Stress–energy–momentum pseudotensor: Difference between revisions

change dash in "Landau–Lifshitz" ; remove self-link
(change dash in "Landau–Lifshitz" ; remove self-link)
In the theory of [[general relativity]], a '''stress-energy-momentum pseudotensor''', such as the '''Landau-LifshitzLandau–Lifshitz pseudotensor''', is an extension of the non-gravitational [[stress-energy tensor]] which incorporates the [[energy-momentum]] of gravity. It allows the [[energy-momentum]] of a system of gravitating matter to be defined. In particular it allows the total of matter plus the gravitating energy-momentum to form a [[conserved current]] within the framework of [[general relativity]], so that the ''total'' energy-momentum crossing the [[hypersurface]] (3-dimensional boundary) of ''any'' compact [[space-time]] [[hypervolume]] (4-dimensional submanifold) vanishes.
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-LifshitzLandau–Lifshitz pseudotensor==
The use of the Landau-LifshitzLandau–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 7-5062-4256-7 chapter 11, section #96</ref> allows the energy-momentum conservation laws to be extended into [[general relativity]]. Subtraction of the matter [[stress-energy-momentum tensor]] from the combined pseudotensor results in the gravitational stress-energy-momentum pseudotensor.
#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-LifshitzLandau–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-LifshitzLandau–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 [[Levi-Civita connection]]; only the first derivative terms in the metric survive and these vanish where the frame is locally inertial around any chosen point. As a result the entire pseudotensor vanishes locally (again, around any chosen point) <math>t_{LL}^{\mu \nu} = 0</math>, which demonstrates the delocalisation of gravitational energy-momentum.<ref name="LL"/>
===Cosmological constant===
When the [[Landau-LifshitzLandau–Lifshitz pseudotensor]] was formulated it was commonly assumed that the [[cosmological constant]],<math>\Lambda \,</math> , was zero. Nowadays [[accelerating universe|we don't make that assumption]], and expression needs the addition of a <math>\Lambda \,</math> term, giving:
:<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 Landau-LifshitzLandau–Lifshitz pseudotensor:
*[[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>Landau-LifshitzLandau–Lifshitz equation 96.9 </ref>
*[[Christoffel_symbols|Affine 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>Landau-LifshitzLandau–Lifshitz equation 96.8 </ref>
This definition of energy-momentum is covariantly applicable not just under Lorentz transformations, but also under general coordinate transformations.