Fusion809

Equations of General Relativity in the presence of Electromagnetic Field

$\partial _{\nu }F^{\mu \nu }=\mu _{0}j^{\mu }$

$\partial _{\sigma }F_{\mu \nu }+\partial _{\nu }F_{\sigma \mu }+\partial _{\mu }F_{\nu \sigma }=0$

$R_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }R+\Lambda g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }$
${\frac {d^{2}x^{\mu }}{d\tau ^{2}}}+\Gamma _{\sigma \nu }^{\mu }{\frac {dx^{\sigma }}{d\tau }}{\frac {dx^{\nu }}{d\tau }}={\frac {q}{m}}F_{\nu }^{\mu }{\frac {dx^{\nu }}{d\tau }}$

Linearized General Relativity with an Electromagnetic Field

${\bar {h}}_{\mu \nu }=h_{\mu \nu }-{\frac {1}{2}}\eta _{\mu \nu }h$

$\Box {\bar {h}}_{\mu \nu }={\frac {4\pi G}{\mu _{0}c^{4}}}\left(4\eta ^{\alpha \beta }F_{\mu \alpha }F_{\beta \nu }-\eta _{\mu \nu }\eta ^{\alpha \gamma }\eta ^{\beta \delta }F_{\beta \gamma }F_{\alpha \delta }\right)-{\frac {16\pi G}{c^{4}}}T_{\mu \nu }^{M}$

${\frac {d^{2}x^{\mu }}{d\tau ^{2}}}+{\frac {1}{2}}\eta ^{\mu \sigma }\left(\partial _{\beta }h_{\sigma \alpha }+\partial _{\alpha }h_{\sigma \beta }-\partial _{\sigma }h_{\alpha \beta }\right){\frac {dx^{\alpha }}{d\tau }}{\frac {dx^{\beta }}{d\tau }}={\frac {q}{m}}\left(\eta ^{\mu \sigma }-h^{\mu \sigma }\right)F_{\nu \sigma }{\frac {dx^{\nu }}{d\tau }}$

Schwarzschild Geodesics

$\mu \equiv {\frac {GM}{c^{2}}}$
$ds^{2}=c^{2}\left(1-{\frac {2\mu }{r}}\right)dt^{2}-{\frac {dr^{2}}{1-{\frac {2\mu }{r}}}}-r^{2}d\theta ^{2}-r^{2}\sin ^{2}\theta d\phi ^{2}$
${\mathcal {L}}=c^{2}\left(1-{\frac {2\mu }{r}}\right)\left({\frac {dt}{d\tau }}\right)^{2}-{\frac {1}{1-{\frac {2\mu }{r}}}}\left({\frac {dr}{d\tau }}\right)^{2}-r^{2}\left({\frac {d\theta }{d\tau }}\right)^{2}-r^{2}\sin ^{2}{\theta }\left({\frac {d\phi }{d\tau }}\right)^{2}$