From Modified Newtonian Gravity to a Maxwell-Type Gravitational Interaction

This paper continues a classical-mechanical approach in which a modified Newtonian gravitational model was previously developed within a four-dimensional geometrical framework. In that construction, classical kinetic energy leads naturally to a Minkowski metric and an action-based formulation of the equations of motion. Consequently, the present paper extends this model, relating the exponential modification of the central Newtonian field to the factor \((1-v^{2}/c^{2})\). The resulting modified central force \(\mathbf{F}_{Ne}\) provides the retrograde contribution to the observed perihelion advance. A complementary velocity-dependent gravitomagnetic interaction \(\mathbf{F}_{gm}\) is derived from the conditions: \[ \hat{r}^{2}d_{t}\varphi = r^{2}e^{2\varrho/r}d_{t}\varphi = \mathrm{const.} \quad \text{and} \quad \mathbf{F}_{gm} \cdot \mathbf{v} = 0. \] Together, these interactions reproduce the observed perihelion advance. The corresponding scalar and vector potentials yield a Maxwell-type gravitational formulation, with the gravitomagnetic interaction incorporated into the action through a term linear in velocity. The total gravitational interaction is then examined in the limiting case of a light ray. The gravitostatic contribution vanishes as \(v \rightarrow c\), while the gravitomagnetic interaction \(\mathbf{F}_{gm}\) gives the deflection angle \(\Delta\theta = 4GM/(bc^{2})\), corresponding to approximately \(1.75\) arcsec for a grazing ray. Thereafter, a classical-mechanical interpretation of the Michelson--Morley null result is also obtained: the round-trip propagation times along the parallel and transverse arms are both \(2h/c\). These results extend the model without assuming the invariance of the speed of light as an independent postulate.