Revisiting Maxwell's Equations for Observers: Direct Derivation of Doppler and Sagnac Effect

Maxwell's equations accurately describe the propagation of electromagnetic (EM) waves. However, the conventional formulation implies invariance to observers, posing challenges in explaining how the wave is perceived by different observers. For example, Doppler effect shows that observers perceive the same EM wave differently. By employing mathematical transformations, we derive a general form of Maxwell’s wave equations by incorporating a "time scaling" factor to account for observer perception. The original form is shown to be a special case when static. The Doppler effect and Sagnac effect are directly derived from Maxwell’s equations. All the results are consistent with established experiments. Our findings offer a fresh perspective, promising a deeper understanding and unification of EM phenomena. We extend Maxwell’s equations to describe not only the propagation of an EM wave, but also how it will be measured by different observers.