A formalism of Gravitation based on a Physical Field Strength

TL;DR

This paper introduces a gauge-theoretic formalism of gravity treating it as a physical force with true degrees of freedom, unifying classical and quantum perspectives and explaining dark energy and dark matter effects.

Contribution

It presents a novel gauge-invariant formalism of gravity based on a tensor field strength, extending the notion of translations and incorporating a scalar field for dark energy and matter effects.

Findings

Gravity is described as a genuine force with physical degrees of freedom.

The formalism reduces to General Relativity when gravitational coupling vanishes.

Dark energy and dark matter effects are explained through the extended gauge degrees of freedom.

Abstract

We propose a reformulation of gravitation in which the gravitational interaction is treated as a genuine force rather than an inertial effect arising from spacetime geometry. Within this framework, the difference between the affine connection and a flat reference connection defines a tensor $\mathrm{K}^\mu{}_{\alpha\beta}$, identified as the gravitational field strength. This object cannot be eliminated by coordinate transformations, demonstrating that gravity possesses true physical degrees of freedom. The formalism introduces vector fields $\xi_a{}^\mu$ that extend the notion of infinitesimal translations to curved spacetime and naturally yield a gauge-invariant field strength $\mathfrak{F}^{\xi a}{}_{\mu\nu}$. The dynamics of the gravitational field are governed by a Lagrangian of Yang--Mills type with an additional scalar degree of freedom $\phi^{2}$, corresponding to the Newtonian potential. In the limit of vanishing gravitational coupling $\mathfrak{g}\to0$, the theory reduces to General Relativity, while for nonzero $\mathfrak{g}$ it constitutes an $\mathrm{SU(2)\times U(1)}$ gauge theory of gravity. The framework provides a unified description in which dark energy emerges as the self-interaction energy of the $\phi$ field, and dark-matter-like effects arise from the extended gravitational degrees of freedom. This formulation offers a consistent bridge between classical and quantum descriptions of gravity and clarifies the conceptual foundations of the gravitational interaction.