A dynamical approach to General Relativity based on proper time

Abstract: This work places the invariant $ds^2$ at the center of the gravitational interaction, interpreting it not as a purely geometric object but as the differential of proper time, endowed with direct physical meaning. Starting from the extension of Fermat's principle to massive particles--namely, the requirement that freely falling bodies follow trajectories that extremize proper time, which for timelike motion corresponds to a local maximum--and invoking the universality of Galilean free fall, we derive the form of $ds^2$ in a static gravitational field. Lorentz invariance then provides the natural framework to extend this result to systems involving moving matter. The invariant derived through this procedure matches the weak-field limit of General Relativity formulated in the harmonic gauge. Within this linearized regime, we show that the structure of the theory already contains the seeds of its non-linear completion: any dynamically consistent extension to strong gravitational fields necessarily involves the Ricci tensor. From this viewpoint, Einstein's field equations appear not as a postulated geometric law, but as the unique covariant closure required to ensure energy momentum conservation and the self consistency of the gravitational interaction.