An inductive approach to $f(R)$ gravity and its application to extended sources

In a previous work (Phys. Rev. D 110, 064042), a new family of $f(R)$ gravitational Lagrangians was constructed inductively using only Solar System physics. Specifically, the modified Einstein equations were solved perturbatively on a Schwarzschild background, and the corresponding Lagrangian was reconstructed a posteriori from the solution. Classical Solar System tests were then employed to constrain the fundamental length scale quantifying the deviation from general relativity. In the first part of this work, the construction of the model is critically reviewed, emphasizing the generality of the approach and the applicability to different modified gravity frameworks. In the second part, the modified Newtonian potential produced by an extended source is derived within the new inductive $f(R)$ model, discussing the consistency of the Newtonian limit. In this context, the application to galactic dynamics emerges as an intriguing perspective.