Mecânica newtoniana, equações de Maxwell e as transformações de Lorentz

Abstract

This work develops a mathematical and conceptual analysis of the transition from Newtonian Mechanics to Special Relativity, emphasising the formal structures underpinning each theory and the reasons that make the relativistic reformulation of space and time necessary. Aimed at an audience with a mathematical background, it seeks not only to present the physical results, but also to illuminate the algebraic and geometric structures that enable this theoretical reformulation. In the first part, Classical Mechanics is presented in the light of vector formalism and analytic geometry, discussing the concepts of absolute space, absolute time, laws of motion, reference frames, and conserved quantities. Subsequently, Electromagnetism is introduced through Maxwell’s equations, the interpretation of light as an electromagnetic wave, and the description of electric and magnetic fields. This analysis highlights the incompatibility between Galilean relativity and the invariance of Maxwell’s equations, a conflict empirically reinforced by the Michelson–Morley experiment. From this conceptual crisis, the work proceeds to the second part, devoted to the formulation of Special Relativity. Einstein’s postulates, Lorentz transformations, and the structure of space-time are introduced, presenting fundamental relativistic quantities such as proper time, four-vectors, the electromagnetic tensor, and the Minkowski metric. It is shown how the classical concepts of momentum, energy, and force are reformulated within the new theoretical framework, as well as the covariant form of Maxwell’s equations, which acquire elegance and geometric coherence within the tensor formalism. Finally, some applications and consequences of modern physics are discussed, highlighting phenomena and technologies whose understanding depends directly on Relativity, such as global positioning systems (GPS), high-energy processes in particle physics, properties of electromagnetic fields and waves, and the foundations of contemporary astrophysics. Thus, the work demonstrates that Special Relativity does not replace Classical Physics, but rather generalises it in a consistent manner, preserving it as a limiting case and integrating mechanical and electromagnetic descriptions within a single conceptual and mathematical framework. In conclusion, this work emphasises that Special Relativity emerged not merely as a theoretical correction, but as a natural consequence of a deeper mathematical conception. By presenting the concepts and derivations with formal precision, it is hoped that this work contributes to a clearer mathematical understanding of this conceptual leap.