Brans-Dicke Wormholes in Non-Vacuum Spacetime

Analytical wormhole solutions in Brans-Dicke theory in the presence of matter are presented. It is shown that the wormhole throat must not be necessarily threaded with exotic matter. PACS number(s): 04.20Jb, 04.50+h ∗E-mail: [email protected] 1 The field equations of General Relativity, being local in character, admit solutions with nontrivial topology. Among these, wormholes have been extensively studied [1]. Their most salient feature is that an embedding of one of their spacelike sections in Euclidean space displays two asymptotically flat regions joined by a throat. The interest on wormholes is twofold. From the point of view of the euclidean path integral formulation of quantum gravity, Coleman [2] and Giddins and Strominger [3], among others, have shown that the effect of wormholes is to modify low energy coupling constants and to provide probability distributions for them. In particular, Coleman [4] showed that, in the dilute wormhole approximation, the probability distribution for universes is infinitely peaked at Λ = 0, rendering all other values of the cosmological constant improbable. On the purely gravitational side, the interest has been recently focused on traversible wormhole [1,5–7]. Most of the efforts are directed to study static configurations [9] that must have a number of specific properties in order to be traversible. The most striking of these properties is the violation of the energy conditions [10]. It implies that the matter that generates the wormhole is exotic [1], viz. its energy density is negative, as seen by static observers. Geometrically, this is a direct consequence of the singularity theorems of Hawking and Penrose [11]. Although we do not know of any such exotic material to date, quantum field theory might come to the rescue [12]. Finally, we should mention yet another proposal related to whormholes. It has been shown [5,13] that a nonstatic wormhole’s throat can be transformed into a time tunnel. Physical effects in this type of space-times have been studied in [14]. Wormhole solutions have also been discussed in alternative theories of gravity, such as R+R theories [15], Moffat’s nonsymmetric theory [16], Einstein-Gauss-Bonnet theory [17], and Brans-Dicke theory [18]. In the last case, static wormhole solutions were found in vacuum, the source of gravity being the scalar field. Dynamical solutions are discussed in [19]. The aim of this paper is to look for static wormhole solutions of Brans-Dicke theory in a general setting, i.e. in the presence of matter that obeys a generic equation of state. Following the conventions of [20], the field equations of Brans-Dicke theory are 2 Rμν = 8π Φ ( Tμν − ω + 1 2ω + 3 T gμν ) + ω Φ;μΦ;ν Φ2 + Φ;μ;ν Φ (1) Φ;μ = 8π 2ω + 3 T (2) The assumption of a static spacetime entails that it is possible to choose a metric and a scalar field such that gμν,t = 0 Φ,t = 0 gti = 0 (3) (i = r, θ, φ). We further require spherical symmetry, so that the line element can be written in Schwarzschild form: ds = −edt + edr + r(dθ + sin θdφ) (4) For the stress-energy tensor of matter we choose T tt = −ρ(r) T r r = −τ(r) T θ θ = T φ φ = p(r) (5) and zero otherwise. Finally we adopt the following equation of state for matter: