Electromagnetic quasinormal modes of a Schwarzschild black hole in a superconducting plasma

Abstract

We formulate a generally covariant extension of London electrodynamics in four-dimensional curved spacetime and investigate the propagation of electromagnetic perturbations in a superconducting medium surrounding a black hole. The ordinary London relation is promoted to a gauge-invariant four-dimensional constitutive equation involving the electromagnetic four-potential and the phase of the superconducting condensate. When combined with the covariant Maxwell equations, the resulting system is equivalent, in the London (unitary) gauge, to a Proca theory with an effective photon mass set by the inverse London penetration depth. We derive the corresponding Schrödinger-like wave equations for electromagnetic perturbations in the axial and polar sectors of the Schwarzschild spacetime in the presence of the superconducting plasma. The axial sector is governed by a single Regge--Wheeler-type equation whose effective potential contains the London mass term, whereas the polar sector contains two coupled physical degrees of freedom. Quasinormal-mode boundary conditions are formulated for the Schwarzschild geometry. The London penetration depth shifts the oscillation frequencies, modifies the damping rates, and can generate long-lived quasibound configurations when the effective field mass is sufficiently large. This formalism provides a framework for studying electromagnetic relaxation in curved superconducting media, compact objects containing superconducting phases, and effective massive-vector fields around black holes.