On the stability of exceptional Brans–Dicke wormholes

In our previous papers we have analyzed the stability of vacuum and electrovacuum static, spherically symmetric space-times in the framework of the Bergmann–Wagoner–Nordtvedt class of scalar-tensor theories (STT) of gravity. In the present paper, we continue this study by examining the stability of exceptional solutions of the Brans–Dicke theory with the coupling constant \(\omega =0\) that were not covered in the previous studies. Such solutions describe neutral or charged wormholes and involve a conformal continuation: the standard conformal transformation maps the whole Einstein-frame manifold \({\mathbb {M}}_\textrm{E}\) to only a part of the Jordan-frame manifold \({\mathbb {M}}_\textrm{J}\), which has to be continued beyond the emerging regular boundary S, and the new region maps to another manifold \({\mathbb {M}}_\textrm{E}\) \({}_{-}\). The metric in \({\mathbb {M}}_\textrm{J}\) is symmetric with respect to S only if the charge q is zero. Our stability study concerns radial (monopole) perturbations, and it is shown that the wormhole is stable if \(q \ne 0\) and unstable only in the symmetric case \(q=0\).