User:Fusion809

Equations of General Relativity in the presence of Electromagnetic Field
$$\partial_{\nu} F^{\mu\nu} =\mu_{0} j^{\mu} $$ $$ \partial_{\sigma} F_{\mu\nu}+\partial_{\nu} F_{\sigma\mu}+\partial_{\mu} F_{\nu\sigma}=0 $$ $$ R_{\mu\nu}-\frac{1}{2} g_{\mu\nu} R+\Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} $$ $$ \frac{d^2 x^{\mu}}{d\tau^2} +\Gamma^{\mu}_{\sigma\nu} \frac{dx^{\sigma}}{d\tau} \frac{dx^{\nu}}{d\tau} =\frac{q}{m} F^{\mu}_{\nu} \frac{dx^{\nu}}{d\tau} $$

Linearized General Relativity with an Electromagnetic Field
$$ \bar{h}_{\mu\nu}=h_{\mu\nu} -\frac{1}{2} \eta_{\mu\nu} h $$ $$ \Box \bar{h}_{\mu \nu} = \frac{4 \pi G}{\mu_{0} c^4} \left(4 \eta^{\alpha\beta} F_{\mu\alpha} F_{\beta\nu}-\eta_{\mu\nu} \eta^{\alpha\gamma} \eta^{\beta\delta} F_{\beta\gamma} F_{\alpha\delta}\right)-\frac{16\pi G}{c^{4}} T^M_{\mu\nu}$$ $$ \frac{d^{2} x^{\mu}}{d\tau^{2}}+\frac{1}{2} \eta^{\mu\sigma} \left(\partial_{\beta} h_{\sigma\alpha} +\partial_{\alpha} h_{\sigma\beta}-\partial_{\sigma} h_{\alpha\beta}\right) \frac{d x^{\alpha}}{d\tau} \frac{d x^{\beta}}{d\tau}=\frac{q}{m} \left(\eta^{\mu\sigma}-h^{\mu\sigma}\right) F_{\nu\sigma} \frac{d x^{\nu}}{d\tau} $$

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