Intra--Galactic thin shell wormhole and its stability

0.2 MB·English

by  Ivana Bochicchio

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 sym-
metric 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 an-
alyzed 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 <Rsmtaabxle <a0 <rH1 (30) σ(a)=−rA+ ea24C , (39)
if the cosmologicalconstant is positive and k >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<=kk<CkC case, the zero of the Weierstrass function aI is a simple
k>kC 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 ∀ 0<a<a and
Z
∀ a>a buttherespectivebehaviorsofthewormholeare
Z
different; ii)when 0<k<kc, the function Φ(a) has two
simple zeros at a¯1 and a¯2 (i.e. there are two inversion
pointsa¯1anda¯2). Theexpansionispossible∀ 0<a≤a¯1
and ∀a ≥ a¯2; iii) when k > 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≤a is possible.
I
uation: if a = a and a < a , the wormhole goes on
0 Z 0 Z
6
expanding asymptotically approaching the static model
with a as radius. On the contrary, for a > 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¯ <a <a¯ ,sinceinthiscasetheWeierstrass Ouranalysisincluded,amongothermathematicallyal-
1 0 2
function is negative: the initial radius of the wormhole lowed values, also the well accepted physical values of k
8
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and P. Salgado, Phys. Rev. D bf79, 024005 (2009).
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× ×
Rev. D68,064004(2003);F.S.N.LoboandP.Crawford,
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without the hypothesis of dark matter. For the mass of [11] L.A. Anchordoqui, S.P. Bergliaffa and D. F. Torres,
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