Lorentz Transformations and the Interval Invariance Requirement

Abstract

The Lorentz transformations (LT) are conventionally derived from the requirement of interval invariance in Minkowski spacetime. However, this requirement alone is insufficient to uniquely determine the LT. Through straightforward matrix analysis, we demonstrate that an infinite family of transformations — including real, complex, and special Galilean forms — also leave the interval invariant. The physical and mathematical meaning of these transformations is clarified by distinguishing between alibi (active) and alias (passive) interpretations, corresponding respectively to Galilean translations in 3D space and rotations in 4D spacetime. We show that the LT emerge as a special case when the time coordinate is formally taken as imaginary, which simplifies the description of Maxwell's equations but does not impose a fundamental speed limit. Thus, the LT should be understood as a convenient alias transformation projecting 3D Galilean motion into a 4D formalism, rather than as a unique consequence of interval invariance.