  The detection method of gravitational waves (GW) using electromagnetic (EM)
cavities has garnered significant attention in recent years. This paper
thoroughly examines the analysis for the perturbation of the EM field and
raises some issues in the existing literature. Our work demonstrates that the
rigid condition imposed on the material, as provided in the literature, is
inappropriate due to its reliance on a gauge-dependent quantity that cannot be
controlled experimentally. Instead, we incorporate elasticity into the material
and revise the governing equations for the electric field induced by GWs,
expressing them solely in terms of gauge-invariant quantities. Applying these
equations to a cylindrical cavity with the TM010 mode, we present the GW
antenna pattern for the detector.
