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

TL;DR

This paper develops a covariant gauge theory of gravity that unifies the first-order (Palatini) and second-order (metric) formalisms, establishing conditions for metric compatibility through Hamiltonian dynamics.

Contribution

It introduces a gauge-theoretic framework linking the Palatini and metric formalisms of gravity via conjugate momenta and Hamiltonian formalism, clarifying their correspondence.

Findings

Establishes a covariant gauge theory of gravity without torsion.

Shows conditions under which the metric formalism emerges from the gauge theory.

Demonstrates the role of Hamiltonian independence in metric compatibility.

Abstract

This is a shortened version of an invited talk at the XIII International Workshop "Lie Theory and its Applications in Physics", June 17-23, Varna, Bulgaria. 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. By the metric conjugate momenta, a correspondence between the Affine-Palatini formalism and the metric formalism is established. For, 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 equations of motion correspond to the solution is the metric formalism.