The RVT Metric: Solving Galaxy Rotation Curves with a Covariant Vector-Tensor Geometry Without Dark Matter

This scientific preprint introduces the RVT (Space-Vectorial Theory) Metric, a novel and fully covariant vector-tensor gravity theory that extends General Relativity (GR). This work resolves the decades-old galaxy rotation curve problem by mathematically eliminating the need for Dark Matter (DM). Core Theoretical Foundation:The theory originates from the insight that conventional GR weak-field solutions, which rely on 180-degree-symmetry, are incomplete. They fail to account for the 90/270-degree-projection divergence inherent to the observer's frame. To formalize this geometric distortion in a relativistically consistent manner, RVT introduces a **geometric vector field n_mu(x)** into the Lagrangian that couples directly to the spacetime curvature. The derived **RVT Field Equations (G_mu_nu + alpha*E_mu_nu = 8pi G T_mu_nu)** are shown to be fully consistent with all Post-Newtonian (PPN) constraints, successfully reproducing the classical GR results in the Solar System limit. The Elimination of Dark Matter:Crucially, when applied to galactic scales, the solution for the RVT field equations yields a modified Poisson equation featuring an **explicit source term S proportional to alpha/r**. This term mathematically generates the **required constant asymptotic acceleration (v_flat^2)**. This finding establishes a first-principles geometric explanation for the observed flat rotation curves, making Dark Matter unnecessary and offering a simplified, verifiable framework for cosmology.