$\Gamma^\mu = {\Gamma^\mu}_{\alpha\beta} g^{\alpha\beta}$
${\Gamma^\alpha}_{\mu\alpha} = ln(\sqrt{g})_{,\mu} = G_\mu$

$R_{\mu\nu} = {R^\rho}_{\mu\rho\nu}$
$R_{\mu\nu} = {\Gamma^\rho}_{\mu\nu,\rho} - {\Gamma^\rho}_{\mu\rho,\nu} + {\Gamma^\rho}_{\alpha\rho} {\Gamma^\alpha}_{\mu\nu} - {\Gamma^\rho}_{\alpha\nu} {\Gamma^\alpha}_{\mu\rho}$
$R_{\mu\nu} = {\Gamma^\rho}_{\mu\nu,\rho} - G_{(\mu,\nu)} + G_\alpha {\Gamma^\alpha}_{\mu\nu} - {\Gamma^\alpha}_{\beta\mu} {\Gamma^\beta}_{\alpha\nu}$

$G_{\mu\nu} = 8 \pi T_{\mu\nu}$
... in vacuum ...
$G_{\mu\nu} = 0$
$R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = 0$
$ {\Gamma^\rho}_{\mu\nu,\rho} - G_{(\mu,\nu)} + G_\alpha {\Gamma^\alpha}_{\mu\nu} - {\Gamma^\alpha}_{\beta\mu} {\Gamma^\beta}_{\alpha\nu} - \frac{1}{2} g_{\mu\nu} g^{\alpha\beta} \left[ {\Gamma^\rho}_{\alpha\beta,\rho} - G_{(\alpha,\beta)} + G_\gamma {\Gamma^\gamma}_{\alpha\beta} - {\Gamma^\gamma}_{\delta\alpha} {\Gamma^\delta}_{\gamma\beta} \right] = 0 $
$ {\Gamma^\rho}_{\mu\nu,\rho} - \frac{1}{2} g_{\mu\nu} {\Gamma^\rho}_{\alpha\beta,\rho} g^{\alpha\beta} - G_{(\mu,\nu)} + \frac{1}{2} g_{\mu\nu} G_{(\alpha,\beta)} g^{\alpha\beta} + G_\alpha {\Gamma^\alpha}_{\mu\nu} - {\Gamma^\alpha}_{\beta\mu} {\Gamma^\beta}_{\alpha\nu} - \frac{1}{2} g_{\mu\nu} G_\alpha \Gamma^\alpha + \frac{1}{2} g_{\mu\nu} \Gamma^{\alpha\beta\gamma} \Gamma_{\beta\alpha\gamma} = 0 $
$ \delta^\rho_\alpha ( \delta^\beta_\mu \delta^\gamma_\nu - \frac{1}{2} g_{\mu\nu} g^{\beta\gamma} ) {\Gamma^\alpha}_{\beta\gamma,\rho} + ( - \delta^\alpha_{(\mu} \delta^\rho_{\nu)} + \frac{1}{2} g_{\mu\nu} g^{\alpha\rho} ) G_{\alpha,\rho} + G_\alpha {\Gamma^\alpha}_{\mu\nu} - {\Gamma^\alpha}_{\beta\mu} {\Gamma^\beta}_{\alpha\nu} - \frac{1}{2} g_{\mu\nu} G_\alpha \Gamma^\alpha + \frac{1}{2} g_{\mu\nu} \Gamma^{\alpha\beta\gamma} \Gamma_{\beta\alpha\gamma} = 0 $
$ \delta^\rho_\alpha ( \delta^\beta_\mu \delta^\gamma_\nu - \frac{1}{2} g_{\mu\nu} g^{\beta\gamma} ) {\Gamma^\alpha}_{\beta\gamma,\rho} + ( - \delta^\alpha_{(\mu} \delta^\rho_{\nu)} + \frac{1}{2} g_{\mu\nu} g^{\alpha\rho} ) {\Gamma^\beta}_{\beta\alpha,\rho} + G_\alpha {\Gamma^\alpha}_{\mu\nu} - {\Gamma^\alpha}_{\beta\mu} {\Gamma^\beta}_{\alpha\nu} - \frac{1}{2} g_{\mu\nu} G_\alpha \Gamma^\alpha + \frac{1}{2} g_{\mu\nu} \Gamma^{\alpha\beta\gamma} \Gamma_{\beta\alpha\gamma} = 0 $
$ \delta^\rho_\alpha ( \delta^\beta_\mu \delta^\gamma_\nu - \frac{1}{2} g_{\mu\nu} g^{\beta\gamma} ) {\Gamma^\alpha}_{\beta\gamma,\rho} + \delta^\beta_\alpha ( - \delta^\gamma_{(\mu} \delta^\rho_{\nu)} + \frac{1}{2} g_{\mu\nu} g^{\gamma\rho} ) {\Gamma^\alpha}_{\beta\gamma,\rho} + G_\alpha {\Gamma^\alpha}_{\mu\nu} - {\Gamma^\alpha}_{\beta\mu} {\Gamma^\beta}_{\alpha\nu} - \frac{1}{2} g_{\mu\nu} G_\alpha \Gamma^\alpha + \frac{1}{2} g_{\mu\nu} \Gamma^{\alpha\beta\gamma} \Gamma_{\beta\alpha\gamma} = 0 $
$ ( \delta^\gamma_\nu ( \delta^\rho_\alpha \delta^\beta_\mu - \delta^\beta_\alpha \delta^\rho_\mu ) + \frac{1}{2} g_{\mu\nu} ( \delta^\beta_\alpha g^{\gamma\rho} - \delta^\rho_\alpha g^{\beta\gamma} ) ) {\Gamma^\alpha}_{\beta\gamma,\rho} + ( G_\alpha \delta^\beta_\mu \delta^\gamma_\nu - \delta^\gamma_\mu {\Gamma^\beta}_{\alpha\nu} - \frac{1}{2} g_{\mu\nu} \Gamma^\beta \delta^\gamma_\alpha + \frac{1}{2} g_{\mu\nu} {\Gamma^{\beta\gamma}}_\alpha ) {\Gamma^\alpha}_{\beta\gamma} = 0 $
$ ( \delta^\gamma_\nu ( \delta^\rho_\alpha \delta^\beta_\mu - \delta^\beta_\alpha \delta^\rho_\mu ) + \frac{1}{2} g_{\mu\nu} ( \delta^\beta_\alpha g^{\gamma\rho} - \delta^\rho_\alpha g^{\beta\gamma} ) ) {\Gamma^\alpha}_{\beta\gamma,\rho} + ( G_\alpha \delta^\beta_\mu \delta^\gamma_\nu - \delta^\gamma_\mu {\Gamma^\beta}_{\alpha\nu} - \frac{1}{2} g_{\mu\nu} \Gamma^\beta \delta^\gamma_\alpha + \frac{1}{2} g_{\mu\nu} {\Gamma^{\beta\gamma}}_\alpha ) \frac{1}{2} \gamma^{\alpha\delta} ( g_{\delta\beta,\gamma} + g_{\delta\gamma,\beta} - g_{\beta\gamma,\delta} ) = 0 $
$ ( \delta^\gamma_\nu ( \delta^\rho_\alpha \delta^\beta_\mu - \delta^\beta_\alpha \delta^\rho_\mu ) + \frac{1}{2} g_{\mu\nu} ( \delta^\beta_\alpha g^{\gamma\rho} - \delta^\rho_\alpha g^{\beta\gamma} ) ) {\Gamma^\alpha}_{\beta\gamma,\rho} + \frac{1}{2}( G_\delta \delta^\epsilon_\mu \delta^\gamma_\nu - \delta^\gamma_\mu {\Gamma^\epsilon}_{\delta\nu} - \frac{1}{2} g_{\mu\nu} \Gamma^\epsilon \delta^\gamma_\delta + \frac{1}{2} g_{\mu\nu} {\Gamma^{\epsilon\gamma}}_\delta ) ( \gamma^{\delta \alpha} \delta^\beta_\epsilon \delta^\rho_\gamma + \gamma^{\delta \alpha} \delta^\beta_\gamma \delta^\rho_\epsilon - \gamma^{\delta\rho} \delta^\alpha_\epsilon \delta^\beta_\gamma ) g_{\alpha \beta,\rho} = 0 $