83

(|Φ|

)

fl

= ∫ 4 − −163

+ 21 − ∂ Φ

∂ Φ

− (|Φ|2) + fl,

(1)

isacomplex scalar field with the potential

Our goal here is to study equilibrium

where Φ and

denotes

the

action of

the

fluid.

wormhole-plus-fluid solutions. To this end, it is

0,1,2,3

.

, . . . =

in polar Gaussian coordinates

Hereafter2

the Greek indices run over

convenient to take the spherically symmetric metric

right-hand side contains the energy-momentum

2

=

(

0

)

2

−

2

−

2

(Θ

2

+ sin Θ

2

),

can

obtain

the

gravitational

equations

whose

2

By varying (1) with respect to the metric, one

with

a

tensor

We

will

study localized solutions

= −2 (∂ Φ ∂ Φ + ∂ Φ∂ Φ ) +

where

and

are

functions

of

the

radial

presence of the complex ghost scalar field in the

1

coordinate

only.

∂ Φ

∂ Φ

+ +

wormholelike topology, which is ensured by the

+ 2

for the scalar field the harmonic ansatz

1

dependence in the gravitational equations, we use

+(

+ ) − ,

action

(1).

Then,

in

order

to

have no

time

Φ(

, ) = ( )

,

its pressure. In turn, varying (1) with respect to the

which

of

the

where

is the energy density of the fluid and

is

ensures

0

that

the

spacetime

−0

scalar field, one obtains the equation for

,

equations:

1

∂

∂Φ

−

∂ −

∂ = |Φ|2

Φ.\

configurationunderconsideration remainsstatic.As

Φ

a result, we have the following set of Einstein-scalar

− 2 ′′ + ′ 2

+ 12 = 8 4 00

= 8 4 +

21 [−( ′2 + 2 − 2) + ] ,

(2)

′′ + 21

−

′

′

+ ′ + 12 = 8 4

11 = 8 4

− + 21 ( ′2

+ 2

−

2 + ) ,

+ ] ,

(3)

′ ′

+ 21

′′

+ 41

′2

= −8 4

22

= −

8 4 −

+ 21 [−( ′2

−

2 − 2)

(4)

′′

+ 21 ′ + 2 ′ ′

+ 2 −

+ |Φ|2

= 0,

(5)

and the boundary conditions, it is possible to

where the prime denotes differentiation with respect

to

. Depending on the specific form of the potential

obtain

localized

equilibrium

solutions

by

solving

− 2 ′′

+ ′ 2 + 12

= (1 + ) − ′2 − Ω2 − 2

− 2

+ Λ2

4,

(10)

− ′ ′ + ′ + 12 = −+1 + ′2 + Ω2 − 2 −

2 +

Λ2 4

,

,

(11)

′′ + 21

′ ′ + 21 ′′ +

41 ′2 = +1 + ′2 − Ω2 − 2 + 2 − Λ2 4

(12)

one may

′′ + 21

′

+ 2 ′

′ + (Ω2 − − 1 + Λ2)

= 0,

(13)

use= 8any three/of

themin calculations. Here,

configurations2

under consideration.

where

. Since, due to the Bianchi

constant

plays the role of the Arnowitt-Deser-

identities,notallof2 these2equationsareindependent,

Misner (ADM) mass of the wormhole-plus-fluid

we will solve Eqs. (8), (10), (12), and (13), treating

We solve the set of equations (8) and (10)-(13)

the first-order equation (11) as a constraint equation

numerically,usingtheboundary conditions(14) and

to check the accuracy of the computations.

(15)andvaryingthevalueofthebosonfrequency

Ω

When solving the above equations, we use the

in the interval

[the lower limit

symmetric

boundary

conditions

imposed

at

the

corresponds to

the case of real scalar fields and the

center

,

0 ≤ Ω ≤ 1

Ω = 0

(0) =

, (0) = , (0) = , (0) = 1

consideration may be subdivided into two regions:

= 0

nonoscillating asymptotic behavior of the scalar

(14)

field; see Eq. (17)]. In doing so, the systems under

(i) the internal one, where both the fluid and the

(all first-order derivatives are supposed to be zero at

scalar field are present; (ii) the external one, where

the center). The constraint equation (11) then yields

only the scalar field is present. Correspondingly, the

=

Ω

−1+(Λ/2)

−.

the external ones at the edge of the fluid,

.

2

2

−

2

solutionsintheexternalregioncanbeobtainedfrom

1

zero. The internal solutions should be matched with

(15)

Eqs. (10)-(13), where the function

is set to be

Using

this

expression

and

expanding

the

This is done by equating the corresponding

=

2

,

one

can find

from

Eq.

(10)

the

boundary ofthe

can

≈

. In turn, the integration constant

value

of the2second derivative of

values

at the center

+ 1/2

the

vicinity

of

the

center

as

be determined from the

and their derivatives. The

function

in

of the functions

,

,

( ) = 0

solutions, proceeding

the

asymptotic

from

2

= 2 {2Ω

− [1 + (

+ 1)]}.

fluid

is defined by thecondition

2

−

asymptotically flat.

2

(16)

requirement that the external solutions must be

asinthecaseofbosonstars[22],assuch

aparameter,Ω

typical solutions for the scalar field function

, the

one

can

take

. Then,

the other

two

parameters

fluid function

, and the metric function

. The

Thus we have threeparameters –

,

, and

,

In the left panels of figure 1, we show a set of

should

be

chosen in such a way as to ensure

value of the

one of which can be chosen arbitrarily. For instance,

and

. In this sense, we

limiting value of the physically acceptable interval.

and

limiting

case

is

undistinguishable

since

boson frequency

represents the

with an eigenvalue problem for the parameters

regular

solutions do

exist

not

= 0.5

but

are

→

0

→

,

,

for

all

asymptotic flatness of the spacetime, when

Ω = 1

(see

restricted by some critical values of

will deal′

Note that

for

the

chosen

value

another

It is also useful to write out the asymptotic

Ω = 0

behavior of the solutions:

= crit

and

Ω

.

2 2

′

2

below). An interesting feature of the system is the

In the right panels of figure 1, we exhibit the

→

1

−

,

→

,

→

1 −

,

possibility of the presence of the maximum of the

solutions for the metric function

. Its behavior

fluid density shifted with respect to the center of the

→

exp −

2

for 0 ≤ Ω <

configuration; see figure 1(a).

of the

expansion

strongly

depends

on

the

sign

1 − Ω

= 0.5

< 1,

→ 3

3

for Ω

= 1,

is on either side of the center of the

4

1

exp − 8| 2|

double throat when the minimum value

given

coefficient

from (16). Namely,

for

the

(17)

,

Σthe2 configurations always possess a

where

1, 2

, and

3

are integration constants and

min{ ( )}

2

< 0

2

2

. Note that the integration

possibilities: (i) if

, there is one equator (a

2

Ω

/√1 − Ω

= −1 +

configuration.

In

this

case

there

are

two

X

2.0

Λ=1

17.5

Λ=1

1.5

17.0

1.0

16.5

0.5

×6.67

16.0

0

xb

th

×10

15.5