Covariant Gravitational Field Equations of Self-Variation Theory

We develop the gravitational sector of Self-Variation Theory (SVT), a covariant framework in which gravitational dynamics emerge from the self-variation of intrinsic physical quantities and their associated spacetime equilibration structure. Starting from the SVT gravitational potential, we derive the corresponding field intensity, propagation properties, and an effective static spherically symmetric spacetime geometry. In the infinite-range regime, the theory reproduces the standard weak-field parametrized post-Newtonian limit with γ=1 and β=1, yielding agreement with General Relativity in classical tests such as perihelion precession, light deflection, and Shapiro time delay. We further investigate the large-scale gravitational behavior predicted by the theory. For small negative values of the dimensionless parameter k, the SVT field generates extended quasi-flat galactic rotation curves without requiring a dark matter halo. The rotational dynamics are governed by a single emergent length scale Q, which is determined by the baryonic mass and characteristic velocity scale of the system. The onset and extent of the flat rotation regime occur at a radius satisfying Rmax=1.2Q, providing a direct mass–velocity–radius relation for disk galaxies. These results suggest that Self-Variation Theory provides a covariant geometric framework capable of unifying local relativistic gravitational phenomena with large-scale galactic dynamics through a single emergent structural scale.