A Geometric Interpretation of Cumulative Proper Time in General Relativity

This work presents an exploratory and mathematically consistent geometric interpretation of gravitational time dilation within General Relativity. Instead of treating time dilation solely as a local change in clock rates, the paper emphasizes the cumulative nature of proper time as a geometric effect arising from evolving spacetime curvature. The framework is formulated entirely within standard General Relativity, using the ADM formalism and the lapse function to characterize accumulated proper-time displacement along spacetime trajectories. An exact solution of Einstein’s field equations is provided to demonstrate the internal consistency of the approach. Physical examples, including atomic clocks and relativistic timekeeping scenarios, are discussed to illustrate the interpretation. This manuscript does not propose new physical laws, exotic matter, or mechanisms for time travel. Rather, it offers a conceptual reorganization of well-established relativistic results that may provide additional clarity in the understanding of time accumulation and spacetime geometry.