Simple Gravity: A Scalar Field Formulation of Light Propagation in the Weak-Field Regime

This work develops a scalar formulation of gravity focused on the propagation of light in the weak-field regime. The framework is based on an exponential response of local temporal and spatial intervals to a gravitational potential Φ. This structure naturally leads to an effective refractive index n (Φ) = e^-2Φ/c², allowing gravitational effects on light to be described within an optical framework while preserving local invariance of the speed of light. From this formulation, gravitational redshift, light deflection, and Shapiro time delay are derived in agreement with standard weak-field results. An exact description of light trajectories is also obtained, reinforcing the internal consistency of the approach. The results support an interpretation of gravity as a scalar field modifying the properties of an effective medium, providing an alternative to geometric descriptions based on spacetime curvature. This work is part of an ongoing series developing the Simple Gravity framework.