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In the theory of [[general relativity]], a '''stress–energy–momentum pseudotensor''', such as the '''Landau–Lifshitz pseudotensor''', is an extension of the nongravitational [[stress–energy tensor]] which incorporates 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.
==Landau–Lifshitz pseudotensor==
The use of the Landau–Lifshitz combined matter+gravitational stress–energy–momentum [[pseudotensor]]<ref name="LL">[[Lev Davidovich Landau]] & [[Evgeny Mikhailovich Lifshitz]], ''The Classical Theory of Fields'', (1951), Pergamon Press, ISBN 7506242567 chapter 11, section #96</ref> allows the
===Requirements===
#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.
#The Landau–Lifshitz pseudotensor appears to include second derivative terms in the metric, but in fact the explicit second derivative terms in the pseudotensor cancel with the implicit second derivative terms contained within the [[Einstein tensor]], <math>G^{\mu \nu}\,</math>. This is more evident when the pseudotensor is directly expressed in terms of the metric tensor or the [[LeviCivita connection]]; only the first derivative terms in the metric survive and these vanish where the frame is locally inertial at any chosen point. As a result the entire pseudotensor vanishes locally (again, at any chosen point) <math>t_{LL}^{\mu \nu} = 0</math>, which demonstrates the delocalisation of gravitational
===Cosmological constant===
::<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</ref>
This definition of
==Einstein pseudotensor==
:<math>(({T_{\mu}}^{\nu} + {t_{\mu}}^{\nu})\sqrt{g})_{,\nu} = 0 .</math>
Clearly this pseudotensor for gravitational
==See also==
