Notes on Einstein Field Equation

Metric tensor:

    \[g_{\mu\nu} = g_{\nu\mu}\]

    \[g^{\mu\sigma} g_{\nu\sigma} = \delta^\mu_\nu\]

    \[g^{\mu\nu} g_{\mu\nu} = \delta^\mu_\mu = D\]

(D: dimension of the spacetime)

Christoffel symbol:

    \[\Gamma^\lambda_{\mu\nu} = \frac{1}{2} g^{\lambda\sigma}(\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\sigma\mu} - \partial_\sigma g_{\mu\nu})\]

Riemann tensor:

    \[{R^\rho}_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma}\]

Ricci tensor:

    \[R_{\mu\nu} = {R^\lambda}_{\mu\lambda\nu}\]

Ricci scalar (curvature scalar):

    \[R = {R^\mu}_\mu = g^{\mu\nu} R_{\mu\nu}\]

Trace of energy-momentum tensor:

    \[T = {T^\mu}_\mu = g^{\mu\nu} T_{\mu\nu}\]

Einstein field equation:

    \[R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8 \pi T_{\mu\nu}\]

or

    \[R_{\mu\nu} = 8 \pi \left(T_{\mu\nu} - \frac{1}{2} T g_{\mu\nu} \right)\]

Torsion tensor (have nothing to do with energy-momentum tensor T_{\mu\nu}):

    \[{T^\lambda}_{\mu\nu} = \Gamma^\lambda_{\mu\nu} - \Gamma^\lambda_{\nu\mu} = 2 \Gamma^\lambda_{[\mu\nu]}\]

Properties of the Riemann tensor (R_{\rho\sigma\mu\nu}=g_{\rho\lambda} {R^{\lambda}}_{\sigma\mu\nu}):

    \[R_{\rho\sigma\mu\nu} = -R_{\sigma\rho\mu\nu}\]

(antisymmetric in first two indices)

    \[R_{\rho\sigma\mu\nu} = -R_{\rho\sigma\nu\mu}\]

(antisymmetric in last two indices)

    \[R_{\rho\sigma\mu\nu} = R_{\mu\nu\rho\sigma}\]

(invariant under interchange of the first and last pair of indices)

    \[R_{\rho\sigma\mu\nu} + R_{\rho\mu\nu\sigma} + R_{\rho\nu\sigma\mu} = 0\]

or

    \[R_{\rho[\sigma\mu\nu]} = 0\]

    \[R_{[\rho\sigma\mu\nu]} = 0\]

Properties of the Ricci tensor:

    \[R_{\mu\nu} = R_{\nu\mu}\]

Relation between R and T:

    \[R = -8 \pi T\]

Einstein tensor:

    \[G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu}\]

so Einstein field equation can be rewritten as:

    \[G_{\mu\nu} = 8 \pi T_{\mu\nu}\]

Geodesic equation:

    \[\frac{d^2 x^\mu}{{d\lambda}^2} + \Gamma^\mu_{\rho\sigma} \frac{d x^\rho}{d \lambda} \frac{d x^\sigma}{d \lambda} = 0\]

Covariant derivative:

    \[\nabla_\sigma V^\mu = \partial_\sigma V^\mu + {\Gamma_\sigma}^\mu_\lambda V^\lambda\]

    \[\nabla_\sigma W_\nu = \partial_\sigma W_\nu + {\Gamma_\sigma}^\lambda_\nu W_\lambda\]

    \[\nabla_\sigma {T^{\mu_1 \mu_2 \ldots \mu_k}}_{\nu_1 \nu_2 \ldots \nu_k} = \partial_\sigma {T^{\mu_1 \mu_2 \ldots \mu_k}}_{\nu_1 \nu_2 \ldots \nu_k}\\ ~~~~ + {\Gamma_\sigma}^{\mu_1}_{\lambda} {T^{\lambda \mu_2 \ldots \mu_k}}_{\nu_1 \nu_2 \ldots \nu_k} + {\Gamma_\sigma}^{\mu_2}_{\lambda} {T^{\mu_1 \lambda \ldots \mu_k}}_{\nu_1 \nu_2 \ldots \nu_k} + \ldots\\ ~~~~ - {\Gamma_\sigma}^{\lambda}_{\nu_1} {T^{\mu_1 \mu_2 \ldots \mu_k}}_{\lambda \nu_2 \ldots \nu_k} - {\Gamma_\sigma}^{\lambda}_{\nu_2} {T^{\mu_1 \mu_2 \ldots \mu_k}}_{\nu_1 \lambda \ldots \nu_k} - \ldots\]

Energy-momentum tensor:

    \[\nabla_\mu T^{\mu\nu}=0\]

November 24th, 2010 | 荒唐 | Categories: 数学, 物理 | Comments: No Comments