Weyssenhoff spinning fluid in Riemann-Cartan spacetime: direct variation with respect to the spin density of the fluid

Abstract

The authors' original version of a Lagrangian based spinning fluid parametrises the spin density in terms of a set of tetrads which are then varied. This formalism necessarily introduces the spin module function of Halbwachs (1960) which is not varied. First they show that this spin module function is related to the component of the spin vector in the locally Lorentz (anholonomic) frame and is therefore covariantly constant by construction. Incorporating this constraint in their Lagrangian, they then develop a consistent variational principle for the spinning fluid in which the spin density (in the anholonomic frame) is directly varied. The resulting field equations give the same improved energy-momentum tensor, relationship between the torsion and the spin, and Fermi-Walker transport of the spin tensor, in complete agreement with formulation in terms of the tetrads alone.