A Dynamical Approach to General Relativity Based on Proper Time

This work places the invariant ds2 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 ds2 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 nonlinear 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.