One of the most fascinating features of Einstein’s theory of General Relativity (GR) consists in the fact that spacetime may be curved and topologically nontrivial, describing intriguing objects like black holes and wormholes, that is, spacetime structures that link together two separate universes. In contrast to black holes, the occurrence of wormholes is speculative, and so far, there is no observational evidence for the existence of such structures; however, their existence is in accordance with GR, provided the introduction of “exotic” matter to support the throat, that is, matter whose stress-energy-momentum tensor violates the (averaged) null energy condition. One possibility to obtain such a behaviour is to consider phantom scalar fields, i.e., scalar fields that have a negative kinetic energy. At this point, a pressing question regarding the relevance of static wormholes in GR concerns their dynamical stability under small perturbations. In the first part of my talk, I will retrace some fundamental assumptions of GR and of the scalar field theory, reintroducing the concepts of spacetime, observer, coordinate system, metric and (phantom) scalar fields; I will also provide a mathematical rigorous definition of wormhole spacetime metric and present some known wormhole models with spherical symmetry. The second part of my talk is devoted to the linear stability analysis of wormhole configurations: I will show how it is possible to reduce this study to the spectral analysis of a Schrödinger-type operator (selfadjoint, in a suitably defined Hilbertian framework) which appears in a master equation describing small perturbations of the wormhole in any coordinate system. This general method is then applied to some wormhole families, showing their linear instability.
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