Thin shells around traversable wormholes

0.33 MB·English

by  Francisco S. N. Lobo

Thin shells around traversable wormholes
Francisco S. N. Lobo∗
Centro de Astronomia e Astrof´ısica da Universidade de Lisboa,
Campo Grande, Ed. C8 1749-016 Lisboa, Portugal
Applying the Darmois-Israel thin shell formalism, we construct static and dynamic thin shells
aroundtraversablewormholes. Firstly,byapplyingthecut-and-pastetechniqueweapplyalinearized
stability analysis to thin-shell wormholes in the presence of a generic cosmological constant. We
find that for large positive values of the cosmological constant, i.e., the Schwarzschild-de Sitter
solution, theregions ofstability significantly increaserelatively totheSchwarzschild case, analyzed
by Poisson and Visser. Secondly, we construct static thin shell solutions by matching an interior
wormhole solution to a vacuum exterior solution at a junction surface. In the spirit of minimizing
the usage of exotic matter we analyze the domains in which the weak and null energy conditions
are satisfied at the junction surface. The characteristics and several physical properties of the
surface stresses are explored, namely, we determine regions where the sign of tangential surface
4
0 pressureispositiveandnegative(surfacetension). Anequationgoverningthebehavioroftheradial
0 pressure across the junction surface is deduced. Specific dimensions of the wormhole, namely, the
2 throat radius and the junction interface radius, are found by taking into account the traversability
conditions, and estimates for thetraversal time and velocity are also determined.
n
a
J

8
1 I. INTRODUCTION

1
v Interestin traversablewormholes,ashypotheticalshortcutsinspacetime, hasbeenrekindledby the classicalpaper
3 by Morris and Thorne [1]. The subject has also served to stimulate research in several branches, for instance, the
8 violation of the energy conditions [2, 3], closed timelike curves and the associated difficulties in causality violation
0 [2, 4, 5], and superluminal travel [6], amongst others.
1
As the violation of the energy conditions is a problematic issue, depending on one’s point of view, it is useful to
0
minimize the usage of exotic matter. For instance, an elegant way of achieving this, is to construct a simple class of
4
wormhole solutions using the cut and paste technique, implemented by Visser [2, 7, 8], in which the exotic matter is
0
/ concentratedatthe wormholethroat. The surfacestress-energytensorcomponents ofthe exoticmatter atthe throat
c
are determined, invoking the Darmois-Israel formalism [9, 10, 11]. These thin-shell wormholes are extremely useful
q
as one may apply a stability analysis for the dynamical cases, either by choosing specific surface equations of state
-
r [12,13,14],orbyconsideringalinearizedstabilityanalysisaroundastaticsolution[15,16],inwhichaparametrization
g
of the stability of equilibrium is defined, so that one does not have to specify a surface equation of state. In [16],
:
v the generalizationofthe Poisson-Visserlinearizedanalysis[15] for thin-shell wormholeswasdone inthe presence of a
i
X non-vanishingcosmologicalconstant. It wasfound that for largepositive values ofΛ, i.e., the Schwarzschild-deSitter
solution, the regions of stability significantly increase relatively to the Schwarzschild case analyzed by Poisson and
r
a Visser.
As an alternative to these thin-shell wormholes, one may also consider that the exotic matter is distributed from
the throat to a radius a, where the solution is matched to an exterior vacuum spacetime. Several simple cases were
analyzed in [1], but one may invoke the Darmois-Israelformalism to consider a broader class of solutions. Thus, the
thin shell confines the exotic matter threading the wormhole to a finite region, with a delta-function distribution of
the stress-energy tensor on the junction surface. The motivation of this analysis resides in minimizing the usage of
exoticmatterthreadingtheinteriorwormholesolution,whichinprinciplemaybe madearbitrarilysmall[17,18],and
imposing that the surface stresses of the thin shell obey the energy conditions. One may generalize and systematize
[19] the particular case of a matching with a constant redshift function and a null surface energy density on the
junction boundary, studied in [20]. A similar analysis for the plane symmetric case, with a negative cosmological
constant, is done in [21]. The plane symmetric traversable wormhole is a natural extension of the topological black
holesolutionsfoundbyLemos[22,23,24,25],uponadditionofexoticmatter. Theseplanesymmetricwormholesmay
beviewedasdomainwallsconnectingdifferentuniverses. Theymayhaveplanartopology,anduponcompactification
of one or two coordinates, cylindrical topology or toroidal topology, respectively.
∗Electronicaddress: flobo@cosmo.fis.fc.ul.pt
2
This paper is organized as follows. In Section II the Darmois-Israel thin shell formalism is briefly reviewed. In
SectionIII, a linearizedstability analysisis studied inthe presenceofa genericcosmologicalconstant[16]. InSection
IVwematchaninteriorwormholespacetimetoanexteriorvacuumsolutionandfindspecificregionswheretheenergy
conditions at the junction are obeyed [19]; we also analyze the physical properties and characteristics of the surface
stresses, namely, we find domains where the tangential surface pressure is positive or negative (surface tension); we
deduce an expression governing the behavior of the radial pressure across the junction boundary; and finally, we
determine specific dimensions ofthe wormholeby takinginto accountthe traversabilityconditionsandfindestimates
for the traversaltime and velocity. Finally, in Section V, we conclude.
II. OVERVIEW OF THE DARMOIS-ISRAEL FORMALISM
Consider two distinct spacetime manifolds, and , with metrics given by g+ (xµ) and g− (xµ), in terms
of independently defined coordinate systems xµMa+nd xµ.MT−he manifolds are boundedµbνy h+ypersurfaµcνes−Σ and Σ ,
+ − + −
respectively, with induced metrics g+ and g−. The hypersurfaces are isometric, i.e., g+(ξ)=g−(ξ)=g (ξ), in terms
ij ij ij ij ij
of the intrinsic coordinates, invariant under the isometry. A single manifold is obtained by gluing together
+
M M
and at their boundaries, i.e., = , with the natural identification of the boundaries Σ=Σ =Σ .
− + − + −
ThMe three holonomic basis vectoMrs e M= ∪∂/M∂ξi tangent to Σ have the following components eµ = ∂xµ/∂ξi,
(i) (i)|± ±
which provide the induced metric on the junction surface by the following scalar product
g =e e =g eµ eν . (1)
ij (i)· (j) µν (i) (j)|±
We shallconsider atimelike junctionsurface Σ,defined by the parametricequationofthe formf(xµ(ξi))=0. The
unit normal 4 vector,nµ, to Σ is defined as
−
∂f ∂f −1/2 ∂f
n = gαβ , (2)
µ ± ∂xα ∂xβ ∂xµ
(cid:12) (cid:12)
(cid:12) (cid:12)
with nµnµ =+1 and nµeµ(i) =0. The Israel forma(cid:12)(cid:12)lism requires(cid:12)(cid:12)that the normals point from M− to M+ [10].
The extrinsic curvature, or the second fundamental form, is defined as K =n eµ eν , or
ij µ;ν (i) (j)
∂2xµ ∂xα ∂xβ
K± = n +Γµ± . (3)
ij − µ ∂ξi∂ξj αβ ∂ξi ∂ξj
(cid:18) (cid:19)
NotethatforthecaseofathinshellK isnotcontinuousacrossΣ,sothatfornotationalconvenience,thediscontinuity
ij
in the second fundamental form is defined as κ =K+ K−.
ij ij − ij
Now, the Lanczos equations follow from the Einstein equations for the hypersurface, and are given by
1
Si = (κi δi κk), (4)
j −8π j − j k
where Si is the surface stress-energy tensor on Σ.
j
The first contracted Gauss-Kodazzi equation or the “Hamiltonian” constraint
1
G nµnν = (K2 K Kij 3R), (5)
µν ij
2 − −
with the Einstein equations provide the evolution identity
SijK = [T nµnν Λ/8π] . (6)
ij µν
− −
The convention [X] X+ X− and X (X+ +X− )/2 is used.
Σ Σ Σ Σ
≡ | − | ≡ | |
The second contracted Gauss-Kodazzi equation or the “ADM” constraint
G eµ nν =Kj K, , (7)
µν (i) i|j − i
with the Lanczos equations gives the conservation identity
Si = T eµ nν . (8)
j|i µν (j)
h i
3
In particular, considering spherical symmetry considerable simplifications occur, namely κi = diag κτ ,κθ,κθ .
j τ θ θ
The surface stress-energy tensor may be written in terms of the surface energy density, σ, and the surface pressure,
p, as Si =diag( σ,p,p). The Lanczos equations then reduce to (cid:0) (cid:1)
j −
1
σ = κθ , (9)
−4π θ
1
p = (κτ +κθ ). (10)
8π τ θ
III. CUT AND PASTE TECHNIQUE: THIN-SHELL WORMHOLES WITH A COSMOLOGICAL
CONSTANT
In this section we shall construct a class of wormhole solutions, in the presence of a cosmological constant, using
the cut-and-paste technique. Consider the unique spherically symmetric vacuum solution, i.e.,
−1
2M Λ 2M Λ
ds2 = 1 r2 dt2+ 1 r2 dr2+r2(dθ2+sin2θdφ2). (11)
− − r − 3 − r − 3
(cid:18) (cid:19) (cid:18) (cid:19)
If Λ>0, the solution is denoted by the Schwarzschild-deSitter metric. For Λ<0, we havethe Schwarzschild-antide
Sitter metric, and of course the specific case of Λ=0 is reduced to the Schwarzschild solution. Note that the metric
(1) is not asymptotically flat as r . Rather, it is asymptotically de Sitter, if Λ > 0, or asymptotically anti-de
→ ∞
Sitter, if Λ<0. But, considering low values of Λ, the metric is almost flat in the range M r 1/√Λ. For values
≪ ≪
below this range, the effects of M dominate, and for values above the range, the effects of Λ dominate, as for large
values of the radial coordinate the large-scale structure of the spacetime must be taken into account.
The specific case of Λ = 0 is reduced to the Schwarzschild solution, with a black hole event horizon at r = 2M.
b
Consider the Schwarzschild-de Sitter spacetime, Λ > 0. If 0 < 9ΛM2 < 1, the factor g(r) = (1 2M/r Λr2/3)
− −
possesses two positive real roots, r and r , corresponding to the black hole and the cosmological event horizons of
b c
the de Sitter spacetime, respectively, given by
r = 2Λ−1/2 cos(α/3), (12)
b
r = 2Λ−1/2 cos(α/3+4π/3), (13)
c
where cosα 3MΛ1/2, with π <α<3π/2. In this domain we have 2M <r <3M and r >3M.
b c
For the Sc≡hw−arzschild-antide Sitter metric, with Λ<0, the factor g(r) = 1 2M/r+ Λr2/3 has only one real
− | |
positive root, r , given by
b
(cid:0) (cid:1)
1/3
3M 1 1
r = 3 1+ 1+ + 3 1 1+ , (14)
b (cid:18)|Λ|(cid:19) vu s 9|Λ|M2 vu −s 9|Λ|M2 
u u
t t 
corresponding to a black hole event horizon, with 0<r <2M.
b
A. The cut-and-paste construction
Giventhis,wemayconstructawormholesolution,usingthecut-and-pastetechnique[2,7,8]. Considertwovacuum
solutions with Λ and remove from each spacetime the region described by
Ω r a a>r , (15)
± ± b
≡{ ≤ | }
where a is a constant and r is the black hole event horizon, corresponding to the Schwarzschild-de Sitter and
b
Schwarzschild-anti de Sitter solutions, equation (12) and equation (14), respectively. The removal of the regions
results in two geodesically incomplete manifolds, with boundaries given by the following timelike hypersurfaces
Σ r =a a>r . (16)
± ± b
≡{ | }
Identifying these two timelike hypersurfaces,Σ =Σ , results in a geodesically complete manifold, with two regions
+ −
connectedbyawormholeandtherespectivethroatsituatedatΣ. Thewormholeconnectstworegions,asymptotically
de Sitter or anti-de Sitter, for Λ>0 and Λ<0, respectively.
4
The intrinsic metric at Σ is given by
ds2 = dτ2+a2(τ)(dθ2+sin2θdφ2), (17)
Σ −
where τ is the proper time as measured by a comoving observer on the wormhole throat.
B. The surface stresses
Theimpositionofsphericalsymmetryissufficienttoconcludethatthereisnogravitationalradiation,independently
of the behavior of the wormhole throat. The position of the throat is given by xµ(τ,θ,φ)=(t(τ),a(τ),θ,φ), and the
respective 4-velocity is
1 2M/a Λa2/3+a˙2
Uµ = − − ,a˙,0,0 , (18)
1 2M/a Λa2/3
p − − !
where the overdot denotes a derivative with respect to τ.
Theunitnormaltothethroatmaybedeterminedbyequation(2)orbythecontractions,Uµn =0andnµn =+1,
µ µ
and is given by
a˙
nµ = , 1 2M/a Λa2/3+a˙2,0,0 . (19)
1 2M/a Λa2/3 − −
(cid:18) − − (cid:19)
p
Usingequation(3),equation(11)andequation(19),thenon-trivialcomponentsoftheextrinsiccurvaturearegiven
by
1
Kθ± = 1 2M/a Λa2/3+a˙2 , (20)
θ ±a − −
pM/a2 Λa/3+a¨
Kτ± = − , (21)
τ ± 1 2M/a Λa2/3+a˙2
− −
p
Thus, the Einstein field equations, equations (9)-(10), with equations (20)-(21), then provide us with the following
surface stresses
1
σ = 1 2M/a Λa2/3+a˙2, (22)
−2πa − −
1 1p M/a 2Λa2/3+a˙2+aa¨
p = − − . (23)
4πa 1 2M/a Λa2/3+a˙2
− −
We also verify that the above equations imply thpe conservation of the surface stress-energy tensor
a˙
σ˙ = 2(σ+p) (24)
− a
or
d(σA) dA
+p =0, (25)
dτ dτ
where A=4πa2 is the area of the wormhole throat. The first term represents the variation of the internal energy of
the throat, and the second term is the work done by the throat’s internal forces.
C. Linearized stability analysis
Equation (22) may be recast into the following dynamical form
2M Λ
a˙2 a2 (2πσa)2 = 1, (26)
− a − 3 − −
5
which determines the motion of the wormhole throat. Considering an equation of state of the form, p = p(σ), the
energy conservation, equation (24), can be integrated to yield
1 dσ
ln(a)= . (27)
−2 σ+p(σ)
Z
This result formally inverted to provide σ = σ(a), can be finally substituted into equation (26). The latter can also
be written as a˙2 = V(a), with the potential defined as
−
2M Λ
V(a)=1 a2 (2πσa)2 . (28)
− a − 3 −
Onemayexplorespecificequationsofstate,butfollowingthePoisson-Visseranalysis[15],weshallconsideralinear
perturbation around a static solution with radius a . The respective values of the surface energy density and the
0
surface pressure, at a , are given by
0
1
σ = 1 2M/a Λa2/3, (29)
0 −2πa − 0− 0
0
q
1 1 M/a 2Λa2/3
p = − 0− 0 . (30)
0
4πa0 1−2M/a0−Λa20/3
One verifies that the surface energy density is alwapys negative, implying the violation of the weak and dominant
energy conditions. One may verify that the null and strong energy conditions are satisfied for a 3M for a generic
0
≤
Λ, for a generic cosmologicalconstant [16].
Linearizingaroundthestablesolutionata=a ,weconsideraTaylorexpansionofV(a)arounda tosecondorder,
0 0
which provides
1
V(a)=V(a )+V′(a )(a a )+ V′′(a )(a a )2+O (a a )3 , (31)
0 0 0 0 0 0
− 2 − −
where the prime denotes a derivative with respect to a, d/da. (cid:2) (cid:3)
Definetheparameterη(σ)=dp/dσ =p′/σ′. Thephysicalinterpretationofηisdiscussedin[15],and√ηisnormally
interpreted as the speed of sound. Evaluated at the static solution, at a = a , using equations (29)-(30), we readily
0
find V(a )=0 and V′(a )=0, with V′′(a ) given by
0 0 0
V′′(a )= 2 2M + Λa2+ M/a0−Λa20/3 2 +(1+2η ) 1 3M , (32)
0 −a20" a0 3 0 1(cid:0)−2M/a0−Λa20(cid:1)/3 0 (cid:18) − a0 (cid:19)#
where η =η(σ ).
0 0
The potential V(a), equation (31), is reduced to
1
V(a)= V′′(a )(a a )2+O (a a )3 , (33)
0 0 0
2 − −
so that the equation of motion for the wormhole throat presents the(cid:2)following(cid:3)form
1
a˙2 = V′′(a )(a a )2+O (a a )3 , (34)
0 0 0
−2 − −
to the order of approximationconsidered. If V′′(a )<0 is verified,(cid:2)then the p(cid:3)otential V(a ) has a localmaximum at
0 0
a , where a small perturbation in the wormhole throat’s radius will provoke an irreversible contraction or expansion
0
of the throat. Thus, the solution is stable if and only if V(a ) has a local minimum at a and V′′(a )>0, i.e.,
0 0 0
3M 1 3M/a +3M2/a2 ΛMa
η 1 < − 0 0− 0 . (35)
0 − a − 2(1 2M/a Λa2/3)
(cid:18) 0 (cid:19) − 0− 0
We need to analyze equation (35) for several cases. The right hand side of equation (35) is always negative [16],
while the left hand side changes sign at a = 3M. Thus, one deduces that the stability regions are dictated by the
0
following inequalities
1 3M/a +3M2/a2 ΛMa
η < − 0 0− 0 , a >3M (36)
0 −2(1 2M/a Λa2/3)(1 3M/a ) 0
− 0− 0 − 0
1 3M/a +3M2/a2 ΛMa
η > − 0 0− 0 , a <3M. (37)
0 −2(1 2M/a Λa2/3)(1 3M/a ) 0
− 0− 0 − 0
One may analyze severalcases.
6
1. De Sitter spacetime.
For the de Sitter spacetime, with M =0 and Λ>0, equation (35) reduces to
1
η < , for 0<a < 3/Λ. (38)
0 −2(1 Λa2/3) 0
− 0
p
The stability region is depicted in the left plot of figure 1.
2. Anti-de Sitter spacetime.
For the anti-de Sitter spacetime, with M =0 and Λ<0, equation (35) gives
1
η < , for a >0, (39)
0 −2(1+ Λ a2/3) 0
| | 0
and the respective stability region is depicted in the right plot of figure 1.
0
0
–0.1
–1 Stable Stable
–0.2
η–2 η–0.3
De Sitter solution –0.4 Anti-de Sitter solution
–3 (Λ> 0) (Λ< 0)
–0.5
–4 0.2 0.4 λ 0.6 0.8 1 –0.6 1 2 δ 3 4 5
FIG. 1: We have defined λ = a0/(3/Λ)1/2 and δ = a0/(3/|Λ|)1/2, respectively. The regions of stability are depicted in the
graphs, below thecurves for thede-Sitterand theanti-deSitter solutions, respectively.
3. Schwarzschild spacetime.
This is the particular case of the Poisson-Visseranalysis [15], with Λ=0, which reduces to
1 3M/a +3M2/a2
η < − 0 0 , a >3M, (40)
0 0
−2(1 2M/a)(1 3M/a )
0
− −
1 3M/a +3M2/a2
η > − 0 0 , a <3M. (41)
0 0
−2(1 2M/a)(1 3M/a )
0
− −
The stability regions are shown in the left plot of figure 2.
4. Schwarzschild-anti de Sitter spacetime.
For the Schwarzschild-antide Sitter spacetime, with Λ<0, we have
1 3M/a +3M2/a2+ Λ Ma
η < − 0 0 | | 0 , a >3M (42)
0 −2(1 2M/a + Λ a2/3)(1 3M/a ) 0
− 0 | | 0 − 0
1 3M/a +3M2/a2+ Λ Ma
η > − 0 0 | | 0 , a <3M (43)
0 −2(1 2M/a + Λ a2/3)(1 3M/a ) 0
− 0 | | 0 − 0
The regions of stability are depicted in the right plot of figure 2, considering the value 9ΛM2 =0.9. In this case,
| |
the black hole event horizon is given by r 1.8M. We verify that the regions of stability decrease, relatively to the
b
≃
Schwarzschildcase.
7
6 6
Stable Λ= 0 Stable 9 | Λ | M 2 = 0.9
4 4
2 2
η η
0 0
–2 –2
Stable
Stable
–4 –4
–6 2.5 3 3α.5 4 4.5 5 –6 2 2.5 3 α3.5 4 4.5 5
FIG. 2: We have defined α = a0/M. The regions of stability are depicted for the Schwarzschild and the Schwarzschild-anti
de Sitter solutions, respectively. Imposing the value 9|Λ|M2 =0.9 in the Schwarzschild-anti de Sitter case, we verify that the
stability regions decrease relatively to theSchwarzschild solution.
5. Schwarzschild-de Sitter spacetime.
For the Schwarzschild-de Sitter spacetime, with Λ>0, we have
1 3M/a +3M2/a2 ΛMa
η < − 0 0− 0 , a >3M (44)
0 −2(1 2M/a Λa2/3)(1 3M/a ) 0
− 0− 0 − 0
1 3M/a +3M2/a2 ΛMa
η > − 0 0− 0 , a <3M. (45)
0 −2(1 2M/a Λa2/3)(1 3M/a ) 0
− 0− 0 − 0
The regions of stability are depicted in figure 3 for increasing values of 9ΛM2. In particular, for 9ΛM2 = 0.7 the
black hole and cosmologicalhorizons are given by r 2.33M and r =4.71M,respectively. Thus, only the interval
b c
2.33<a /M <4.71istakenintoaccount,asshownin≃the rangeoftherespectiveplot. Analogously,for9ΛM2 =0.9,
0
we find r 2.56M and r =3.73M. Therefore only the range within the interval 2.56<a /M <3.73 corresponds
b c 0
≃
to the stability regions, also shown in the respective plot.
We verify that for largevalues ofΛ, or largeM, the regionsof stability aresignificantly increased, relativelyto the
Λ=0 case.
IV. A THIN SHELL AROUND A TRAVERSABLE WORMHOLE
As an alternative to the thin-shell wormhole, constructed using the cut-and-paste technique analyzed above, one
mayalsoconsiderthattheexoticmatterisdistributedfromthethroatr toaradiusa,wherethesolutionismatched
0
to an exterior vacuum spacetime. Thus, the thin shell confines the exotic matter threading the wormhole to a finite
region, with a delta-function distribution of the stress-energy tensor on the junction surface.
We shall match the interior wormhole solution [1]
dr2
ds2 = e2Φ(r)dt2+ +r2(dθ2+sin2θdφ2), (46)
− 1 b(r)/r
−
to the exterior vacuum solution
−1
2M Λ 2M Λ
ds2 = 1 r2 dt2+ 1 r2 dr2+r2(dθ2+sin2θdφ2), (47)
− − r − 3 − r − 3
(cid:18) (cid:19) (cid:18) (cid:19)
at a junction surface, Σ. Thus, the intrinsic metric to Σ is given by
ds2 = dτ2+a2(dθ2+sin2θdφ2). (48)
Σ −
Note thatthe junctionsurface,r=a,is situatedoutside theeventhorizon,i.e., a>r ,to avoidablackhole solution.
b
8
6 6
Stable Λ 2 Λ 2
9 M = 0.3 9 M = 0.5
4 4 Stable
2 2
η η
0 0
–2 –2
Stable Stable
–4 –4
–6 3 α4 5 6 –6 2.5 3 3.5 4α 4.5 5 5.5 6
6 6
9 Λ M 2 = 0.7 9 Λ M2 = 0.9
4 4
Stable
Stable
2 2
η η
0 0
–2 –2
Stable Stable
–4 –4
–6 2.5 3 3α.5 4 4.5 –6 2.6 2.8 3 α3.2 3.4 3.6
FIG. 3: We have defined α=a0/M. The regions of stability for the Schwarzschild-de Sitter solution, imposing 9ΛM2 = 0.3,
9ΛM2 =0.5, 9ΛM2 =0.7 and 9ΛM2 =0.9, respectively. The regions of stability are significantly increased, relatively to the
Λ=0 case, for increasing values of 9ΛM2.
Thus, using equation (3), the non-trivial components of the extrinsic curvature are given by
M Λa
Kτ + = a2 − 3 , (49)
τ
1 2M Λa2
− a − 3
q
b(a)
Kτ − = Φ′(a) 1 , (50)
τ − a
r
and
1 2M Λ
Kθ+ = 1 a2 , (51)
θ a − a − 3
r
1 b(a)
Kθ− = 1 . (52)
θ a − a
r
The Einsteinequations,equations (9)-(10), withthe extrinsic curvatures,equations(49)-(52), thenprovideus with
the following expressions for the surface stresses
1 2M Λ b(a)
σ = 1 a2 1 , (53)
−4πa r − a − 3 −r − a !
1 1 M 2Λa2 b(a)
p = − a − 3 ζ 1 , (54)
8πa 1 2M Λa2 − r − a 
− a − 3
q 
with ζ = 1+aΦ′(a). If the surface stress-energy terms are null, the junction is denoted as a boundary surface. If
surface stress terms are present, the junction is called a thin shell, which is represented in figure 4.
9
t t
nµ
(ext) (int) (ext) (int)
Σ Σ
Identify
Surface
stresses
(S i )
j
r a a r
0 0
FIG. 4: Two copies of static timelike hypersurfaces, Σ, embedded in asymptotic regions, separating an interior wormhole
solution from an exteriorvacuumspacetime. Bothcopies are identifiedat thewormhole throat, r0. Thesurface stressesreside
on Σ, and members of thenormal vector field, nµ, are shown.
The surface mass of the thin shell is given by
b(a) 2M Λ
M =4πa2σ =a 1 1 a2 . (55)
shell
r − a −r − a − 3 !
One may interpret M as the total mass of the system, in this case being the total mass of the wormhole in one
asymptotic region. Thus, solving equation (55) for M, we finally have
b(a) b(a) M Λ
M = +M 1 shell a3. (56)
shell
2 r − a − 2a !− 6
A. Energy conditions at the junction surface
The junction surface may serveto confine the interior wormholeexotic matter to a finite region,which in principle
maybe made arbitrarilysmall,andone mayimposethatthe surfacestress-energytensorobeysthe energyconditions
at the junction, Σ [2, 26].
We shall only consider the weak energy condition (WEC) and the null energy condition (NEC). The WEC implies
σ 0 and σ+p 0, and by continuity implies the null energy condition (NEC), σ+p 0.
≥ ≥ ≥
From eqs. (53)-(54), we deduce
1 b(a) 1 3M
σ+p= (2 ζ) 1 − a . (57)
8πa − r − a − 1 2M Λa2
− a − 3
 q 
WeshallnextfinddomainsinwhichtheNECissatisfied,byimposingthatthesurfaceenergydensityisnon-negative,
σ 0, i.e., 1 b(a)/a 1 2M/a Λa2/3.
≥ − ≥ − −
p p
1. Schwarzschild solution
Consider the Schwarzschild solution, Λ =0 and we impose that the surface energy density is non-negative, σ 0.
≥
For the particular case of ζ 1, from equation (57) we verify that σ+p 0 is readily satisfied for a.
≤ ≥ ∀
For 1<ζ <2, the NEC is verified in the following region
ζ 1
2M <a 2M − 2 . (58)
≤ ζ 1
(cid:18) − (cid:19)
10
For convenience, by defining a new parameter ξ =2M/a, equation (58) takes the form
ζ 1
− ξ <1. (59)
ζ 1 ≤
− 2
For ζ = 2, the NEC is satisfied for ξ 2/3, i.e., a 3M. For ζ > 2, we need to impose the NEC in the region of
≥ ≤
equation (59); with σ+p<0 for ξ <(ζ 1)/(ζ 1/2).
− −
2. Schwarzschild-de Sitter solution
FortheSchwarzschild-deSitterspacetime,Λ>0,weshallonceagainimposeanon-negativesurfaceenergydensity,
σ 0. Consider the definitions β =9ΛM2 and ξ =2M/a.
≥
For ζ <2 the condition σ+p 0 is readily met for β β , with β given by
0 0
≥ ≤
27 ξ2 1
β = (1 ζ)+ ζ ξ . (60)
0
4 (2 ζ) − − 2
− (cid:20) (cid:18) (cid:19) (cid:21)
Choosing a particular example, for instance ζ = 0.5, consider figure 5. The region of interest is shown below the
−
solid line, which is given by β =27ξ2(1 ξ)/4. The case of ζ = 0.5 is depicted as a dashed curve, and the NEC is
r
− −
obeyed to the right of the latter.
For ζ = 2, then the NEC is verified for β and ξ 2/3, i.e., r < a 3M, with r given by equation (12). This
b b
∀ ≥ ≤
analysis is depicted in figure 5, to the right of the dashed curve, represented by ξ =2.
For the case of ζ > 2, the condition σ+p 0 needs to be imposed in the region β β β ; and σ+p < 0 for
0 r
≥ ≤ ≤
β < β . The specific case of ζ = 5 is depicted as a dashed curve in figure 5. The NEC needs to be imposed to the
0
right of the respective curve.
1
0.8
0.6
β
0.4 ζ=2
ζ= − 0.5 ζ=5
0.2
0 0.2 0.4 ξ 0.6 0.8 1
FIG. 5: Analysis of the null energy condition for the Schwarzschild-de Sitter spacetime. We have considered the definitions
β =9ΛM2 and ξ =2M/a. Only the region below the solid line is of interest. We have considered specific examples, and the
NEC is obeyed to theright of each respective dashed curves,ζ =−0.5, ζ =2 and ζ =5. Seetext for details.
3. Schwarzschild-anti de Sitter solution
Considering the Schwarzschild-anti de Sitter spacetime, Λ < 0, once again a non-negative surface energy density,
σ 0, is imposed. Consider the definitions γ =9ΛM2 and ξ =2M/a.
≥ | |
For ζ 1 the condition σ+p 0 is readily met for γ and ξ. For 1<ζ <2, the NEC is satisfied in the region
≤ ≥ ∀ ∀
γ γ , with γ given by
0 0
≥
27 ξ2 1
γ = (1 ζ)+ ζ ξ . (61)
0
4 (2 ζ) − − 2
− (cid:20) (cid:18) (cid:19) (cid:21)
The particular case of ζ = 1.8 is depicted in figure 6. The region of interest is delimited by the ξ-axis and the area
to the left of the solid curve, which is given by γ =27ξ2(ξ 1)/4. Thus, the NEC is obeyed above the dashed curve
r
−
represented by the value ζ =1.8.