Mauricio Cataldo et al- Evolving Lorentzian wormholes supported by phantom matter and cosmological...

8/3/2019 Mauricio Cataldo et al- Evolving Lorentzian wormholes supported by phantom matter and cosmological constant

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arXiv:0812.4436v1

[gr-qc]23Dec2008

APS/123-QED

Evolving Lorentzian wormholes supported by phantom matter and cosmological

constant

Mauricio Cataldo

Departamento de Fsica, Facultad de Ciencias, Universidad del BoBo,Avenida Col lao 1202, Casilla 5-C, Concepcion, Chile.

Sergio del Campo

Instituto de Fsica, Facultad de Ciencias,Pontificia Universidad Catolica de Valparaso,

Avenida Brasil 2950, Valparaso, Chile.

Paul Minning and Patricio Salgado

Departamento de Fsica, Facultad de Ciencias Fsicas y Matematicas,Universidad de Concepcion, Casilla 160-C, Concepcion, Chile.

(Dated: December 23, 2008)

In this paper we study the possibility of sustaining an evolving wormhole via exotic matter made ofphantom energy in the presence of a cosmological constant. We derive analytical evolving wormholegeometries by supposing that the radial tension of the phantom matter, which is negative to theradial pressure, and the pressure measured in the tangential directions have barotropic equations of

state with constant state parameters. In this case the presence of a cosmological constant ensuresaccelerated expansion of the wormhole configurations. More specifically, for positive cosmologicalconstant we have wormholes which expand forever and, for negative cosmological constant we havewormholes which expand to a maximum value and then recolapse. At spatial infinity the energydensity and the pressures of the anisotropic phantom matter threading the wormholes vanish; thusthese evolving wormholes are asymptotically vacuum -Friedmann models with either open or closedor flat topologies.

PACS numbers: 04.20.Jb, 04.70.Dy,11.10.Kk

I. INTRODUCTION

It is well known that in 1917 Einstein added a constantterm into his field equations of general relativity allow-ing static cosmological models for the Universe. Thisterm, called cosmological constant, would create a repul-sive gravitational force that does not depend on positionnor on time. Later, after Hubble discovered that theuniverse is expanding, Einstein discarded this constantterm, as the biggest blunder of his life.

However, recent observations are indicating that a cos-mological constant may play an essential role in the cos-mic expansion of the Universe [1]. According to the stan-dard cosmology the total energy density of the Universeis dominated today by both dark matter and dark energydensities. The dark energy component in general is con-

sidered as a kind of vacuum energy homogeneously dis-tributed with negative pressure. The observational dataprovide compelling evidence for the existence of dark en-ergy which dominates the present day Universe and ac-

[email protected]@[email protected]@udec.cl

celerates its expansion.

In principle, any matter content which violates thestrong energy condition and possesses a positive energydensity and a negative pressure, may cause the dark en-ergy effect of repulsive gravitation. So the main problemof modern cosmology is to identify this form of dark en-ergy that dominates the universe today. Dark energycomposed of just a cosmological term is fully consis-tent with existing observational data. A cosmologicalconstant can be associated with a time independent darkenergy density. Another candidate is the phantom mat-ter, whose energy density increases with the expansion ofthe universe [2].

In general relativity one can also consider other grav-itational configurations where a cosmological constant,and even the phantom matter, may play an importantrole. For example the presence of the cosmological con-stant allows charged and noncharged black hole geome-tries in 2+1 gravity [3]. On the other hand wormholes,as well as black holes, are an extraordinary consequenceof Einsteins equations of general relativity and, dur-ing recent decades, there has been a considerable in-terest in the field of wormhole physics. For instance,Lorentzian wormhole configurations [4, 5] may be sup-ported by phantom matter [7, 8, 9] or by a cosmologicalconstant [10, 11, 12].
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Two separate directions emerged: one relating to staticLorentzian wormholes and the other concerned withtime-dependent Lorentzian ones [13, 14, 15]. In bothcases the interest has been focused on traversable worm-holes, which have no horizons, allowing two-way passagethrough them. However, most of the efforts are directedto study static configurations which are traversable. Themost striking property of such a wormhole is the violation

of energy conditions. This implies that the matter sup-porting the traversable wormholes is exotic [4, 5], whichmeans that it has very strong negative pressures, or eventhat the energy density is negative, as seen by static ob-servers.

One interesting aspect that we shall consider here is thepossibility of sustaining traversable wormhole spacetimesvia exotic matter made out of phantom energy. The lat-ter is considered to be a possible candidate for explainingthe late time accelerated expansion of the Universe [16].This phantom energy has a very strong negative pres-sure and violates the null energy condition, so becom-ing a most promising ingredient to sustain traversable

wormholes. Notice however that in this case we shalluse the notion of phantom energy in a more extendedsense since, strictly speaking, the phantom matter is ahomogeneously distributed fluid, and here it will be aninhomogeneous and anisotropic fluid [6, 7, 9]. It is inter-esting to note that in Ref. [17] is considered the possi-bility that the enormous negative pressure in the centerof a dark energy star may, in principle, imply a topologychange, consequently opening up a tunnel and convert-ing the dark energy star into a wormhole. It should bementioned that the stability of some specific wormholestatic configurations also has been considered [18].

The theoretical construction of wormholes is usuallyperformed by using the method where, in order to havea desired metric, one is free to fix the form of the metricfunctions, such as the redshift and shape functions, oreven the scale factor for evolving wormholes. In this wayone may have a redshift function without horizons, orwith a desired asymptotic. Unfortunately, in this casewe can obtain expressions for the energy and pressuredensities which are physically unreasonable.

For constructing wormholes another method can beused. One may impose conditions on the stress-energytensor threading the wormhole configuration. For ex-ample, one can impose the socalled selfdual condition = t = 0, on the matter supporting the wormhole,where = Tuu and t = (T

1

2

T g)uu arethe energy density measured by a static observer [19].One can also prescribe the matter content by specifyingthe equations of state of the radial and of the tangen-tial pressures or by choosing a more specific matter field,such as, for example, a classical scalar field. Notice that,as a simple model of phantom matter, one may considera classical scalar field with a negative kinetic energy term(the so called ghost or phantom scalar field). Static [20]and nonstatic wormholes [21] were constructed with thehelp of such a ghost scalar field. In all these here de-

scribed cases one can solve the Einstein field equationsin order to find all the metric functions.

In this paper we shall consider the radial and tangen-tial pressures of the matter threading the wormhole toobey barotropic equations of state with constant stateparameters coupled to a cosmological constant. Morespecifically, we shall find all evolving wormhole geome-tries which have their radial and tangential pressures

proportional to the energy density in the presence of acosmological constant.The outline of the present paper is as follows: In Sec.

II, we briefly review some important aspects of evolvingwormholes and write the Einstein equations for matterconfigurations considered here. In Sec. III, we obtainthe general metrics which describe evolving wormholeswith anisotropic pressures obeying barotropic equationsof state with constant state parameters in the presence ofa cosmological constant. In Sec. IV, the properties of theobtained wormhole geometries are studied and discussed.

II. EVOLVING LORENTZIAN WORMHOLES

AND THE EINSTEIN FIELD EQUATIONS

The metric ansatz of Morris and Thorne [4] for thespacetime which describes a static Lorentzian wormholeis given by

ds2 = e2(r)dt2 + dr2

1 b(r)r+ r2(d2 + sin2d2), (1)

where (r) is the redshift function, and b(r) is the shapefunction since it controls the shape of the wormhole. Inorder to have a wormhole the functions b(r) and (r)must satisfy the general constraints discussed by Morris

and Thorne in Ref. [4]. These constraints provide a min-imum set of conditions which lead, through an analysisof the embedding of the spacelike slice of (1) in a Eu-clidean space, to a geometry featuring two asymptoticallyflat regions connected by a bridge. Although asymptoti-cally flat wormhole geometries have been extensively con-sidered in the literature, asymptotically antide Sitterwormholes are also of particular interest [22].

Now, the evolving wormhole spacetime may be ob-tained by a simple generalization of the original Morrisand Thorne metric (1) to a time-dependent metric givenby

ds2 =

e2(t,r)dt2 +

a(t)2

dr2

1 b(r)r+ r2(d2 + sin2d2)

, (2)

where a(t) is the scale factor of the universe. Note thatthe essential characteristics of a wormhole geometry arestill encoded in the spacelike section. It is clear that ifb(r) 0 and (t, r) 0 the (2) metric becomes theflat Friedmann-Robertson-Walker (FRW) metric and, asa(t) const and (t, r) = (r) it becomes the staticwormhole metric (1).


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Now we shall write the Einstein equations for the (2)metric. In order to simplify the analysis and the physicalinterpretation we now introduce the proper orthonormalbasis as

ds2 = (t)(t) + (r)(r) + ()() + ()(), (3)where the basis oneforms () are given by

(t) = e(t,r)dt; (r) = a(t) dr1 b(r)r

;

() = a(t) rd; () = a(t) rsind. (4)

In general, for the metric (2) one might introduce amatter source described by an imperfect fluid, with itsfourvelocity u() oriented along the tetrad (t). Thusfor the spacetime (2) one might consider a stress-energytensor of an imperfect fluid containing the coefficients ofbulk viscosity, shear viscosity and heat conduction (forthe considered metric we might have energy flux in theradial direction), where all of these transport coefficients

might be functions oft and/or ofr. Of course in this casethe energymomentum tensor has also nondiagonal en-tries. However, as we have pointed out above, we shalluse the notion of phantom energy in a slightly more ex-tended sense: We shall consider this traditionally homo-geneous and isotropic exotic source to be generalized toan inhomogeneous and anisotropic fluid, but still witha diagonal energy-momentum tensor (as is usually con-sidered in phantom cosmologies). This means that, forthe tetrad basis (4), the only nonzero components of theenergymomentum tensor T()() are the diagonal termsT(t)(t), T(r)(r), T()() and T()(), which are given by

T(t)(t) = (t, r), T(r)(r) = pr(t, r) =

(t, r),

T()() = T()() = pl(t, r), (5)

where the quantities (t, r), pr(t, r), (t, r)(= pr(t, r)),and pl(t, r)(= p(t, r) = p(t, r)) are respectively the en-ergy density, the radial pressure, the radial tension perunit area, and the lateral pressure as measured by ob-servers who always remain at rest at constant r, , .Thus for the evolving spherically symmetric wormholemetric (2) the Einstein equations with cosmological con-stant

R()() R

2g()() = T()() g()()

are given by

3e2(t,r)H2 +b

a2r2= (t, r) + , (6)

e2(t,r)

2a

a+ H2

b

a2r3+ 2e2(t,r)H

t+ (7)

2

r2a2(r b)

r= pr(t, r) ,

e(t,r)

2a

a+ H2

+

b rb2a2r3

+ (8)

2e(t,r)H

t+

1

2a2r2(2r b rb)

r+

1

a2r(r b)

r

2+

2

r2

= pl(t, r) ,

2e(t,r)

r b(r)r

ra = 0, (9)

where = 8G, H = a/a, and an overdot and a primedenote differentiation d/dt and d/dr respectively.It is direct to see that, for the diagonal energy

momentum tensor (5), Eq. (9) gives some constraintson relevant metric functions which separate the worm-hole solutions into two branches: one static branch givenby the condition a = 0 and another non-static branch for/r = 0. This latter condition implies that the redshiftfunction can only be a function of t, i.e. (t, r) = f(t)so, without any loss of generality, by rescaling the timecoordinate we can set (t, r) = 0.

It is interesting to note that the static branch with(r) = const allows one to have isotropic pressures forthe matter threading the wormhole (see for example [7]).As we shall see below, since for the non-static branch wemust fulfill the condition (t, r) = 0, the matter pressurescannot be isotropic at all.

In what follows we shall restrict our discussion to evolv-ing wormhole geometries, so we must consider the non-static branch. Thus, as we stated above, without any lossof generality we shall require (t, r) = 0. By using theconservation equation T; = 0, we have that

t+ H(3 + pr + 2pl) = 0, (10)

2(pl pr)r

=2(pl + )

r=

prr

. (11)

From these equations we see that for an isotropic pres-sure, i.e. pl = pr = p, we have to require pr/r = 0,so the pressure will depend only on time t, obtaining thestandard cosmological conservation equation + 3H( +

p) = 0. If we want to study wormholes with pressuresdepending on both variables t and r, we must consideronly anisotropic pressures, thus requiring pr = pl .

In the following we are interested in studying mattersources threading the wormhole described by barotropicequations of state with constant state parameters. Thuswe shall require that

(t, r) =

pr(t, r) =

r (t, r),

pl(t, r) =

l(t, r), (12)

where r and l are constant state parameters. Clearly,the requirement (12) with r = l = allows us toconnect the evolving wormhole spacetime (2) with thestandard FRW cosmologies, where the isotropic pressuredensity is expressed as p = , with constant state pa-rameter .

Now, with the help of the conservation equations (10)and (11) we can easily solve the Einstein equations (6)


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(9). Note that in these equations the cosmological con-stant is not present, so we can solve the field equationsfollowing the procedure of Ref. [9]. From the structureof these conservation equations we see that one can writethe energy density in the form (t, r) = t(t)r(r). Thusfrom the conservation equations (10) and (11) we obtain

t(t) = C1a(3+r+2l ), (13)

r(r) = C2r2(lr)/r , (14)

where C1 and C2 are integration constants, obtaining forthe full energy density

(t, r) = C r2(lr)/r a(3+r+2l ), (15)

where we have introduced a new constant C in order toredefine C1 and C2.

Now, by subtracting Eqs. (7) and (8), and using the fullenergy density (15), we obtain the differential equation

(l r) C r2(lr)/ra(3+r+2l )

=3b rb

2a2r3. (16)

It is straightforward to see that in order to have a solu-tion for the shape function b = b(r) we must impose theconstraint

r + 2l + 1 = 0 (17)

on the state parameters r and l , thus obtaining for theshape function

b(r) = C3r3 C r r1/r , (18)

where C3 is a new integration constant. Notice thatthe constraint (17) implies that the radial and tangen-tial pressures are given by

pr = r, pl = 12 (1 + r), (19)

so the energy density and pressures satisfy the followingrelation:

+ pr + 2pl = 0. (20)

Now, from Eqs. (6), (15), (18) and taking into accountthe constraint (17) we obtain the following master equa-tion for the scale factor:

3H2 = 3C3a2

+ . (21)

In the following paragraph, with the help of this equation,we shall determine the scale factor, and then the specificform of the energy density (15).

Lastly, note that there is another branch of sphericallysymmetric solutions to Eqs. (6)(8). By adding theseequations and taking into account Eqs. (12) and (15) weobtain the equation

6a

a 2 = (1 + r + 2l) C r2(lr)/r a(3+r+2l).

(22)

Now, if we take r = l = , we obtain a differentialequation for a(t) and which has a first integral givenby

3

a

a

2+

Const

a2= Ca(t)3(1+) + . (23)

This differential equation is the standard Friedmann

equation for a FRW universe filled with an ideal fluid(with p(t) = (t)) and a cosmological constant. Theterm Const/a2 is related to the curvature term while theenergy density scales as a3(1+).

It is worth noticing that, if we take 1 + r + 2l = 0(see Eq. (17)), we obtain from Eq. (22) the differentialequation 3a/a = which has a first integral of the formof the master equation (21).

III. WORMHOLE SOLUTIONS

Now we shall study the specific wormhole configu-

rations which should be allowed by the master equa-tion (21). The case = 0 (which implies that a(t) =C3 t + Const) was considered before in Ref. [9] so weshall restrict ourselves here to the case = 0. To startwith, we shall consider first the static case of a de Sitterwormhole.

A. The C3 = 0 case: The de Sitter wormhole

First, let us consider the case C3 = 0 in Eq. (21). Inthis case the solution is given by

a(t) = a0e/3 t, (24)and is allowed only for > 0. Thus the metric (2) takesthe form

ds2 = dt2 + a20e2

/3 t (25)

dr2

1 + Cr r(1+1/r)+ r2(d2 + sin2d2)

,

describing contracting and expanding wormholes. If wechoose the plus sign in Eq. (24) we obtain an inflation-ary scale factor, which gives an exponential expansionfor an inflating wormhole. This kind of wormholes was

considered before by Roman in Ref. [11]; however, in thiswork the resulting properties are associated with a stress-energy tensor of a general form, without imposing a spe-cific equation of state among the energy density and thepressures. In this case, for an inflating wormhole, we ob-tain from Eq. (12) that the anisotropic pressures take theform (19) where the energy density is given by

(t, r) r(1+3r)/r

e2

/3 t. (26)


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grr

b(r) = r0

r

r0

1/r

+kr30

r

r0

31

r

r0

(1+3r)/r,

a2(t)g1rr = 1 rr0(1+r)/r

kr20

r

r0

21

r

r0

(1+3r)/r, (33)

respectively. This implies that the wormhole throat is lo-cated at r0. In this case the energy density of the matterthreading the wormhole takes the following form:

(t, r) =kr20 1

r20ra(t)2

r

r0

(1+3r)/r. (34)

Clearly, in order to have an evolving wormhole we mustrequire r < 1 or r > 0 (in both of these cases,in the grr metric component, (1 + r)/r > 0 and(1 + 3r)/r > 0), implying that the phantom energycan support the existence of evolving wormholes in thepresence of a cosmological constant. As before, for a mat-ter made of phantom energy with r < 1, we have thatat spatial infinity its energy density vanishes. Thus thewormhole configurations with k = 1, 1 (or equivalentlyC3 = 0) are asymptotically vacuum -Friedmann mod-els with open and closed topologies respectively with thecorresponding scale factors of the table II. For r > 0we have the same behavior.

IV. PROPERTIES OF THE SOLUTIONS AND

DISCUSSION

Now we shall consider some characteristic propertiesof the above found evolving wormhole geometries. Es-sentially, as we shall see, these wormholes coupled to acosmological constant have the same properties as havethe evolving wormholes with = 0 discussed in Ref. [9],since the cosmological constant directly controls the be-havior of the scale factor a(t) and not the shape functionb(r) of the spacelike section of the metric (2). However itmust be noticed that in this case we have a richer variety

of topologies since now we can consider evolving worm-hole configurations with k = 0 and k = 1, while only thecase k = 1 for = 0. Effectively, for example, from themetric (30) we can see that for wormholes supported byphantom matter at spatial infinity (r ) we have thefollowing asymptotic metric:

ds2 dt2 +a2

(t)

dr2

1 kr2 + r2(d2 + sin2d2)

. (35)

This metric has slices t = const which are 3spaces ofconstant curvature: open for k = 1, flat for k = 0and closed for k = 1. This implies that the asymptoticmetric (35) is foliated with spaces of constant curvature.

Now the presence of the cosmological constant ofcourse indicates that these wormholes are not asymp-totically flat. If we calculate the Riemann tensor for themetric (30), we find that its independent nonvanishing

components are

R()(r)(r)() = R()(r)(r)() =

a2 + k + 12C(1 + r) r(1+3r)/r

a2,

R()()()() =a2 + k 12Cr r(1+3r)/r

a2,

R(t)()(t)() = R(t)()(t)() = R(t)(r)(t)(r) =a

a. (36)

From these expressions and the scale factors given in Ta-ble II we see that at spatial infinity, i.e. r , theseRiemann tensor components do not vanish for a worm-hole with r < 1 or r > 0, except for the case = 0 asstated in Ref. [9]. In conclusion, regardless of the energydensity (31) vanishing for r , the Riemann tensorcomponents do not vanish, due to the presence of thecosmological constant .

Let us now consider the energy conditions. It is wellknown that, in all cases, the violation of the weak energycondition (WEC) is a necessary condition for a staticwormhole to exist. In general, for the energymomentumtensor (5), WEC reduces to the following inequalities:

(t, r) 0, (t, r) + pr(t, r) 0,(t, r) + p

l

(t, r)

0, (37)

which may be rewritten by using the expressions (12)and (17) as follows:

(t, r) 0, (1 + r) (t, r) 0,(1 r) (t, r) 0. (38)

Thus, for the found gravitational configurations (forwhich we must require r < 1 or r > 0 in order tohave a wormhole), Eqs. (34) and (38) imply that for:

k = 0 o r k = 1 and > 0, we must requirer 0) and > 0, we mayrequire r < 1 for r20 < 1 (and the WEC is alwaysviolated) or require 0 < r < 1 for r20 > 1 (and theviolation of WEC is avoided). Unfortunately thislatter case must be ruled out for the considerationof evolving wormhole configurations as we shall seebelow.


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Note that for the case k = 1 the value r0 = 1 impliesthat = 0 and b(r) = r3, so we have in this case ahomogeneous and isotropic closed FRW cosmology.

In order to dilucidate whether or not the shape of thewormhole is maintained, we must check the fulfillment ofthe flaring out condition, which is given by

d2r

dz2=

b br2b2

=b br

2a(t0)b2> 0, (39)

where r = a(t0)r, b(r) = a(t0)b(r) (for a detailed dis-cussion see Ref. [9]). Taking into account the shape func-tion b(r) of Eq. (33) we obtain

d2r

dz2=

r

r0

1/r

2Drr3(r/r0)1/r + (1 + r)r0(kr20 1)2r(kr3(r/r0)1/r + r0(1 kr20))2

. (40)

Clearly if k = 0, 1 this constraint is satisfied for theentire range of the radial coordinate r. If k = 1 this

constraint is always violated for r0 > 1, and for r0 < 1it may be satisfied. Thus we can have wormholes withk = 1 only for r0 < 1 and then WEC is violated.

On the other hand, from Eq. (22) and the con-straint (17) we conclude that the expansion of the worm-hole is accelerated due to the presence of the cosmologi-cal constant. So this family of evolving wormholes, sup-ported by an anisotropic phantom energy, may expandwith acceleration for k = 1, 0, 1. More specifically, fromthe form of the scale factors of Table II, we conclude thatfor the cases with > 0 the wormhole expands forever,and for < 0 the wormhole expands to a maximum valueand then recolapses.

The shape of a wormhole is determined by b(r) asviewed, for example, in an embedding diagram in a flat3dimensional Euclidean space R3. To construct in ourcase such a diagram of a wormhole we can follow the pro-cedure described in Ref. [9]. In order to have a good em-bedding we must impose the condition b(r) 0. Clearlyfrom Eq. (33) we see that for k = 0 and k = 1 the shapefunction b(r) is positive for all r > r0. For k = 1the shape function is positive for r0 < r < rmax where

rmax = r0

1 + 1r20

r/(1+3r).

Let us now consider the tidal forces experienced by atraveller while crossing this kind of wormholes. We re-call that the tidal acceleration must not exceed one Earthgravity, i.e. g = 9.8 m/s2, in order for wormhole travelto be at all convenient for human beings [4]. The calcu-lus of these tidal forces may be simplified by introduc-ing a new orthonormal reference frame which moves ata constant speed v with respect to observers who alwaysremain at rest at constant r, and (see Ref. [9] andalso cites therein).

Thus, for the generic metric (2) (with (t, r) = 0), andby considering the size of the travellers body (i.e. headto feet) to be 2 (m), the Riemann tensor components

in this new basis are constrained to be

|R1010| = aa gc2 2 m 1(108m)2 , (41)

and

|R2020 | = |R3 030 | =2 aa 2

2

2a2r3

2a2r3 b + rb g

c2 2 m 1

(108m)2,

(42)

where = 1/

1 2 with = v/c.In this case the radial tidal constraint (41) can be re-

garded as directly constraining the acceleration of theexpansion of the wormhole, while the lateral tidal con-straint (42) can be regarded as constraining the speed vof the traveller while crossing the wormhole.

In particular, the evolving wormholes considered in

this paper evolve with scale factors shown in Table II.This implies that the expansion is not accelerated (i.e.a = 0) only for the cases = 0, thus satisfying the con-straint (41). Now, by taking into account Eq. (22), weconclude that for all = 0 cases of Table II, the radialtidal constraint (41) implies the following constraint onthe cosmological constant:3

gc2 2 m 1(108m)2 . (43)Note that this constraint on the cosmological constant isvalid for all values of k = 1, 0, 1.

Now, by using the Einstein equation (8) with Eqs. (17)and (22), the lateral tidal constraint (42) may be rewrit-ten as follows:3 4 12 22(1 + r)

gc2 2 m 1(108m)2 .Since constraint (41) implies that the cosmological con-stant 1016 m2 and the motion of the traveller benonrelativistic (i.e. v < < c, 1), the above con-straint may be rewritten as

1

2 v

c2

(1 + r)

g

c2 2 m 1

(108m)2.

(44)

Thus the lateral tidal constraint (42) can be regardedmore exactly as constraining both the speed v of the trav-eller and the energy density of the matter threading thewormhole. By taking into account the expression for theenergy density (34), we may rewrite Eq. (44) as follows:v

2(1 + r)(kr20 1)

r20ra(t)2

r

r0

(1+3r)/r


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As we can see, this constraint is more severe at the worm-hole throat, thus evaluating it at r = r0 we obtainv2(1 + r)(kr20 1)r20ra(t)2

0. For example for the case k = 0and a(t) = a0e

/3 t we have that the constraint (46) issaturated for

te0 =1

2

/3ln

v

2(1 + r)

gr20ra20

,

with v2(1+r)gr20ra

20

> 1, thus for any t te0 > 0 the men-

tioned lateral tidal constraint will be satisfied.

V. ACKNOWLEDGEMENTS

This work was supported by CONICYT throughGrants FONDECYT N0 1080530 and 1070306 (MC, SdCand PS), and by Direccion de Investigacion de la Uni-versidad del BoBo (MC). SdC also was supportedby PUCV grant N0 123.787/2008 and P.S. and P.M.by Universidad de Concepcion through DIUC Grant N0

208.011.048-1.0.

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