Landau-Lifshitz stress-energy pseudotensor

This Mathematica notebook explains the derivation of the Landau-Lifshitz pseudotensor, a mathematical object used to describe the energy and momentum of a gravitational field in general relativity. Although not a true tensor, as it depends on the choice of coordinates and cannot be localized, the pseudotensor has useful properties such as being symmetric, having zero divergence in flat spacetime, and being conserved in asymptotically flat spacetime. The concept of the stress-energy-momentum (SEM) tensor is introduced, which is a true tensor that combines energy, mass, momentum, and their fluxes and stresses. Derived from the action principle, the SEM tensor satisfies conservation law in flat spacetime. However, in curved spacetime, it is not conserved due to the presence of a gravitational field. A pseudotensor that locally has zero divergence is then found and corrected for curvature effects. The pseudotensor can be obtained in terms of second derivatives of the metric tensor, which characterizes the geometry of spacetime. It is constructed from the Einstein tensor, related to the Ricci tensor and Ricci scalar, which are in turn related to the Riemann tensor containing all information about spacetime curvature. A computer algebra system called xTensor is used to perform calculations and simplify expressions. Another way to obtain the pseudotensor is in terms of first derivatives of the metric tensor. This form is more convenient for some applications such as calculating gravitational radiation emitted by a system. The pseudotensor can also be obtained in terms of first derivatives of the metric density, a modified version of the metric tensor that includes a factor of $\sqrt{-g}$ where $g$ is the determinant of the metric tensor. Preferred by some authors because it makes integration over curved spacetime easier, this factor is related to the Jacobian determinant of coordinate transformations and affects the volume element. Another way to obtain the pseudotensor is in terms of Christoffel symbols, which describe spacetime curvature. Related to first derivatives of the metric tensor, they are used to define covariant derivatives invariant under coordinate transformations. xTensor is used to perform calculations and compare them with EinS. The notebook concludes with remarks on the physical meaning and interpretation of the pseudotensor and mentions alternative definitions and approaches for finding a pseudotensor that can describe energy and momentum of a gravitational field in general relativity.