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 1980's 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, : ∂μ , not ∇μ ) vanishes so that we have a conserved expression for the total stress–energy–momentum. (This is required of any conserved current.)

Definition

Landau & Lifshitz showed that there is a unique construction that satisfies these requirements, namely where:

Verification

Examining the 4 requirement conditions we can see that the first 3 are relatively easy to demonstrate:

Cosmological constant

When the Landau–Lifshitz pseudotensor was formulated it was commonly assumed that the cosmological constant, \Lambda ,, was zero. Nowadays, that assumption is suspect, and the expression frequently gains a \Lambda term, giving: 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–Lifshitz pseudotensor: This definition of energy–momentum is covariantly applicable not just under Lorentz transformations, but also under general coordinate transformations.

Einstein pseudotensor

This pseudotensor was originally developed by Albert Einstein. Paul Dirac showed that the mixed Einstein pseudotensor satisfies a conservation law Clearly this pseudotensor for gravitational stress–energy is constructed exclusively from the metric tensor and its first derivatives. Consequently, it vanishes at any event when the coordinate system is chosen to make the first derivatives of the metric vanish because each term in the pseudotensor is quadratic in the first derivatives of the metric. However it is not symmetric, and is therefore not suitable as a basis for defining the angular momentum.