content_image/415394.jpg content_image/415393.jpg content_image/415396.jpg content_image/415397.jpg content_image/415392.jpg content_image/415390.jpg content_image/415389.jpg content_image/415391.jpg content_image/415398.jpg content_image/415375.jpg content_image/415384.jpg content_image/415374.jpg content_image/415376.jpg content_image/415380.jpg content_image/415379.jpg
metric given by 

<img src='content_image/415394.jpg'>

Throughout this work, we consider the Brans class I so- lution, which corresponds to setting the gauge β − ν = 0. Thus, the field equations yield the following solutions 

<img src='content_image/415393.jpg'>

<img src='content_image/415396.jpg'>

<img src='content_image/415397.jpg'>

where α$_{0}$ , β$_{0}$ , B , C , and ϕ$_{0}$ are constants. Note that the asymptotic flatness condition imposes that α$_{0}$ = β$_{0}$ = 0, as can be readily verified from Eqs. (5) and (6). 

In order to analyze traversable wormholes in vacuum Brans-Dicke theory, it is convenient to express the space- time metric in the original Morris-Thorne canonical form [1]: 

<img src='content_image/415392.jpg'>

where Φ( R ) and b ( R ) are the redshift and shape func- tions, respectively. To be a wormhole solution, several properties are imposed [1], namely, the throat is located at R = R$_{0}$ and b ( R$_{0}$ ) = R$_{0}$ . A flaring out condition of the throat is imposed, i.e., [ b ( R ) − Rb $^{′}$( R )] /b $^{2}$( R ) > 0, which reduces to b $^{′}$( R$_{0}$ ) < 1 at the throat, where the prime here denotes a derivative with respect to R . The condi- tion 1 − b ( R ) /R ≥ 0 is also imposed. To be traversable, one must demand the absence of event horizons, so that Φ( R ) must be finite everywhere. 

Confronting the Morris-Thorne metric with the isotropic metric (4), the radial coordinate r → R is rede- fined as 

<img src='content_image/415390.jpg'>

so that Φ( R ) and b ( R ) are given by 

<img src='content_image/415389.jpg'>

(12) respectively. The wormhole throat condition b ( R$_{0}$ ) = R$_{0}$ imposes the minimum allowed r -coordinate radii r ± 0 given by 

<img src='content_image/415391.jpg'>

<img src='content_image/415398.jpg'>

The values R ± 0 can be obtained from Eq. (10) using Eq. (13). Note that R → ∞ as r → ∞ , so that b ( R ) /R → 0 as R → ∞ . The condition b ( R ) /R ≤ 1 is also verified for all R ≥ R ± 0 . The redshift function Φ( R ) has a singularity at r = r$_{S}$ = B , so that the minimum allowed values of r ± 0 must necessarily exceed r$_{S}$ = B . It can also be verified from Eq. (10) that r ± 0 ≥ B which implies R ± 0 ≥ 0. The energy density and the radial pressure of the wormhole material are given by [6] 

respectively, where Z is defined as 

Adding Eqs. (14) and (15), one arrives at 

<img src='content_image/415375.jpg'>

which will be analyzed in the NEC violation below. 

In [6], the authors considered negative energy densities, which consequently violates the weak energy condition (WEC). Now, Eq. (14) imposes the following condition: 

which can be rephrased as 

by taking into account Eq. (8). Note that the function C ( ω ) is still unspecified. 

However, it is important to emphasize that negative energy densities are not a necessary condition in worm- hole physics. The fundamental ingredient is the viola- tion of the NEC, ρ + p$_{r}$ < 0, which is imposed by the flaring out condition [1]. To find the general restriction for ρ + p$_{r}$ < 0 at the throat r$_{0}$ , amounts to analyzing the factor in square brackets in Eq. (17), namely, the condition λC ( r 2 0 + B $^{2}$) + 2 Br$_{0}$ ( C + 1 − λ $^{2}$) > 0. Using Eqs. (8) and (13), the latter condition is expressed as: 

<img src='content_image/415384.jpg'>

where posed by the term in the square root, in square brackets, 

<img src='content_image/415374.jpg'>

<img src='content_image/415376.jpg'>

<img src='content_image/415380.jpg'>

<img src='content_image/415379.jpg'>

