Theoretical construction of stable traversable wormholes

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by  Peter K. F. Kuhfittig

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. Theinteriortraversablewormholesolutionisjoined completing the construction.
[1] M.S. Morris and K.S. Thorne, “Wormholes in spacetime pp. 391-404, 2004.
and their use for interstellar travel: A tool for teaching [10] F.S.N.Lobo,“Stabilityofphantomwormholes,”Physical
generalrelativity,”AmericanJournal ofPhysics,vol.56, Review D, vol. 71, no. 12, Article ID 124022, 9 pages,
no. 5, pp.395-412, 1988. 2005.
[2] E. Poisson and M. Visser, “Thin-shell wormholes: Lin- [11] H. Shinkai and S.A. Hayward, “Fate of the first
earization stability,” Physical Review D, vol. 52, no. 12, traversible wormhole: Black-hole collapse orinflationary
pp.7318-7321, 1995. expansion,”Physical Review D,vol.66,no.4,ArticleID
[3] J.P.S. Lemos, F.S.N. Lobo, and S.Q. de Oliveira, 044005, 9 pages, 2002.
“Morris-Thorne wormholes with cosmological constant,” [12] K.A. Bronnikov, G. Clement, C.P. Constantinidis, and
Physical Review D, vol. 68, no. 6, Article ID 064004, 15 J.C.Fabris,“Structureandstabilityofcoldscalar-tensor
pages, 2003. black holes,” Physics Letters A, vol. 243, no. 3, pp.121-
[4] F.S.N.Lobo,“Surfacestressesonathinshellsurrounding 127, 1998.
atraversablewormhole,”ClassicalandQuantumGravity, [13] J.A. Gonzales, F.S. Guzman, and O. Sarbach, “Insta-
vol. 21, no. 21, pp.4811-4832, 2004. bility of wormholes supported by a ghost scalar field. I.
[5] F.S.N. Lobo, “Stable dark energy stars,” Classical and Linear stability analysis,” Classical and Quantum Grav-
Quantum Gravity, vol. 23, no. 5, pp. 1525-1541, 2006. ity, vol. 26, no. 1, Article ID 015010, 14 pages, 2009.
[6] F.S.N. Lobo and P. Crawford, “Stability analysis of dy- [14] J.A. Gonzales, F.S. Guzman, and O. Sarbach, “Insta-
namic thin shells,” Classical and Quantum Gravity, vol. bility of wormholes supported by a ghost scalar field.
22, no. 22, pp.4869-4885, 2005. II. Nonlinear evolution,” Classical and Quantum Grav-
[7] J.P.S. Lemos and F.S.N. Lobo, “Plane symmetric thin- ity, vol. 26, no. 1, Article ID 015011, 20 pages, 2009.
shell wormholes: solutions and stability,” Physical Re- [15] E.F. Eiroa and C. Simeone, “Stabilty of Chaplygin gas
view D, vol. 78, no. 4, Article ID 044030, 9 pages, 2008. thin-shellwormholes,” Physical Review D,vol.76,no.2,
[8] M. Ishak and K. Lake, “Stability of transparent spheri- Article ID 024021, 8 pages, 2007.
callysymmetricthinshellsandwormholes,”PhysicalRe- [16] R.A.KonoplyaandC.Molina,“Theringingwormholes,”
view D, vol. 65, no. 4, Article ID 044011, 6 pages, 2002. Physical Review D, vol. 71, no. 12, Article ID 124009, 5
[9] F.S.N.LoboandP.Crawford,“Linearizedstabilityanal- pages, 2005.
ysis of thin-shell wormholes with a cosmological con- [17] S.V.SushkovandY.-Z.Zhang,“Scalarwormholesincos-
stant,” Classical and Quantum Gravity, vol. 21, no. 2, mological setting and their instability,” Physical Review
5
D, vol. 77, no. 2, Article ID 024042, 8 pages, 2008. eralrelativity,”NuovoCimento,vol.44B,pp.1-14,1966,
[18] S. Sushkov, “Wormholes supported by a phantom en- (corrections in vol. 48B).
ergy,” Physical Review D, vol. 71, no. 4, Article ID [21] M. Visser, Lorentzian Wormholes: From Einstein to
043520, 5 pages, 2005. Hawking, (New York: American Institute of Physics,
[19] F.S.N. Lobo, “Chaplygin traversable wormholes,” Phys- 1995), chapter14.
ical Review D,vol.73, no.6, ArticleID064028, 9pages, [22] L. Schwartz, The´orie des distributions, (Paris: Hermann
2006. & Cie, 1950).
[20] W.Israel,“Singularhypersurfacesandthinshellsingen-