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Irreducible decompositions of nonmetricity, torsion, curvature and Bianchi identities in metric-affine spacetimes

The irreducible components of the curvature under the Lorentz group are of direct physical relevance in the four-dimensional Riemannian geometry of general relativity. The same is true for both curvature and torsion in the four-dimensional Riemann-Cartan geometry of the Poincare gauge theory of gravitation. In the latter theory a knowledge of these irreducible components is also extremely useful when setting up the Lagrangian and searching for exact solutions. The author deals with an n-dimensional metric-affine spacetime of arbitrary signature. In such a spacetime the connection is no longer metric so that there is an additional geometric object-the nonmetricity. The irreducible decompositions of nonmetricity, torsion and curvature under the pseudo-orthogonal group as well as those of the corresponding Bianchi identities are derived. Because of the increasing use made of it in the literature, the exterior form notation is used throughout.