弯曲时空中的麦克斯韦方程组

广义相对论中真空中的电磁理论的方程为

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

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

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

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

其中${\displaystyle g^{\alpha \beta }\,}$ 是度规张量${\displaystyle g_{\alpha \beta }\,}$ 的倒数，而${\displaystyle g\,}$ 是度规张量的行列式，${\displaystyle A_{\alpha }\,}$ 是电磁场的四维势，${\displaystyle F^{\alpha \beta }\,}$ 是电磁场的四维协变张量，${\displaystyle D^{\mu \nu }\,}$ 是位移电流张量，${\displaystyle f_{\mu }\,}$ 是洛伦兹力的密度，${\displaystyle J_{\mu }\,}$ 是四维电流密度。尽管方程组中使用了偏导数，这些方程仍然在任意曲面坐标变换下是协变的。也就是说如果将偏导数换成协变导数，引入的附加项会自动消去从而保持形式不变。

电磁场的四维势${\displaystyle A_{\alpha }\,,}$ 是一个协变矢量，它的坐标变换规则为

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

电磁场是一个协变的二阶反对称张量，它用电磁势可以定义为

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

为证明它的洛伦兹不变性，我们对其进行坐标变换

${\displaystyle {\bar {F}}_{\alpha \beta }\,=\,{\frac {\partial {\bar {A}}_{\beta }}{\partial {\bar {x}}^{\alpha }}}\,-\,{\frac {\partial {\bar {A}}_{\alpha }}{\partial {\bar {x}}^{\beta }}}\,}$ 

${\displaystyle =\,{\frac {\partial }{\partial {\bar {x}}^{\alpha }}}\left({\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}A_{\gamma }\right)\,-\,{\frac {\partial }{\partial {\bar {x}}^{\beta }}}\left({\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}A_{\delta }\right)\,}$ 

${\displaystyle =\,{\frac {\partial ^{2}x^{\gamma }}{\partial {\bar {x}}^{\alpha }\,\partial {\bar {x}}^{\beta }}}A_{\gamma }\,+\,{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}{\frac {\partial A_{\gamma }}{\partial {\bar {x}}^{\alpha }}}\,-\,{\frac {\partial ^{2}x^{\delta }}{\partial {\bar {x}}^{\beta }\,\partial {\bar {x}}^{\alpha }}}A_{\delta }\,-\,{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial A_{\delta }}{\partial {\bar {x}}^{\beta }}}\,}$ 

${\displaystyle =\,{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial A_{\gamma }}{\partial x^{\delta }}}\,-\,{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}{\frac {\partial A_{\delta }}{\partial x^{\gamma }}}\,}$ 

${\displaystyle =\,{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}\,\left({\frac {\partial A_{\gamma }}{\partial x^{\delta }}}\,-\,{\frac {\partial A_{\delta }}{\partial x^{\gamma }}}\right)\,}$ 

${\displaystyle =\,{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}\,F_{\delta \gamma }\,.}$ 

这一定义暗示了电磁场张量满足关系

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

这一关系包含了法拉第电磁感应定律和磁场的高斯定理，因此也叫做法拉第-高斯方程。具体而言，

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

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

虽然这个关系包含了64个分量方程，但只有四个是独立的。借助电磁场张量的反对称性，可知只有${\displaystyle \lambda ,\mu ,\nu \,}$ 等于1,2,3或2,3,0或3,0,1或0,1,2时的方程是彼此独立的。

法拉第-高斯方程有时也写作下面的形式

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

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

其中按照惯例用分号表示协变导数，逗号表示偏导数，方括号表示反对称形式。电磁场张量的协变导数为

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

其中${\displaystyle \Gamma _{\beta \gamma }^{\alpha }\,}$ 是克里斯托费尔符号，两个下标是对称的。

位移电场${\displaystyle \mathbf {D} \,,}$ 和附属磁场${\displaystyle \mathbf {H} \,,}$ 构成一个反对称的逆变二阶张量。真空中这个张量为

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

注意这个方程是电磁理论中唯一有度规（即引力）存在的方程。并且这一方程具有尺度不变性，即将度规乘以一个常数不会改变方程的形式。也就是说，引力只能通过改变全局坐标中的光速来影响相应的电磁场，由于光在引力场中会发生偏折，其效应等同于质量的引力场增加了周围时空的折射率，从而影响了对应的位移电场和附属磁场。

更一般地，当物质中的磁化-极化张量不为零时，我们有

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

电磁位移张量的坐标变换规则为

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

其中使用了雅可比行列式。如果磁化-极化张量存在，它和电磁位移张量具有同样的变换规则。

电磁位移张量的散度被定义为电流，在真空中

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

如果磁化-极化张量存在，则上式替换为

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

这一方程包含了高斯定理和安培环路定理。

无论磁化-极化张量是否存在，电磁位移张量是反对称的这一事实暗示了电流是一个守恒量：

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

这个二阶偏导数为零是由于偏导是对易的。

电流的安培-高斯定义并不能决定它的大小，因为电磁四维势的大小并未确定。相反地，通常的步骤是使电流等于一个用其他场表示的表达式（主要是电荷），然后解得电磁位移、电磁场和电磁势。

电流是一个逆变的矢量，因而其变换规则是

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

变换规则的验证如下：

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

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

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

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

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

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

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

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

下面只需证明

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

这是微分学中一条已知定理的应用：

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

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

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

洛伦兹力的密度是一个协变矢量：

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

如果在一个测试粒子上的作用力只有引力和电磁力，则有

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

其中${\displaystyle p^{\alpha }\,}$ 是粒子的四维动量，${\displaystyle t\,}$ 是将粒子世界线参数化的任意时间坐标。式中第一项含有克里斯托费尔符号，表示引力项，第二项含有电荷，表示电磁力项。

方程具有洛伦兹不变性，对时间坐标变换的验证方法为将方程乘以${\displaystyle {\frac {dt}{d{\bar {t}}}}\,}$ 并使用链式法则。

对空间坐标变换，通过对克里斯托费尔符号进行坐标变换

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

我们得到

${\displaystyle {\frac {d{\bar {p}}_{\alpha }}{dt}}\,-\,{\bar {\Gamma }}_{\alpha \gamma }^{\beta }\,{\bar {p}}_{\beta }\,{\frac {d{\bar {x}}^{\gamma }}{dt}}\,-\,q\,{\bar {F}}_{\alpha \gamma }\,{\frac {d{\bar {x}}^{\gamma }}{dt}}\,}$ 

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

${\displaystyle =\,{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}\,\left({\frac {dp_{\delta }}{dt}}\,-\,\Gamma _{\delta \zeta }^{\epsilon }\,p_{\epsilon }\,{\frac {dx^{\zeta }}{dt}}\,-\,q\,F_{\delta \zeta }\,{\frac {dx^{\zeta }}{dt}}\right)\,+\,}$ 

${\displaystyle {\frac {d}{dt}}\left({\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}\right)\,p_{\delta }\,-\,\left({\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\eta }}}\,{\frac {\partial ^{2}x^{\eta }}{\partial {\bar {x}}^{\alpha }\partial {\bar {x}}^{\gamma }}}\right)\,{\frac {\partial x^{\epsilon }}{\partial {\bar {x}}^{\beta }}}\,p_{\epsilon }\,{\frac {\partial {\bar {x}}^{\gamma }}{\partial x^{\zeta }}}\,{\frac {dx^{\zeta }}{dt}}\,}$ 

${\displaystyle =\,0\,+\,{\frac {d}{dt}}\left({\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}\right)\,p_{\delta }\,-\,{\frac {\partial ^{2}x^{\epsilon }}{\partial {\bar {x}}^{\alpha }\partial {\bar {x}}^{\gamma }}}p_{\epsilon }\,{\frac {d{\bar {x}}^{\gamma }}{dt}}\,=\,0\,}$ 

在真空中经典电磁场的拉格朗日量（在密度意义下的单位为焦耳/米³）是一个标量：

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

其中${\displaystyle F^{\alpha \beta }=g^{\alpha \gamma }F_{\gamma \delta }g^{\delta \beta }\,.}$ 。这里的四维电流项应理解为各种对电流有贡献的场的总和。

如果我们将自由电流与束缚电流分离开来，拉格朗日量则变为

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

作为爱因斯坦引力场方程的源，电磁场的应力-能量张量是一个协变的对称张量

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

并且它是无迹的

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

这是由于电磁场的传播速度在不同参考系下是不变的。

为了表达能量和动量的守恒律，电磁应力-能量张量的最佳表示方法是

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

对于上式可以证明

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

并可重写为

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

这个方程的意义是，电磁场能量的减少相当于电磁场对引力场以及通过洛伦兹力对物质所做的功，类似地，电磁场动量的减少率相当于作用在引力场上的电磁力以及作用在物质上的洛伦兹力。

守恒律的推导过程为

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

${\displaystyle {\frac {1}{\mu _{0}}}\,F_{\mu \alpha }\,g^{\alpha \beta }\,F_{\beta \gamma ;\nu }\,g^{\gamma \nu }\,{\sqrt {-g}}\,}$ 

${\displaystyle =\,-{\frac {1}{\mu _{0}}}(F_{\mu \alpha ;\nu }F^{\alpha \nu }\,-\,{\frac {1}{2}}F_{\sigma \alpha ;\mu }F^{\alpha \sigma }){\sqrt {-g}}\,}$ 

${\displaystyle =\,-{\frac {1}{\mu _{0}}}((-F_{\nu \mu ;\alpha }-F_{\alpha \nu ;\mu })F^{\alpha \nu }\,-\,{\frac {1}{2}}F_{\sigma \alpha ;\mu }F^{\alpha \sigma }){\sqrt {-g}}\,}$ 

${\displaystyle =\,-{\frac {1}{\mu _{0}}}(F_{\mu \nu ;\alpha }F^{\alpha \nu }-F_{\alpha \nu ;\mu }F^{\alpha \nu }\,+\,{\frac {1}{2}}F_{\sigma \alpha ;\mu }F^{\sigma \alpha }){\sqrt {-g}}\,}$ 

${\displaystyle =\,-{\frac {1}{\mu _{0}}}(F_{\mu \alpha ;\nu }F^{\nu \alpha }-{\frac {1}{2}}F_{\alpha \nu ;\mu }F^{\alpha \nu }){\sqrt {-g}}\,}$ 

${\displaystyle =\,-{\frac {1}{\mu _{0}}}(-F_{\mu \alpha ;\nu }F^{\alpha \nu }\,+\,{\frac {1}{2}}F_{\sigma \alpha ;\mu }F^{\alpha \sigma }){\sqrt {-g}}\,}$ 

所得的的结果是为零的，因为它等于自身的负值（对照推导的第二步）。

从电磁理论的狭义相对论形式可以借助电磁场张量修改得到非齐次的电磁波方程

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

其中

${\displaystyle R_{acbd}\,}$ 

是黎曼张量的协变形式，而${\displaystyle \Box \,}$ 是达朗贝尔算符在协变导数情形下的推广。

波方程可以写成四维势的形式（参见ref 2, p. 569）

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

其中

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

是里奇张量。而这里假设了洛伦茨规范在弯曲时空中的推广具有下面的形式

${\displaystyle {A^{a}}_{;a}=0}$ 

这个波方程与平直时空中的波方程具有十分类似的形式，除了导数被替换为协变导数，以及在表示源的项中多了一项描述时空曲率的项。这个方程和弯曲时空中的洛伦兹力也有相似之处，这里的四维势${\displaystyle {A^{a}}\,}$ 相当于洛伦兹力中的四维坐标。

在考虑麦克斯韦方程组本身与背景时空无关时，时空度规在广义相对论中被认为是受电磁场影响的动态变化的变量，这导致了电磁波方程和麦克斯韦方程组是非线性的。这一点可以从爱因斯坦场方程中的曲率张量依赖于应力-能量张量看出。

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

其中

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

是爱因斯坦张量，${\displaystyle G\,}$ 是万有引力常数，${\displaystyle R\,}$  （标量曲率）是里奇张量的迹。应力-能量张量来源于粒子的应力和能量，但同时也来源于电磁场，这造就了非线性。

• 电磁波方程
• 非齐次的电磁波方程
• 麦克斯韦方程组在狭义相对论中的形式
• 麦克斯韦方程组的历史
• 广义相对论的理论动机
• 广义相对论的数学方法
• 电真空解

• 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. 

• 弯曲时空中的麦克斯韦场