Conservation Laws in General Relativity

TLDR

Conservation laws in general relativity are studied via the transformation properties of the Lagrangian, revealing that the mixed‑index energy‑momentum complex requires a symmetric counterpart, which Landau and Lifshitz constructed. The authors construct two hierarchies of complexes, \(T_{(n)\mu}^{\ \nu}\) and \(\mathcal{T}_{(n)}^{\mu\nu}\), whose tensor‑density weights lead, for \(n=0\), to the canonical and Landau‑Lifshitz complexes, and by demanding that the energy‑momentum complex generate coordinate transformations and that total energy and momentum form a free vector, they identify the canonical complex as the correct description of field plus matter energy and momentum, while a similar requirement for angular momentum yields a distinct complex from \(\mathcal{T}_{(-1)}^{\mu\nu}\). The resulting angular‑momentum complex differs from both the Landau‑Lifshitz and Bergmann‑Thomson constructions, confirming the canonical complex as the appropriate energy‑momentum descriptor.

Abstract

The conservation laws are examined from the transformation properties of the Lagrangian. The energy-momentum complex obtained has mixed indices, ${{T}_{\ensuremath{\mu}}}^{\ensuremath{\nu}}$, whereas a symmetric quantity ${\mathcal{T}}^{\ensuremath{\mu}\ensuremath{\nu}}$ 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 ${{T}_{(n)\ensuremath{\mu}}}^{\ensuremath{\nu}}$ and ${{\mathcal{T}}_{(n)}}^{\ensuremath{\mu}\ensuremath{\nu}}$ 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 ${{T}_{\ensuremath{\mu}}}^{\ensuremath{\nu}}$ and the Landau-Lifshitz ${\mathcal{T}}^{\ensuremath{\mu}\ensuremath{\nu}}$, 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 ${{T}_{\ensuremath{\mu}}}^{\ensuremath{\nu}}$ 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 ${{\mathcal{T}}_{(\ensuremath{-}1)}}^{\ensuremath{\mu}\ensuremath{\nu}}$. 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.