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is precisely condition (19). Thus, a necessary condition for vacuum Brans-Dicke wormholes is the existence of negative effective energy densities. However, we empha- size that it is condition (20), i.e., the violation of the NEC at the throat, that generic vacuum Brans-Dicke worm- holes should obey. 

A specific choice of C ( ω ) considered extensively in the literature, is the Agnese-La Camera function [4] given by 

Using this function, it was shown that static wormhole solutions in vacuum Brans-Dicke theory only exist in a narrow interval of the coupling parameter [6], namely, − 3 / 2 < ω < − 4 / 3. However, we point out that this re- sult is only valid for vacuum solutions and for the specific choice of C ( ω ) considered by Agnese and La Camera [4]. As mentioned in the Introduction, relationship (21) was derived on the basis of a post-Newtonian weak field ap- proximation, and it is important to emphasize that there is no reason for it to hold in the presence of compact ob- jects with strong gravitational fields. The choice given by (21) is a tentative example and reflects how crucially the wormhole range for ω depends on the form of C ( ω ). Ev- idently, different forms for C ( ω ) different from Eq. (21) would lead to different intervals for ω . 

Note that in [7], the negative values of the coupling parameter ω were extended to arbitrary positive val- ues of omega, i.e., ω < ∞ , in the context of two-way traversable wormhole Brans solutions (we refer the reader to Ref [7] for specific details). An interesting exam- ple was provided in Ref. [17], in the context of grav- itational collapse in the Brans-Dicke theory, where the choice C ( ω ) ∼ − ω − 1 / 2 was analyzed. More specifically, the authors in [6] considered C ( ω ) = − qω − 1 / $^{2}$, with q < 0 so that C ( ω ) > 0. Thus, the constraint (19) is satisfied only if ω > 4 /q $^{2}$. However, we will be interested in solu- tions which include the value ω = 0, in order to find an equivalence with the f ( R ) solutions found in [16]. The specific choices we consider below possess the following limits, C ( ω ) → 0, λ ( ω ) → 1 as ω → ∞ , in order to recover the Schwarzschild exterior metric in standard co- ordinates. 

Another issue that needs to be mentioned is that the above-mentioned interval imposed on ω was also obtained by considering negative energy densities. In principle, the violation of the WEC combined with an adequate choice of C ( ω ) could provide a different viability and less restrictive interval (including the value ω = 0) from the case of − 3 / 2 < ω < − 4 / 3 considered in [6]. In this context, we consider below different forms of C ( ω ) that allow the value ω = 0 in the permitted range. Thus, to satisfy the constraint (19), both factors C ( ω ) and [1 − ωC ( ω ) / 2] should both be positive, or both negative. 

Consider the following specific choice 

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where a is a real constant. The requirement that λ 2 > 0, i.e., Eq. (8), is satisfied. The function C ( ω ) is positive for all real ω , and the second term, in square brackets, of Eq. (19), is positive everywhere for a 2 > 1 / 16. Therefore, for this case, condition (19) is satisfied for all ω . For a 2 < 1 / 16, [1 − ωC ( ω ) / 2] has two real roots, namely, ω 0 ± = (1 ± √$_{1}$ − 16 a ) / 4; the lesser value is positive and thus both the second term and condition (19) will be positive at ω = 0. Thus, if a 2 < 1 / 16, the condition (19) is satisfied for ω ∈ R − [ ω 0 $_{−}$; ω 0 $_{+}$]. Figure 1 depicts condition (19) (depicted as a solid curve), i.e., negative energy densities, and condition (20) (depicted as the dashed curves), i.e., the violation of the NEC, for a = 1. For the latter, only the cases of ( s, t ) = (0 , 1) and ( s, t ) = (1 , 1) of condition (20) are allowed; and are depicted in Fig. 1 by the small and large peaks, respectively. 

In the limiting case, C ( ω ) → 0, λ ( ω ) → 1 as ω → ∞ , one simply recovers the Schwarzschild exterior metric in standard coordinates. This can be verified from Eqs. (11) and (12), which impose b ( R ) = 2 M and b $^{′}$| r 0 = 0. However, in this limit, the inequality (20) is violated, and there are no traversable wormholes. 

Consider a second specific choice given by 

The requirement that λ 2 > 0, i.e., Eq. (8), is also sat- isfied. This function, for A > 0, is positive for all ω . Therefore, in order to satisfy condition (19), the restric- tion (1 − ωC ( ω ) / 2) > 0 is imposed. We verify that if 0 < A < 2 exp(1 / 2), then (1 − ωC ( ω ) / 2) > 0 for all ω , so that conditions (19) and (20) are both satisfied. If A > 2 exp(1 / 2), then the second term (1 − ωC ( ω ) / 2) will have two real positive roots, i.e., ω$_{0}$$_{,}$$_{1}$ > 0. For this choice of A , we have the following range of allowed ω : R − ] ω$_{0}$, ω$_{1}$ [. Moreover, since ω$_{0}$ > 0, the value ω = 0 will always be in the set of allowed values. 

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