Hydrodynamic Analog of the Klein Paradox: Vacuum Instability and Pair Production in a Linear Elastic Medium

The Klein Paradox—the anomalous scattering of relativistic fermions off a high potential step—signals the limit of the single-particle interpretation of the Dirac equation. While Quantum Field Theory resolves this via pair production, the microscopic mechanism is often obscured by abstract formalism. In this work, we investigate this phenomenon through the framework of Analog Gravity and Condensed Matter Physics. We utilize a hydrodynamic model wherein a relativistic particle is treated as a localized elastic excitation (defect) within a continuous linear medium. We demonstrate that when the external stress (potential) exceeds the medium’s binding energy threshold (V > 2mc²), the system undergoes a mechanical instability analogous to dielectric breakdown. This instability naturally generates modes with inverted topological winding, which we identify as antiparticles. By solving the boundary conditions for this elastic system, we reproduce the transmission coefficients of Hansen and Ravndal and recover the Schwinger limit for pair production rates. This approach provides a clear pedagogical model based on continuum mechanics to visualize vacuum decay processes, suggesting that the ”paradox” is simply the elastic response of a medium under supercritical stress.