Gauge-invariant spherical linear perturbations of wormholes in Einstein gravity minimally coupled to a self-interacting phantom scalar field

Recently, there has been quite a lot of interest in static, spherical wormhole spacetimes and the question of their stability with respect to time-dependent perturbations. The consideration of linearized perturbations usually leads to a master wave equation with effective potential, which can then be analyzed using standard tools from quantum mechanics. However, in the wormhole case, particular care must be taken with the gauge conditions when formulating the master equation. A poor coordinate choice, based for example on fixing the areal radial coordinate, may lead to singularities at the throat, which complicate the stability analysis, or might even lead to erroneous conclusions regarding the stability of the underlying wormhole configuration. In this work, we present a general method for deriving a gauge-invariant wave system of linearized perturbation equations in the spherically symmetric case, assuming that the matter supporting the wormhole is a phantom scalar field—that is, a self-interacting scalar field whose kinetic energy has the reversed sign. We show how this system can be decoupled and reduced to a single master wave equation with a regular potential, with no intermediate steps involving singularities at the throat. Two applications of our formalism are given. First, we rederive the master equation for the linearly perturbed Ellis-Bronnikov wormhole using our new, singularity-free method. As a second application, we derive the master equation describing the linear perturbations of a certain anti–de Sitter wormhole, provide a detailed analysis of the spectral properties of the underlying operator, and prove that, as in the Ellis-Bronnikov case, this wormhole is linearly unstable and possesses a single unstable mode. In the final part of the paper, we consider a wormhole with de Sitter–type ends, whose spacetime presents horizons and admits a nonstatic extension beyond them; for this system, we derive partial results of linear instability.