Conservation Laws in General Relativity

Abstract

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.