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). Mathematical developments in the 1980s have allowed pseudotensors to be understood as sections of jet bundles, thus providing a firm theoretical foundation for the concept of pseudotensors in general relativity.
Landau–Lifshitz pseudotensor 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.
Requirements Landau and Lifshitz were led by four requirements in their search for a gravitational energy momentum pseudotensor, t LL μ ν {\displaystyle t_{\text{LL}}^{\mu \nu }} :
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. t LL μ ν = t LL ν μ {\displaystyle t_{\text{LL}}^{\mu \nu }=t_{\text{LL}}^{\nu \mu }} , (to conserve angular momentum) that, when added to the stress–energy tensor of matter, T μ ν {\displaystyle T^{\mu \nu }} , its total ordinary 4-divergence (∂μ, not ∇μ) vanishes so that we have a conserved expression for the total stress–energy–momentum. (This is required of any conserved current.) 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. If gravitational energy is a function of its force field, as is usual for other forces, then the associated gravitational pseudotensor should also vanish locally.
Definition Landau and Lifshitz showed that there is a unique construction that satisfies these requirements, namely
t LL μ ν = − 1 κ G μ ν + 1 2 κ ( − g ) ( ( − g ) ( g μ ν g α β − g μ α g ν β ) ) , α β {\displaystyle t_{\text{LL}}^{\mu \nu }=-{\frac {1}{\kappa }}G^{\mu \nu }+{\frac {1}{2\kappa (-g)}}\left((-g)\left(g^{\mu \nu }g^{\alpha \beta }-g^{\mu \alpha }g^{\nu \beta }\right)\right)_{,\alpha \beta }}
where:
Gμν is the Einstein tensor (which is constructed from the metric) gμν is the inverse of the metric tensor, gμν g = det(gμν) is the determinant of the metric tensor. g < 0, hence its appearance as − g {\displaystyle -g} .
, α β = ∂ 2 ∂ x α ∂ x β {\textstyle {}_{,\alpha \beta }={\frac {\partial ^{2}}{\partial x^{\alpha }\partial x^{\beta }}}} are partial derivatives, not covariant derivatives κ = 8πG/c4 is the Einstein gravitational constant G is the Newtonian constant of gravitation
Verification Examining the 4 requirement conditions we can see that the first 3 are relatively easy to demonstrate:
Since the Einstein tensor, G μ ν {\displaystyle G^{\mu \nu }} , is itself constructed from the metric, so therefore is t LL μ ν {\displaystyle t_{\text{LL}}^{\mu \nu }}
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