Gauge theory of Gravity based on the correspondence between the $1^{st}$ and the $2^{nd}$ order formalisms

David Benisty, Eduardo I. Guendelman

Published 2018-08-06Version 1

A covariant canonical gauge theory of gravity free from torsion is studied. Using a metric conjugate momentum and a connection conjugate momentum, which takes the form of the Riemann tensor, a gauge theory of gravity is formulated, with form-invariant Hamiltonian. Through the introduction of the metric conjugate momenta, a correspondence between the Affine-Palatini formalism and the metric formalism is established, since when the dynamical gravitational Hamiltonian $\tilde{H}_{Dyn}$ does not depend on the metric conjugate momenta, a metric compatibility is obtained from the equation of motions and the energy momentum is covariant conserved. When the gravitational Hamiltonian $\tilde{H}_{Dyn}$ is depend on the metric conjugate momentum, an extension to the metric compatibility comes from the equation of motions and the energy momentum covariant conservation violates. For a sample of the $\tilde{H}_{Dyn}$ which consists a quadratic term of the connection conjugate momentum, the effective Lagrangian has the Einstein Hilbert term with a quadratic Riemann term in the second order formalism. A bouncing inflation, in the context of cosmological solutions of this action is briefly discussed.