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Concept
Stress–energy–momentum pseudotensor
Summary
In the theory of general relativity, a stress–energy–momentum pseudotensor, such as the Landau–Lifshitz pseudotensor, is an extension of the non-gravitational stress–energy tensor that 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 (such as Erwin Schrödinger) have objected to this derivation on the grounds that pseudotensors 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 bundles, which are now recognized as perfectly valid objects in general relativity.
The Landau–Lifshitz pseudotensor, a stress–energy–momentum pseudotensor for gravity, when combined with terms for matter (including photons and neutrinos), allows the energy–momentum conservation laws to be extended into general relativity.
Landau and Lifshitz were led by four requirements in their search for a gravitational energy momentum pseudotensor, :
that it be constructed entirely from the metric tensor, so as to be purely geometrical or gravitational in origin.
that it be index symmetric, i.e. , (to conserve angular momentum)
that, when added to the stress–energy tensor of matter, , its total 4-divergence vanishes (this is required of any conserved current) so that we have a conserved expression for the total stress–energy–momentum.
that it vanish locally in an inertial frame of reference (which requires that it only contains first order and not second or higher order derivatives of the metric). This is because the equivalence principle requires that the gravitational force field, the Christoffel symbols, vanish locally in some frames.
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