Simple Gravity: A Nonlinear Scalar Field Approach to Gravitation

This work proposes a nonlinear extension of gravitation based on the hypothesis that the energy of the gravitational field itself acts as an additional source of the field. Starting from the Poisson equation of Newtonian gravity, a term proportional to the square of the gradient of the potential is introduced, properly scaled by 1/c², reflecting the equivalence between mass and energy. The formulation is constructed consistently from a Lagrangian, ensuring the physical structure of the theory. In the weak-field regime, the resulting solution presents a correction to the gravitational potential proportional to 1/r², which induces an additional term in the force of order 1/r³. It is shown that this modification is sufficient to reproduce classical effects in the weak-field regime traditionally attributed to general relativity, such as the perihelion precession of Mercury and the deflection of light in gravitational fields. The theory maintains a simple scalar structure, offering a conceptually simpler alternative description in the weak-field regime.