Energy-momentum and angular momentum densities in gauge theories of gravity

Abstract

In the $\bar{\mbox{\rm Poincar\'{e}}}$ gauge theory of gravity, which has been formulated on the basis of a principal fiber bundle over the space-time manifold having the covering group of the proper orthochronous Poincar\'{e} group as the structure group, we examine the tensorial properties of the dynamical energy-momentum density ${}^{G}{\mathbf T}_{k}{}^{\mu}$ and the ` ` spin" angular momentum density ${}^{G}{\mathbf S}_{kl}{}^{\mu}$ of the gravitational field. They are both space-time vector densities, and transform as tensors under {\em global} $SL(2,C)$- transformations. Under {\em local} internal translation, ${}^{G}{\mathbf T}_{k}{}^{\mu}$ is invariant, while ${}^{G}{\mathbf S}_{kl}{}^{\mu}$ transforms inhomogeneously. The dynamical energy-momentum density ${}^{M}{\mathbf T}_{k}{}^{\mu}$ and the ` ` spin" angular momentum density ${}^{M}{\mathbf S}_{kl}{}^{\mu}$ of the matter field are also examined, and they are known to be space-time vector densities and to obey tensorial transformation rules under internal $\bar{\mbox{\rm Poincar\'{e}}}$ gauge transformations. The corresponding discussions in extended new general relativity which is obtained as a teleparallel limit of $\bar{\mbox{\rm Poincar\'{e}}}$ gauge theory are also given, and energy-momentum and ` ` spin" angular momentum densities are known to be well behaved. Namely, they are all space-time vector densities, etc. In both theories, integrations of these densities on a space-like surface give the total energy-momentum and {\em total} (={\em spin}+{\em orbital}) angular momentum for asymptotically flat space-time. The tensorial properties of canonical energy-momentum and ` ` extended orbital angular momentum" densities are also examined.