First-Order Perturbations of Covariant Maxwell Equations in Gravitational Waves

We present a systematic theoretical framework for investigating first-order electromagnetic (EM) perturbations induced by gravitational waves (GWs). Beginning with the covariant Maxwell equations, we derive the complete first-order perturbation equations in terms of both the EM field tensor and the four-potential, demonstrating their equivalence alongside the residual gauge invariance under the Lorenz gauge condition.