Stress–energy–momentum pseudotensor: Difference between revisions

 
*[[Metric tensor]] version:
:<math>(-g)(t_{LL}^{\mu \nu} =+ \frac{c^4}{16\piLambda G}((\sqrt{-g}g^{\mu \nu}),_}{8\alphapi G}() = \sqrtfrac{-gc^4}g^{16\alphapi \betaG}),_{\beta} - ((\sqrt{-g}g^{\mu \alpha nu}),_{\alpha }(\sqrt{-g}g^{\nualpha \beta}),_{\beta} +- </math>
::<math>- (\sqrt{-g}g^{\mu \alpha }),_{\alpha }(\sqrt{-g}g^{\nu \beta}),_{\beta} +\frac{1}{2}g^{\mu \nu}g_{\alpha \beta}(\sqrt{-g}g^{\alpha \sigma }),_{\rho }(\sqrt{-g}g^{\rho \beta }),_{ \sigma }-</math>
::<math>-(g^{\mu \alpha }g_{\beta \sigma }(\sqrt{-g}g^{\nu \sigma }),_{\rho }(\sqrt{-g}g^{\beta \rho }),_{\alpha }+g^{\nu \alpha }g_{\beta \sigma}(\sqrt{-g}g^{\mu \sigma }),_{\rho }(\sqrt{-g}g^{\beta \rho }),_{\alpha })+</math>
::<math>+g_{\alpha \beta }g^{ \sigma \rho }(\sqrt{-g}g^{\mu \alpha }),_{ \sigma }(\sqrt{-g}g^{\nu \beta }),_{\rho }+\,</math>
::<math>+\frac{1}{8}(2g^{\mu \alpha }g^{\nu \beta }-g^{\mu \nu}g^{\alpha \beta })(2g_{ \sigma \rho }g_{\lambda \omega}-g_{\rho \lambda }g_{ \sigma \omega})(\sqrt{-g}g^{ \sigma \omega}),_{\alpha }(\sqrt{-g}g^{\rho \lambda }),_{\beta })</math><ref>Landau-Lifshitz equation 96.9 </ref>
*[[Christoffel_symbols|Affine connection]] version:
:<math>t_{LL}^{\mu \nu} + \frac{\Lambda g^{\mu \nu}}{8\pi G}= \frac{c^4}{16\pi G}((2\Gamma^{ \sigma }_{\alpha \beta }\Gamma^{\rho }_{ \sigma \rho }-\Gamma^{ \sigma }_{\alpha \rho }\Gamma^{\rho }_{\beta \sigma }-\Gamma^{ \sigma }_{\alpha \sigma }\Gamma^{\rho }_{\beta \rho})(g^{\mu \alpha }g^{\nu \beta }-g^{\mu \nu}g^{\alpha \beta })+</math>
::<math>+g^{\mu \alpha }g^{\beta \sigma }(\Gamma^{\nu}_{\alpha \rho }\Gamma^{\rho }_{\beta \sigma }+\Gamma^{\nu}_{\beta \sigma } \Gamma^{\rho }_{\alpha \rho } - \Gamma^{\nu}_{ \sigma \rho } \Gamma^{\rho }_{\alpha \beta } - \Gamma^{\nu}_{\alpha \beta } \Gamma^{\rho }_{ \sigma \rho })+</math>
::<math>+g^{\nu \alpha }g^{\beta \sigma }(\Gamma^{\mu}_{\alpha \rho }\Gamma^{\rho }_{\beta \sigma }+\Gamma^{\mu}_{\beta \sigma } \Gamma^{\rho }_{\alpha \rho } - \Gamma^{\mu}_{ \sigma \rho } \Gamma^{\rho }_{\alpha \beta } - \Gamma^{\mu}_{\alpha \beta } \Gamma^{\rho }_{ \sigma \rho })+</math>