0}\,J^{\sigma }}$

## 介质中麦克斯韦方程组的协变形式

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

${\displaystyle {\mathcal {M}}^{\mu \nu }={\begin{pmatrix}0&-P_{x}c&-P_{y}c&-P_{z}c\\P_{x}c&0&M_{z}&-M_{y}\\P_{y}c&-M_{z}&0&M_{x}\\P_{z}c&M_{y}&-M_{x}&0\end{pmatrix}}}$

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

${\displaystyle {\mathcal {D}}^{\mu \nu }={\begin{pmatrix}0&D_{x}c&D_{y}c&D_{z}c\\-D_{x}c&0&H_{z}&-H_{y}\\-D_{y}c&-H_{z}&0&H_{x}\\-D_{z}c&H_{y}&-H_{x}&0\end{pmatrix}}.}$

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

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

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

${\displaystyle \mathbf {J} _{\text{free}}=\sigma \mathbf {E} \,}$
${\displaystyle \mathbf {P} =\epsilon _{0}\chi _{e}\mathbf {E} \,}$
${\displaystyle \mathbf {M} =\chi _{m}\mathbf {H} \,}$

## 拉格朗日量

${\displaystyle {\mathcal {L}}\,=\,{\mathcal {L}}_{\mathrm {field} }+{\mathcal {L}}_{\mathrm {int} }=-{\frac {1}{4\mu _{0}}}F^{\alpha \beta }F_{\alpha \beta }+A_{\alpha }J^{\alpha }\,.}$

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

## 广义相对论中的推广

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

## 参考文献

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.
• R. P. Feynman, F. B. Moringo, and W. G. Wagner. Feynman Lectures on Gravitation. Addison-Wesley. 1995. ISBN 0-201-62734-5.