A Rigorous Comparative Study of Galilean Relativity and Newtonian Relativity: Mathematical Foundations, Physical Interpretations, and the Transition to Modern Physics

This research paper, titled "A Rigorous Comparative Study of Galilean Relativity and Newtonian Relativity: Mathematical Foundations, Physical Interpretations, and the Transition to Modern Physics," is written by Mirza Adnan Mohtashim and dated May 8, 2026. It offers a detailed theoretical examination of the classical framework of Galilean and Newtonian relativity alongside Einstein's special relativity, exploring how the two approaches differ and connect. The work traces the historical development from Galileo's principle of relativity and Newton's laws through the crisis caused by Maxwell's electromagnetism and the Michelson-Morley experiment, leading to the emergence of the Lorentz transformations. It systematically derives both sets of transformation equations, compares their predictions on concepts such as simultaneity, time, space, velocity addition, length contraction, causality, and spacetime geometry, and demonstrates how Galilean relativity naturally arises as the low-velocity approximation of the relativistic framework. The paper also discusses group-theoretic structures, physical interpretations, experimental confirmations, and the broader implications for general relativity, quantum field theory, and our understanding of the universe. Overall, it presents a clear narrative of the profound paradigm shift from absolute space and time to the unified Minkowski spacetime while highlighting the continued practical validity of classical mechanics at everyday speeds.