On the stability of exceptional Brans–Dicke wormholes

Abstract 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 $$\omega =0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> 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 $${\mathbb {M}}_\textrm{E}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mtext>E</mml:mtext> </mml:msub> </mml:math> to only a part of the Jordan-frame manifold $${\mathbb {M}}_\textrm{J}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mtext>J</mml:mtext> </mml:msub> </mml:math> , which has to be continued beyond the emerging regular boundary S, and the new region maps to another manifold $${\mathbb {M}}_\textrm{E}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mtext>E</mml:mtext> </mml:msub> </mml:math> $${}_{-}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mo>-</mml:mo> <mml:mrow/> </mml:mmultiscripts> </mml:math> . The metric in $${\mathbb {M}}_\textrm{J}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mtext>J</mml:mtext> </mml:msub> </mml:math> 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 $$q \ne 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and unstable only in the symmetric case $$q=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> .