Energy–momentum and angular momentum densities in the Poincaré gauge theory of gravity

Abstract

In the $\overline{\rm {Poincar\acute{e}}}$ gauge theory of gravity, we examine the energy–momentum and angular momentum densities of the source of the gravitational field (gauge potentials $A^{k}{}_{\mu }, \, A^{kl}{}_{\mu }$). There are two types of these quantities, the Hilbert-type densities ${\boldsymbol \Theta}_{k}{}^{\mu },\, {\boldsymbol \Phi}_{kl}{}^{\mu }$, which appear in the right-hand sides of the gravitational field equations, and $\, ^{M}{\boldsymbol T}_{k}{}^{\mu }, \, ^{M}{\boldsymbol S}_{kl}{}^{\mu }$, which are the densities (Noether currents) of the generators of $T^{4}\underline{\otimes }SL(2,C)$ transformations. We show that these are identical to each other, respectively, i.e., ${\boldsymbol \Theta}_{k}{}^{\mu }\equiv \, ^{M}{\boldsymbol T}_{k}{}^{\mu },\, {\boldsymbol \Phi}_{kl}{}^{\mu }\equiv \, ^{M}{\boldsymbol S}_{kl}{}^{\mu }$, in parallel with the cases of the electromagnetic and Yang–Mills fields.