Conservation Laws in General Relativity

The conservation laws are examined from the transformation properties of the Lagrangian. The energy-momentum complex obtained has mixed indices, T_μ^ν, whereas a symmetric quantity T^μν is required for the definition of angular momentum. Such a symmetric quantity has been constructed by Landau and Lifshitz. In the course of examining the relationship between these quantities, two hierarchies of complexes T_(n)μ^ν and T_(n)^μν are constructed. Under linear coordinate transformations the former are tensor densities of weight (n+1) and the latter of weight (n+2). For n=0 these reduce to the canonical T_μ^ν and the Landau-Lifshitz T^μν, respectively.

By requiring the energy-momentum complex to generate the coordinate transformations, and the total energy and momentum to form a free vector, one can identify the canonical complex T_μ^ν as the appropriate quantity to describe the energy and momentum of the field plus matter. Similarly, by requiring the total angular momentum to behave as a free antisymmetric tensor, one can construct, in the usual manner, an appropriate quantity from T_(-1)^μν. The angular momentum complex so defined differs from that proposed by Landau and Lifshitz as well as from an independent construction by Bergmann and Thomson.