Bowl Geometry and the Geodesic Structure of Gravitational Propagation

This work presents a geometric interpretation of gravitational propagation based on an effective “bowl” geometry. In this framework, gravity is not treated as a force acting in a fixed background space, but as a flow following extremal paths (geodesics) in an effective geometry derived from a time-depth model. At small scales, propagation appears isotropic and reproduces the inverse-square law. At larger scales, however, the geometry induces alignment of propagation paths (channeling), reducing transverse spreading. This leads to a transition in the effective cross section from disk-like to ring-like behavior, modifying the scaling of gravitational strength from 1/r² to 1/r. The model introduces a natural transition scale r₀, which scales with mass as r₀ ∼ √M, providing a geometric explanation for deviations from standard inverse-square behavior without introducing additional mass components. This paper focuses on the geometric foundation of the model.