Coherent multipath wave response on Reissner-Nordstr\"{o}m analogue surface

Abstract

To uncover how the intrinsic metric of a relativistic compact object governs macroscopic wave phenomena, we establish a theoretical framework mapping the charge dependent spatial geometry of a Reissner-Nordstr\"{o}m (RN) black hole onto the coherent response of an analogue curved surface. By solving an exact spatial geodesic boundary value problem on an isometrically embedded equatorial slice, we extract the discrete multi-loop path-length spectrum and convert this geometric backbone into a physical wave field via a finite-path surface Huygens-Fresnel construction. We analytically compute the arbitrary order winding trajectories alongside their high winding accumulation limits, demonstrating that the analogue charge acts as a precise physical dial that reconfigures the event horizon throat and fundamentally reorganizes the discrete path sequence. Furthermore, we find that this underlying geometric deformation uniquely dictates the macroscopic interference, revealing that steady state spatial fringes, spectral resonance combs, and transient temporal echo ladders are intrinsically unified physical projections of a single charge controlled path spectrum. This systematic parameter to response methodology establishes a rigorous theoretical bridge between strong field gravitational lensing and tabletop transformation optics, providing a highly tunable blueprint for future multi domain analogue gravity experiments.