Affine theory of gravitation

Research Article

Abstract

We propose a theory of gravitation, in which the affine connection is the only dynamical variable describing the gravitational field. We construct a simple dynamical Lagrangian density that is entirely composed from the connection, via its curvature and torsion, and is a polynomial 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 proportional to 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 physical construction of the metric as a polynomial function of the connection, and explains dark energy as an intrinsic property of spacetime.

Keywords

Affine connection Torsion tensor Cosmological constant  Dark energy 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of New HavenWest HavenUSA

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