A new theoretical framework has been developed to investigate first-order electromagnetic (EM) perturbations induced by gravitational waves (GWs). Starting from the covariant Maxwell equations, researchers have derived the complete first-order perturbation equations for both the EM field tensor and the four-potential, demonstrating their equivalence and the residual gauge invariance under the Lorenz gauge condition. This work provides a systematic basis for understanding how GWs can interact with and modify existing electromagnetic fields in the universe.

The study obtained explicit first-order expressions for the induced electric and magnetic fields, as well as for the associated EM energy-momentum tensor. As an illustration, the interaction between a plane EM wave and a GW in the transverse-traceless gauge was analytically evaluated. This analysis is crucial for quantifying the effects of GWs on electromagnetic phenomena, which could have implications for the detection and characterization of these cosmic waves.

The results demonstrate that the maximum modulus of the coupling coefficient is on the order of $10^2$. Quantitatively, this means that a typical astrophysical gravitational wave with a dimensionless strain of $h_0 \sim 10^{-21}$ generates a first-order electromagnetic response on the order of $10^{-19}$ relative to the amplitude of the incident field. This finding establishes a concrete magnitude for the interaction, which could guide future experiments or the interpretation of astrophysical observations where both GWs and EM fields are present. The ability to predict these perturbations is an important step towards a complete understanding of the interconnectedness between gravity and electromagnetism.