Geometric Series of the Lorentz Factor in Special Relativity: Algebraic Structure and Energy-Momentum Decomposition

This paper presents a systematic algebraic exploration of the Lorentz factor raised to the second power within the framework of special relativity. By expressing the squared Lorentz factor as an infinite geometric series, this study examines the resulting term-by-term expansions for the total relativistic energy squared and the relativistic momentum-energy term. Through explicit mathematical derivations, this paper demonstrates how multiplying the infinite geometric series expansion of the squared Lorentz factor by mass and velocity parameters yields an exact series representation of Einstein's energy-momentum invariant relation, a foundational relation that famously served as the classical springboard for Paul Dirac's formulation of the relativistic wave equation. It discusses non-relativistic limits, higher-order corrections, and the divergence behavior as particle speeds approach the speed of light.