General class of vacuum Brans-Dicke wormholes Francisco S. N. Lobo ∗ and Miguel A. Oliveira † Centro de Astronomia e Astrof´ısica da Universidade de Lisboa, Campo Grande, Ed. C8 1749-016 Lisboa, Portugal (Dated: October 30, 2018) Recently, traversable wormhole geometries were constructed in the context of f ( R ) gravity. The latter is equivalent to a Brans-Dicke theory with a coupling parameter ω = 0, which is apparently excluded from the narrow interval, − 3 / 2 < ω < − 4 / 3, extensively considered in the literature of static wormhole solutions in vacuum Brans-Dicke theory. However, this latter interval is only valid for a specific choice of an integration constant of the field equations derived on the basis of a post-Newtonian weak field approximation, and there is no reason for it to hold in the presence of compact objects with strong gravitational fields. In this context, we construct a general class of vacuum Brans-Dicke wormholes that include the value of ω = 0. PACS numbers: 04.20.Jb, 04.50.Kd Introduction : Wormholes are hypothetical tunnels in spacetime and in classical general relativity are supported by exotic matter, which involves a stress energy tensor that violates the null energy condition (NEC) [1]. Several candidates have been proposed in the literature, such as solutions in Einstein-Gauss-Bonnet theory [2]; wormholes on the brane [3]; solutions in Brans-Dicke theory [4–8], which will be further explored in this brief report; worm- hole solutions in semiclassical gravity [9]; exact wormhole solutions using conformal symmetries [10]; solutions sup- ported by equations of state responsible for the cosmic ac- celeration [11]; and NEC respecting geometries were fur- ther explored in conformal Weyl gravity [12]; the possi- bility of distinguishing wormhole geometries by using as- trophysical observations of the emission spectra from ac- cretion disks was also explored [13], etc (see Refs. [14, 15] for more details and [15] for a recent review). Recently, traversable wormhole geometries in the con- text of f ( R ) modified theories of gravity were also con- structed [16]. The matter threading the wormhole was imposed to satisfy the energy conditions, so that it is the effective stress-energy tensor containing higher or- der curvature derivatives that is responsible for the NEC violation. Thus, the higher order curvature terms, inter- preted as a gravitational fluid, sustain these nonstandard wormhole geometries, fundamentally different from their counterparts in general relativity. Furthermore, we note that f ( R ) modified theories of gravity are equivalent to a Brans-Dicke theory with a coupling parameter ω = 0, and a specific potential related to the function f ( R ) and its derivative. However, the value ω = 0 is apparently excluded from the interval, − 3 / 2 < ω < − 4 / 3, of the coupling parameter, extensively considered in the litera- ture of static wormhole solutions in vacuum Brans-Dicke theory. In Brans-Dicke theory, analytical wormhole solutions where G µν is the Einstein tensor and □ 2 ≡ ϕ ; ρ ; $_{ρ}$. It is useful to work in isotropic coordinates, with the where R is the curvature scalar, ω is a constant dimen- sionless coupling parameter, and ϕ is the Brans-Dicke scalar. We adopt the convention 8 πG = c = 1 through- out this work. The above action provides the following field equations: General class of Brans wormholes : The matter-free ac- tion in Brans-Dicke theory is given by In this context, we construct a general class of vacuum Brans-Dicke wormholes that include the value of ω = 0, and thus constructing a consistent bridge with the worm- hole solutions in f ( R ) gravity found in [16]. Further- more, we present the general condition for the existence of Brans-Dicke wormhole geometries based on the NEC violation, and show that the presence of effective nega- tive energy densities is a generic feature of these vacuum solutions. were constructed [4–6]. It was shown that static worm- hole solutions in vacuum Brans-Dicke theory only ex- ist in a narrow interval of the coupling parameter [6], namely, − 3 / 2 < ω < − 4 / 3. However, this result is only valid for vacuum solutions and for a specific choice of an integration constant of the field equations given by C ( w ) = − 1 / ( ω + 2). The latter relationship was derived on the basis of a post-Newtonian weak field approxima- tion, and it is important to emphasize that there is no reason for it to hold in the presence of compact objects with strong gravitational fields. S = 1 2 ∫ d $^{4}$x ( − g ) 1 2 [ ϕ R − ϕ − $^{1}$ω ( ϕ ) g $^{µν}$ϕ$_{,µ}$ ϕ$_{,ν}$ ] , (1) G µν = − ω ϕ 2 ( ϕ$_{,µ}$ϕ$_{,ν}$ − 1 2 g$_{µν}$ϕ$_{,σ}$ϕ ,σ ) − 1 ϕ ( ϕ$_{;}$$_{µ}$ϕ$_{;}$$_{ν}$ − g$_{µν}$ □ $^{2}$ϕ ) , (2) □ $^{2}$ϕ = 0 , (3) $^{∗}$Electronic address: flobo@cii.fc.ul.pt $^{†}$Electronic address: miguel@cosmo.fis.fc.ul.pt arXiv:1001.0995v2 [gr-qc] 22 Mar 2010