Notes on Einstein Field Equation
Metric tensor:
![\[g_{\mu\nu} = g_{\nu\mu}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-bf25cec1e9336d6657d6cfad8b97e78d_l3.png "Rendered by QuickLaTeX.com")
![\[g^{\mu\sigma} g_{\nu\sigma} = \delta^\mu_\nu\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-add06c201725feb52ba25f17f5ebb125_l3.png "Rendered by QuickLaTeX.com")
Christoffel symbol:
![\[\Gamma^\lambda_{\mu\nu} = \frac{1}{2} g^{\lambda\sigma}(\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\sigma\mu} - \partial_\sigma g_{\mu\nu})\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-fc9cf87d77bbcc1db3d7f602d3cbdec5_l3.png "Rendered by QuickLaTeX.com")
Riemann tensor:
![\[{R^\rho}_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-8615683bb4f9892bb08a793ddc19c053_l3.png "Rendered by QuickLaTeX.com")
Ricci tensor:
![\[R_{\mu\nu} = {R^\lambda}_{\mu\lambda\nu}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-54c270e059255d9fbe26c759b00c7450_l3.png "Rendered by QuickLaTeX.com")
Ricci scalar (curvature scalar):
![\[R = {R^\mu}_\mu = g^{\mu\nu} R_{\mu\nu}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-cb900533d20b4eb072429ba59a831c26_l3.png "Rendered by QuickLaTeX.com")
Trace of energy-momentum tensor:
![\[T = {T^\mu}_\mu = g^{\mu\nu} T_{\mu\nu}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-e85df44a1ce64f5597fd80d785f25f97_l3.png "Rendered by QuickLaTeX.com")
Einstein field equation:
![\[R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8 \pi T_{\mu\nu}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-f68b805353ea805103e8b5eb363be680_l3.png "Rendered by QuickLaTeX.com")
or
![\[R_{\mu\nu} = 8 \pi \left(T_{\mu\nu} - \frac{1}{2} T g_{\mu\nu} \right)\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-cd0a3fec4904284d5c82cd17fde65a94_l3.png "Rendered by QuickLaTeX.com")
Torsion tensor (have nothing to do with energy-momentum tensor
![\[T_{\mu\nu}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-a7fba67e3cf4eba0f19bcc7f7f92f1d9_l3.png "Rendered by QuickLaTeX.com")
):
![\[{T^\lambda}_{\mu\nu} = \Gamma^\lambda_{\mu\nu} - \Gamma^\lambda_{\nu\mu} = 2 \Gamma^\lambda_{[\mu\nu]}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-84ecd588822b946f0530780758384f96_l3.png "Rendered by QuickLaTeX.com")
Properties of the Riemann tensor (
![\[R_{\rho\sigma\mu\nu}=g_{\rho\lambda} {R^{\lambda}}_{\sigma\mu\nu}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-e94aab518022788cc132c4bd91d60948_l3.png "Rendered by QuickLaTeX.com")
):
![\[R_{\rho\sigma\mu\nu} = -R_{\sigma\rho\mu\nu}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-ae1137165edae17b2239ca781250f867_l3.png "Rendered by QuickLaTeX.com")
(antisymmetric in first two indices)
![\[R_{\rho\sigma\mu\nu} = -R_{\rho\sigma\nu\mu}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-03d8c9799abc4c3488832803352c532e_l3.png "Rendered by QuickLaTeX.com")
(antisymmetric in last two indices)
![\[R_{\rho\sigma\mu\nu} = R_{\mu\nu\rho\sigma}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-f905b8157d97c3c5457976e1675c64c9_l3.png "Rendered by QuickLaTeX.com")
(invariant under interchange of the first and last pair of indices)
![\[R_{\rho\sigma\mu\nu} + R_{\rho\mu\nu\sigma} + R_{\rho\nu\sigma\mu} = 0\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-2c2af50926183fc3ab3ca9dbffd84a5d_l3.png "Rendered by QuickLaTeX.com")
or
![\[R_{\rho[\sigma\mu\nu]} = 0\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-abc6abf3e06fc25f4b43d5c508f4fb0b_l3.png "Rendered by QuickLaTeX.com")
![\[R_{[\rho\sigma\mu\nu]} = 0\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-d976f6ce5b5bbd46ed914774006e0f7d_l3.png "Rendered by QuickLaTeX.com")
Properties of the Ricci tensor:
![\[R_{\mu\nu} = R_{\nu\mu}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-cdf40f2a74b266eeee193830d722f662_l3.png "Rendered by QuickLaTeX.com")
Relation between R and T:
![\[R = -8 \pi T\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-de68a1e787a80f5a638f3cafa3e4971c_l3.png "Rendered by QuickLaTeX.com")
Einstein tensor:
![\[G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-71c2151a2cd9b7d031b1afa44a3143d7_l3.png "Rendered by QuickLaTeX.com")
so Einstein field equation can be rewritten as:
![\[G_{\mu\nu} = 8 \pi T_{\mu\nu}\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-4327c9d53b3adc5e9c555a4b099ef82d_l3.png "Rendered by QuickLaTeX.com")
Geodesic equation:
![\[\frac{d^2 x^\mu}{{d\lambda}^2} + \Gamma^\mu_{\rho\sigma} \frac{d x^\rho}{d \lambda} \frac{d x^\sigma}{d \lambda} = 0\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-098351283cb78f35ae6abd6d6c19eab6_l3.png "Rendered by QuickLaTeX.com")
Covariant derivative:
![\[\nabla_\sigma V^\mu = \partial_\sigma V^\mu + {\Gamma_\sigma}^\mu_\lambda V^\lambda\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-1c6e54c4c5ef06d14d84a801426f8c26_l3.png "Rendered by QuickLaTeX.com")
![\[\nabla_\sigma W_\nu = \partial_\sigma W_\nu + {\Gamma_\sigma}^\lambda_\nu W_\lambda\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-91efe1519229b5392babc2e7bfb80585_l3.png "Rendered by QuickLaTeX.com")
![\[\nabla_\sigma {T^{\mu_1 \mu_2 \ldots \mu_k}}_{\nu_1 \nu_2 \ldots \nu_k} = \partial_\sigma {T^{\mu_1 \mu_2 \ldots \mu_k}}_{\nu_1 \nu_2 \ldots \nu_k}\\ ~~~~ + {\Gamma_\sigma}^{\mu_1}_{\lambda} {T^{\lambda \mu_2 \ldots \mu_k}}_{\nu_1 \nu_2 \ldots \nu_k} + {\Gamma_\sigma}^{\mu_2}_{\lambda} {T^{\mu_1 \lambda \ldots \mu_k}}_{\nu_1 \nu_2 \ldots \nu_k} + \ldots\\ ~~~~ - {\Gamma_\sigma}^{\lambda}_{\nu_1} {T^{\mu_1 \mu_2 \ldots \mu_k}}_{\lambda \nu_2 \ldots \nu_k} - {\Gamma_\sigma}^{\lambda}_{\nu_2} {T^{\mu_1 \mu_2 \ldots \mu_k}}_{\nu_1 \lambda \ldots \nu_k} - \ldots\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-1c751ecdc007d52ebf05f157c12bdd3b_l3.png "Rendered by QuickLaTeX.com")
Energy-momentum tensor:
![\[\nabla_\mu T^{\mu\nu}=0\]](http://zhblog.engic.org/wp-content/ql-cache/quicklatex.com-b4ee195f3336f3fdb994fff03786b006_l3.png "Rendered by QuickLaTeX.com")
插件比较
