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, ${T_{\mu }}^{\nu }$, whereas a symmetric quantity ${\mathcal{T}}^{\mu \nu }$ 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 ${T_{\mathrm{}\left(n\right)\mathrm{}\mu }}^{\nu }$ and ${{\mathcal{T}}_{\mathrm{}\left(n\right)\mathrm{}}}^{\mu \nu }$ are constructed. Under linear coordinate transformations the former are tensor densities of weight ($n+1\mathrm{}$) and the latter of weight ($n+2\mathrm{}$). For $n=0\mathrm{}$ these reduce to the canonical ${T_{\mu }}^{\nu }$ and the Landau-Lifshitz ${\mathcal{T}}^{\mu \nu }$, 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 ${T_{\mu }}^{\nu }$ 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 ${{\mathcal{T}}_{\mathrm{}(-1\mathrm{})\mathrm{}}}^{\mu \nu }$. 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.