N-dimensional static and evolving Lorentzian wormholes with cosmological constant

0.2 MB·English

by  Mauricio Cataldo

N-dimensional static and evolving Lorentzian wormholes with cosmological constant
Mauricio Cataldo
∗
Departamento de F´ısica, Facultad de Ciencias, Universidad del B´ıo–B´ıo,
Avenida Collao 1202, Casilla 5-C, Concepci´on, Chile.
Paola Meza and Paul Minning
† ‡
Departamento de F´ısica, Facultad de Ciencias F´ısicas y Matema´ticas,
Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile.
(Dated: January 27, 2011)
Abstract: WepresentafamilyofstaticandevolvingsphericallysymmetricLorentzianwormhole
solutions in N+1 dimensional Einstein gravity. In general, for static wormholes, we require that at
1
1 least the radial pressure has a barotropic equation of state of the form pr = ωrρ, where the state
0 parameter ωr is constant. On the other hand, it is shown that in any dimension N ≥ 3, with
φ(r)=Λ=0 and anisotropic barotropic pressure with constant state parameters, static wormhole
2
configurations are always asymptotically flat spacetimes, while in 2+1 gravity there are not only

n asymptoticallyflatstaticwormholesandalsomoregeneralones. Inthiscase,themattersustaining
a thethree-dimensionalwormholemaybeonlyapressurelessfluid. Inthecaseofevolvingwormholes
J
with N ≥ 3, the presence of a cosmological constant leads to an expansion or contraction of the

6 wormhole configurations: for positive cosmological constant we have wormholes which expand for-
2 everand,fornegativecosmological constantwehavewormholeswhichexpandtoamaximumvalue
and then recollapse. In the absence of a cosmological constant the wormhole expands with con-
]
stantvelocity,i.ewithoutaccelerationordeceleration. In2+1dimensionstheexpandingwormholes
c
q always havean isotropic and homogeneous pressure, depending only on thetime coordinate.
-
r
g PACSnumbers: 04.20.Jb,04.70.Dy,11.10.Kk
[

1 I. INTRODUCTION spacetimes, which also have attracted the attention of
v
the community. The interest is related for example to
4
black hole physics [2], wormhole physics, Kaluza-Klein
3 It is well known the interest of studying gravitational
gravity, multidimensional and string/brane cosmology,
0 fields in spacetimes with arbitrary dimensions, with or
5 without cosmological constant. The theoretical proper- among others. It is interesting to note for example, that
. there is a lack of uniqueness for black holes in higher
1 ties of these multidimensional gravitational fields could
dimensions, unlike the four dimensional counterparts.
0 be quite different from one dimension to another and so
Specifically, the higher dimensional rotating black hole
1 itisofmuchinteresttogetaninsightintohowthespace-
1 time dimension may influence the gravitational dynam- metric [3] is not unique, unlike the Kerr geometry in
: (3+1) dimensions [4].
v ics. For example, recently, there was a lot of interest in
The showed interest in higher dimensional spacetimes
i low-dimensional gravity. The discovery of the existence
X can be extended also to the study of wormhole physics,
of three-dimensional black hole solutions represents one
r which has rapidly grown into an active area of research.
of the main advances for low-dimensional gravity theo-
a
Euclidean wormholes have been studied by Gonzales-
ries. Whilein(3+1)-dimensionalgravityblackholesolu-
Diaz and by Jianjun and Sicong [5] for example. The
tions there exist with or without cosmological constant,
Lorentzian ones have been studied in the context of the
in (2+1) dimensions the cosmological constant plays a
N-dimensional Einstein gravity [6] and Einstein-Gauss-
clueroleintheirexistence. Fromthepointofviewofthe
Bonnet theory of gravitation [7]. Wormholes in the con-
equilibriumconfigurationsofstarsthe number ofdimen-
text of brane worlds are discussed in [8], while the con-
sions ofspacetime alsocaninfluence this equilibrium. In
struction of thin-shell electrically charged wormholes in
Ref. [1] the authors show that dimensionality does in-
d-dimensionalgeneralrelativitywithacosmologicalcon-
crease the effect of mass but not the contribution of the
stant is discussed in Ref. [9].
pressure, which is the same in any dimension.
Evolving higher dimensional wormholes have been
The efforts also have been directed to the extension
studied in Refs. [10] and [11]. The authors of Ref. [10]
of the analysis of (3+1)-solutions to higher dimensional
study wormholesolutionstoEinsteingravitywithanar-
bitrary number of time dependent compact dimensions
and a matter-vacuum boundary. On the other hand,
in Ref. [11] the authors consider non-static wormholes,
∗[email protected]
†[email protected] mainly in 2+1 and 3+1 dimensions, with the required
‡[email protected] matter satisfying the weak energy conditions. The au-
2
thors explore several different scale factors and derive The signs refer to the two asymptotically flat regions
±
the corresponding consequences. whichareconnectedby the wormhole. The equality sign
In this paper, we shall obtain a family of static and in (2) holds only at the throat.
evolving spherically symmetric wormhole solutions, in Constraint 4: Asymptotic flatness condition, i.e. as
N+1 dimensional gravity, in the presence of a cosmo- l (or equivalently, r ) then b(r)/r 0 and
→ ±∞ → ∞ →
logical constant and by imposing at least a barotropic φ(r) 0.
→
equation of state, with constant state parameter, on the Notice that these constraints provide a minimum set
radial pressure. of conditions which lead, through an analysis of the em-
The organization of the paper is as follows: In Sec. II beddingofthespacelikesliceof(1)inaEuclideanspace,
we give some characterization of Lorentzian wormholes. to a geometry featuring two asymptotically flat regions
In Sec. III the field equations for evolving wormholes connected by a bridge [15].
in N+1 gravity are formulated. In Sec. IV and Sec. V Inthispaperwearenotincludingconsiderationsabout
thestaticandnon-staticN+1dimensionalwormholesare the traversability constraints discussed by Morris and
discussed respectively for N 3. In Sec. VI the 2+1- Thorne [13].
≥
dimensionalstaticandnon-staticwormholesaretreated.
Finally, in Sec. VII we conclude with some remarks.
III. N+1–DIMENSIONAL LORENTZIAN
WORMHOLES
II. CHARACTERIZATION OF LORENTZIAN
WORMHOLES
A. The metric and the matter source
Onthepurelygravitationalside,theinterestonworm-
Theevolvingsphericallysymmetricwormholeinhigher
hole geometries has been mainly focused on Lorentzian
dimensions may be obtained by a simple generalization
wormholes,andwasespeciallystimulatedbythepioneer-
of the original Morris and Thorne metric [13] to a time-
ing work of Morris, Thorne and Yurtsever [12], where
dependent metric given by
static, sphericallysymmetricLorentzianwormholeswere
defined and considered to be an exciting possibility for dr2
constructing time machine models with these exotic ob- ds2 = e2φ(t,r)dt2+a2(t) +r2dΩ2 =
jects. The metric ansatz of Morris and Thorne [13] for − 1− b(rr) N−1!
the spacetime which describes a static Lorentzianworm- N 1
−
hole is given by θ(t)θ(t)+θ(r)θ(r)+ θ(θi)θ(θi). (4)
−
i=1
dr2 X
ds2 = e2φ(r)dt2+ +r2dΩ2, (1)
− 1 b(r) 2 wherea(t)isthescalefactoroftheuniverse,andθ(µ) are
− r the proper orthonormalbasis whose one-forms are given
where dΩ2 =dθ2+sin2θdϕ2 and the functions φ(r) and by,
2
b(r) are referred to as redshift function and shape func-
θ(t) =eφ(t,r)dt,
tion respectively.
MorrisandThornehavediscussedindetailthegeneral
constraints on the functions b(r) and φ(r) which make a dr
wormhole [13, 14]: θ(r) =a(t) ,
Constraint 1: A no–horizon condition, i.e. eφ(r) is fi- 1 b(r)
− r
nite throughout the space–time in order to ensure the q
absence of horizons and singularities.
Constraint 2: Minimum value of the r–coordinate, i.e. θ(θ1) =a(t)rdθ ,
1
atthethroatofthe wormholer =b(r)=b , b beingthe
0 0
minimum value of r.
Constraint 3: Finiteness of the proper radial distance, θ(θ2) =a(t)rsinθ dθ ,...,
1 2
i.e.
b(r) N 2
1, (2) −
r ≤ θ(θN−1) =a(t)r sinθidθN 1.
−
i=1
(for r b ) throughout the space–time. This is required Y
0
≥
in order to ensure the finiteness of the proper radial dis- It must be noticed that the metric (4) includes static
tance l(r) defined by wormholes determined by the condition a(t) = a =
0
const. The constant a may be absorbed by redefin-
0
r dr ing the shape function in the following form: b(r)
l(r)=± 1 b(r)/r. (3) r a2(r b(r)). 7→
Zb0 − − 0 −
p
3
In general, for the metric (4), one might introduce where κ = 8πG, H = a˙/a and an overdot and a prime
a matter source described by an imperfect fluid. Of denote differentiation d/dt and d/dr respectively. Using
courseinthis casethe energy-momentumtensorhasalso the conservation equation Tµ =0 we have that,
∇µ ν
non-diagonal entries. However, we shall use the notion
of phantom energy in a slightly more extended sense: ∂ρ
+H(p +(N 1)p +Nρ) = 0, (11)
r l
We shall consider this traditionally homogeneous and ∂t −
isotropic exotic source to be generalized to an inhomo- ∂pr ∂φ N 1
+ (ρ+p ) − (p p ) = 0. (12)
geneous and anisotropic fluid, but still with a diagonal ∂r ∂r r − r l− r
energy-momentum tensor. This means that the only
Weshallstudymattersourcesdescribedwithatleast
nonzero components of the energy-momentum tensor in
a barotropic equation of state for p (t,r). Thus we can
this basis are r
write for the radial pressure
T =ρ(t,r), T =p (t,r)= τ(t,r), (5)
(t)(t) (r)(r) r − pr(t,r)=ωrρ(t,r), (13)
and
where ω is a constantstate parameter. One canrequire
r
thesameforp (t,r). Thusinsomecasesweshallconsider
T =...=T =p (t,r). (6) l
(θ1)(θ1) (θN−1)(θN−1) l solutions with a lateral pressure given by
where the quantities ρ(t,r), p (t,r), τ(t,r)(= p (t,r)),
r − r pl(t,r)=ωlρ(t,r), (14)
and p (t,r)(= p (t,r)) are respectively the energy den-
l θi
sity, the radialpressure,the radialtensionper unit area, where ω is a constant state parameter.
l
and the lateral pressure as measured by observers who In order to find solutions to the field equations we
always remain at rest at constant r, θi. can see that the equation (8) plays a fundamental role.
For the diagonal energy-momentum tensor (5) and (6),
Eq. (8) implies that the solutions are separated into
B. The Einstein field and the conservation two branches: one static branch given by the condi-
equations tion H = a˙/a = 0 and another non-static branch for
∂φ(t,r)/∂r =0.
Forthe evolvingsphericallysymmetricwormholemet-
ric(4)theEinsteinfieldequationswithcosmologicalcon-
stant Λ are given by IV. STATIC N+1 WORMHOLE SOLUTIONS
N(N −1)e−2φ(t,r)H2 + In general, for the static case, we shall suppose that
2 the shift and the shape functions, energy density and
(N 1) pressures are functions of the radial coordinate r, and
+ 2a−2r3 (rb′+(N −3)b) = κρ(t,r)+Λ, (7) that only the radial pressure has a barotropic equation
of state given by p (r) = ω ρ(r). Then, the required
r r
conditionforthestatic branchH =a˙/a=0implies that
(N 1)e φ(t,r)H r b(r)∂φ the field equations take the following form:
−
− − =0, (8)
a r ∂r
r (N 1)
− (rb′+(N 3)b) = κρ(r)+Λ, (15)
2r3 −
N 2 a¨ ∂φ
(N 1)e−2φ(t,r) − H2+ H
− − 2 a − ∂t −
(cid:18) (cid:19) (N 1)(N 2)b (N 1)(r b(r))dφ
− − + − −
(N 1)(N 2) (N 1)(r b(r))∂φ − 2r3 r2 dr
− − b+ − − =
− 2r3a2 r2a2 ∂r =κω ρ(r) Λ, (16)
r
−
=κp (t,r) Λ, (9)
r
−
rb +(2N 5)b (2N 4)rdφ N 2
′
− − − −
N 2 a¨ ∂φ − 2r2 dr − 2r3 ×
−(N −1)e−2φ(t,r) 2− H2+ a −H ∂t − (r b) d2φ dφ 2
rb′+(2(cid:18)N −5)b−(2N −4)r∂(cid:19)φ (rb′+(N −4)b)+ −r dr2 +(cid:18)dr(cid:19) ! (17)
− 2a2r2 ∂r −
=κp (r) Λ.
N 2 l −
− (rb′+(N 4)b) +
−2r3a2 − In this case the conservation equation takes the form
(r b) ∂2φ ∂φ 2
+ − + =κp (t,r) Λ, (10) dρ dφ N 1
a2r ∂r2 (cid:18)∂r(cid:19) ! l − ωrdr + dr (1+ωr)ρ− r− (pl−ωrρ)=0. (18)
4
Inordertosolvethe fieldequationswecanconsideronly ρ(r) fromEq.(15), andthe shape functionb(r) (givinga
Eqs. (15),(16) and (18). Thus we have four unknown restrictedformofφ(r))ortheredshiftfunctionφ(r)(giv-
functions of r, i.e. ρ(r), p (r), b(r) and φ(r), for three ing a restrictedform ofb(r)) froma differential equation
l
field equations. In order to construct solutions one can obtained from Eqs. (15) and (16). Thus, from Eqs. (15)
consider restricted choices for b(r) or φ(r). So, we shall and(16)we find that the shape function may be written
construct solutions by finding the lateral pressure p (r) in the form
l
from the conservation equation (18), the energy density
Cr2ω−rN−(N−3) 2r2ω−rN−(N−3) (N−2)(ωr+1) 2φ(r)
b(r)= e2φ(r)/ωr + ωr(N −1)e2φ(r)/ωr Z (cid:16)(N −1)φ′+Λ(1+ωr)r(cid:17)r ωr e ωr dr, (19)
or equivalently the redshift function as
(ω (N 3)+N 2)(N 1)b(r) 2 (1+ω )Λr3+(N 1)ω rb
r r r
φ(r)= − − − − − ′. (20)
2(N 1)(r b(r))r
Z − −
Now we shall consider specific static wormhole solu- (N 3) < α < 1. In order to make this space-time
− −
tions. a traversable wormhole we must require that eφ(r) = 1
which implies that N(1+ω ) 2 3ω +ω α = 0, or
r r r
− −
equivalentlyα= 2+3ωr−N(1+ωr). Finally,themetricand
A. b(r)∼rα solution the energy density takeωtrhe following form:
First, let us consider the case b(r) = r0(r/r0)α with dr2
Λ=0. This choice permits us to consider the possibility ds2 =dt2 r2dΩ2 , (24)
of having an asymptotically flat space-time. By putting − 1 r0 (1+ωrω)r(N−2) − N−1
− r
these expressions into Eq. (20) we have that the shift
(cid:0) (cid:1)
function takes the form
r −1+α N(1+ωr)2−(12−−α3)ωr+ωrα κρ(r)= (N −1)(2−N) r0 N(1+ωωrr)−2 . (25)
eφ(r) = 1 , (21) 2ω r 2 r
−(cid:18)r0(cid:19) ! r 0 (cid:16) (cid:17)
Among p (r)=ωρ(r), we have for the lateral pressure
then the metric and the energy density are given by r
ds2 =−(cid:18)1−(cid:16)rr0(cid:17)1−α(cid:19)N(1+ωr)−1−2−α3ωr+ωrα dt2+ κpl(r)= (N −2)2(ωωrrr+02 N −2)(cid:16)rr0(cid:17)N(ωrω+r1)−2. (26)
dr2
+r2dΩ2 , (22)
1 r0 1−α N−1 Notethatforωr <−1andωr >0wehaveasymptotically
− r flat wormholes, while for 1 < ωr < 0 this does not
−
(cid:0) (cid:1) occur. Ontheotherhand,fordimensionsN 3wehave
≥
apositiveenergydensityifω <0,andanegativeenergy
(N 1)(α+N 3) r 3 α r
κρ(r)= − − 0 − , (23) density for ω > 0. Thus we conclude that for ω >
2r 2 r r r
0 0 we have asymptotically flat wormholes with negative
(cid:16) (cid:17)
respectively. Note that by making N = 3 we obtain the energy density, while for ωr < 1 (or 1 < ωr < 0)
− −
3+1-wormholesolution discussed by Lobo in Ref. [16]. asymptoticallyflatwormholessupportedbyanenergyof
From the metric (22) we conclude that we have phantom(orquintessence)type,sincewehaveanegative
an asymptotically flat space-time if 1 α > 0 and radial pressure pr. In this case, for N 3, the energy
Nhav(1e+thωatr)e−φ(r2)−31ωarn+dωb(rrα)/>r 0,0sfinorceri−n thi.sHcoawseevweer dalewnasiytsythρe→ine0qufaolritryN→(1∞+ωsrin)ce2f<or0ω(rN≥<(1+−ω1r()ωr2>>00))
thisN-dimensio→nalspace-timeis→anon-tra→ver∞sableworm- takes place. − −
holesinceaneventhorizonislocatedatr =r . Itisinter- Non asymptotically flat solutions may be obtained by
0
esting tonote thatfor N 3 this non-traversableworm- requiringΛ=0. Thesespacetimesareexpressedthrough
≥ 6
holemayhaveapositiveenergydensitybyrequiringthat hypergeometric functions.
5
B. φ(r)=0 solution C. Pressure with constant state parameters
Now we shall consider the generalstatic solution for a Nowweshallconsiderthecaseφ(r)=0andbarotropic
6
barotropic radial pressure of the form (13) with φ(r) = pressure with constant state parameters (13) and (14).
0. Clearly, this is the more natural choice for the shift This means that now we have three field equations for
function,inordertohaveafiniteeφ(r) throughoutallthe three unknown functions b(r), φ(r) and ρ(r).
space–time. Thus by putting φ(r) = 0 into Eq. (19) we Byputtingp (r)=ω ρ(r)intoEq.(18)weobtainthat
l l
find that the energy density is given by
b(r)=r r 2ω−rN−(N−3)+ 2Λ(ωr+1) r3 , ρ(r)=Cr(N−1)ω(ωrr−ωl) e−(1+ωωrr)φ(r), (32)
0
r (ω N +N 2)(N 1)
(cid:18) 0(cid:19) r − −
(27) where C is an integration constant. From Eq. (15), and
by taking into account the Eq. (32), we find that
and from Eqs. (16) and (18) we obtain for the energy
densityandthelateralpressurethefollowingexpressions: (N−1)(ωr−ωl)
κC r ωr +Λ rN 1
(1+ωr)φ(r) −
κρ(r)= (N −1)(2−N) r0 N(ωrω+r1)−2 + b(r)=2r−N+3 (cid:18) e ωNr 1 (cid:19) dr+
2ω r2 r Z −
r 0 (cid:16) (cid:17) 2Λ C1r−N+3, (33)
, (28)
(N(ωr+1)−2) where C1 is a new integrationconstant. Inorderto have
thegeneralsolutionforthiscasewecanfindthefunction
(N 2)(ω +N 2) r N(ωr+1)−2 φ(r) by solving the differentialequation (16), by putting
κpl(r)= − 2ωrr2 − r0 ωr + into it the expressions (32) and (33). Unfortunately, the
r 0 obtaineddifferentialequationforφ(r)istoocomplicated,
(cid:16) (cid:17)
2Λω
r so we shall give a particular solution by considering a
. (29)
(N(ωr+1) 2) restricted choice of the shift function φ(r).
−
Let us consider the case
Clearly for Λ=0 we obtain the previous asymptotically
flat wormhole solution given by Eqs. (24)-(26). For Λ = r α
0 we do not have two asymptotically flat regions, an6d eφ(r) = . (34)
r
may have wormholes with two asymptotically de Sitter (cid:18) 0(cid:19)
regionsortwoasymptoticallyanti-deSitterregions,since This choice clearly may ensure the absence of horizons
as r the cosmological term dominates. In other and singularities for 0 < r r < , since the shift
→ ∞ 0 ≤ ∞
words,forverylargevaluesoftheradialcoordinater the function is finite throughoutthe space-time. In this case
large-scalecurvatureofthespacetimemustbetakeninto the field equations require that Λ = 0, so in order to
account[17]. Onthe otherhand, itis remarkablethatin have a shift function of the form (34) the cosmological
thiscasewecanhavepositiveenergydensitynotonlyfor constant must vanish. Thus the metric takes the form
ω <0,andalsoforpositivevaluesofthestateparameter
ωrr by requiring that Λ>−(N−41ω)r(r202−N)(N(ωr+1)−2). ds2 =− rr0 N−2dt2+ 1+ 1 dr2 C +
It is interesting to note that if we require that the (cid:16) (cid:17) ωr − r N−2
lateral pressure has also a barotropic equation of state r0
given by Eq. (14), i.e. p (r) = ω ρ(r), then we obtain (cid:0)r2(cid:1)dΩ2N 1, (35)
l l −
that the cosmological constant must vanish, i.e. Λ = 0,
where C is a constant of integration, the energy density
and the dimensional constraint
is given by
N(ω +1) ω +ω =2 (30)
l − l r (N 1)(N 2)
κρ(r) = − − , (36)
mustbefulfilled. Inthiscasethelateralpressuremaybe − 2
2r2ω r
written as 0 r r0
(cid:16) (cid:17)
2 N ωr and the constraint
p (r)= − − ρ(r)=
l
N 1
(cid:18) − (cid:19) N +Nωr+2 4ωr
(2−N −2ωωrr)2(2−N) rr0 N(ωrω+r1)−2. (31) ωl = − 2(N −1)− (37)
r 0
(cid:16) (cid:17) was used. Clearly for dimensions N 3 we have a pos-
Thus, static wormhole configurations with eφ(r) =1 and itive energy density if ω < 0, and≥a negative energy
r
anisotropic barotropic pressure with constant state pa- density for ω >0.
r
rametersarealways,inanydimensionN =3,asymptot- One can rewrite the wormhole metric (35) in a more
ically flat spacetimes. 6 appropriate form. From the condition g 1(r = r ) = 0,
r−r 0
6
we obtain for the integration constant C = 1 + 1/ω . given by
r
Then, we have for the metric component g 1(r) = (1+
1/ωr)(1 1/(r/r0)(N−2)), and the wormhr−orle throat is ds2 = r0 dt2+ dr2 +r2dΩ2, (43)
located a−t r . − r 4 C 2
Note that0this wormhole is not asymptotically flat, so (cid:16) (cid:17) − − rr0
in order to this wormhole connects two different asymp- where the isotropic pressure and the energy density are
totically flat regions we need to match this solution to given p= 1ρ(r)= 1 l, and as we stated above
an exterior N-dimensional vacuum spacetime, i.e. to the −5 −κr02 rr0 2
N+1-dimensional Schwarzschild solution [3]. this solution is not a wormhole.
(cid:0) (cid:1)
This wormholespacetime hasan interestingfeature to
be remarked: if we rescale the coordinate time t and the
V. EVOLVING N+1 WORMHOLE SOLUTIONS
radial coordinate r we can rewrite the metric to
ds2 = r˜0 N−2dt2+ dr2 + Now we shallconsider the non-static branchof the so-
− r 1 1 lutions. As we stated above in order to have non-static
(cid:18) (cid:19) − r˜r0 N−2 wormholes we must require ∂φ(t,r)/∂r = 0. This con-
1 dition implies that the redshift function can only be a
1+ (cid:0)r2(cid:1)dΩ2 , (38)
(cid:18) ωr(cid:19) N−1 tfuhnecttiimonecoofotr,dii.ne.atφe(tt,,sro)w=itfh(otu),taanndyltohsesnowfgeecnaenrarlietsycawlee
This metric has a solid angle deficit, which depends on shallrequireφ(t,r)=f(t)=0. Inthis case we areinter-
the value of the state parameter. ested in solutions having a barotropic anisotropic pres-
It is interesting to note that if we want an isotropic sure with constant state parameters given by Eqs. (13)
solutionwiththeshiftfunctionoftheform(34)weobtain and (14).
from Eq. (37), by putting ω =ω , that Now, from the conservation equation (12) we obtain
r l
ω = 2−N. (39) ρ(t,r)=ρ0a−(ωr+(N−1)ωl+N)rNω−r1(ωl−ωr), (44)
2+N
where ρ is an integration constant, and by subtracting
0
Thus the metric is given by
equations (9) and (10) we have that
ds2 =−(cid:16)rr0(cid:17)N−2dt2+ 2−4N −dr2rr0CN−2 +r2dΩ2N−1, N2a−2r32(rb′−3b)= κρ0(aω(rωr−+(ωNl−)r1)Nωω−lr+1(Nω)l−ωr). (45)
(cid:0) (cid:1) (40) It is straightforward to see that in order to have a
solution for the shape function b = b(r) the following
where the energy density and the isotropic pressure are
constraint must be imposed;
given
ω +(N 1)ω +N =2, (46)
(N 1)(N +2) r − l
ρ(r) = − , (41)
2 onthestateparametersω andω ,thusobtainingforthe
2κr2 r r l
0 r0 shape function and the energy density
(N 1)(cid:16)(2 (cid:17)N)
p(r) = 2−κr2 r−2 . (42) b(r) = C1r3− (N 2κ2ρ)(0Nωr 1)r−Nω−r2−(N−3), (47)
0 r0 − −
This energy density is always p(cid:16)osit(cid:17)ive. In order to have r−N−2ω+rNωr
a wormhole we must require C = 24N, ensuring that ρ(t,r) = ρ0 a2 , (48)
g 1(r = r ) = 0. However, it can be−shown that in this
r−r 0 whereC isanintegrationconstant. Now,fromequation
casethe metric componentg ispositivefor0<r <r 1
rr 0 0 (7) we obtain the following differential equation for the
and negative for r > r , so the metric (40) does not
0 scale factor
represent a wormhole spacetime for N 3.
≥
Let us note that a four dimensional spherically sym-
2Λa2
metric static wormhole solution with a shift function of a˙ = C . (49)
1
the form (34) and isotropic pressure was considered by ±sN(N 1) −
−
Lobo in Ref. [16]. However the solution given by Eq.
Thesolutionforthisequationdependsonthesignsofthe
(32) of the Ref. [16] does not have isotropic pressure of
cosmologicalconstantΛandtheintegrationconstantC ,
the form p=ωρ. As we can see from the N-dimensional 1
as we display in Table I.
solution(40)-(42)theself-consistentfour-dimensionalso-
α
lution with a shift function of the form eφ = r is Now, it must be noted that the radial coodinate in this
r0
(cid:16) (cid:17)
7
a(t) C1 Λ r −N(1+ωωrr)−2
a0e± N(N2Λ−1)t =0 >0 kr21−(cid:18)r0(cid:19) , (53)
C1N(N−1) sinp −2Λ t+φ <0 <0
2Λ N(N−1) 0  
respectively. This implies that the wormhole throat is
q−C1N2Λ(N−1)sinh(cid:16)q N(N2Λ−1)t+φ(cid:17)0 <0 >0 located at r0, and the energy density is given by
q C1N2(ΛN−1)cosh (cid:16)qN(N2Λ−1)t+φ0(cid:17) >0 >0 κρ(t,r)= (N −1)(N −2)(kr02−1) r0 N(1+ωωrr)−2 .
q (cid:16)q (cid:17) 2ω r2a2(t) r
TABLE I: The table shows the possible scale factors derived r 0 (cid:16) (cid:17) (54)
from Eq. (49).
It is clear that for N 3 the cosmological constant di-
≥
rectly controls the behavior of the scale factor a(t) and
not the shape function b(r), which mainly is controlled
solution may be rescaled in order to absorb the integra- by the state parameter ω . In order to have an evolv-
r
tion constant C . In this case the metric is given by ing wormhole we must require ω < 1 or ω > 0 (in
1 r r
−
both of these cases, in the g metric component (53),
ds2 = dt2+a2(t) rr
(N 2)(1+ω )/ω > 0 and (N(1+ω ) 2)/ω > 0),
− × r r r r
− −
dr2 implying that the phantom energy can support the ex-
1−kr2+ (N2κ1ρ)0(Nωr 2)r−(N−2ω)(r1+ωr) +r2dΩ2N−1!,(50) imstoelnocgeicaolf ceovnoslvtainngt,waonrdmthhoelesenienrgtyhedepnresisteyncveanoifshaescoast-
− −
r . On the other hand, clearly for ω < 1 or
wfohreCre1k>=0.0Wfoer sCu1m=ma0r,izke=all−p1osfosirbCle1sc<al0e afancdtokrs=fo1r sωlric→e>s∞0t =thecomnsettrwich(ic5h0)araetNsp-adtimialeninsifionniatyl s(prarc→es∞o−f)choans-
the found wormhole solutions in Table II.
stantcurvature: openfork=-1,flatfork=0andclosed
a(t) k Λ for k = 1. This implies that for r the metric (50)
→ ∞
const 0 0 is foliated with spaces of constant curvature.
t+const −1 0 Now some words about the energy conditions. It is
a e± N(N2Λ−1)t 0 >0 well known that, in all cases, the violation of the weak
0 energy condition (WEC)
−N(N−1)sinp −2Λ t+φ −1 <0
2Λ N(N−1) 0
ρ 0, ρ+p 0,
qN(N2Λ−1)sinh(cid:16)qN(N2Λ−1)t+φ0(cid:17) −1 >0 ≥ ρ+pr ≥ 0, (55)
l ≥
qN(N−1) cosh(cid:16)q 2Λ t+φ (cid:17) 1 >0
2Λ N(N−1) 0 is a necessary condition for a static wormhole to exist.
q (cid:16)q (cid:17) In the case of our non-static solution we must require
TABLE II: The table shows all the possible scale factors for ω < 1 or ω >0 in order to have evolving wormholes.
r r
thegeneral solution (50) ofan evolvingLorentzian wormhole Thus−ingeneral,fork =0 ork = 1andρ>0,wemust
inN+1dimensionswiththeradialtensionandthetangential −
require ω < 1, so the WEC is always violated (this is
r
pressure having barotropic equations of state with constant −
independent of the value of the cosmological constant),
state parameters.
while for k = 1 (in this case Λ > 0) and ρ > 0, we may
require ω < 1 for r2 < 1 (and the WEC is always
Nowweshallrewritethewormholemetric(50)inamore violated)ror re−quire 0 0< ωr < 1 for r02 > 1 (and the
appropiate form. From the condition g 1(r = r ) = 0, violation of WEC is avoided). Unfortunately this latter
r−r 0
case must be ruled out for the consideration of evolving
we obtain for the integration constant ρ
0
wormhole configurations.
(N 1)(N 2)(kr2 1) (N−2)(1+ωr) For N = 3 we obtain the evolving wormhole solutions
ρ = − − 0− r ωr , (51)
0 2κω 0 discussed in Refs. [18] and [19], where the traversabil-
r
itycriteriaforthese fourdimensionalwormholesarealso
yieldingfortheshapefunctionandthemetriccomponent
considered.
g
rr
N−2+(N−3)ωr
r − ωr VI. 2+1 EVOLVING LORENTZIAN
b(r) = r +
0
r WORMHOLES
(cid:18) 0(cid:19)
N(1+ωr)−2
kr3 1 r − ωr , (52) As we can see from evolving N+1 wormhole solutions
 −(cid:18)r0(cid:19)  the shape function b(r) in Eq. (47) is not well defined
for N = 2. On the other hand, by studying more ac-
 (N−2)(1+ωr) 
a2(t)g 1 = 1 r − ωr curately the field equations for evolving wormholes in
r−r − r − any dimensions (7)-(10) we conclude that the nature of
(cid:18) 0(cid:19)
8
such wormholes for N = 2 and N 3, are quite dif- On the other hand, for the shape function given by
ferent. Effectively, in this case the c≥ondition φ(r) = 0 b(r) = r (r/r )α and φ(r) = 0 the solution has the
0 0
6
mustberequired. Thus,thepressuresustainingthetree- form(21)-(23)with N =2,obtaining a2+1-dimensional
dimensionalevolvingwormholesmustbealwaysisotropic non-traversable wormhole with an event horizon located
and homogeneous, i.e. of the form p (t) = p (t) = p(t), at r . In this case the energy density is always negative,
r l 0
while for N 3 the pressure must be always inhomoge- tothecontraryoftherealpossibilityofhavingapositive
≥
neous and anisotropic, i.e. given by p (t,r) and p (t,r). energy density for N 3.
r l
≥
Inotherwords,itcanbeseenfromthefieldequations(7)- It must be noted that three-dimensional static worm-
(10)thatforN 3therequirementsφ(r)=0,p =p (t) hole configurations are discussed by Perry and Mann in
r r
≥
and p = p (t) immediately implies that the shape func- Ref. [20], where the constraints on the field equations to
l l
tionb(r) mustvanish, andthenwe mustconsiderapres- obtain wormholes are presented and further constraints
sure of the form p (t,r) and p (t,r). For N = 2 clearly on traversibility are discussed.
r l
this does not occur.
So now we shall discuss separately wormhole space-
times in 2+1 dimensions. B. Non-static three-dimensional branch
Inthreedimensionalgravitythemetricforanevolving
wormhole is given by Let us now discuss the 2+1-non-static branch with
∂φ(t,r)/∂r =0. In this case the Einstein field equations
dr2 are given by
ds2 = e2φ(t,r)dt2+a2(t) +r2dθ2 . (56)
− 1 b(r) ! (rb b)
− r H2+ ′− =κρ+Λ. (59)
2r3a2
The field equations may be directly obtained by putting
N = 2 into Eqs. (7)-(12). As in the N-dimensional case
a¨
the solutions are separatedinto two branches: one static =κp Λ. (60)
r
− a −
branch given by the condition a(t) = const and another
non-static branch for ∂φ(t,r)/∂r =0.
a¨
=κp Λ. (61)
l
− a −
A. Static three-dimensional branch
∂ρ
+H(p +p +2ρ) = 0. (62)
r l
In general, for the static branch the solutions may be ∂t
obtained by putting N =2 into Eqs. (19) and (20), and ∂pr (pl pr)
− = 0. (63)
then we can impose a restricted form of the shape func- ∂r − r
tion b(r) or the redsfiht function eφ(r). For φ(r) = 0 we
It’s clear from Eqs. (60) and (61) that only an isotropic
obtainthatthe pressureisisotropicandconstant,andin
pressure is permitted, then we shall write p = p = p.
thepresenceofthecosmologicalconstanttakesthe value r l
Thus from Eq. (63) we obtain that ∂p/∂r = 0, which
κp = κp = Λ. In this case the energy density is given
r l implies that the pressure has the general form p = p(t).
by
In the following we shall discuss the cases p=const and
p=p(t).
rb b
′
κρ(r)= − Λ. (57)
2r3 −
Note that a static wormhole sustained by a pressureless 1. Case p=const and ρ(t,r)
fluidis only possiblein2+1(i.e. Λ=0). For dimensions
N 3, the requirements φ(r) = 0 and p = p = 0 im- Byputtingp=constintoEqs.(60)and(62)weobtain
r l
≥
plies that the shape function must vanish. Let us now for the scale factor and the energy density
consideraspecificasymptoticallyflatwormholegivenby
b(r) = r (r/r )α. Thus the pressureless fluid has an en- a(t)=C1sin( κp Λ t)+C2cos( κp Λ t), (64)
0 0 − −
ergy density given by ρ (r)
p ρ(t,r)= 0 pp, (65)
a2 −
(α 1)
κρ(r)= − Λ. (58) respectively. Now by putting the scale factor andenergy
2r02(r/r0)3−α − density from Eqs. (64) and (65) into Eq. (59) we obtain
the following expression for the shape function:
In this case, we must require α < 1 in order to have a
three-dimensional wormhole, thus for Λ = 0 the energy b(r)
density is always negative. For Λ = 0 we can demand =r2 C12+C22 (Λ κp)+
r −
that Λ < α 1 < 0 in order to ha6ve a positive energy
2−r2 (cid:0) (cid:1)
0 2κ rρ (r)dr+C, (66)
density for r r0. 0
≥ Z
9
whereC isanintegrationconstant. Bygivingarestricted for α = 2, where BesselJ and BesselY are the Bessel
6 −
formofρ (r) wecanobtainthe shape function. One can functions of the first and second kinds respectively. In
0
alsoimpose arestrictedformofb(r) andobtainthe form order to have a traversable wormhole we must give
of the function ρ (r) with the help of the expression a restricted form of the shape function satisfying all
0
wormhole constraints, as for example, b(r) = r (r/r )α.
0 0
κρ (r)= 1 b(r) ′+ C12+C22 (κp Λ), Clearly in this case from Eq. (71) we have F(r) < 0 for
0 2r r − α < 1, thus the energy density (70) may be negative or
(cid:18) (cid:19)
(cid:0) (cid:1) (67) positive during the evolution. This mainly depends on
the relation between the terms F(r)/a2 and H2. As an
where ′ = d/dr. Let us consider a specific wormhole exampleletusconsiderthe caseofapowerlawscalefac-
solution given by b(r)=r0(r/r0)α. Thus we obtain tor a(t) = tβ. In this case the energy density will be
given by
(α 1)
κρ (r)= − + C12+C22 (κp Λ). (68)
As w0e know2αr02(<r/1r0f)o3r−hαavin(cid:0)g b(r)/r (cid:1)1, an−d in order κρ(r)= 2r02(r(/αr−0)31−)αt2β + βt22 −Λ. (73)
≤
to have a positive energy density we must require that
Clearly in this case for Λ = 0 there exist values of the
κp > −2r02((αC−12+1)C22) +Λ. Note that in this case even for parameterswhichensureth6epositivityoftheenergyden-
Λ=0 we can have ρ(r)>0. sity during all evolution of the scale factor. For Λ = 0
It is interesting to note that this wormhole, for every wealwayscanfind avalue ofthe cosmictime t=t >0,
0
slice a(t)=a0 =const, and for p=Λ=0 with α=1/2 where the energy density vanishes, thus ρ(t,r) may be
reproduces the asymptotically flat and static wormhole negative for 0<t<t or t>t .
0 0
solution discussed in Ref. [20]. It must be noticed that the case p(t) and ρ(t) must be
excluded from consideration since in this case we must
requirethat b(r)=0, andthen we cannothave a worm-
2. Case p=p(t) and ρ(t,r)
hole configuration.
By direct integration of Eq. (62) we have that
VII. DISCUSSION
F(r) 2κ da(t)
κρ(t,r)= a(t)p(t) dt. (69)
a2(t) − a2(t) dt
Z (cid:18) (cid:19) Inthis paper wehaveobtainedN+1-dimensionalsolu-
Since p(t) is an arbitrary function, this implies that the tionsfortheEinsteinfieldequationswhichdescribestatic
generalformoftheenergydensityisρ(t,r)= F(r)+C(t). and evolving spherically symmetric Lorentzian worm-
a2(t)
holes. In general, for static wormholes, we require that
By putting the expression p(t) = a¨(t)/a(t) + Λ into
− at least the radial pressure has a barotropic equation of
Eq. (69) we obtain finally
stateoftheformp =ω ρ,wherethestateparameterω
r r r
F(r) a˙ 2 isconstant,andforevolvingwormholeswealsorequirea
κρ(t,r)= + Λ. (70) barotropicequationofstatep =ω ρ withconstantstate
a2(t) a − l l
(cid:18) (cid:19) parameter ω for the lateral pressure.
l
By substituting ρ(t,r) from the above equation in Forstaticwormholesitisshownthat,inanydimension
Eq. (59) we obtain that N 3, with φ(r) = Λ = 0 and anisotropic barotropic
≥
pressure with constant state parameters, they are al-
1 b(r) ′ waysasymptoticallyflatspacetimes,whilein2+1gravity
F(r)= . (71)
2r r the static wormholes may have more general asymptotic
(cid:18) (cid:19)
spaces. In this case, the matter sustaining the three-
Clearly, for having a solution we must give a restricted
dimensional wormhole may be only a pressureless fluid.
form of the pressure in order to find the form of the
The nature of evolving wormholes in 2+1-dimensions
scale factor. For example, for p(t) = p = const we ob-
and N+1-dimensions, with N 3, are quite different.
tain the discussed above solution (64), (65) and (67). If ≥
For evolving wormholes in any dimensions the condition
we consider the pressure given by p(t) = tα, then the
φ(r)=0 must be required. However, this constraint im-
scale factor takes the form a(t) = C t1/2(1+√1 4C) +
1 − plies that the pressures sustaining the tree-dimensional
C2t1/2(1−√1−4C) for α= 2, and wormhole configurations must be always homogeneous,
−
i.e. only must depend on the cosmological time, while
√Ct1/2α+1
a(t)=C1 √tBesselJ (α+2)−1,2 + for N ≥ 3 these pressures must be always inhomoge-
α+2 ! neous, i.e. of the form p(t,r), in order to have an evolv-
ing wormhole spacetime. This canbe seen from the field
√Ct1/2α+1
C2 √tBesselY (α+2)−1,2 α+2 !, (72) epqru=atpiorn(ts)(a7n)d-(1p0l)=(fpolr(tN). T≥hi3s)inbmyerdeiqauteirlyingimφp(lrie)s=tha0t,
10
theshapefunctionb(r)mustvanish. Inthecaseofevolv- pending only on the time coordinate.
ing wormholes with N 3, the presence of a cosmologi-
≥
cal constant leads to an expansion or contraction of the
wormhole configurations: for positive cosmological con-
stant we have wormholes which expand forever and, for
VIII. ACKNOWLEDGEMENTS
negativecosmologicalconstantwehavewormholeswhich
expand to a maximum value and then recollapse. In the
absenceofacosmologicalconstantthewormholeexpands ThisworkwassupportedbyCONICYTthroughGrant
with constant velocity, i.e without acceleration or decel- FONDECYT No. 1080530 (MC), PhD. scholarship No.
eration. In 2+1 dimensions the expanding wormholes 21080709(PM), and by Direcci´on de Investigaci´on de la
always have an isotropic and homogeneous pressure, de- Universidad del B´ıo-B´ıo(MC).
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