User:Fusion809

Equations of General Relativity in the presence of Electromagnetic Field

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

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

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

Linearized General Relativity with an Electromagnetic Field

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

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

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

Schwarzschild Geodesics

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