The transition between special and general relativity reveals a conceptual gap, particularly when dealing with inertial frames and accelerated observers. However, many of the mathematical tools of differential geometry - typically associated with general relativity - can already be applied within the framework of special relativity to provide a complete description of gravity-free physics.
The first part of this bachelor’s thesis aims to explore and clarify the transition between both theories of relativity, offering insights that may also serve didactic purposes. Central to this endeavor is the replacement of the loosely defined concept of an inertial frame of reference with a rigorous definition of pseudo-orthonormal bases. Building on this foundation, the structure of Minkowski space and passive Lorentz transformations are examined through their relationships to these bases.
The formalism is then extended within the second part of this thesis to include more general, non-inertial coordinate systems. A rigorous formulation of active Lorentz transformations is developed, grounded in the properties established for passive transformations. Since active transformations correspond to actual physical changes in a system, this approach enables the generalization and relativistic derivation of Newton’s second law. This framework allows to easily distinguish between real and fictitious forces and remains even applicable in the Newtonian limit.
