The dynamical structure of gravitational theories with GL(4,R) connections

Abstract

We investigate here GL (4,R)-gauge theories of gravity based on variational principles. The components of tetrad fields eμ(α), the components of metrics g(α)(β), and the components of connections Γλ(α)(β) are taken as the gravitational potentials. Matter potentials are the components of GL (4,R)-tensor fields φΣ. We derive the conservation laws for a general theory, that is, the Belinfante–Rosenfeld and Bianchi identities, and find minimal systems of independent variational equations. The natural GL (4,R)-covariant Hamiltonian formulation of the theory induces a GL (3,R)-covariant Hamiltonian formulation related to a chosen slicing of space-time. The Hamiltonian field equations corresponding to this formulation describe the dynamics of the system. We determine 20 symplectic constraints, 20 gauge transformations, and 20 gauge variables generic for a general gravitational Lagrangian. As an example, we consider the Gl (4,R)–Einstein theory in vacuum as well as in the presence of a vector field and find the complete canonical formulation in both cases.