On the equivalence of the Einstein-Hilbert and the Einstein-Palatini formulations of general relativity for an arbitrary connection

TL;DR

This paper demonstrates the equivalence of Einstein-Hilbert and Einstein-Palatini formulations of general relativity for any connection, using Hamiltonian methods and without assuming symmetric connections, also relating to Einstein-Cartan theory with fermions.

Contribution

It provides a comprehensive Hamiltonian analysis proving the equivalence of the two formulations without gauge fixing and clarifies the gauge character of the projective transformation.

Findings

Established the equivalence of Einstein-Hilbert and Einstein-Palatini formalisms.

Revealed the gauge nature of the projective transformation.

Connected the formalism to Einstein-Cartan theory with fermionic matter.

Abstract

In the framework of the Einstein-Palatini formalism, even though the projective transformation connecting the arbitrary connection with the Levi Civita connection has been floating in the literature for a long time and perhaps the result was implicitly known in the affine gravity community, yet as far as we know Julia and Silva were the first to realise its gauge character. We rederive this result by using the Rosenfeld-Dirac-Bergmann approach to constrained Hamiltonian systems and do a comprehensive self contained analysis establishing the equivalence of the Einstein-Palatini and the metric formulations without having to impose the gauge choice that the connection is symmetric. We also make contact with the the Einstein-Cartan theory when the matter Lagrangian has fermions.