4 Conclusion : Recently, in the context of f ( R ) modified theories of gravity, traversable wormhole geometries were constructed. As f ( R ) gravity is equivalent to a Brans- Dicke theory with a coupling parameter ω = 0, one may be tempted to find these solutions inconsistent with the permitted interval, − 3 / 2 < ω < − 4 / 3, extensively con- sidered in the literature of static wormhole solutions in vacuum Brans-Dicke theory. Thus the choice provided by Eq. (21), in addition to the WEC and NEC violation, re- flects how crucially the range of ω depends on the form of C ( ω ), and we have shown that adequate choices of C ( ω ) provide different viability regions and less restrictive in- tervals, that include ω = 0. In this context, we have constructed a more general class of vacuum Brans-Dicke wormholes that include the value of ω = 0, proving the consistency of the solutions constructed in f ( R ) gravity. Furthermore, we deduced the general condition for the existence of vacuum Brans-Dicke wormhole geometries, and have shown that the presence of effective negative energy densities is a generic feature of these vacuum solu- tions. It will also be interesting to generalize this analysis in Brans-Dicke theory in the presence of matter. Work along these lines is presently underway. Figure 2 depicts condition (19) (depicted as a solid curve), i.e., negative energy densities, and condition (20) (depicted as dashed curves), i.e., the violation of the NEC for A = 3 exp(1 / 2). 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. 2 by the smaller and larger peaks, respectively. FIG. 2: Plot of the energy conditions for C ( ω ) = A exp( − ω $^{2}$/ 2), with A = 3 exp(1 / 2). In particular, the WEC expressed by condition (19) is given by the solid line; and the NEC, expressed by the condition (20), is given by the dashed curve. For the latter, only the cases of ( s, t ) = (0 , 1) and ( s, t ) = (1 , 1) of condition (20) are allowed; and are depicted by the smaller and larger peaks, respectively. Acknowledgements : MO acknowledges financial sup- port from a grant attributed by Centro de Astronomia e Astrof´ısica da Universidade de Lisboa (CAAUL), and financed by Funda¸c˜ ao para a Ciˆ encia e Tecnologia (FCT). --------- NEC _____ WEC –2 0 2 4 6 8 10 –3 –2 –1 0 1 2 3 w [1] M. S. Morris and K. S. Thorne, Am. J. Phys. 56 , 395 (1988). [2] B. Bhawal and S. Kar, Phys. Rev. D 46 , 2464-2468 (1992); G. Dotti, J. Oliva, and R. Troncoso, Phys. Rev. D 75 , 024002 (2007). [3] L. A. Anchordoqui and S. E. P Bergliaffa, Phys. Rev. D 62 , 067502 (2000); K. A. Bronnikov and S.-W. Kim, Phys. Rev. D 67 , 064027 (2003); M. La Camera, Phys. Lett. B573 , 27-32 (2003); F. S. N. Lobo, Phys. Rev. D 75 , 064027 (2007). [4] A. G. Agnese and M. La Camera, Phys. Rev. D 51 , 2011 (1995). [5] L. A. Anchordoqui, S. E. Perez Bergliaffa and D. F. Tor- res, Phys. Rev. D 55 , 5226 (1997). [6] K. K. Nandi, B. Bhattacharjee, S. M. K. Alam and J. Evans, Phys. Rev. D 57 , 823 (1998). [8] A. Bhattacharya, I. Nigmatzyanov, R. Izmailov and K. K. Nandi, Class. Quant. Grav. 26 (2009) 235017. [9] S. V. Sushkov, Phys. Lett. A164 , 33-37 (1992); A. A. Popov, Phys. Rev. D 64 , 104005 (2001); D. Hochberg, A. Popov and S. V. Sushkov, Phys. Rev. Lett. 78 , 2050 (1997); N. R. Khusnutdinov and S. V. Sushkov, Phys. Rev. D 65 , 084028 (2002); A. R. Khabibullin, N. R. Khusnutdinov and S. V. Sushkov, Class. Quant. Grav. 23 627-634 (2006); N. R. Khusnutdinov, Phys. Rev. D 67 , 124020 (2003); R. Garattini and F. S. N. Lobo, [7] K. K. Nandi, A. Islam and J. Evans, Phys. Rev. D 55 , 2497 (1997); [17] T. Matsuda, Prog. Theor. Phys. 47 , 738 (1972). [16] F. S. N. Lobo and M. A. Oliveira, Phys. Rev. D 80 , 104012 (2009). [15] F. S. N. Lobo, “Exotic solutions in General Relativ- ity: Traversable wormholes and ’warp drive’ spacetimes,” arXiv:0710.4474 [gr-qc]. [14] J. P. S. Lemos, F. S. N. Lobo and S. Quinet de Oliveira, Phys. Rev. D 68 , 064004 (2003). [13] T. Harko, Z. Kovacs and F. S. N. Lobo, Phys. Rev. D 78 , 084005 (2008); T. Harko, Z. Kovacs and F. S. N. Lobo, Phys. Rev. D 79 , 064001 (2009). [12] F. S. N. Lobo, Class. Quant. Grav. 25 , 175006 (2008). [10] C. G. Boehmer, T. Harko and F. S. N. Lobo, Phys. Rev. D 76 , 084014 (2007); C. G. Boehmer, T. Harko and F. S. N. Lobo, Class. Quant. Grav. 25 , 075016 (2008). [11] S. Sushkov, Phys. Rev. D 71 , 043520 (2005); F. S. N. Lobo, Phys. Rev. D 71 , 084011 (2005); F. S. N. Lobo, Phys. Rev. D 71 , 124022 (2005); F. S. N. Lobo, Phys. Rev. D 73 , 064028 (2006); F. S. N. Lobo, Phys. Rev. D 75 , 024023 (2007); P. F. Gonz´ alez-D´ıaz, Phys. Rev. D 68 , 084016 (2003); P. F. Gonz´ alez-D´ıaz, Phys. Rev. Lett. 93 071301 (2004); P. F. Gonz´ alez-D´ıaz and J. A. Jimenez-Madrid, Phys. Lett. B596 16-25 (2004). Class. Quant. Grav. 24 , 2401 (2007); R. Garattini and F. S. N. Lobo, Phys. Lett. B 671 , 146 (2009).