Stress–energy–momentum pseudotensor

The Landau–Lifshitz pseudotensor, a stress–energy–momentum pseudotensor for gravity, when combined with terms for matter (including photons and neutrinos), allows the energy–momentum conservation laws to be extended into general relativity. Requirements Landau and Lifshitz were led by four requirements in their search for a gravitational energy momentum pseudotensor, t_\text{LL}^{\mu \nu}: \begin{align} (-g)\left(t_\text{LL}^{\mu \nu} + \frac{\Lambda g^{\mu \nu}}{\kappa}\right) = \frac{1}{2 \kappa}\bigg[&\left(\sqrt{-g}g^{\mu \nu}\right)_{,\alpha}\left(\sqrt{-g}g^{\alpha \beta}\right)_{,\beta} - \left(\sqrt{-g}g^{\mu \alpha}\right)_{,\alpha}\left(\sqrt{-g}g^{\nu \beta}\right)_{,\beta} + {} \\ &\frac{1}{8}\left(2g^{\mu \alpha}g^{\nu \beta}-g^{\mu \nu}g^{\alpha \beta}\right)\left(2g_{\sigma \rho}g_{\lambda \omega}-g_{\rho \lambda}g_{\sigma \omega}\right)\left(\sqrt{-g}g^{\sigma \omega}\right)_{,\alpha}\left(\sqrt{-g}g^{\rho \lambda}\right)_{,\beta} - {} \\ &\left(g^{\mu \alpha}g_{\beta \sigma}\left(\sqrt{-g}g^{\nu \sigma}\right)_{,\rho}\left(\sqrt{-g}g^{\beta \rho}\right)_{,\alpha}+g^{\nu \alpha}g_{\beta \sigma}\left(\sqrt{-g}g^{\mu \sigma}\right)_{,\rho}\left(\sqrt{-g}g^{\beta \rho}\right)_{,\alpha}\right) + {} \\ &\left.\frac{1}{2}g^{\mu \nu}g_{\alpha \beta}\left(\sqrt{-g}g^{\alpha \sigma}\right)_{,\rho}\left(\sqrt{-g}g^{\rho \beta}\right)_{,\sigma} + g_{\alpha \beta}g^{\sigma \rho}\left(\sqrt{-g}g^{\mu \alpha}\right)_{,\sigma}\left(\sqrt{-g}g^{\nu \beta}\right)_{,\rho}\right] \end{align} • Affine connection version: \begin{align} t_\text{LL}^{\mu \nu} + \frac{\Lambda g^{\mu \nu}}{\kappa} = \frac{1}{2 \kappa}\Big[ &\left(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}\right)\left(g^{\mu \alpha}g^{\nu \beta} - g^{\mu \nu}g^{\alpha \beta}\right) + {}\\ &\left(\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}\right)g^{\mu \alpha}g^{\beta \sigma} + \\ &\left(\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}\right)g^{\nu \alpha}g^{\beta \sigma} + \\ &\left.\left(\Gamma^{\mu}_{\alpha \sigma} \Gamma^{\nu}_{\beta \rho} - \Gamma^{\mu}_{\alpha \beta} \Gamma^{\nu}_{\sigma \rho}\right)g^{\alpha \beta}g^{\sigma \rho}\right] \end{align} This definition of energy–momentum is covariantly applicable not just under Lorentz transformations, but also under general coordinate transformations. == Einstein pseudotensor ==

This pseudotensor was originally developed by Albert Einstein. Paul Dirac showed that the mixed Einstein pseudotensor {t_\mu}^\nu = \frac{1}{2 \kappa \sqrt{-g}} \left( \left(g^{\alpha\beta}\sqrt{-g}\right)_{,\mu} \left(\Gamma^\nu_{\alpha\beta} - \delta^\nu_\beta \Gamma^\sigma_{\alpha\sigma}\right) - \delta_\mu^\nu g^{\alpha\beta} \left(\Gamma^\sigma_{\alpha\beta} \Gamma^\rho_{\sigma\rho} - \Gamma^\rho_{\alpha\sigma} \Gamma^\sigma_{\beta\rho}\right)\sqrt{-g} \right) satisfies a conservation law \left(\left({T_\mu}^\nu + {t_\mu}^\nu\right)\sqrt{-g}\right)_{,\nu} = 0 . Clearly this pseudotensor for gravitational stress–energy is constructed exclusively from the metric tensor and its first derivatives. Consequently, it vanishes at any event when the coordinate system is chosen to make the first derivatives of the metric vanish because each term in the pseudotensor is quadratic in the first derivatives of the metric tensor field. However it is not symmetric, and is therefore not suitable as a basis for defining the angular momentum. == See also ==