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(→Metric and affine connection versions: Ha  and another) 
(change dash in "Landau–Lifshitz" ; remove selflink) 

In the theory of [[general relativity]], a '''stressenergymomentum pseudotensor''', such as the '''
Some people object to this derivation on the grounds that [[pseudotensor]]s are inappropriate objects in general relativity, but the conservation law only requires the use of the 4[[divergence]] of a pseudotensor which is, in this case, a tensor (which also vanishes). Also, most pseudotensors are sections of [[jet bundle]]s, which are perfectly valid objects in GR.
==
The use of the
===Requirements===
#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
#The
===Cosmological constant===
When the
:<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 and affine connection versions===
Landau & Lifshitz also provide two equivalent but longer expressions for the
*[[Metric tensor]] version:
::<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>
*[[Christoffel_symbolsAffine connection]] version:
:<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>
This definition of energymomentum is covariantly applicable not just under Lorentz transformations, but also under general coordinate transformations.
