Variational formalism and field equations of tetrad gravitation theory in the Weyl-Cartan space

A variational formalism of tetrad gravitation theory is developed in the Weyl-Cartan space with independent variations in the tetrad coefficients, metric tensor components, and affine connectivity coefficients that considers the Weyl condition imposed on the nonmetricity based on the method of undetermined Lagrange multipliers. The gravitational field equations are derived for the Lagrangian comprising all possible quadratic convolutions of curvature, torsion, and nonmetricity tensors in addition to the linear component. Differential identities are obtained for the general gravitational Lagrangian in the Weyl-Cartan space. © Springer Science+Business Media, Inc. 2006.