A formalism of Gravitation based on a Physical Field Strength

TL;DR

The paper reframes gravity as a genuine gauge force, introducing the tensor $K^{\mu}{}_{\alpha\beta} = \Gamma^{\mu}{}_{\a\u0061\u03b3\u0062} - \widehat{\Gamma}^{\mu}{}_{\alpha\beta}$ as the coordinate-invariant gravitational field strength and extending infinitesimal translations to curved spacetime via vector fields $\xi_a{}^{\mu}$ to yield a gauge-invariant curvature $\mathfrak{F}^{ξa}{}_{\mu\nu}$. It demonstrates that gravity can be cast as a Yang–Mills–type theory with an $SU(2)\times U(1)$ gauge group, supplemented by a scalar $φ^{2}$ representing the Newtonian potential, such that $\mathfrak{g}\to0$ recovers General Relativity while $\mathfrak{g}\neq0$ yields a gauge-theoretic gravity with potential dark-energy and dark-matter phenomenology. The framework provides a natural mechanism for dark energy via the self-energy of $φ^{2}$ and explains dark-matter–like effects through the extended gravitational degrees of freedom, all while remaining compatible with GR in the weak-field limit. It also lays out a path toward quantization via standard gauge-theory techniques, suggesting a promising route to a quantum theory of gravity grounded in a covariant, Yang–Mills–type action.

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 $\mathrm{K}^μ_{αβ}$, 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 $ξ_a{}^μ$ that extend the notion of infinitesimal translations to curved spacetime and naturally yield a gauge-invariant field strength $\mathfrak{F}^{ξa}{}_{μν}$. The dynamics of the gravitational field are governed by a Lagrangian of Yang--Mills type with an additional scalar degree of freedom $φ^{2}$, 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 $φ$ 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.