Electromagnetic Energy Invariance via Time-Phase Separation of Maxwell's Equations

In classical treatments of electromagnetic wave propagation, combining Faraday's and Amp`ere–Maxwell's equations typically yields an oscillatory instantaneous Poynting vector, even though the time-averaged energy flux remains constant. Conventional tensor and complex-formalism approaches resolve this issue using phase considerations implicitly. This work proposes an alternative interpretation that introduces an explicit time-phase difference of π/2 between Faraday's and Amp`ere–Maxwell's equations, yielding a combined field representation in which the energy density remains invariant at all times during propa- gation. The formulation reproduces electromagnetic invariants within a real (non-complex) framework and emphasizes energy consistency under both passive Lorentz transformations (moving observers) and active transformations (moving charges). This approach offers a conceptually accessible complement to tensor and complex-vector methods, enabling a broader understanding of classical electromagnetism without replacing existing formalism. Keywords: Electromagnetism; Riemann–Silberstein vector; signed-Euclidean geometry; Lorentz invariance; electromagnetic invariance; charge radiation.