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 ${\mathbb M}_E$ to only a part of the Jordan-frame manifold ${\mathbb M}_J$, which has to be continued beyond the emerging regular boundary S, and the new region maps to another manifold ${\mathbb M}_{E-}$. The metric in ${\mathbb M}_J$ 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 $q=0$