Yang Mills Theory of Gravity

TL;DR

This paper reformulates general relativity in a self-dual formalism akin to Yang-Mills theory, using spin currents and space-time decomposition to derive equations of motion without Einstein's equations.

Contribution

It introduces a novel approach to solve Einstein's equations using spin currents in a self-dual formalism, connecting gravity with gauge theories.

Findings

Derived equations of motion using spin currents

Identified conjugate momentum and energy density in the formalism

Extended the framework to include Yang-Mills Lagrangian

Abstract

The canonical formulation of general relativity is based on decomposition space--time manifold $M$ into $ R\times \Sigma$, this decomposition has to preserve the invariance of general relativity, invariance under general coordinates, and local Lorentz transformations. These symmetries associate with conserved currents that are coupled to gravity. In this paper, we try to solve the equations of motion of general relativity in self-dual formalism using only the spin currents(Lorentz currents), in static case, and without needing using the Einstein's equation, that makes the general relativity similar to Yang-Mills theory of gauge fields. We give an example, matter located at a point, so we have spherical symmetric system. Then we add Yang--Mills Lagrangian to general relativity Lagrangian. Finally we use the decomposition of the space--time manifold $M=R\times \Sigma$ to find that $\Sigma^{0a}_i$ is a conjugate momentum of $A^{i}_a$ and $\Sigma^{ab}_i F_{ab}^i$ is energy density.