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(clean up, typos fixed: , → , (2) using AWB) 

[[Lev Davidovich LandauLandau]] & [[Evgeny Mikhailovich LifshitzLifshitz]] were led by four requirements in their search for a gravitational energy momentum pseudotensor, <math>t_{LL}^{\mu \nu}\,</math>:<ref name="LL"/>
# that it be constructed entirely from the [[Metric tensor (general relativity)metric tensor]], so as to be purely geometrical or gravitational in origin.
# that it be index symmetric
# that, when added to the [[stress–energy tensor]] of matter, <math>T^{\mu \nu}\,</math>, its total 4[[divergence]] vanishes (this is required of any [[conserved current]]) so that we have a conserved expression for the total stress–energy–momentum.
# that it vanish locally in an [[inertial frame of reference]] (which requires that it only contains first and not second or higher [[derivative]]s of the metric). This is because the [[equivalence principle]] requires that the gravitational force field, the [[Christoffel symbols]], vanish locally in some frame. If gravitational energy is a function of its force field, as is usual for other forces, then the associated gravitational pseudotensor should also vanish locally.
where:
* ''G''<sup>
* ''g''<sup>
* ''g'' = det(''g''<sub>
* <math>,_{\alpha \beta} = \frac{\partial^2}{\partial x^{\alpha} \partial x^{\beta}}\,</math> are [[partial derivative]]s, not [[covariant derivative]]s.
===Verification===
Examining the 4 requirement conditions we can see that the first 3 are relatively easy to demonstrate:
#Since the Einstein tensor, <math>G^{\mu \nu}\,</math>, is itself constructed from the metric, so therefore is <math>t_{LL}^{\mu \nu} </math>
#Since the Einstein tensor, <math>G^{\mu \nu}\,</math>, is symmetric so is <math>t_{LL}^{\mu \nu} </math> since the additional terms are symmetric by inspection.
#The Landau–Lifshitz pseudotensor is constructed so that when added to the [[stress–energy tensor]] of matter, <math>T^{\mu \nu}\,</math>, its total 4[[divergence]] vanishes: <math>((g)(T^{\mu \nu} + t_{LL}^{\mu \nu}))_{,\mu} = 0 </math>. This follows from the cancellation of the Einstein tensor, <math>G^{\mu \nu}\,</math>, with the [[stress–energy tensor]], <math>T^{\mu \nu}\,</math> by the [[Einstein field equations]]; the remaining term vanishes algebraically due the commutativity of partial derivatives applied across antisymmetric indices.
===Cosmological constant===
When the Landau–Lifshitz pseudotensor was formulated it was commonly assumed that the [[cosmological constant]],<math>\Lambda \,</math>
:<math>t_{LL}^{\mu \nu} =  \frac{c^4}{8\pi G}(G^{\mu \nu}+\Lambda g^{\mu \nu}) + \frac{c^4}{16\pi G (g)}((g)(g^{\mu \nu}g^{\alpha \beta}  g^{\mu \alpha}g^{\nu \beta}))_{,\alpha \beta}</math>
This is necessary for consistency with the [[Einstein field equations]].
*[[Metric tensor]] version:
:<math>(g)(t_{LL}^{\mu \nu} + \frac{c^4\Lambda g^{\mu \nu}}{8\pi G}) = \frac{c^4}{16\pi G}((\sqrt{g}g^{\mu \nu}),_{\alpha }(\sqrt{g}g^{\alpha \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
*[[
:<math>t_{LL}^{\mu \nu} + \frac{c^4\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>
::<math>+g^{\alpha \beta }g^{ \sigma \rho}(\Gamma^{\mu}_{\alpha \sigma } \Gamma^{\nu}_{\beta \rho }  \Gamma^{\mu}_{\alpha \beta } \Gamma^{\nu}_{ \sigma \rho }))</math><ref>Landau–Lifshitz equation 96.8
This definition of energymomentum is covariantly applicable not just under Lorentz transformations, but also under general coordinate transformations.
