2 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] 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 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]: Adding Eqs. (14) and (15), one arrives at respectively, where Z is defined as ρ = − 4 B $^{2}$r$^{4}$Z$^{2}$[( C + 1) 2 − λ $^{2}$] λ $^{2}$( r 2 − B $^{2}$)4 , (14) p$_{r}$ = − 4 Br $^{3}$Z2 λ $^{2}$( r 2 − B $^{2}$)$_{4}$[ λC ( r 2 + B $^{2}$) − Br ( C 2 1 + λ $^{2}$)] , (15) ρ + p$_{r}$ = − 4 Br $^{3}$Z2 λ $^{2}$( r 2 − B $^{2}$)$_{4}$[ λC ( r $^{2}$+ B $^{2}$)+2 Br ( C +1 − λ $^{2}$)] , (17) Z ≡ ( r − B r + B ) ( C +1) /λ . (16) 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: [ C ( ω ) + 1] 2 > λ $^{2}$( ω ) , (18) C ( ω ) [ 1 − ωC ( ω ) 2 ] > 0 , (19) 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: ( − 1) s + t +1 [ ( − 1) $^{s}$( C + 1) + ( − 1) t √ C ( 1 − ωC 2 ) ] × × C (1 − ωC/ 2) √ (4 + 2 ω ) C 2 + 4( C + 1) > 0 , (20) where s, t = 0 , 1. Note that a necessary condition im- posed by the term in the square root, in square brackets, Confronting the Morris-Thorne metric with the isotropic metric (4), the radial coordinate r → R is rede- fined as so that Φ( R ) and b ( R ) are given by (12) respectively. The wormhole throat condition b ( R$_{0}$ ) = R$_{0}$ imposes the minimum allowed r -coordinate radii r ± 0 given by Φ( R ) = α$_{0}$ + 1 λ { ln [ 1 − B r ( R ) ] − ln [ 1 + B r ( R ) ]} , (11) b ( R ) = R { 1 − [ λ [ r $^{2}$( R ) + B $^{2}$] − 2 r ( R ) B ( C + 1) λ [ r $^{2}$( R ) + B $^{2}$] ] $^{2}$} , R = re β 0 (1 + B/r ) 2 ( 1 − B/r 1 + B/r ) Ω , Ω = 1 − C + 1 λ , (10) r ± 0 = α $^{±}$B , α ± = (1 − Ω) ± √ Ω(Ω − 2) . (13) ds 2 = − e 2Φ( R $^{)}$dt2 + dR 2 1 − b ( R ) /R + R $^{2}$( dθ 2 + sin $^{2}$θdψ$^{2}$) (9) e α ( r ) = e α 0 ( 1 − B/r 1 + B/r )$^{1}$ λ , (5) ds 2 = − e 2 α ( r $^{)}$dt2 + e 2 β ( r $^{)}$dr2 + e 2 ν ( r $^{)}$r$^{2}$( dθ 2 + sin $^{2}$θdψ$^{2}$) . (4) metric given by e β ( r ) = e β 0 (1 + B/r ) 2 ( 1 − B/r 1 + B/r )$^{λ$_{−}$}$$^{C$_{−}$}$$^{1}$ λ , (6) ϕ ( r ) = ϕ$_{0}$ ( 1 − B/r 1 + B/r )$^{C}$ λ , (7) λ 2 ≡ ( C + 1) 2 − C ( 1 − ωC 2 ) > 0 , (8) 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.