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Theoretical construction of stable traversable wormholes Peter K.F. Kuhfittig Department of Mathematics Milwaukee School of Engineering Milwaukee, Wisconsin 53202-3109 USA Itisshowninthispaperthatitispossible,atleastinprinciple,toconstructatraversablewormhole that is stable to linearized radial perturbations by specifying relatively simple conditions on the shape and redshift functions. PAC numbers: 04.20.Jb,04.20.Gz II. TRAVERSABLE WORMHOLES Using units in which c = G = 1, the interior wormhole 0 1 geometry is given by the following metric [1]: 0 dr2 2 ds2 = e2Φ(r)dt2+ +r2(dθ2+sin2θdφ2). (1) − 1 b(r)/r n I. INTRODUCTION − u The motivation for this idea is the Schwarzschild line J element 3 Wormholes may be defined as handles or tunnels in the spacetime topology linking widely separated regions of ds2 = 1 2M dt2+ dr2 +r2(dθ2+sin2θdφ2). c] our Universe or of different universes altogether. That −(cid:18) − r (cid:19) 1−2M/r q such wormholes may be traversable by humanoid trav- (2) - elers was first conjectured by Morris and Thorne [1] in In Eq. (1), Φ = Φ(r) is referred to as the redshift func- r g 1988. To hold a wormhole open, violations of certain tion,whichmustbeeverywherefinitetopreventanevent [ energy conditions must be tolerated. horizon. The function b = b(r) is usually referred to as 2 the shape function. The minimum radius r = r0 is the Another frequently discussed topic is stability, that throat of the wormhole, where b(r ) = r . To hold a v 0 0 3 is, determining whether a wormhole is stable when sub- wormholeopen, the weak energy condition (WEC) must 2 jected to linearized perturbations around a static solu- be violated. (The WEC requires the stress-energy ten- 4 tion. Much of the earlier work concentrated on thin- sor T to obey T µαµβ 0 for all time-like vectors αβ αβ 3 shell Schwarzschild wormholes using the cut-and-paste and, by continuity, all null≥vectors.) As a result, the 3. technique [2]. In this paper we are more interested shape function must obey the additional flare-outcondi- 0 in constructing wormhole solutions by matching an in- tion b′(r ) < 1 [1]. For r > r , we must have b(r) < r, 0 0 9 terior traversable wormhole geometry with an exterior while limr→∞b(r)/r = 0 (asymptotic flatness). Well 0 Schwarzschild vacuum solution and examining the junc- away from the throat both Φ and b need to be adjusted, v: tionsurface. (Forfurtherdiscussionofthisapproach,see as we will see. i Refs. [3–8].) A linearized stability analysis of thin-shell TheneedtoviolatetheWECwasfirstnotedinRef.[1]. X wormholes with a cosmologicalconstant can be found in Awell-knownmechanismforthisviolationistheCasimir r Ref. [9],whileRef. [10]discussesthestabilityofphantom effect. Other possibilities are phantom energy [18] and a wormholes. In other, related,studies, the Ellis drainhole Chaplygin traversable wormholes [19]. wasfoundtobeunstabletonon-linearperturbations[11] Since the interior wormhole solution is to be matched butstabletolinearperturbations[12]. AccordingtoRefs. with an exterior Schwarzschild solution at the junction [13, 14], however, such wormholes are actually unstable surface r = a, denoted by S, our starting point is the to both types of perturbations. Darmois-Israel formalism [20, 21]: if K is the extrinsic ij curvatureacrossS (alsoknownas the secondfundamen- Aratherdifferentapproachtostabilityanalysisispre- tal form), then the stress-energy tensor Si is given by sented in Ref. [15]. In Ref. [16], an example of a sta- j the Lanczos equations: ble traversable wormhole connecting two branes in the Randall-Sundrum model is considered, while Ref. [17] 1 Si = [Ki ] δi [K] , (3) discusses the instability of scalar wormholes in a cosmo- j −8π j − j logical setting. where [X] = limr→a+X (cid:0) limr→a−X =(cid:1) X+ X−. So The purpose of this paper is to show that it is in prin- [Kij] = Ki+j −Ki−j, which−expresses the discon−tinuity in ciple possible to construct a traversable wormhole that the second fundamental form, and [K] is the trace of is stable to linearized radial perturbations. The condi- [Ki ]. j tions onthe redshift and shape functions at the junction In terms of the energy-density σ and the surface pres- surface are surprisingly simple. sure ,Si =diag( σ, , ).TheLanczosequationsnow P j − P P 2 yield where m = 4πa2σ is the mass of the junction surface, s which is a thin shell in Ref. [10]. 1 σ = [Kθ ] (4) When linearized around a static solution at a = a , −4π θ 0 the solution is stable if, and only if, V(a) has a local and minimum value of zero at a=a , that is, V(a )=0 and 0 0 ′ ′′ V (a ) = 0, and its graph is concave up: V (a ) > 0. 1 0 0 = [Kτ ]+[Kθ ] . (5) For V(a) in Eq. (14), these conditions are met [6]. P 8π τ θ Since the junction surface S is understood to be well (cid:0) (cid:1) A dynamic analysis can be obtained by letting the ra- away from the throat, we expect σ to be positive. diusr =abeafunctionoftime,asinRef.[2]. According Eq. (10) then implies that b(a) < 2M, rather than to Lobo [10], the components of the extrinsic curvature b(a) = 2M, which the Schwarzschild line element (2) are given by might suggest. (One can also say that the interior and M +a.. exterior regions may be separated by a thin shell. The Kτ+ = a2 , (6) reasonfor this is that in its most generalform, the junc- τ 1 2M +a.2 tionformalismjoinstwodistinctspacetimemanifoldsM+ − a andM− withmetricsgivenintermsofindependentlyde- q fined coordinate systems xµ and xµ [6].) What needs to Φ′ 1 b(a) +a.2 +a.. a.2[b(a)−ab′(a)] be emphasized is that eve+n if b(a)−< 2M, b(a) can be Kτ− = − a − 2a[a−b(a)] , (7) arbitrarilyclose to 2M withoutaffecting the aboveanal- τ (cid:16) 1 (cid:17)b(a) +a.2 ysis. In particular, V(a0) = 0 and V′(a0) = 0 even if q − a lima→a0−b(a)=2M,since,bythed′′efinitionofleft-hand and limit, b(a)<2M. The condition V (a0)>0 should now ′′ be written V (a )>0. 0 Kθ+ = 1 1 2M +a.2, (8) − θ a − a r III. THE LINE ELEMENT Kθ− = 1 1 b(a) +a.2. (9) θ a − a Given our aim, the construction of a stable wormhole, r ′ ourmainrequirementcannowbestatedasfollows: apart For future use let us also obtain σ : from from the usual conditions at the throat, we require that σ = 1 (Kθ+ Kθ−)= b = b(r) be an increasing function of r having a contin- −4π θ − θ uous second derivative and reaching a maximum value ′ at some r = a. In other words, we require that b(r) 1 1 2M +a.2 1 b(a) +a.2 , (10) approach zero continuously as r a (Fig. 1). Keep- − 4πa r − a −r − a ! ing in mind the Schwarzschild lin→e element (2), we let one can calculate σ. 1 1 3M +a.2 aa.. σ′ = . = − a − a 4πa2  1 2M +a.2 − a 1q3b(a) + b′(a) +a.2 aa.. − 2a 2 − . (11) − 1 b(a) +a.2  − a q  AgainfollowingLobo[10],rewritingEq.(10)inthe form 2M 2M .2 b(a) .2 1 +a = 1 +a 4πσa (12) − a − a − r r ro a will yield the following equation of motion: .2 a +V(a)=0. (13) Here V(a) is the potential, which can be put into the following convenient form: FIG.1: Theinteriorshapefunctionattainsamaximumvalue at r=a. 1b(a)+M m2 M 1b(a) 2 V(a)=1 2 s − 2 , (14) − a − 4a2 − m b(r) = 2M for r > a since M = 1b(a). In this manner, (cid:18) s (cid:19) 2 3 ′ both b(r) and b(r) remain continuous across the junc- So by Eq. (17). tion surface r =a. It follows directly from Eq. (10) that σ = 0 at r = a. It is also desirable to have = 0 at d2 r = a, thereby making Sij = 0. To this end wPe choose dr2gµν(a±)= Φ(r) so that d2 d2 Θ(r a) g (a+)+Θ[ (r a)] g (a ). ′ M − dr2 µν − − dr2 µν − Φ(a )= . (15) − a(a 2M) − Up to the second derivatives, then, the δ-function does [Of course, Φ(r) must still be finite at the throat, while not appear, in agreement with Visser [21]: by adopting Φ(a ) = Φ(a+).] For r > a, Φ(r) = 1ln(1 2M), so Gaussiannormalcoordinates,thetotalstress-energyten- tahnadt−KΦθ′(+a−)K=θ−Φ=′(a0+a)t.rW=ea.noSwo ha=ve0K2atττ+r−=−Kabτrτ−yE=qs0. smoar,ywbheicwhriatltseondienptehnedsfoornmthTe se=coδn(dη)dSeriv+atiΘve(sη)oTf+gµν+, θ − θ P µν µν µν (5)-(9), as desired. Θ( η)T−, thereby showing the δ-function contribution FortheabovechoiceofΦ(r),theresultinglineelement at−the lµoνcation of the thin shell; here σ is necessarily is greater than zero. In our situation, however, S = 0 at ij dr2 the boundarysurface,sothat,onceagain,the δ-function ds2 = e2Φ(r)dt2+ +r2(dθ2+sin2θdφ2), does not appear. − 1 b(r)/r − Even more critical in the stability analysis is the need r a, to study the secondderivativeofV(a) in Eq.(14). Since ≤ V′′(a) involvesm′′ =(4πa2σ)′′, let us firstuseEq.(4)to s dr2 write σ′ in the following form: ds2 = e2Φ(r)dt2+ +r2(dθ2+sin2θdφ2), − 1 b(a)/r − r >a. (16) σ′ = 1 Θ(r a) d Kθ++Θ[ (r a)] d Kθ− . −4π − dr θ − − dr θ (cid:18) (cid:19) Note especially that As long as b(r) is an increasing function without the as- d d d d ′ g (a )= g (a+) and g (a )= g (a+). sumed maximum value at r = a, σ will have a jump tt tt rr rr dr − dr dr − dr ′′ discontinuity at r = a. So σ is equal to δ(r a) times (17) ′ − the magnitude of the jump [22]. If b(a ) = 0, on the Since the components of the stress-energy tensor are 0 other hand, the calculations leading to Eq. (11) show equal to zero at S, the junction is a boundary surface, ′ that σ is continuous at a = a . It follows that there is ratherthanathinshell [10],andK iscontinuousacross 0 ij ′′ no δ-function in the expression for V (a ). S. 0 Without the δ-function, one cannot simply declare 4πa2σ to be the (finite) mass of the spherical surface IV. STABILITY r = a, since the thickness of an ordinary surface is undefined. (It is quite another matter to assert that dm = 4πσa2da, which can indeed be integrated over a As noted at the end of Sec. II, V(a ) = 0 and ′ 0− finite interval.) V (a0−) = 0 even if lima→a0−b(a) = 2M, since b(a) < Returning to Eq. (11), when σ′ is evaluated at the 2M. In Sec. III we saw that in the absence of surface ′ ′ static solution, then b(a ) = 0 implies that σ (a ) = 0. stresses our junction is a boundary surface, rather than 0 0 ′ So σ approaches zero, its minimum value, continuously a thin shell: since b(r) goes to zero continuously as asa a ,and,asaconsequence,σ >0intheopenin- r a , b(r) continues smoothly at r = a to become → 0− 2M→to−the right of a (Fig. 1). This implies that the terval(a0−ǫ,a0);hereǫisarbitrarilysmall,butfinite(as opposedtoinfinitesimal). Asaresult,σisapproximately usual thin-shell formalism using the δ-function is not di- constant, but nonzero, in the boundary layer extending rectly applicable. To show this, suppose we write the from r = a ǫ to r = a . So for a (a ǫ,a ), derivatives in Eq. (17) in the following form: m = 4πa2σ0is−a positive co0nstant, but o∈ne th0at−can0be s d g =Θ(r a) d g+ (r)+Φ[ (r a)]g− (r), made as small as we please. Referring back to Eq. (14), dr µν − dr µν − − µν we now find the second derivative of V, making use of ′ where Θ is the Heaviside step function. Then by the the condition b(a0)=0. Since ms is fixed, we get product rule, 1b′′(a ) b(a )+2M d2 V′′(a )= 2 0− 0− g (a )= 0− − a − (a )3 dr2 µν ± 0− 0− d2 d2 3m2s + b′′(a0−)[M − 21b(a0−)]. (18) Θ(r−a)dr2gµν(a+)+Θ[−(r−a)]dr2gµν(a−) − 2(a0−)4 m2s d d +δ(r a) gµν(a+) gµν(a ) . Sincems isarbitrarilysmall,butnonzero,thethirdterm − dr − dr − on the right-hand side is arbitrarily close to zero, while (cid:18) (cid:19) 4 ′′ the last term is equal to zero. From V (a ) > 0, we toanexteriorSchwarzschildsolutionatthejunctionsur- 0 − obtain face r = a, where Φ and b must meet the conditions discussed in Sec. III. Our main conclusion is that the 2[b(a )+2M] b′′(a )< 0− . wormhole is stable to linearized radial perturbations if 0− − (a0 )2 b=b(r) satisfies the following condition at the static so- − lution a = a : b′′(a ) < 8M/a2, where M is the total Using our arbitrary ǫ, we can also say that 0 0 − 0 mass of the wormhole in one asymptotic region. ′′ Since the curve b = b(r) is concave down, b (a ) < 0, ′′ 2[b(a0 ǫ)+2M] 0 b (a0−ǫ)<− (a− ǫ)2 . but its curvature has to be sufficiently large in absolute 0− value to overtake 8M/a20 = 4b(a0)/a20. This condition is The continuity of b(r) and a2 now implies that simple enough to suggest that the form of b(r) can be easily adjusted “by hand.” 8M A function that meets the condition locally can also ′′ b (a )< . (19) 0 − a2 be obtained by converting the above inequality to the 0 differential equation This is the stability criterion. 4b(r) ′′ b (r)+ = λ, r2 − V. DISCUSSION whereλis a smallpositiveconstant. Confiningourselves to the interval (a ,a ], a solution is 1 0 This paper discusses the stability of Morris-Thorne and 1 other traversable wormholes, each having the metric b(r)=c√rsin √15 lnr λr2. given by Eq. (1), where Φ(r) and b(r) are the redshift 2 − (cid:18) (cid:19) and shape functions, respectively. The shape function is assumed to satisfy the usual flare-out conditions at the Totheleftofa ,b(r)canbejoinedsmoothlytoafunction 1 throat, while the redshift function is assumed to be fi- thatmeetstherequiredconditionsatthethroat,thereby nite. 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