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

Abstract

In the $\overline{\mathrm{Poincar}\mathrm{\'e}}$ 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 ${}^{G}{\mathbf{T}}_{k}^{\ensuremath{\mu}}$ and the ``spin'' angular momentum density ${}^{G}{\mathbf{S}}_{\mathrm{kl}}^{\ensuremath{\mu}}$ of the gravitational field. They are both space-time vector densities, and transform as tensors under global $\mathrm{SL}(2,C)$ transformations. Under local internal translation, ${}^{G}{\mathbf{T}}_{k}^{\ensuremath{\mu}}$ is invariant, while ${}^{G}{\mathbf{S}}_{\mathrm{kl}}^{\ensuremath{\mu}}$ transforms inhomogeneously. The dynamical energy-momentum density ${}^{M}{\mathbf{T}}_{k}^{\ensuremath{\mu}}$ and the ``spin'' angular momentum density ${}^{M}{\mathbf{S}}_{\mathrm{kl}}^{\ensuremath{\mu}}$ 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 $\overline{\mathrm{Poincar}\mathrm{\'e}}$ gauge transformations. The corresponding discussions in extended new general relativity which is obtained as a teleparallel limit of $\overline{\mathrm{Poincar}\mathrm{\'e}}$ 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. The tensorial properties of canonical energy-momentum and ``extended orbital angular momentum'' densities are also examined.