A Pedagogical Revisit of Perihelion Precession, Light Bending, and Stellar Equilibrium Within Curved Space Newtonian Framework

We develop a pedagogical, time-independent, potential-based curved 3D spatial nonprior geometric Newtonian framework under weak-field approximation which provides an analytically transparent description of perihelion precession and light bending in gravitationally curved space, using methods accessible from standard classical mechanics and without invoking tensor calculus. The approach offers an alternative interpretive route to these familiar phenomena, complementing their usual presentation in terms of spacetime curvature in General Relativity. The present framework naturally accommodates additional contributions to the perihelion advance arising from the gravitational interactions between a planet and other celestial bodies beyond the Sun. Upon incorporating gravitationally curved space geometry to the intrinsic stellar equilibrium-where gravitational collapse is counterbalanced by quantum mechanical degeneracy pressure-the proposed framework predicts a characteristic mass scale of approximately 0.68 ⊙ for Newtonian White Dwarfs, associated with a stable radius of about 0.014 ⊙. In the case of Newtonian Neutron Stars, the framework yields an approximate characteristic mass of 1.75 ⊙ , corresponding to a stable radius of 10.7km, illustrating the utility of the approach in astrophysical contexts. Our study further motivates the incorporation of all relevant interaction-potentials, gravitational and non-gravitational, into the line element of a 4D curved spacetime, thereby enabling a generalized description of wavelength-shifts-including gravitational redshift as a special case, and a formulation of spacetime compactness scale that reduces to the Schwarzschild radius in the weak-field limit.