A formalism of Gravitation based on a Physical Field Strength

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.