Static Spherically Symmetric Wormholes with Isotropic Pressure

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[6] non- fields and [5] Maxwell linear be linear source nature: may electromagnetic fluid wormholes of A example supporting for fields. stresses matter anisotropic specific with from arise can field 4]. 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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 analyze 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 radial 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 . (9) ideal fluid (a string gas). − r − IV. ON SPHERICAL WORMHOLES WITH 1 r − 0 ISOTROPIC PRESSURE AND LINEAR This metric represents a spacetime of constant curvature, EQUATION OF STATE for which the pressure and energy density are given by 2 p = ρ/3 = 1/κr0, and it is a particular case of the Since, for a vanishing redshift function, the only so- well-known− static− Einstein universe (for which we have lution allowed by the isotropic pressure condition (6) is 2 2 κρ = 3/r0 Λ and κp = 1/r0 +Λ, where Λ is the the spacetime (9), we shall now on take into considera- cosmological− constant). − tion a non-vanishing redshift function in all considered −1 The inverse of the radial metric component grr van- wormhole solutions. ishes at r = r0, as we would expect for wormholes. How- In this section, in order to follow with our study, we ever, for r > r0 the radial metric component grr becomes shall impose on the isotropic pressure the linear equation negative, so the solution is valid only for 0 r r0. This of state implies that there is no zero-tidal-force wormhole≤ ≤ solutions sustained everywhere by an isotropic perfect fluid. p = ωρ, (11) In other words, any zero-tidal-force wormhole must be filled by a single fluid with anisotropic pressure. There- where the state parameter ω is a constant. Thus from fore, in order to generate spherically symmetric worm- Eq. (5) we have holes, sustained by a single matter source with isotropic pressure, we must consider spacetimes with φ(r) = const. ωρ′ = (1 + ω)ρφ′ (12) Notice that this result does not mean that it6 is not − possible to have a zero-tidal-force wormhole sustained by and then the energy density is given by a perfect (ideal) fluid at spatial infinity. For an explicit example let us consider the spacetime − 1+ω φ(r) ρ(r)= Ce ω , 2 2 2 dr 2 2 2 2 ds = dt α r dθ + sin θdφ . (10) where C is an integration constant. If φ(r) = const we − r 1 − obtain that energy density is constant, in agreement with r0 − the result of the previous section. For α> 0 the metric covers the range r0 r< and de- From Eqs. (2), (7) and (12) the following master dif- scribes a wormhole spacetime. The energy≤ density,∞ radial ferential equation for the shape function b(r) is obtained:

1 2r2ω(1 + ω)(r b)b′b′′′ +4ω ω + r2(r b)b′′2 + ωr((1 + ω)rb′ + (5ω 3)b +2r 6ωr)b′b′′ − − 2 − − − − 1 5 16 8 3b′2 ω + (1 + ω)rb′ 1+ ω2 + ω b + ωr =0. (13) 3 − 3 3 3

In general it is hard to find analytical solutions to while for ω = 1 we have that eφ(r) = 1 Ar2 with Eq. (13). Nevertheless, one can make some checks to p = ρ = 3A.− − − − prove the correctness of the above equation. Notice, On the other hand, the first term of Eq. (13) vanishes for example, that Eq. (13) is fulfilled identically for 3 for ω = 0 and ω = 1. Thus for ω = 0 we obtain b(r)= b(r) = Ar . Thus from Eq. (12) we obtain that ω = 1 A, e2φ(r) = 1 A/r−, ρ = p = 0, i.e. the Schwarzschild or φ(r) = const. For the latter case p = ρ/3 = −A, − − − solution. It is well known that the Schwarzschild solution 4 may be interpreted as a non-traversable wormhole. (4). It becomes clear that this solution is defined by a For ω = 1 we obtain the Kottler solution, i.e. b(r)= redshift function of general form given by A + Br3, e−2φ(r) =1 A/r Br2, ρ = p =3B [13]. We turn next to often− used− power-law− form of the shape β φ(r) r function in wormhole spacetimes: b(r) = A/rn. This e = , (16) r0 choice ensures that for r and n > 1 the M- → ∞ − T constraint b(r)/r 1 is satisfied. Thus, by putting with β a constant. In the following section we shall dis- n ≤ b(r) = A/r into the master equation (13) we find that cuss the solution generated by field equations (2)-(4) with this equation is satisfied if n = 3 (for arbitrary ω) or the restricted choice (16). n = 5/3 (and ω = 3). − The− obtained negative− values of the n-parameter are less than 1. This implies that there is no spherically − V. ON SPHERICAL WORMHOLE WITH symmetric traversable wormholes characterized by a ra- φ r r β e ( ) = dial metric component given by g−1 =1 (r /r)n+1 (with r0 rr − 0 n> 1), and sustained by isotropic pressure sources with a linear− equation of state (11), independently of the form In order to show the correctness of the conclusion of of the redshift function φ(r). The implications of this the previous section, we must provide the static spheri- result tell us that we must consider more general forms cally symmetric solution with the specific choice of the of the shape function b(r) and/or of the equation of state power-law redshift function (16). In Ref. [14] Tolman of the isotropic pressure p(ρ). provides explicit analytical solutions for static spheres of Lastly, let us note that this conclusion is not in fluids with isotropic pressure. For our purpose, it is con- agreement with the result obtained in subsection III- venient to consider the solution V, obtained in Sec. 4 B of the Ref. [4], where the author discusses the non- of Ref. [14], which justly take into account the metric asymptotically flat wormhole given by component gtt having the form of Eq. (16). We shall re- obtain the Tolman solution by assuming that the redshift 2 3−α dr function is given by Eq. (16). 2 2ω( 1+ω ) 2 ds = (r/r0) dt 1−α − 1 (r0/r) − By putting Eq. (16) into Eq. (7) we find for the shape − function r2 dθ2 + sin2 θdφ2 , (14)

3−α β − β 1 r0 β (β 2) − (2 +1)( 3+ ) 1+β p = ωρ = 2 , (15) b(r)= 2 − r C r , (17) −8πr0 r β 2 β 1 − − − where α = 1/ω (see Eqs. (32) and (33) of Ref. [4]). No- where C is a constant of integration. It becomes clear tice that the− equation of state (15) has the linear form of that this form of the shape function is more general than Eq. (11) and the shape function just the form not allowed the discussed above power-law shape function b(r) = by the master Eq. (13), therefore this non-asymptotically A/rn, as we would expect. Thus, the metric, energy ﬂat solution is not consistent with Einstein equations (2)- density and pressure are provided by

2β 2 2 r 2 dr 2 2 2 2 ds = dt 2(β2−2β−1) r dθ + sin θdφ , (18) r0 − − − β (β−2) ˜ r 1+β 1 2 + C − β −2 β−1 r0 − − 2 −2 2 β 1+β r 1+β (β 2) β κρ(r) = C˜ (2 β +1)(β 3)(1 + β)−1 r−2 + − , (19) r − ( 2 β 1+ β2) r2 0 − − 2 − − −2 2 β 1+β r 1+β β2 κp(r) = (2 β + 1) C˜ r−2 , (20) r − ( 2 β 1+ β2) r2 0 − −

˜ −2(β2−2β−1)/(1+β) respectively, where C = Cr0 . Note that This solution exhibits a solid angle deﬁcit, and does not the pressure is isotropic but not of barotropic type. We describe a wormhole geometry. have such an equation of state only for C˜ = 0 and β = 1/2, obtaining for the latter value gtt = r0/r, −1 −˜ 1 By requiring the standard wormhole condition b(r0)= grr = Cr0/r 4, and p = ρ/5 = κr2 , i.e. ω = 1/5. − − − r0 on Eq. (17) we obtain for the metric (18) the following 5 expression:

2β 2 2 2 r 2 (1+2β β ) dr ds = dt −2(1+2β−β2) r0 − 1+β − 1 r − r0 r2 dθ2+ sin2 θdφ2 . (21)

In order to the metric (21) describes a traversable 1+2β−β2 wormhole the condition 1+β < 0 must be required, implying that the parameter β varies in the ranges β > 1 + √2 or 1 <β < 1 √2. However, in this case 2− − 1+2β β < 0, so the radial metric component grr becomes− negative, and then the metric (21) does not describes a wormhole solution. The metric component grr does not change its sign if 2 we require 1+2β−β > 0 and 1+2β β2 > 0. This implies 1+β − that the parameter β varies in the ranges 1 √2 <β< 1+√2. In this case the radial coordinate varies− from zero to a maximum value rmax = r0 > 0, hence the solution FIG. 1: Plot shows the behaviour of rmax of Eq. (28) for corresponds to a fluid sphere of radius r0 with isotropic β ≤ −1 and r0 = 1. The maximum value is reached at β = pressure. −4.745695219 where rmax = 1.232835973. For β → −∞ we Nevertheless, as we shall see in the following subsec- have rmax → 1. tion, the considered Tolman solutions can describe wormhole geometries fulfilling the required conditions grr > 0 and r r0. It is interesting to note that this geometry describes a ≥ spacetime with a solid angle deficit (or excess). This can be seen directly by making the rescaling ̺2 = (β2 2β − − A. A truly wormhole geometry 1)r2. Then, the metric (22) becomes

2β 2 In order to show that the metric (21) may correctly 2 ̺ 2 d̺ ds = dt 2(1+2β−β2) describe a Lorentzian spacetime for r r0 let us rewrite ̺0 − 1+β − ≥ ̺ 1 it in the following form: ̺0 − ̺2 r 2β (β2 2 β 1) dr2 dθ2 + sin2 θdφ2 . (25) 2 2 (β2 2 β 1) ds = dt −2(1+2β−−β2) r0 − 1+β − − − r 1 r0 − This new form of the metric (22) shows explicitly the r2 dθ2 + sin2 θdφ2 . (22) presence of a solid angle deficit for <β< 1 √3 or 1+ √3 <β< , and a solid angle−∞ excess for− 1 It becomes clear that this metric describes a Lorentzian √3 <β < 1 √2∞ or 1+ √2 <β < 1+ √3. These− 2 spacetime for r r if β2 2 β 1 > 0 and 1+2β−β > topological defects− vanish for β = 1 √3, obtaining a ≥ 0 − − 1+β ± 0, which implies that the condition β < 1 must be non-flat asymptotic spacetime. required. − The geometrical properties and characteristics of these For the metric (22) the energy density and the isotropic solutions can be explored through the embedding dia- pressure are given by grams, which helps to visualize the shape and the size π of slices t = const, θ = 2 of the metric (22) by using a 2(β2−2β−1) − 1+β standard embedding procedure in ordinary three dimen- r (2β + 1)(β 3) r0 sional Euclidean space. In general, in order to embed κρ = − + (23) π (1 + β)(β2 2β 1)r2 two dimensional slices t = const, θ = 2 of the generic − − metric (1) the equation β(β 2) − , (β2 2β 1)r2 dz(r) 1 2− − − − = (26) − 2(β 2β 1) r 1+β dr 1 r (2β + 1) b(r) r0 − κp = (24) q (β2 2β 1)r2 − − − is used for the lift function z(r) [1]. Thus, for slices t = β2 const, θ = π of the metric (22) we obtain for the first . 2 (β2 2β 1)r2 derivative of the lift function − − 6

that the embedding of considered spacetimes in ordinary three dimensional Euclidean space has a finite size since it extends from r0 up to a maximum radial value rmax. Effectively, the radial coefficient grr of the metric (22) implies that the denominator of the fraction is positive for any r > r0, thus the numerator must be also positive. 2(β2−2β−1) β+1 Then the condition β(β 2) r 1 must be − r0 ≤ required, implying that r0 r rmax, where ≤ ≤ − β+1 r = r (β(β 2)) 2(β2−2β−1) . (28) max 0 − Notice that for any value of the β-parameter we have dz(r) dz(r) that dr r0 = and dr rmax = 0. It can be shown that for β| = ∞1 this r | = r , as well as for β − max 0 → we have that rmax r0. The sphere rmax has a −∞maximum value for β = →4.745695219 where it takes the − value rmax = 1.232835973 r0. In Fig. (1) we show the FIG. 2: Plots of the embedding function z(r) for various val- behaviour of rmax for β 1 and r0 = 1. ues of the β-parameter are shown. The throat width has been In order to study≤− the shape and the size of set to r = 1. The embedding of each slice t = const, θ = π , 0 2 slices t = const, θ = π of the metric (22) β = const of the wormhole (22) extends from the throat 2 we shall consider for the β-parameter values r0 = 1 to rmax > r0. The heigh of the z-function in- β = 2, 3, 4.745695219, 10, 15, 30, 100 < 1. creases with decreasing β-parameter. It becomes clear that − − − − − − − − as β → −∞ the rmax → r0 = 1. These embeddings are shown in Fig. (2). For a full visualization of the surfaces the diagrams must be ro- tated about the vertical z-axis. We conclude from these diagrams that for β < 1 the metric (22) has typical wormhole shapes, i.e. presence− of a global minimum at r = r0, where the throat of the wormhole is located. The radial extension for all embeddings is finite as we would expect, and the heigh of the z-function increases with decreasing β-parameter. In Fig. (3) we show three-dimensional wormhole embedding diagrams for the values of the β-parameter 2, 3, 4.745695219, 10. It becomes clear that all embedding− − − surfaces flare− outward. This can be seen also from the fact that for the metric (22) the inverse of the embedding function r(z) satisfies

d2r/dz2 = 2β r(β2 2β 1)2 r FIG. 3: The ﬁgure shows three dimensional wormhole em- − − − r0 2 , (29) bedding diagrams for β = −2, −3, −4.745695219, −10. The 2β2 2(2β+1) heighs of the diagrams increase with decreasing β-parameter. 2 r 1+β 2 r 1+β r0(1 + β) r (β 2β) r The throat width is the same for all diagrams. 0 − − 0 ! implying that at the throat we have that

2 2 1 2(1+2β−β2) d r/dz r0 = , (30) β+1 | −r0(1 + β) β(β 2) r dz(r) r0 = v − − . (27) which is positive for any β < 1. Thus, the required u 2(1+2 β−β2) dr u β+1 − u r 1 ﬂare-out condition for the wormhole throat is satisﬁed [1]. u r0 u − Note that Eq. (30) implies that for β we have t d2r/dz2 0. By taking into account→ that−∞ we have dz(r) r0 This expression implies that at the throat = , also that| r → r if β , then we may conclude dr ∞ max 0 and it vanishes for β = 1 √2 > 1, so we have the that the shape→ of the wormhole→ −∞ embedding becomes a ± − standard behaviour of this derivative at the wormhole cylinder of radius r0 for big negative values of the β- throat. On the other hand, we can see from Eq. (27) parameter. 7

In conclusion, the metric (22) describes a wormhole ge- dition pr = pl must be required for radial and lateral ometry with isotropic pressure wich extends from r = r0 pressures. to r = . Since these wormholes are not asymptotically We show that it is not possible to sustain a zero-tidal- ∞ flat, and the embedding in ordinary three dimensional force wormhole by a perfect fluid, thus a single fluid Euclidean space extends from r0 to rmax, we may match threading a zero-tidal-force wormhole must be necessar- them, as an interior spacetime, to an exterior vacuum ily anisotropic. This implies that if we want to generate spacetime at the finite junction surface r = rmax. spherically symmetric wormholes, sustained by a single From Eqs. (23) and (24) we conclude that at the throat matter source with isotropic pressure, we must consider we have for the energy density that ρ(r0) < 0 if 3 < spacetimes with φ(r) = const. − 6 β < 1 and ρ(r0) 0 for β 3, while for the pressure Also we discuss the possibility of having isotropic fluids we obtain− p(r ) =≥ 1/r2 <≤−0. On the other hand we 0 − 0 with a linear equations of state. In particular, we show have that that a wormhole with a power-law shape function cannot

2− − be supported by an ideal fluid with a linear equation of − 2(β 2β 1) r 1+β state. Therefore we consider more general forms for the 2 r (2β + 1)(β 1) ρ + p = 0 − (31) shape function and the equation of state of the isotropic (1 + β)(β2 2β 1)r2 − − − pressure. 2β In this manner, we generate and discuss the general , (β2 2β 1)r2 solution for a non-asymptotically flat family of static − − wormholes characterized by a redshift function given by 2 then at the throat ρ + p = 2 is fulfilled. Thus for Eq. (16). The obtained wormhole solutions exhibit al- (1+β)r0 β < 1 we have that always ρ + p < 0, which allows us ways a solid angle deficit and do not satisfy energy con- to conclude− that the energy conditions are not satisfied ditions. The embeddings of these spacetimes in ordinary at the wormhole throat. three dimensional Euclidean space have a finite size since Since we are interested in studying static wormhole they extend from r0 up to a maximum radial value rmax configurations, for which we must require β < 1, we given by Eq. (28). However, notice that the metric (22), − or equivalently the metric (25), is well behaved for r r , conclude that we have wormholes only exhibiting a solid ≥ 0 angle deficit, for which we obtain that 0 < 1 < including the sphere rmax. For r rmax wormhole slices β2−2β−1 ≥ 1/2. The polar and azimuthal angles are restricted to cannot be embedded in an ordinary Euclidean space. In- 0 θ π and 0 φ < 2π, respectively, therefore 0 stead, a space with indefinite metric must be used. It ≤ ≤ ≤ ≤ is interesting to note that one may match such a non- θ˜ π/√2 and 0 φ<˜ √2π, where θ˜ = θ/ β2 2β 1 ≤ ≤ − − asymptotically flat wormhole, as an interior spacetime, and φ˜ = φ/ β2 2β 1. − − p to an exterior vacuum spacetime at the finite junction p surface r = rmax. VI. CONCLUSIONS

The study of spherically symmetric traversable worm- VII. ACKNOWLEDGEMENTS holes in General Relativity has been mostly focused in sources with anisotropic pressures. In this work we This work was supported by Dirección de Investigación present and discuss static spherical wormhole spacetimes de la Universidad del B´ıo-B´ıothrough grants N0 DIUBB supported by a single perfect fluid, for which the con- 140708 4/R and N0 GI 150407/VC.

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