Intra–Galactic thin shell wormhole and its stability Ivana Bochicchio∗ and Ettore Laserra† Dipartimento di Matematica, Universita´ degli Studi di Salerno, Via Ponte Don Melillo, 84084 Fisciano (SA), Italy In this paper, we construct an intra-galactic thin shell wormhole joining two copies of identical galacticspacetimesdescribedbytheMannheim-KazanasdeSittersolutioninconformalgravityand studyitsstabilityundersphericalperturbations. Weassumethethinshellmaterial asaChaplygin gas and discuss in detail thevalues of therelevant parameters underwhich thewormhole is stable. WestudythestabilityfollowingthemethodbyEiroaandwealsoqualitativelyanalyzethedynamics through themethod of Weierstrass. Wefind that thewormhole is generally unstablebut thereis a small intervalin radius for which thewormhole is stable. PACSnumbers: 04.20.-q,04.20.Jb,98.80.-k 3 1 0 INTRODUCTION The equation of state of the material making the shell 2 is left to one’s choice. We choose the equation of state n of the Chaplygin gas because the free parameter in it Wormholesaretopologicalhandlesconnectingtwodis- a can encompass different types of matter. The Chaply- J tant regions of spacetime. These objects are solutions of gin gas has recently considered by several authors (see, the Einstein equations and have been known for a long 8 for example, Refs. [14, 17] for a wormhole with this time [1, 2], but they are intensely investigated by physi- model for the matter and Refs. [18, 19] to deeply un- ] cists in recent times (see [3] for an extensive discussion) c derstand the reasons for its introduction). A physically aftertheseminalworkbyMorrisandThorne[4]in1988. q meaningful spherically symmetric galactic spacetime is - We can mention only a few with the understanding that r the Mannheim-Kazanas de Sitter solution (MKdS) [20] g the list is by no means exhaustive. Dynamical worm- in conformal gravity. This solution has the special fea- [ holes were discovered and studied in Refs. [5–7] and ture that it does notrequire the hypothesis of darkmat- wormholesincosmologicalsettingswerecontemplatedin 1 ter in the galactic halo region, yet predicts the observed variousworks([8,9]andreferencestherein), withpartic- v flat rotation curves and also allows the computation of 2 ularattentionbeingpaidtowormholeswithcosmological the sizes of galaxies[21, 22] without requiring additional 3 constantΛ,whichareasymptoticallyde Sitterorantide 5 Sitter according to the sign of Λ [10]. These geometrical assumptions. We choose two copies of the MKdS solu- 1 tionforourconstructionofintra-galacticwormhole. The constructionsaredesignedtoallowtravelerstopassfrom . paper is organized as follows: The next section details 1 one regionofspacetime to anotherorevenfromoneuni- 0 how to construct the thin-shell wormhole from spheri- verse to another. It is also to be noted that wormholes 3 callysymmetricspacetimefollowingthemethodbyEiroa are as valid solutions of Einstein’s theory of gravity as 1 [23], which include equations that should be satisfied by : are the black holes in the sense that no experiment has v the throat radius and the energy conservation. Then we so far ruled out the existence of macroscopic wormhole i work out the expression for the potential energy in or- X structures in classical gravity. In the high energy ex- der to construct the Weierstrass equation. In Sect. 3, r periments, such as LHC, microscopic wormholes are a a distinctpossibility. Ontheotherhand,itwouldbeofin- the formalism is applied to a particular spherically symmetric metric, namely, the MKdS metric. We assume terest to know if there could exist wormholes connecting that thin-shell matter is in the form of a Chaplygin gas. two spacetimes of spherical symmetric galaxies. We are Under these conditions, we consider the equations that interestedhereintraversablewormholesbutthesituation give the possible radii of the wormhole and determine isthat,ahumanlytraversablewormholesolutioncovered their stability under spherical perturbations. In Sec 4., bysingleregularcoordinatechartisgenerallyunavailable we present the general Weierstrass criterion and in Sec except in a few cases such as the Ellis wormhole [1] and 5., we investigate the dynamical behavior of thin shell. inaparticularclassofsolutionintheBrans-Dicketheory Certain specific values of relevant parameters are also [11]. Thus, one has to use the cut and paste technology considered. Finally, in Sec. 6, the conclusions of this developed by Visser [3, 12], where one joins two copies work are summarized. of spacetimes across a throat threaded by a thin-shell of matter, to construct a traversable wormhole. Some re- centarticlesrelatedtothin-shellwormholescanbefound in Refs. [13–15]. Note that in [15] a solution describing a wormholeshell joining two identicalLemaitre-Tolman- Bondi (LTB) universes is analyzed (see [16] and refer- ences therein for some remarks on these Universes). 2 PRELIMINARY EQUATIONS and f(a)+a˙2 2a¨+f′(a) h′(a) The spherically symmetric line element in polar coor- p=pθ =pϕ = p 8π (cid:20)f(a)+a˙2 + h(a)(cid:21). (7) dinates (t,r,θ,ϕ) is b b Inparticular,for static wormholeswith a as radius,the 0 ds2 = f(r)dt2+f(r)−1dr2+h(r)(dθ2+sin2θdϕ2) (1) surface energy density and the pressure are obtained by − the previous equations putting a˙ = 0. 0 where h(r) is alwayspositive andf(r) is a positive func- Moreover,thenegativesignin(6)plustheflare–outcon- tion for a given radius. ditionh′(a)>0impliesσ <0,indicatingthatthematter Consider now a wormholeshell Σ at r =a which joins at the throat is ”exotic”. two identical copies of the space time. Thin shell worm- In order to introduce the equationof state for the exotic holes are not creatednaturally as the result of a dynam- matter at the throat, it is possible to suppose that such ical process after the big bang but are to be constructed matter is a generalized Chaplygin gas. For this gas, the by cut and paste method as follows. Thus we choose a pressurehasoppositesigntotheenergydensity,resulting radius a greaterthan the event horizonradius r (if the H a positive pressure: geometry(1)hasany)andcuttwoidenticalcopiesofthe region with r a A ≥ p = , (8) σ α ± = Xα =(t,r,θ,ϕ)/r a , (2) | | M { ≥ } where A>0 and 0<α 1.When α = 1 the Chaplygin ≤ and paste them at the two-dimensional hypersurface gas equation of state is recovered. Replacing Eqs. (6) and (7) in (8), it is possible to obtain the differential Σ Σ± = X/F(r) = r a=0 . (3) equation that should be satisfied by the throat radius ≡ { − } of the thin shell wormhole (see [23]). In particular, for In this way, a new manifold M = M+∪M− is created static wormholes, it becomes: with two mouths on each side. If h′(r) > 0 (condition of flare out), this construction [f′(a )h(a )+f(a )h′(a )][h′(a )]α 0 0 0 0 0 creates a geodesically complete manifold representing a (9) wormhole with two regions connected by a throat of ra- 2A[4πh(a0)]α+1[f(a0)](1−α)/2 =0, − dius a, where the surface of minimal area is located (see with the condition a > r if the original metric has 0 H [23]). Since the spacetime is not asymptotically flat, the horizons. resulting wormhole will have the topology of a dumbell. Moreover,fromEqs. (6)and(7),theenergyconservation In order to know the geometry on the thin shell con- equation reads [23]: necting the two sides, we define on the shell coordinates ξi = (τ,θ,ϕ),withτ thepropertimeontheshell. More- d d a˙ f(a)+a˙2 (σ )+p A = [h′(a)]2 2h(a)h′′(a) , over,usingtheSen-Darmois-Israelformalism[24]andin- dτ A dτ − p2h(a) (cid:8) (cid:9) troducing the unit normal to Σ in , it is possible to (10) M obtainthefollowingexpressionforthesecondfundamen- where = 4πh(a) is the area of the wormhole throat, tal form (or extrinsic curvature) [23]: d (σ )AtheinternalenergychangeofthethroatandpdA dτ A dτ the work done by the internal forces of the throat; the h′(a) K± = K± = f(a)+a˙2 , (4) r.h.s. representsthe flux. Hence Eq. (10)canbe written θθ ϕϕ ±2h(a) p as (see [23]) bb bb σ K± = f′(a)+2a¨ , (5) h(a)σ′+h′(a)(σ+p)+(cid:8)[h′(a)]2−2h(a)h′′(a)(cid:9)2h′(a) =0. ττ ±2h f(a)+a˙2 (11) bb p By the equation of state, p is a function of σ, thus Eq. where the orthonormalbasis {eτ =eτ,eθ =a−1eθ,eϕ = (11) is a first order differential equation for which an (asinθ)−1eϕ} has been adoptedband whbere a prime abnd unique solution with a given initial condition always ex- the dot represent, respectively, the derivative with re- ists. Then Eq. (11) can be integrated to obtain σ(a). spect to r and τ. Replacingσ(a)inEq. (6),thedynamicsofthewormhole Introducing the surface stress-energy tensor S = throat is completely determined by the single equation ij diag(σ,p ,p ),whereσisthesurfaceenergydensityand θ ϕ bb a˙2 = V(a), (12) p ,p arethetransversepressures,theLanczosequations θ ϕ b b − give [23]: b b with f(a)+a˙2h′(a) h(a) 2 σ = ; (6) V(a) = f(a) 16π2 σ(a) . (13) −p 4π h(a) − (cid:20)h′(a) (cid:21) 3 Equation(12) is the Weierstrass equationand it will analyzed in the Sects. 4 and 5 in order to study the dy- namics of our constructed wormhole. Finally, in order to study the stability of the static wormhole one has to r2 =2 γ2+3k cosθ + γ , (20) consider the second derivative of the potential.[26] Pre- H r 9k2 3 3k cisely,observingthatthewormholeisstableunderradial perturbation if and only if V′′(a ) > 0, it will be useful 0 where r2 >r1, recall the following expression [23]: H H V′′(a0) = θ = arctan( 2 ∆1/q1), (21) f′′(a )+ (α−1)[f′(a0)]2 + (1−α)h′(a0) + αh′′(a0) f′(a ) − p− 0 2f(a0) h 2h(a0) h′(a0) i 0 +(α+1) h′′(a0) αh′(a0) 2 f(a ). 8Mγ3+γ2+36kMγ 108k2M2+4k (cid:20) h(a0) −(cid:16) h(a0) (cid:17) (cid:21) 0 ∆1 = − 108k4− , (22) (14) The next section will be devoted to constructing the 2γ3 9kγ+54k2M thin-shell wormhole in the MKdS spacetime. First, we q = − − . (23) will investigatethe eventhorizonsof the geometry in or- 1 27k3 der to find the right interval of radial coordinate. Next, we willemploy Eiroa’sconsiderationsusing a recentnew result on the maximal stable galactic size. Then, after fHrL fHrL fHrL the analysis of stability of static wormhole through the standard potential approach, we will exploit the Weier- r1H r2H r (cid:143)rH r ~rH r strassmethodto classifyalsothe dynamicsofthe MKdS thin wormhole. FIG. 1: The three pictures represent the graph of the metric function f(r) = 1− 2M +γr−kr2 for different r value of k. The red curve describes f(r) when 0 < k < MANNHEIMW-KOARZMAHNOALSE-DE SITTER η1+η2. ThetwohorizonsaremarkedbyrH1 andrH2. The greencurverepresent themetricfunctionfork<η1−η2. Insuch case the geometry has onlyone horizon placed at ThesphericallysymmetricMannheim-Kazanas-deSit- r¯H. The blue curve is f(r) for 0 > k > η1 −η2. The ter (MKdS) metric line element in polar coordinates unique horizon is marked by r˜H. (t,r,θ,ϕ) and in units G = 1, vacuum speed of light c=1, is Whenkisnegativeandsuchthatk <η η theevent 1 2 − horizon is placed at 2M ds2 = 1 +γr kr2 dt2 −(cid:18) − r − (cid:19) (15) q q γ + 1 2M +γr kr2 −1dr2+r2dΩ2 r¯H = r3 − 22 − ∆2+r3 ∆2− 22 − 3k . (24) − r − p p | | (cid:0) (cid:1) where M is the mass, k( Λ) is of the order of the cos- Onthe contrary,when0>k>η η the eventhorizon ≡ 3 1− 2 mologicalconstantandγ >0istheconformalparameter is placed at in the (MKdS) solution of Weyl gravity . If the cosmological constant is positive and k > η +η , γ2 3k θ+4π γ 1 2 r˜ = 2 − | |cos (25) where H r 9k2 3 − 3k | | 9Mγ+1 η = , (16) where 1 54M2 2√ ∆ 2 θ = arctan− − , (26) 108M2(8Mγ+1)γ2+(9Mγ+1)2 q η = , (17) 2 2 p 108M2 the function 108k2M2+4k (1+9Mγ) γ2(1+8γM) ∆ = | | − (27) f(r) = 1 2M +γr kr2 (18) 2 108k4 − r − and is always negative, so we take 0 < k < η +η . In this 1 2 case the geometry has two horizons, which are placed at 2γ3 9k γ 54k2M q = − | | − . (28) 2 27k3 γ2+3k θ+4π γ r1 =2 cos + , (19) H r 9k2 3 3k All these considerations are summarized in Fig. 1 4 The flare out condition holds (h′(r) = 2r > 0) in the cubic in a , hence it can be solved analytically. In the 0 constructed wormhole,hence the spacetime is a geodesi- latter case, one obtains: cally complete manifoldwithtwo regionsconnectedby a 2M¯ throat of radius a, at which the surface of minimal area 1 +γ¯a k¯a2 =0 , (36) − a 0− 0 is located (see [23]). Combining the idea of [23] with the 0 recentnewresultsbyNandiandBhadra[22]onthemax- where max imal stable limit R indicating the size of a galaxy, M 3 stable max M¯ = , γ¯ = γ , k¯=2k+8Aπ2 . (37) we take the radius a greater than R . This limit is 2 2 stable computed from the generic condition for stability, viz., Following the above strategy to locate the horizons and substituting the values of the constants, we can find the g(r) = 2f′2(r) f(r)f′′(r) 3f(r)f′(r)/r <0 . (29) − − solution of Eq. (36) as different allowed values of the static throat radii. Theimportanceofthislimitisthatitshouldberegarded In addition, in the MKdS metric, [h′(a)]2 asthetestableupperlimitonthesizeofagalaxy. Obser- − 2h(a)h′′(a) = 0; hence the flux term in Eq. (10) is zero, vations on galactic sizes have so far respected this limit. max so (10) takes the form of a conservationequation. Thus, TheveryfactthatthereexistsafinitelimitR distin- stable in the case of Chaplygin gas (α=1), it becomes: guishes conformaltheory from some dark matter models because in the latter there is no such limit. These facts 2Aa σ′(a)a2+2σ(a)a =0 . (38) distinguish the present thin shell wormhole from similar − σ(a) others. This equation can be easily integrated. As a general in- Hence, the surface of thin-shell exotic matter (the tegral one obtains [27]: throat) should be limited within the radius range rH2 η +η . where C is the integration constant. Now, replacing Eq. 1 2 On the other hand, if the cosmological constant is nega- (39) into Eq. (13), we can write the explicit expression tive and k satisfy the condition k < η η , the throat for the Weierstrass equation: 1 2 − radius a must satisfy 0 a˙2 = f(a)+8aAπ2+8π2e2C max − a2 (40) a0 >Rstable >r¯H . (31) =ka2 γa 1+ 2M +8aAπ2+8π2e2C . − − a a2 Finally, when 0 > k > η η the radius is placed as 1 2 For sakeof convenience,let’s choose the integrationcon- − follows stant C = log(8π2) and the ”constant of state” A = − 2 max 1 . In such a way, Eq. (40) becomes a0 >Rstable >r˜H . (32) 8π2 2M 1 By replacingEq. (18) inEqs. (6) and(7), andconsid- a˙2 = ka2+a(1 γ) 1+ + . (41) − − a a2 eringthe staticcase,the energydensityandthe pressure Finally, be replacing Eq. (18) in Eq. (14) we obtain at the throat become 2 σ0 =−p−ka30+2πγaa020√+a0a0−2M , (33) V′′(a0)= −4k+ aγ0 + 2((cid:16)α−−12aM)0(cid:18)+2aMa200−(γ2−kkaa0+0)γ+(cid:19)1(cid:17) (42) and +2(α+1)(2M+a0(ak30a20−γa0−1))−2M 2a 2M 4ka3+3γa2 ThesolutionsofEq. (36)correspondtostablewormholes p = 0− − 0 0 . (34) 0 8πa √a ka3+γa2+a 2M if V′′(a0)>0. 0 0 − 0 0 0− p Moreover, from Eq. (9) the throat radius a should sat- 0 Chaplygin gas as the exotic matter on the shell Σ: isfy the equation the analysis of stability 2αaα 4ka3+3γa2+2a 2M 0 − 0 0 0− (cid:0) (cid:1) In order to analyze the stability of the constructed 1−α 22α+3a2(α+1)Aπα+1 ka2+γa +1 2M 2 =0 . wormhole we adopt the Chaplygin gas (α = 1) as the − 0 (cid:16)− 0 0 − a0 (cid:17) exotic matter in the shell Σ. Hence Eq. (42) becomes: (35) For a generalized Chaplygin gas, this equation can be 6M a (3a γ+4) V′′ = − 0 0 . (43) solved numerically, while for α=1 (Chaplygin gas) it is a3 0 5 case@aD case@bD c. Finally, when k is negative and such that k <η1 fVH'r'LHrL fVH'r'LHrL η2, f(r) is positive for r greater than the horizo−n andthebehaviorofthewormholeisveryinteresting (see Fig.3). In fact, contrary to the previous two r1H = (cid:143)a0 r ~rH = (cid:143)a0 r cases, there is a small interval where it is stable. When r¯ a a¯ the wormhole is stable; for all H 0 0 ≤ ≤ the other values a a¯ it becomes unstable. The 0 0 FIG.2: InthesefigurestherelationshipbetweenV′′ (see size of the interval (≥r¯ ,a¯ ) is strictly related to k H 0 Eq. (43)) and f(r) is underlined. On the left, V′′ and and it increases with the decreasing of k. f are plotted for the admissible positive values of k. In this case the radii which make f(r)≥0 lead to V′′ being negative. This means instability for the static wormhole GENERALIZATION OF THE WEIERSTRASS shell. Onthe right, the two functionsare plotted forneg- CRITERION TO THE MKDS–WORMHOLE ative k but greater than the value η1 −η2. Also in this case the second derivative of the potential is negative so that the wormhole shell is unstable. The Weierstrass equation Inthiscase,weobservethatthestabilityanalysisiscom- In this section we will present a generalized form of pletelyindependentofthestateconstantAaswellasthe the classical Weierstrass criterion,which can be applied solutionconstantk. ThismeansthatfordeSitteroranti tomanyproblemsoutsideofclassicalmechanicstoo[25]. deSitterUniverseandforfixedvalueofγ andM wecon- Let’s consider the first order differential equation struct the same “stable interval”: a˙2 =Φ(a) . (45) 2+√4+18γM a 0,a¯ = − . (44) 0 0 ∈(cid:18) 3γ (cid:19) Eq. (45) is called Weierstrass equation and the function Φ(a)iscalledWeierstrass function. The Eq. (45)canbe Now we have to compare the admissible values of r for written as f(r) with the stable interval. In such way the stability problem is completely solved as follows: da = Φ(a) , (46) dτ ± p which can be integrated by separating the variables case@cD x da V''HrL τ(a)= +τ , (47) 0 fHrL ±Za0 Φ(a) p where we choose the sign in agreement with the sign ± of the initial rate a˙ . 0 The importance of the Weierstrass approach is mainly r based on the fact that it is possible to obtain the quali- (cid:143)r (cid:143)a H 0 tativebehaviorofthesolutionsofaWeierstrassequation without integrating it. In such qualitative analysis the FIG. 3: In this figure the relationship between V′′ (see zerosofthe Weierstrassfunctionhavealeadingrole: the Eq. (43)) and f(r) is underlined for negative k but less solutionsoftheWeierstrassequationareconfinedinthose than the value η1 −η2. In this case the admissible val- regionsofthea–axiswherea˙2 0. Theseregionsarede- ues of the static radius are divided into two interval: the ≥ smallest one of stability and the greatest one of instabil- fined by the zeros of the Weierstrass function. These ity. Whenr¯H ≤a0 ≤a¯0,V′′ ispositive: thewormholeis zeros are called barriers, because they cannot be crossed stable; for all the other values a0 ≥a¯0, V′′ changes sign by the solutiona(τ). The barrierssplit the rangeof pos- and the wormhole becomes unstable. sible values for a into allowed (in which Φ(a) 0) and ≥ prohibited intervals (in which Φ(a) 0). If a barrier a B ≤ a. When 0 < k < η + η , f(r) is positive in the is a simple zero of Φ such that 1 2 intervalbetween the horizons,but the function V′′ isnegativeforalla0 ∈(rH1,+∞)(seeFig. 2a.): the Φ(aB)=0 , Φ′(aB)6=0 , (48) shell–wormhole is unstable. then it is called an inversion point because the motion b. When0>k >η1−η2, f(r)ispositiveforr greater reverses its course after reaching it. If a barrier aB is a than the horizon, but the function V′′ is negative multiple zero such that for all a (r˜ ,+ ) (see Fig. 2b.): also in this case the0sh∈ell-wHorm∞hole is unstable. Φ(a )=0 , Φ′(a )=0 , (49) B B 6 then aB separates two allowed intervals and it is called aM a soft barrier. A soft barrier is an asymptotic position, ΦHaL Weierstrass Function,Γ£1 because it is reached in an infinite time. In fact, for a= k>0 a theintegralinEq. (47)becomesdiverging. Finallywe k<0 B recall that an asymptotic position is also an equilibrium position. This means that if at the initial time τ , a = 0 0 a(τ ) = a , then a(τ) = a τ. So, once these zeros 0 B B ∀ arefound,thequalitativebehaviorofthesolutionsofthe Weierstrass equation (45) is completely determined. a The Weierstrass criterion for the evolving MKdS a wormhole M FIG.4: TheWeierstrass functionΦ(a)forγ ≤1intwo Inordertoqualitativelystudythebehaviorofdynamic different cases: i) when k >0 there are no barriers: the MKdS wormhole, let us analyze Eq. (41) which deter- wormhole expansion is possible for all a(τ) belonging to mines the evolution of the dynamic wormhole through the admissible values for a; ii) when k < 0 the function the Weierstrass method. We let a a(τ) and consider Φ(a)hasasimplezeroaM. Sinceitisaninversionpoint the quadratic differential equation (→41) the expansion is possible ∀a∈(0,aM] 2 a˙2 = V(a) = f(a)+16π2 h(a)σ(a) ZEROS OF THE WEIERSTRASS FUNCTION − − hh′(a) i (50) AND DYNAMIC BEHAVIOR OF THE =ka2+(1 γ)a 1+ 2M + 1 , WORMHOLE. − − a a2 that is a Weierstrass equation with Weierstrass function Conformal parameter γ ≤1 2M 1 Φ(a)=ka2+(1 γ)a 1+ + (51) The positivity of the coefficient of a in Eq. (51) im- − − a a2 plies a forever expanding wormhole for positive k, while a different and more complex situation is obtained for depending on the parameter a. Equation (50) translates negative k. into two equations When k is positive, the Weierstrass function has no real and positive zeros. Hence, having no barriers, the ddτa =± Φ(a) (52) expansion of the wormhole is monotonic: if the MKdS = pka2+(1 γ)a 1+ 2M + 1 , wormhole is initially expanding, it will go on expanding ±q − − a a2 without limit. On the contrary, when k is negative the Weierstrass function (51) has a simple zero a . Since wherewehavetochoosethesign inagreementwiththe M ± simple zeros are inversion points, the wormhole can ex- sign of the initial rate a˙ . The zeros of the Weierstrass 0 panduntilitsthroatradiusreachesthemaximumexpan- function (51) depend on the sign of k (and hence of Λ) sion radius given by a . After reaching this value, the and on the fixed value of the conformal parameter γ. M radius of wormhole throat decreases. Thus, in a finite These different situations will be analyzed in detail in time the wormhole could collapse. These two cases are the followingsections,wherethe zerosofthe Weierstrass represented in Fig. 4. function are obtained by finding the positive real roots of the fourth degree equation Conformal parameter γ >1 ka4+(1 γ)a3 a2+2Ma+1=0 . (53) − − FornegativevalueofthecoefficientofainEq. (51),the Note that since featuresoftheWeierstrassfunctiondependonthesignof 1 k. For positive value of k, three different situations can Φ(a) = f(a)+a+ , (54) occur: Φ(a) can have no zeros, simple zeros or multiple − a zeros. On the other hand, negative value of k implies if we evaluate f in the zeros of the Weierstrass function, the existence of a soft barrier, hence the behavior of the we obtain positive values. This is in agreement with the wormhole strongly depends on the initial conditions. geometry. When k is positive itis possible to find a criticalvalue k such that when k = k , the Weierstrass function has c c 7 (cid:143)a1 aZ (cid:143)a2 can’t belong to the interval (a¯1,a¯2). These results are ΦHaL summarized in Fig. 5. Weierstrass Function,Γ>1 The lastcase is related to negative values of k. In this 0k<=kkkC one, i.e. an inversion position. Hence, we have Φ(a) 0 ≤ for a a : the evolution is possible for all values 0 ≥ I ≤ a a . Whenγ > 1andk <0,thewormholeisclosed, ≤ I thatis,ifthewormholeshellisinitiallyexpanding,itwill continuetoexpanduntiltheradiusreachesitsmaximum valuea ; thenitwillcontractbackfroma toeventually a I I collapse in a finite time. This case is represented in Fig. 6. (cid:143)a1 aZ (cid:143)a2 ΦHaL a I FIG. 5: The Weierstrass function Φ(a) for γ > 1 and positive k. The behavior of Φ(a) depends on k: i) when Weierstrass Function,Γ>1 k = kc, Φ(a) has a multiple zero at aZ. An equilibrium k<0 position or a limiting position corresponds to it. In the first case, a = a is the unique possible position. In the Z second case, the expansion is possible ∀ 0a buttherespectivebehaviorsofthewormholeare Z different; ii)when 0 kc the Weierstrass function a has no zeros. The expansion ∀a>0 is possible. a I one multiple zero a . In this situation the behavior of Z FIG. 6: The Weierstrass function Φ(a) for negative k the MKdS wormhole depends on the initial conditions: and γ > 1. In this case the function Φ(a) has a simple its evolution is static if the initial radius a0 equals aZ. zero at aI (i.e. there is one inversion point aI) and the On the other hand, the static situation is a limiting sit- expansion ∀ 0 a the Z 0 Z wormhole expands without limit. CONCLUSIONS If k > k , there are no barriers, hence if the MKdS c wormhole is initially expanding (a˙ > 0), it will go on Wehaveconstructedanintra-galacticthinshellworm- 0 expanding without limit. hole joining two copies of identical galactic space times Finally, when 0 < k < k , the Weierstrass function (51) describedbytheMannheim-Kazanas-deSittersolutionin c admits two simple zeros that can be called a¯ and a¯ . conformalgravityandinterpretedthewormholebehavior 1 2 So when the initial radius of the wormhole is less that using analytical methods. The approaches we have used a¯ and the wormhole is initially expanding, it will go on are the Eiroa’s potential approach for stability and the 1 expanding until its radius reaches the maximal expan- Weierstrass method for the dynamic evolution. It was sion value a¯ ; then it will contract back from a¯ . Hence shownthat the wormholeis generallyunstable but there 1 1 it could collapse in a finite time. On the other hand, if isasmallintervalinradiusforwhichthewormholeis sta- the MKdSwormholeis expandingfromthe initialradius ble depending on the different signs of the cosmological greater than a¯ , it will go on expanding. Once more, if constant. 2 a > a¯ and if the expansion velocity is negative, the TheMKdSsolutioncontainstwoarbitraryparameters, 0 2 wormhole is contracting and its radius will continue to γ and k( Λ) that are knownto play prominent roles in ≡ 3 decreaseuntilitreachesthevaluea¯ . Sincea¯ isaninver- the galactic halo. In particular, k provides the global 2 2 sionposition, the wormholewill startto growandit will quadratic potential that is essentially needed to explain continue to grow indefinitely. Note that the evolution of the observed flat rotation curves [21, 22]. In the present the wormhole, either to indefinite expansion or collapse, paper, we have shown how this constant can determine depends strongly on the initial conditions. Moreover, a thestabilityandalsothedynamicalbehaviorofthethin- peculiarsituationisobservedwhentheinitialradiusa is shell wormhole. 0 suchthata¯ k >0 (55) Class.Quant.Grav. 26, 235017 (2009); A. Bhattacharya, − × R. Izmailov, K. K. Nandi Class. Quantum Gravity 26 which, as can be seen, is an admissible physical value (2010); F.S.N. Lobo, M.A. Oliveira, Phys. Rev. D 81, because it falls within the range above. For this value 067501 (2010); N. M. Garcia, Francisco S.N. 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Intra--Galactic thin shell wormhole and its stability

Similar Intra--Galactic thin shell wormhole and its stability