Static spherically symmetric wormholes with isotropic pressure Mauricio Cataldo∗ Departamento de F´ısica, Facultad de Ciencias, Universidad del B´ıo-B´ıo, Avenida Collao 1202, Casilla 15-C, Concepci´on, Chile, and Grupo de Cosmolog´ıa y Gravitaci´on-UBB. Luis Liempi† and Pablo Rodr´ıguez‡ Departamento de F´ısica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile. (Dated: April 18, 2016) In this paper we study static spherically symmetric wormhole solutions sustained by matter sources with isotropic pressure. We show that such spherical wormholes do not exist in the framework of zero-tidal-force wormholes. On the other hand, it is shown that for the often used power-law shape function there is no spherically symmetric traversable wormholes sustained by sources with a linear equation of state p = ωρ for the isotropic pressure, independently of the form of the redshift function φ(r). We consider a solution obtained by Tolman at 1939 for describing static spheres of isotropic fluids, and show that it also may describe wormhole spacetimes with a power-law redshift function, which leads to a polynomial shape function, generalizing a power-law shape function, and inducing a solid angle deficit. PACS numbers: 04.20.Jb, 04.70.Dy,11.10.Kk I. INTRODUCTION and shape functions. The often used equation of state is the linear barotropic equation of state pr = ωρ, which relates the radial pressure with the energy density [2]. In the framework of Einstein General Relativity the From the cosmological setting, such an equation of state study of spherically symmetric traversable wormhole is associated with phantom dark energy if ω < 1, and spacetimes [1] has been mostly focused in matter sources also may sustain traversable static wormholes [3,− 4]. with anisotropic pressures. Mainly this is due to the fact Anisotropic stress contributions to the gravitational that in order to correctly describe a wormhole spacetime field can arise from specific matter fields. A fluid source one needs a redshift function without horizons, or the with anisotropic stresses supporting wormholes may be redshift and the shape functions giving a desired asymp- for example of electromagnetic nature: linear [5] and non- totic. In this way, the theoretical construction of worm- linear Maxwell fields [6] have been considered in the lit- hole geometries is usually performed by assuming a priori erature. An electric field coupled to a scalar field is con- the form of the redshift and the shape functions, in order sidered in Ref. [7]. to have a desired metric. But Einstein’s field equations for spherically symmetric spacetimes imply that the ra- A spatially varying cosmological constant also has been dial and lateral (or transverse) pressures are not equal. In considered in the framework of static wormholes sup- such a way, by imposing restricted choices on the redshift ported by anisotropic matter sources [8]. In Ref. [9] the and the shape functions we will obtain, in general, for the source of the stress-energy tensor supporting the worm- 2 3 hole geometries consists of an anisotropic brown dwarf arXiv:1604.04578v1 [gr-qc] 15 Apr 2016 energy-momentum tensor that T2 = T3 . This condition implies that a wormhole configuration6 is necessarily sup- star which smoothly joins the vacuum and may possess ported by an anisotropic matter source. However, this an arbitrary cosmological constant, while in Ref. [10] method has some limitations since we can obtain for the anisotropic vacuum stress-energy of quantized fields has energy density, radial and lateral pressures algebraic ex- been proposed as source for static wormholes. pressions which are physically unreasonable. Such anisotropic scenarios are obtained also for reg- ular static, spherically symmetric solutions describing One may also follow a more conventional method used wormholes supported by dark matter non-minimally cou- for finding solutions in general relativity, by prescribing pled to dark energy in the form of a quintessence scalar the matter content with specific equations of state for the field [11]. radial and/or the tangential pressures, and then solve Einstein’s field equations in order to find the redshift The main purpose of this paper is to present and dis- cuss static spherically symmetric wormhole spacetimes supported by a single perfect fluid, i.e. a matter source with isotropic pressure. As far as we know, the only ∗[email protected] spherical wormhole solution discussed up to now is the †[email protected] not asymptotically flat wormhole with isotropic pressure ‡[email protected] considered in Ref. [4]. We will discuss this solution in 2 more detail below in Sections IV and V. It must be no- a single imperfect fluid, from the metric (1) and the Ein- ticed that the study of wormhole solutions sustained by stein field equations Gµν = κTµν we obtain a perfect fluid allows us to consider phantom wormholes − sustained by inhomogeneous and isotropic phantom dark b′ κρ(r)= , (2) energy. r2 The paper is organized as follows. In Sec. II we write b φ′ b κp (r)=2 1 , (3) the Einstein equations for static spherically symmetric r − r r − r3 spacetimes. In Sec. III we analyze the possibility of b having zero-tidal-force wormholes sustained by a matter κp (r)= 1 l − r × source with isotropic pressure, while in Sec. IV we an- alyze the possibility of having spherical wormholes sus- b′r + b 2r b′r b φ′′ + φ′ 2 − φ′ − , (4) tained by a fluid with linear equation of state. In Sec. V − 2r(r b) − 2r2(r b) we re-obtain an analytical solution, previously obtained − − by Tolman, which describes static spheres of fluids with where κ = 8πG, ρ is the energy density, and pr and isotropic pressure, and we show that it may describe also pl are the radial and lateral pressures respectively. From µν a non-asymptotically flat wormhole geometry. the conservation equation T ;ν = 0 we obtain the hydro- static equation for equilibrium of the matter sustaining the wormhole II. FIELD EQUATIONS FOR STATIC 2(p p ) SPHERICALLY SYMMETRIC SPACETIMES p′ = l − r (ρ + p )φ′. (5) r r − r It becomes clear that the main condition for having a The spacetime ansatz for seeking static spherically perfect fluid is given by symmetric solutions can be written in Schwarzschild co- ordinates as pr = pl. (6) dr2 2 2φ(r) 2 2 2 2 2 This condition on the radial and lateral pressures allows ds = e dt b(r) r dθ + sin θdφ , (1) − 1 − us to get the following differential equation connecting − r functions φ(r) and b(r): where eφ(r) and b(r) are arbitrary functions of the ra- dial coordinate. In the case of wormholes these functions b′r 3b +2r b′r 3b φ′′ + φ′ 2 − φ′ = − . (7) are referred to as redshift function and shape function − 2r(r b) 2r2(r b) respectively. The essential characteristics of a wormhole − − geometry are encoded in these functions, so in order to We may consider Eq. (7) as a differential equation for one have a wormhole these two functions must satisfy some of these involved functions, by giving the remaining one. general constraints discussed by Morris and Thorne in By supposing that the redshift function φ(r) is given, we Ref. [1, 12]. obtain a first order differential equation for the shape By assuming that the matter content is described by function b(r), whose general solution is given by ′′ ′ ′ 2r2φ +2r2φ 2−3rφ −3 ′ dr ′′ ′ ′ ′′ ′2 ′ r(1+rφ ) 2r2φ +2r2φ 2−3rφ −3 2r rφ + rφ φ e − ′ dr b(r)= − dr + C e r(1+rφ ) , (8) 1+Rrφ′ Z R where C is an integration constant. Eqs. (7) and (8) have an equation of state. Such input equations of state do a general character, in the sense that they do not involve not normally allow for closed form solutions. In arriv- an equation of state for ρ and p. It must be remarked ing to exact solutions, one can solve the field equations that for these static configurations, sustained by isotropic by making an ad hoc assumption for one of the metric perfect fluids, the Einstein field equations are reduced functions or for the energy density. Hence the equation to a set of three independent differential equations (2), of state can be computed from the resulting metric. (3) and (7) for four unknown functions, namely φ(r), b(r), ρ(r) and p(r). Thus, to study solutions to these field equations, restricted choices of one of the unknown functions must be considered. To obtain a realistic stellar model, one can start with 3 α−2 III. ON ZERO-TIDAL-FORCE WORMHOLES 1+α r and lateral pressures are given by κρ = r2 r + WITH ISOTROPIC PRESSURE − 0 0 α−2 α−2 2 1 r 2 α r r2 , κpr = r2 r r2 and κpl = 2r2 r . 0 0 − 0 0 It is well known that a simple class of solutions cor- Since φ(r) = 0 this wormhole has anisotropic pressures responds to zero-tidal-force wormhole spacetimes, which as we would expect. The metric (10) includes non- are defined by the condition φ(r) = φ0 = const [1, 12]. asymptotically flat spacetimes. Note that for α = 2 By putting φ(r) = const into Eq. (8) we obtain b(r) = we have that ρ = 3/r2 +2/r2, p = 1/r2 2/r2 and 3 0 r 0 Cr . By requiring that b(r0)= r0, the spacetime metric 2 − − pl =1/r0, obtaining at spatial infinity a spacetime with takes the form constant curvature and p = p = ρ/3, which means r l − 2 that asymptotically we have a wormhole sustained by an 2 2 dr 2 2 2 2 ds = dt 2 r dθ + sin θdφ .