$\Gamma^\mu = {\Gamma^\mu}_{\alpha\beta} g^{\alpha\beta}$
${\Gamma^\alpha}_{\mu\alpha} = ln(\sqrt{g})_{,\mu} = G_\mu$
$R_{\mu\nu} = {R^\rho}_{\mu\rho\nu}$
$R_{\mu\nu} = {\Gamma^\rho}_{\mu\nu,\rho} - {\Gamma^\rho}_{\mu\rho,\nu} + {\Gamma^\rho}_{\alpha\rho} {\Gamma^\alpha}_{\mu\nu} - {\Gamma^\rho}_{\alpha\nu} {\Gamma^\alpha}_{\mu\rho}$
$R_{\mu\nu} = {\Gamma^\rho}_{\mu\nu,\rho} - G_{(\mu,\nu)} + G_\alpha {\Gamma^\alpha}_{\mu\nu} - {\Gamma^\alpha}_{\beta\mu} {\Gamma^\beta}_{\alpha\nu}$
$G_{\mu\nu} = 8 \pi T_{\mu\nu}$
... 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
$