Stress–energy–momentum pseudotensor: Difference between revisions

boldface per WP:R#PLA
(Undid revision 796718488 by Ciphers (talk) A pseudotensor cannot be called a tensor and "e.g." does not go before "such as")
(boldface per WP:R#PLA)
Some people (such as [[Erwin Schrödinger]]{{citation needed|date=October 2015}}) have objected 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 now recognized as perfectly valid objects in GR.
 
==Landau–Lifshitz pseudotensor<!--'Landau–Lifshitz pseudotensor' and 'Landau-Lifshitz pseudotensor' redirect here-->==
The use of the '''Landau–Lifshitz pseudotensor'''<!--boldface per WP:R#PLA--> (a combined ''matter+'' plus ''gravitational stress–energy–momentum'' [[pseudotensor]])<ref name="LL">[[Lev Davidovich Landau]] &and [[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===
[[Lev Davidovich Landau|Landau]] &and [[Evgeny Mikhailovich Lifshitz|Lifshitz]] 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, i.e. <math>t_{LL}^{\mu \nu} = t_{LL}^{\nu \mu} \,</math>, (to conserve [[angular momentum]])
 
===Definition===
Landau and& Lifshitz showed that there is a unique construction that satisfies these requirements, namely
 
:<math>t_{LL}^{\mu \nu} = - \frac{c^4}{8\pi G}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>