New insights into Weak Energy Condition and wormholes in Brans-Dicke gravity

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.Comment: 13 pages, 3 figures. Adding in Appendix