Emergent Newtonian Gravity from Scalar Potential Geometry Without a Fundamental Coupling Constant

The gravitational constant G is traditionally treated as a fundamental parameter in both Newtonian gravity and General Relativity, yet its physical origin and numerical value are not derivable from deeper dynamical principles. In this work, we present an alternative geometrical derivation of gravitational acceleration in which inverse–square behavior arises from spatial gradients of a scalar potential field associated with local time–rate structure, without introducing a fundamental coupling constant. Gravitational acceleration emerges directly as a = −∇Θ, where Θ(x) represents a scalar temporal potential whose spatial curvature governs motion. Under spherical symmetry and weak–gradient conditions, solutions to Laplace’s equation yield inverse–square scaling as a purely geometric consequence of field propagation. Within this formulation, the quantity historically written as GM appears only as an effective scaling parameter associated with source normalization and unit calibration, not as a fundamental interaction strength. This approach provides a conceptually transparent pathway for introducing gravitational geometry prior to tensor curvature formalism and offers pedagogical continuity between Newtonian and relativistic interpretations of gravity.