Revisiting Weak Energy Condition and wormholes in Brans-Dicke gravity

It is known that the formation of a wormhole typically involves a violation of the Weak Energy Condition (WEC), but the reverse is not necessarily true. In the context of Brans-Dicke gravity, the $\textit{generalized}$ Campanelli-Lousto solution, which we shall unveil in this paper, demonstrates a WEC violation that coincides with the appearance of $\textit{unbounded}$ sheets of spacetime within the "interior" section. The emergence of a wormhole in the "exterior" section is thus only an indirect consequence of the WEC violation. Additionally, we use the generalized Campanelli-Lousto solution to construct a Kruskal-Szekeres diagram, which exhibits a "gulf" sandwiched between the four quadrants in the diagram, a novel feature in Brans-Dicke gravity. Overall, our findings shed new light onto a complex interplay between the WEC and wormholes in the Brans-Dicke theory.