# 弯曲时空中的麦克斯韦方程组

## 摘要

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

## 电磁四维势

${\displaystyle {\bar {A}}_{\beta }={\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}A_{\gamma }\,.}$

## 电磁场四维张量

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

${\displaystyle {\bar {F}}_{\alpha \beta }\,=\,{\frac {\partial {\bar {A}}_{\beta }}{\partial {\bar {x}}^{\alpha }}}\,-\,{\frac {\partial {\bar {A}}_{\alpha }}{\partial {\bar {x}}^{\beta }}}\,}$
${\displaystyle =\,{\frac {\partial }{\partial {\bar {x}}^{\alpha }}}\left({\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}A_{\gamma }\right)\,-\,{\frac {\partial }{\partial {\bar {x}}^{\beta }}}\left({\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}A_{\delta }\right)\,}$
${\displaystyle =\,{\frac {\partial ^{2}x^{\gamma }}{\partial {\bar {x}}^{\alpha }\,\partial {\bar {x}}^{\beta }}}A_{\gamma }\,+\,{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}{\frac {\partial A_{\gamma }}{\partial {\bar {x}}^{\alpha }}}\,-\,{\frac {\partial ^{2}x^{\delta }}{\partial {\bar {x}}^{\beta }\,\partial {\bar {x}}^{\alpha }}}A_{\delta }\,-\,{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial A_{\delta }}{\partial {\bar {x}}^{\beta }}}\,}$
${\displaystyle =\,{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial A_{\gamma }}{\partial x^{\delta }}}\,-\,{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}{\frac {\partial A_{\delta }}{\partial x^{\gamma }}}\,}$
${\displaystyle =\,{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}\,\left({\frac {\partial A_{\gamma }}{\partial x^{\delta }}}\,-\,{\frac {\partial A_{\delta }}{\partial x^{\gamma }}}\right)\,}$
${\displaystyle =\,{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}\,F_{\delta \gamma }\,.}$

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

${\displaystyle \partial _{\lambda }F_{\mu \nu }+\partial _{\mu }F_{\nu \lambda }+\partial _{\nu }F_{\lambda \mu }\,}$
${\displaystyle =\,\partial _{\lambda }\partial _{\mu }A_{\nu }-\partial _{\lambda }\partial _{\nu }A_{\mu }+\partial _{\mu }\partial _{\nu }A_{\lambda }-\partial _{\mu }\partial _{\lambda }A_{\nu }+\partial _{\nu }\partial _{\lambda }A_{\mu }-\partial _{\nu }\partial _{\mu }A_{\lambda }\,=0\,.}$

${\displaystyle F_{[\mu \nu ;\lambda ]}\,=\,F_{[\mu \nu ,\lambda ]}\,=\,{\frac {1}{6}}\left(\partial _{\lambda }F_{\mu \nu }+\partial _{\mu }F_{\nu \lambda }+\partial _{\nu }F_{\lambda \mu }-\partial _{\lambda }F_{\nu \mu }-\partial _{\mu }F_{\lambda \nu }-\partial _{\nu }F_{\mu \lambda }\right)\,}$
${\displaystyle =\,{\frac {1}{3}}\left(\partial _{\lambda }F_{\mu \nu }+\partial _{\mu }F_{\nu \lambda }+\partial _{\nu }F_{\lambda \mu }\right)=0\,}$

${\displaystyle F_{\alpha \beta ;\gamma }\,=\,F_{\alpha \beta ,\gamma }-{\Gamma ^{\mu }}_{\alpha \gamma }F_{\mu \beta }-{\Gamma ^{\mu }}_{\beta \gamma }F_{\alpha \mu }\,}$

## 电磁位移张量

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

${\displaystyle {\mathcal {D}}^{\mu \nu }\,=\,{\frac {1}{\mu _{0}}}\,g^{\mu \alpha }\,F_{\alpha \beta }\,g^{\beta \nu }\,{\sqrt {-g}}\,-\,{\mathcal {M}}^{\mu \nu }\,.}$

${\displaystyle {\bar {\mathcal {D}}}^{\mu \nu }\,=\,{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\beta }}}\,{\mathcal {D}}^{\alpha \beta }\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\,}$

## 电流

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

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

${\displaystyle \partial _{\mu }J^{\mu }\,=\,\partial _{\mu }\partial _{\nu }{\mathcal {D}}^{\mu \nu }=0\,}$

${\displaystyle {\bar {J}}^{\mu }\,=\,{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,J^{\alpha }\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\,.}$

${\displaystyle {\bar {J}}^{\mu }\,=\,{\frac {\partial }{\partial {\bar {x}}^{\nu }}}\left({\bar {\mathcal {D}}}^{\mu \nu }\right)\,=\,{\frac {\partial }{\partial {\bar {x}}^{\nu }}}\left({\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\beta }}}\,{\mathcal {D}}^{\alpha \beta }\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\right)\,}$
${\displaystyle =\,{\frac {\partial ^{2}{\bar {x}}^{\mu }}{\partial {\bar {x}}^{\nu }\partial x^{\alpha }}}\,{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\beta }}}\,{\mathcal {D}}^{\alpha \beta }\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\,+\,{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,{\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial {\bar {x}}^{\nu }\partial x^{\beta }}}\,{\mathcal {D}}^{\alpha \beta }\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\,+\,}$
${\displaystyle {\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\beta }}}\,{\frac {\partial {\mathcal {D}}^{\alpha \beta }}{\partial {\bar {x}}^{\nu }}}\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\,+\,{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\beta }}}\,{\mathcal {D}}^{\alpha \beta }\,{\frac {\partial }{\partial {\bar {x}}^{\nu }}}\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\,}$
${\displaystyle =\,{\frac {\partial ^{2}{\bar {x}}^{\mu }}{\partial x^{\beta }\partial x^{\alpha }}}\,{\mathcal {D}}^{\alpha \beta }\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\,+\,{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,{\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial {\bar {x}}^{\nu }\partial x^{\beta }}}\,{\mathcal {D}}^{\alpha \beta }\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\,+\,}$
${\displaystyle {\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,{\frac {\partial {\mathcal {D}}^{\alpha \beta }}{\partial x^{\beta }}}\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\,+\,{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\beta }}}\,{\mathcal {D}}^{\alpha \beta }\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]{\frac {\partial {\bar {x}}^{\rho }}{\partial x^{\sigma }}}{\frac {\partial ^{2}x^{\sigma }}{\partial {\bar {x}}^{\nu }\partial {\bar {x}}^{\rho }}}\,}$
${\displaystyle =\,0\,+\,{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,{\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial {\bar {x}}^{\nu }\partial x^{\beta }}}\,{\mathcal {D}}^{\alpha \beta }\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\,+\,}$
${\displaystyle {\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,J^{\alpha }\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\,+\,{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,{\mathcal {D}}^{\alpha \beta }\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]{\frac {\partial {\bar {x}}^{\rho }}{\partial x^{\sigma }}}{\frac {\partial ^{2}x^{\sigma }}{\partial x^{\beta }\partial {\bar {x}}^{\rho }}}\,}$
${\displaystyle =\,{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,J^{\alpha }\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\,+\,{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,{\mathcal {D}}^{\alpha \beta }\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\left({\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial {\bar {x}}^{\nu }\partial x^{\beta }}}\,+\,{\frac {\partial {\bar {x}}^{\rho }}{\partial x^{\sigma }}}{\frac {\partial ^{2}x^{\sigma }}{\partial x^{\beta }\partial {\bar {x}}^{\rho }}}\right)\,.}$

${\displaystyle {\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial {\bar {x}}^{\nu }\partial x^{\beta }}}\,+\,{\frac {\partial {\bar {x}}^{\rho }}{\partial x^{\sigma }}}{\frac {\partial ^{2}x^{\sigma }}{\partial x^{\beta }\partial {\bar {x}}^{\rho }}}\,=\,0\,}$

${\displaystyle {\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial {\bar {x}}^{\nu }\partial x^{\beta }}}\,+\,{\frac {\partial {\bar {x}}^{\rho }}{\partial x^{\sigma }}}{\frac {\partial ^{2}x^{\sigma }}{\partial x^{\beta }\partial {\bar {x}}^{\rho }}}\,=\,{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\nu }}}{\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial x^{\sigma }\partial x^{\beta }}}\,+\,{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\sigma }}}{\frac {\partial ^{2}x^{\sigma }}{\partial x^{\beta }\partial {\bar {x}}^{\nu }}}\,}$
${\displaystyle =\,{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\nu }}}{\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial x^{\beta }\partial x^{\sigma }}}\,+\,{\frac {\partial ^{2}x^{\sigma }}{\partial x^{\beta }\partial {\bar {x}}^{\nu }}}{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\sigma }}}\,=\,{\frac {\partial }{\partial x^{\beta }}}\left({\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\nu }}}\,{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\sigma }}}\right)\,}$
${\displaystyle =\,{\frac {\partial }{\partial x^{\beta }}}\left(\,{\frac {\partial {\bar {x}}^{\nu }}{\partial {\bar {x}}^{\nu }}}\right)\,=\,{\frac {\partial }{\partial x^{\beta }}}\left(\mathbf {4} \right)\,=\,0\,.}$

## 洛伦兹力

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

${\displaystyle {\frac {dp_{\alpha }}{dt}}\,=\,\Gamma _{\alpha \gamma }^{\beta }\,p_{\beta }\,{\frac {dx^{\gamma }}{dt}}\,+\,q\,F_{\alpha \gamma }\,{\frac {dx^{\gamma }}{dt}}\,}$

${\displaystyle {\bar {\Gamma }}_{\alpha \gamma }^{\beta }\,=\,{\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\epsilon }}}\,{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}\,{\frac {\partial x^{\zeta }}{\partial {\bar {x}}^{\gamma }}}\,\Gamma _{\delta \zeta }^{\epsilon }\,+{\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\eta }}}\,{\frac {\partial ^{2}x^{\eta }}{\partial {\bar {x}}^{\alpha }\partial {\bar {x}}^{\gamma }}}\,}$

${\displaystyle {\frac {d{\bar {p}}_{\alpha }}{dt}}\,-\,{\bar {\Gamma }}_{\alpha \gamma }^{\beta }\,{\bar {p}}_{\beta }\,{\frac {d{\bar {x}}^{\gamma }}{dt}}\,-\,q\,{\bar {F}}_{\alpha \gamma }\,{\frac {d{\bar {x}}^{\gamma }}{dt}}\,}$
${\displaystyle =\,{\frac {d}{dt}}\left({\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}\,p_{\delta }\right)\,-\,\left({\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\theta }}}\,{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}\,{\frac {\partial x^{\iota }}{\partial {\bar {x}}^{\gamma }}}\,\Gamma _{\delta \iota }^{\theta }\,+\,{\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\eta }}}\,{\frac {\partial ^{2}x^{\eta }}{\partial {\bar {x}}^{\alpha }\partial {\bar {x}}^{\gamma }}}\right)\,{\frac {\partial x^{\epsilon }}{\partial {\bar {x}}^{\beta }}}\,p_{\epsilon }\,{\frac {\partial {\bar {x}}^{\gamma }}{\partial x^{\zeta }}}\,{\frac {dx^{\zeta }}{dt}}\,-\,q\,{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}\,F_{\delta \zeta }\,{\frac {dx^{\zeta }}{dt}}\,}$
${\displaystyle =\,{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}\,\left({\frac {dp_{\delta }}{dt}}\,-\,\Gamma _{\delta \zeta }^{\epsilon }\,p_{\epsilon }\,{\frac {dx^{\zeta }}{dt}}\,-\,q\,F_{\delta \zeta }\,{\frac {dx^{\zeta }}{dt}}\right)\,+\,}$
${\displaystyle {\frac {d}{dt}}\left({\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}\right)\,p_{\delta }\,-\,\left({\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\eta }}}\,{\frac {\partial ^{2}x^{\eta }}{\partial {\bar {x}}^{\alpha }\partial {\bar {x}}^{\gamma }}}\right)\,{\frac {\partial x^{\epsilon }}{\partial {\bar {x}}^{\beta }}}\,p_{\epsilon }\,{\frac {\partial {\bar {x}}^{\gamma }}{\partial x^{\zeta }}}\,{\frac {dx^{\zeta }}{dt}}\,}$
${\displaystyle =\,0\,+\,{\frac {d}{dt}}\left({\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}\right)\,p_{\delta }\,-\,{\frac {\partial ^{2}x^{\epsilon }}{\partial {\bar {x}}^{\alpha }\partial {\bar {x}}^{\gamma }}}p_{\epsilon }\,{\frac {d{\bar {x}}^{\gamma }}{dt}}\,=\,0\,}$

## 拉格朗日量

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

${\displaystyle {\mathcal {L}}\,=\,-{\frac {1}{4\mu _{0}}}\,F_{\alpha \beta }\,F^{\alpha \beta }\,{\sqrt {-g}}\,+\,A_{\alpha }\,J_{\text{free}}^{\alpha }\,+\,{\frac {1}{2}}\,F_{\alpha \beta }\,{\mathcal {M}}^{\alpha \beta }\,.}$

## 电磁应力-能量张量

${\displaystyle T_{\mu \nu }\,=\,-{\frac {1}{\mu _{0}}}(F_{\mu \alpha }g^{\alpha \beta }F_{\beta \nu }\,-\,{\frac {1}{4}}g_{\mu \nu }\,F_{\sigma \alpha }g^{\alpha \beta }F_{\beta \rho }g^{\rho \sigma })\,}$

${\displaystyle T_{\mu \nu }g^{\mu \nu }\,=\,0\,}$

${\displaystyle {\mathfrak {T}}_{\mu }^{\nu }=T_{\mu \gamma }\,g^{\gamma \nu }\,{\sqrt {-g}}.}$

${\displaystyle {{\mathfrak {T}}_{\mu }^{\nu }}_{;\nu }\,+\,f_{\mu }\,=\,0\,}$

${\displaystyle -{{\mathfrak {T}}_{\mu }^{\nu }}_{,\nu }\,=\,-\Gamma _{\mu \nu }^{\sigma }{\mathfrak {T}}_{\sigma }^{\nu }\,+\,f_{\mu }\,}$

${\displaystyle {{\mathfrak {T}}_{\mu }^{\nu }}_{;\nu }\,+\,f_{\mu }\,=\,-{\frac {1}{\mu _{0}}}(F_{\mu \alpha ;\nu }g^{\alpha \beta }F_{\beta \gamma }g^{\gamma \nu }\,+\,F_{\mu \alpha }g^{\alpha \beta }F_{\beta \gamma ;\nu }g^{\gamma \nu }\,-\,{\frac {1}{2}}\delta _{\mu }^{\nu }\,F_{\sigma \alpha ;\nu }g^{\alpha \beta }F_{\beta \rho }g^{\rho \sigma }){\sqrt {-g}}\,+\,}$
${\displaystyle {\frac {1}{\mu _{0}}}\,F_{\mu \alpha }\,g^{\alpha \beta }\,F_{\beta \gamma ;\nu }\,g^{\gamma \nu }\,{\sqrt {-g}}\,}$
${\displaystyle =\,-{\frac {1}{\mu _{0}}}(F_{\mu \alpha ;\nu }F^{\alpha \nu }\,-\,{\frac {1}{2}}F_{\sigma \alpha ;\mu }F^{\alpha \sigma }){\sqrt {-g}}\,}$
${\displaystyle =\,-{\frac {1}{\mu _{0}}}((-F_{\nu \mu ;\alpha }-F_{\alpha \nu ;\mu })F^{\alpha \nu }\,-\,{\frac {1}{2}}F_{\sigma \alpha ;\mu }F^{\alpha \sigma }){\sqrt {-g}}\,}$
${\displaystyle =\,-{\frac {1}{\mu _{0}}}(F_{\mu \nu ;\alpha }F^{\alpha \nu }-F_{\alpha \nu ;\mu }F^{\alpha \nu }\,+\,{\frac {1}{2}}F_{\sigma \alpha ;\mu }F^{\sigma \alpha }){\sqrt {-g}}\,}$
${\displaystyle =\,-{\frac {1}{\mu _{0}}}(F_{\mu \alpha ;\nu }F^{\nu \alpha }-{\frac {1}{2}}F_{\alpha \nu ;\mu }F^{\alpha \nu }){\sqrt {-g}}\,}$
${\displaystyle =\,-{\frac {1}{\mu _{0}}}(-F_{\mu \alpha ;\nu }F^{\alpha \nu }\,+\,{\frac {1}{2}}F_{\sigma \alpha ;\mu }F^{\alpha \sigma }){\sqrt {-g}}\,}$

## 电磁波方程

${\displaystyle \Box F_{ab}\ {\stackrel {\mathrm {def} }{=}}\ F_{ab;}{}^{d}{}_{d}=\,-2R_{acbd}F^{cd}+R_{ae}F^{e}{}_{b}-R_{be}F^{e}{}_{a}+J_{a;b}-J_{b;a}}$

${\displaystyle R_{acbd}\,}$

${\displaystyle \Box A^{a}={{A^{a;}}^{b}}_{b}=-\mu _{0}J^{a}+{R^{a}}_{b}A^{b}}$

${\displaystyle {R^{a}}_{b}\ {\stackrel {\mathrm {def} }{=}}\ {R^{s}}_{asb}}$

${\displaystyle {A^{a}}_{;a}=0}$

## 麦克斯韦方程组在动态时空中的非线性

${\displaystyle G_{ab}={\frac {8\pi G}{c^{4}}}T_{ab}}$

${\displaystyle {G}_{ab}\ {\stackrel {\mathrm {def} }{=}}\ {R}_{ab}-{1 \over 2}{R}g_{ab}}$

## 参考文献

• 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.