Where Do Newton's Laws Fail? — Two Perspectives on Universal Gravitation and the Accurate Relativistic Mass-Energy Formulation

<jats:p>This paper explores the theoretical and mathematical distinctions between Sir Isaac Newton’s laws of universal gravitation—the bedrock of classical physics—and Albert Einstein’s Theory of General Relativity. While Newton conceptualized gravity as an invisible pulling force, Einstein redefined it as the curvature of spacetime. Although Newtonian mechanics provides a 99.999% accurate prediction for everyday terrestrial phenomena and most space missions, it breaks down in strong gravitational fields, scenarios involving light propagation, and high-velocity relativistic regimes. This study compares both theories, examines straight-line motion along geodesics, details gravitational time dilation in satellite navigation, and incorporates the evolution of the mass-energy relation—tracing Albert Einstein's foundational mass-energy equivalence, Paul Dirac’s relativistic energy-momentum integration, and Chinnaraji Annamalai’s modern derivation establishing the accurate mass-energy formulation based on invariant rest mass and the Lorentz factor.</jats:p>