Stress–energy–momentum pseudotensor: Difference between revisions

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==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 |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.
 
===Requirements===
This pseudotensor was originally developed by [[Albert Einstein]].<ref>[[Albert Einstein]] ''Das hamiltonisches Prinzip und allgemeine Relativitätstheorie (The Hamiltonian principle and general relativity).'' Sitzungsber. preuss. Acad. Wiss. 1916, 2, 1111–1116.</ref><ref>[[Albert Einstein]] ''Der Energiesatz in der allgemeinen Relativitätstheorie. (An energy conservation law in general relativity).'' Sitzungsber. preuss. Acad. Wiss. 1918, 1, 448–459</ref>
 
[[Paul Dirac]] showed<ref>P.A.M.Dirac, ''General Theory of Relativity'' (1975), Princeton University Press, quick presentation of the bare essentials of GTR. {{ISBN |0-691-01146-X}} pages 61&mdash;63</ref> that the mixed Einstein pseudotensor
:<math>{t_{\mu}}^{\nu} = \frac{c^4}{16 \pi G \sqrt{-g}} ( (g^{\alpha\beta}\sqrt{-g})_{,\mu} (\Gamma^{\nu}_{\alpha\beta} - \delta^{\nu}_{\beta} \Gamma^{\sigma}_{\alpha\sigma}) - \delta_{\mu}^{\nu} g^{\alpha\beta} (\Gamma^{\sigma}_{\alpha\beta} \Gamma^{\rho}_{\sigma\rho} - \Gamma^{\rho}_{\alpha\sigma} \Gamma^{\sigma}_{\beta\rho})\sqrt{-g} ) </math>
 
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