Affine theory of gravitation

Abstract

We propose a new theory of gravitation, in which the affine connection is the only dynamical variable describing the gravitational field. We construct the simplest dynamical Lagrangian density that is entirely composed from the connection, via its curvature and torsion, and is an algebraic function of its derivatives. It is given by the contraction of the Ricci tensor with a tensor which is inverse to the symmetric, contracted square of the torsion tensor, $k_{\mu\nu}=S^\rho_{\lambda\mu}S^\lambda_{\rho\nu}$. We vary the total action for the gravitational field and matter with respect to the affine connection, assuming that the matter fields couple to the connection only through $k_{\mu\nu}$. We derive the resulting field equations and show that they are identical with the Einstein equations of general relativity with a nonzero cosmological constant, if the tensor $k_{\mu\nu}$ is regarded as the metric tensor. The cosmological constant is simply a constant of proportionality between the two tensors, which together with $c$ and $G$ provides a natural system of units in gravitational physics. This theory therefore provides a physically valid construction of the metric as an algebraic function of the connection, and naturally explains dark energy as an intrinsic property of spacetime. The observed accelerating expansion of the Universe may be the strongest evidence for torsion.