Nonmetricity and Poincare gauge gravity

We show that in a spacetime geometry with general connection, there exist field redefinitions that replace the mixed-symmetry nonmetricity of the asymmetric connection with the sum of torsion and the field strength of special conformal transformations of a Lorentzian connection. In contrast to general relativity, generic action functionals give different results for metric versus Palatini-style variations. We consider some of the resulting issues in Poincar\`e gauge theory, where it is natural to vary both the solder form and the spin connection, and where the action may differ from the Einstein-Hilbert case. Independent variation may introduce nonmetricity as well as torsion. Appending the defining equation of nonmetricity to the Maurer-Cartan equations of the Poincar\`e Lie algebra, we show that the mixed symmetry components of nonmetricity can be absorbed into an altered torsion tensor, restoring local Lorentz symmetry and metric compatibility while including quadratic terms in the redefined curvature. In order to recover the original torsion and nonmetric fields, we replace the definition of nonmetricity by an additional structure equation. We show that the maximal Lie algebra compatible with the enlarged set is isomorphic to the conformal Lie algebra. Therefore, in a Lorentzian conformal geometry the sum of torsion and the field strength of special conformal transformations is given by the mixed symmetry nonmetricity of an equivalent asymmetric system.
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