Composite vacuum Brans-Dicke wormholes

Sergey V. Sushkov1,2,*1Department of Mathematics and Department of Physics, Kazan Federal University, Kremlevskaya Street 18, Kazan 420008, Russia

2Physics Department, CSU Fresno, Fresno, California 93740-8031, USA

Sergey M. Kozyrev†

Scientific Center for Gravitational Wave Study ‘‘Dulkyn,’’ Kazan, Russia(Received 17 September 2011; published 13 December 2011)

We construct a new static spherically symmetric configuration composed of interior and exterior Brans-

Dicke vacua matched at a thin matter shell. Both vacua correspond to the same Brans-Dicke coupling

parameter !; however, they are described by the Brans class I solution with different sets of parameters of

integration. In particular, the exterior vacuum solution has Cextð!Þ � 0. In this case the Brans class I

solution for any ! reduces to the Schwarzschild one being consistent with restrictions on the post-

Newtonian parameters following from recent Cassini data. The interior region possesses a strong

gravitational field, and so the interior vacuum solution has Cintð!Þ ¼ �1=ð!þ 2Þ. In this case the

Brans class I solution describes a wormhole spacetime provided ! lies in the narrow interval �2�ffiffi3

p3 <

!<�2. The interior and exterior regions are matched at a thin shell made from an ordinary perfect

fluid with positive energy density and pressure obeying the barotropic equation of state p ¼ k� with

0 � k � 1. The resulting configuration represents a composite wormhole, i.e. the thin matter shell with

the Schwarzschild-like exterior region and the interior region containing the wormhole throat.

DOI: 10.1103/PhysRevD.84.124026 PACS numbers: 04.20.�q, 04.20.Jb, 04.50.Kd

I. INTRODUCTION

Brans-Dicke theory is the famous prototype of gravita-tional theories alternative to Einstein’s general relativity[1]. The essential feature of Brans-Dicke theory is thepresence of a fundamental scalar field nonminimallycoupled to curvature, and so it and its generalizations,which may include one or several scalar fields, are gener-ally known as scalar-tensor theories. Initially the Brans-Dicke theory was developed as a modified relativistictheory of gravitation compatible with Mach’s principle[2,3]. The current interest in scalar-tensor theories is mani-fold. They arise naturally as the low energy limit of manytheories of quantum gravity such as superstring theories orthe Kaluza-Klein theory. Moreover, Brans-Dicke andscalar-tensor theories have numerous interesting cosmo-logical applications, which include inflationary scenarios,dark energy and dark matter models, etc. (see, for example,[4]). Static solutions in scalar-tensor theories are also ofinterest. In particular, Brans-Dicke wormholes have beenintensively investigated [5–13]. It is worth noticing thatwormhole solutions may appear in the whole ghost rangeof Brans-Dicke theory (i.e. for any !<�3=2; see theinteresting discussion in Ref. [13]).

The action of Brans-Dicke theory is given by1

S ¼ 1

2

Zdx4

ffiffiffiffiffiffiffi�gp �

�R�!�;��;�

�

�þ Sm; (1)

where R is the scalar curvature, � is a Brans-Dicke scalar,! is a dimensionless coupling parameter, and Sm is anaction of ordinary matter (not including the Brans-Dickescalar). The action (1) provides the following field equa-tions:

G�� ¼ 1

�T�� þ !

�2�;��;� � !

2�2g���;��

;�

þ 1

��;�;� � 1

�g���

;�;�; (2a)

�;�;� ¼ T

2!þ 3; (2b)

where G�� ¼ R�� � 12Rg�� is the Einstein tensor, and

T ¼ T�� is the trace of the matter energy-momentum tensor

T��. In 1962 Brans himself [3] constructed static spheri-

cally symmetric vacuum solutions (i.e., with T�� � 0) of

the Brans-Dicke field equations (2). He found four classesof solutions, which are now known as Brans class I, II, III,and IV solutions. However, it is necessary to emphasizethat solutions from different classes are not independent—one can be derived from the other by some analyticaltransformations [14]. For this reason, throughout thiswork we will discuss only the Brans class I solution, whichis the best known spherically symmetric solution of Brans-Dicke theory.The vacuum Brans class I solution given in isotropic

coordinates reads

*sergey_sushkov@mail.ru†Sergey@tnpko.ru1Units 8�G ¼ c ¼ 1 are used throughout the paper.

PHYSICAL REVIEW D 84, 124026 (2011)

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ds2¼�e2�0

�1�B=r

1þB=r

�2=A

dt2

þe2�0

�1þB

r

�4�1�B=r

1þB=r

�2ðA�C�1Þ=A½dr2þr2d�2�;

(3a)

�ðrÞ¼�0

�1�B=r

1þB=r

�C=A

; (3b)

where d�2 ¼ d�2 þ sin2�d’2 is the linear element of theunit sphere, and the radial coordinate r satisfies the condi-tion r > B in order to provide an analyticity of the solution.Generally, the solution depends on five free parameters:�0, �0, �0, B, and C. The sixth parameter A is not free; itobeys the following constraint condition:

A ¼ ½ðCþ 1Þ2 � Cð1� 12!CÞ�1=2 > 0: (4)

In Brans-Dicke theory the metric (3a) represents an exte-rior gravitational field of some spherical distributionof matter. Far from a source of gravity, i.e., in the limitr ! 1, it takes the form

ds2 ¼ �e2�0�1� 2M

rþOðr�2Þ

�dt2

þ e2�0

�1þ 2M

rþOðr�2Þ

�½dr2 þ r2d�2�; (5)

where M ¼ 2B=A is an asymptotic mass measured by adistant observer, and ¼ 1þ C is the post-Newtonianparameter. Because of asymptotic flatness one should set�0 ¼ �0 ¼ 0. The value of can be estimated from therecent conjunction experiment with Cassini spacecraft asj� 1j � 2:3� 10�5 [15,16]. Hence, one gets

jCj � 2:3� 10�5: (6)

Note that, formally, the parameter C does not depend on!,and so one may directly set C ¼ 0 in Eqs. (3) and (4). As aresult, one finds A ¼ 1 and

ds2¼��1�M=2r

1þM=2r

�2dt2þ

�1þM

2r

�4½dr2þr2d�2�; (7a)

�ðrÞ��0¼ const; (7b)

where Eq. (7a) is nothing but the Schwarzschild metric (inisotropic coordinates). It is obvious that the Schwarzschildsolution is perfectly consistent with observational data.However, one should remember that any exterior vacuumBrans-Dicke solution has to be matched to some interiorone. Supposing that the interior Brans-Dicke solutioncorresponds to some reasonable spherical distribution ofmatter, one can get, on the basis of a post-Newtonian weak-field approximation, the following relationship [17]:

Cð!Þ ¼ � 1

!þ 2: (8)

Now the limiting (Schwarzschild) caseCð!Þ ! 0,Að!Þ!1is only possible under the limit j!j ! 1.2 UsingEqs. (6) and (8) one can find the lower boundary for !:j!j � 5� 104. Thus, the consideration based on the post-Newtonian weak-field approximation leads to the conclu-sion that the Brans-Dicke theory can be consistent with the(local) observations only if ! is very large.On the other hand, there is no reason for the relationship

(8) to hold in the presence of compact objects with stronggravitational fields. For example, in the context of gravi-tational collapse in Brans-Dicke theory, Matsuda [23] had

considered Cð!Þ / �!�1=2. The other examples of essen-tially relativistic objects possessing strong gravitationalfields are represented by wormholes. Vacuum Brans-Dicke wormholes with various Cð!Þ were discussed in

the literature. Namely, in Ref. [8] the case Cð!Þ ¼�q!�1=2 with q < 0 had been considered. Also,Lobo and Oliveira [11] discussed two models: Cð!Þ ¼ð!2 þ!2

0Þ�1 and Cð!Þ ¼ � expð�!2=2Þ.In this paper we will accept a more general conjecture.

Namely, we will suppose that the form of Cð!Þ can be, inprinciple, different in various spacetime regions. In otherwords, this means that various spacetime regions can pos-sess different Brans-Dicke vacua. To justify this supposi-tion one can speculate that Brans-Dicke vacuum states areforming due to phase transitions in some generalized the-ory, and the vacuum state formation depends on localvalues of the gravitational field. As a result, one will obtain‘‘bubbles’’ of different Brans-Dicke vacua divided by‘‘walls.’’Applying the conjecture about a variety of Brans-Dicke

vacua, we will consider a simple static spherically sym-metric configuration composed of two different vacua.

II. COMPOSITE VACUUM SOLUTION

Let us consider a static spherically symmetric configu-ration composed of two Brans-Dicke vacua. The spacetimemetric in both interior and exterior regions is given inisotropic coordinates as follows:

ds2 ¼ �e2�ðrÞdt2 þ e2�ðrÞ½dr2 þ r2d�2�; (9)

so that x ¼ ðt; r; �; �Þ. We will assume that the interior isdescribed by the vacuum Brans class I solution:

2Here it should be mentioned that Brans-Dicke theory (and itsdynamic generalization) in the limit j!j ! 1 reduces to generalrelativity [17] (see, also, Refs. [18–22] where the specific case ofa traceless energy-momentum tensor is discussed).

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e�intðrÞ ¼ e�0

�1� B=r

1þ B=r

�1=A

; (10a)

e�intðrÞ ¼ e�0

�1þ B

r

�2�1� B=r

1þ B=r

�ðA�C�1Þ=A; (10b)

�intðrÞ ¼ �0

�1� B=r

1þ B=r

�C=A

; (10c)

where �0, �0, �0, B, and C are free (still undefined)parameters, and A is given by Eq. (4). Note that the radialcoordinate r runs monotonically from B to a, where a > Bis a boundary of the interior region.

As was already mentioned, an exterior region of somespherical gravitating configuration can also be describedby the Brans class I solution provided that C fulfills theconstraint (6). Assuming Cextð!Þ � 0, we obtain the exte-rior Schwarzschild solution:

e�extðrÞ ¼ 1�M=2r

1þM=2r; (11a)

e�extðrÞ ¼�1þM

2r

�2; (11b)

�extðrÞ ¼ 1; (11c)

where M ¼ 2Bext is the Schwarzschild mass. Note that wehave put �0;ext ¼ �0;ext ¼ 0 in order to provide the asymp-

totic flatness. Also, without loss of generality, we put�0;ext ¼ 1. The radial coordinate r within the exterior

region runs from a to infinity. We will suppose that a >M=2; this guarantees that the exterior region does notcontain the event horizon.

So, r is the global radial coordinate monotonically run-ning from B to a in the interior region, and from a toinfinity in the exterior one. The surface �: r ¼ a is a thinshell where the interior and exterior solutions, (10) and(11), should be matched. Since Eqs. (10) and (11) are thevacuum Brans-Dicke solutions, we should conclude that allordinary matter (excluding the Brans-Dicke scalar) is con-centrated at the thin shell �.

Here it is worth noticing that thin-shell Brans-Dickewormholes were studied in the literature [24,25]. Themodels considered in Refs. [24,25] were constructed bythe cut-and-paste method.3 Though our construction seemsto be similar to cut-and-paste wormhole configurations,this similarity has only a formal character. Actually, theessence of the method is the following: One takes two ofthe same copies of spacetime manifolds with appropriateasymptotics, cuts and casts away ‘‘useless’’ regions ofspacetimes (containing horizons, singularities, etc.), andpastes remaining regions. As a result, one obtains a geo-desically complete wormhole spacetime with given asymp-totics (Schwarzschild, Reissner-Nordstrom, Brans-Dicke,etc.) and a throat being a thin shell of exotic matter

violating the null energy condition. In our case, we haveinitially a thin shell made from ordinarymatter, and thenwelook for appropriate interior and exterior Brans-Dicke so-lutions matched at the shell. Note that this approach issimilar to the problem of a thin shell in general relativity(see Ref. [28]). However, the distinction is that the Birkhofftheorem is not valid in Brans-Dicke theory, and so theinterior and exterior Brans-Dicke vacua are not unique.To analyze a thin-shell configuration we will follow the

standard Darmois-Israel formalism [29], also known as thejunction condition formalism. The shell� is a synchronoustimelike hypersurface with intrinsic coordinates i ¼ð�; �; ’Þ. The coordinate � is the proper time on the shell.Generally, a position of � can be a function of the propertime. However, hereafter we will assume að�Þ � a ¼const. Note that the metric (first fundamental form) andthe scalar field should be continuous on �:

�intðaÞ ¼ �extðaÞ; �intðaÞ ¼ �extðaÞ;�intðaÞ ¼ �extðaÞ � 1:

(12)

Substituting Eqs. (10) and (11) into (12) gives

e�0 ¼�1�M=2a

1þM=2a

��1þ B=a

1� B=a

�1=A

; (13a)

e�0 ¼�1þ M

2a

�2�1þ B

a

��2�1þ B=a

1� B=a

�ðA�C�1Þ=A; (13b)

�0 ¼�1þ B=a

1� B=a

�C=A

: (13c)

At the same time, derivatives of the metric and scalar fieldcan be discontinuous. The discontinuity of the metric isusually described in terms of a jump of the extrinsiccurvatureKij. The extrinsic curvature (second fundamental

form) associated with a hypersurface �: FðxÞ ¼ 0 is givenby

Kij ¼ �n

�@2x

@i@j þ ���

@x�

@i

@x�

@j

����������; (14)

where n is the unit normal (nn ¼ 1) to �:

n ¼��������g��

@F

@x�@F

@x�

���������1=2 @F

@x: (15)

The junction conditions in Brans-Dicke theory (general-ized Darmois-Israel conditions) can be obtained by pro-jecting on � the field equations (2) [30]:

� ½Kij� þ ½K� i

j ¼8�

�

�Sij �

S

3þ 2! ij

�; (16)

½�;n� ¼ 8�S

3þ 2!; (17)

where the notation ½Z� ¼ Zextj� � Zintj� stands for thejump of a given quantity Z across the hypersurface �, nlabels the coordinate normal to this surface, and Sij is the

3The first examples of thin-shell wormholes have been givenby Visser [26,27].

COMPOSITE VACUUM BRANS-DICKE WORMHOLES PHYSICAL REVIEW D 84, 124026 (2011)

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energy-momentum tensor of matter on the shell located at�. The quantities K and S are the traces of Ki

j and Sij,

respectively. Note that Eq. (16) is equivalent to

Sij ¼�

8�

�!þ 1

!½K� i

j � ½Kij��: (18)

The jump of the components of the extrinsic curvatureassociated with two sides of the hypersurface FðxÞ ¼r� a ¼ 0 in the spacetime with the metric (9) can befound as

½K��� ¼ ½�0�e��ðaÞ; ½K�

�� ¼ ½K’’� ¼ ½�0�e��ðaÞ: (19)

The surface stress-energy tensor of a perfect fluid is givenby

Sij ¼�� 0 00 p 00 0 p

0@

1A; (20)

where � and p are the surface energy density and thesurface pressure, respectively. Now, Eq. (18) yields

FIG. 1. Contour plots for �ð�;�;!0Þ and pð�;�;!0Þ for !0 ¼ �2:05, �2:5, and �5. Thin solid curves denote lines ofzero level: �ð�;�;!0Þ ¼ 0 (lower line) and pð�;�;!0Þ ¼ 0 (upper line). A corresponding value is positive in the region belowthe zero level line, and so �> 0 and p > 0 in the shadowed region. Thick solid curves denote lines given by the equation of statepð�;�;!0Þ � k�ð�;�;!0Þ ¼ 0, where k ¼ 1, 1

2 ,13 ,

14 ,

15 from top to bottom.

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� ¼ ��ðaÞe��ðaÞ

8�!ð½�0� þ 2ð!þ 1Þ½�0�Þ; (21)

p ¼ �ðaÞe��ðaÞ

8�!ðð!þ 1Þ½�0� þ ð!þ 2Þ½�0�Þ; (22)

where ½�0� ¼ �0extðaÞ � �0

intðaÞ and ½�0� ¼ �0extðaÞ �

�0intðaÞ. The obtained relations express the surface energy

density � and the surface pressure p in terms of jumps offirst derivatives of the metric functions. Substituting theexpressions (10) and (13) for metric coefficients into (21)and (22), we find

� ¼ 1

4�a!ð1þ�Þ2��ð2!ð1��Þ � 2�þ 1Þ

1��2

� �ð2!ð1� �Aþ CÞ � 2�Aþ 2Cþ 1ÞAð1� �2Þ

�; (23)

p ¼ 1

4�a!ð1þ�Þ2��ð!�þ 2�� 1Þ

1��2

� �ð!ð�A� CÞ þ 2�A� 2C� 1ÞAð1� �2Þ

�; (24)

where � ¼ M=2a and � ¼ B=a are convenient dimen-sionless values such that �< 1 and �< 1.

III. MATTER ON THE THIN SHELL

The resulting expressions (23) and (24) give the energydensity and the pressure of matter filling the thin shell � interms of parameters of the model: the coupling parameter!, the shell radius a, the exterior vacuum parameter �(dimensionless Schwarzschild mass), and the interior vac-uum parameters � and C. Note that both � and p areproportional to a�1, and so, without lost of generality, wecan make rescaling � ! �a�1 and p ! pa�1, or, equiv-alently, just put a ¼ 1. To proceed further, we will fix thespecific form of Cð!Þ given by Eq. (8), so that Cð!Þ ¼�1=ð!þ 2Þ. In this case, Eq. (4) yields

Að!Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2!þ 3

2!þ 4

s: (25)

Note that the expression under the square root in Eq. (25) ispositive provided!<�2 or!>�3=2. Hereafter we willrestrict our consideration to the case !<�2, since onlythis case includes Brans-Dicke wormhole configurations.Substituting the given Cð!Þ and Að!Þ into (23) and (24)

yields

�ð�;�;!Þ ¼ 1

4�!ð1þ�Þ2��ð2!ð1��Þ � 2�þ 1Þ

1��2�

�2!þ 4

2!þ 3

�1=2 �ð2!ð1� ~�ð!ÞÞ � 2 ~�ð!Þ þ 1Þ

1� �2

�; (26a)

pð�;�;!Þ ¼ 1

4�!ð1þ�Þ2��ð!�þ 2�� 1Þ

1��2�

�2!þ 4

2!þ 3

�1=2 �ð! ~�ð!Þ þ 2 ~�ð!Þ � 1Þ

1� �2

�; (26b)

where

~�ð!Þ ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2!þ 3

2!þ 4

sþ 1

!þ 2:

Fixing a particular value of ! ¼ !0, we obtain � and p asfunctions of � and �. In Fig. 1 we present a series ofcontour plots for �ð�;�;!0Þ and pð�;�;!0Þ on theð�;�Þ plane for various values of !0. It is worth noticingthat the plots demonstrate that for all !<�2 there aredomains such that both � and p are positive.

Additionally, any reasonable model of matter shouldinclude an equation of state p ¼ pð�Þ imposing somerelation between the energy density � and the pressurep. Hereafter we will consider the barotropic equation ofstate p ¼ k�. Note that the equation-of-state parameter kis non-negative, k � 0, for ordinary matter with positiveenergy density and non-negative pressure. Moreover, thecondition k � 1 guarantees that the speed of sound inmatter medium does not exceed the speed of light. Inparticular, k ¼ 0 for the dust, k ¼ 1=3 for the radiation,and k ¼ 1 for the stiff matter. By using Eqs. (26), the

equation of state can be rewritten as

pð�;�;!Þ � k�ð�;�;!Þ ¼ 0: (27)

For given ! and k this equation provides an additionalrelation between � and � which can be graphically repre-sented as some curve on the ð�;�Þ plane. In Fig. 1 weshow such curves given for different values of! and k. Thegraphical representation illustrates that for any !<�2and k > kminð!Þ, there exists a domain of ð�;�Þ, where� 2 ð0; �maxð!; kÞÞ and � 2 ð0; �maxð!; kÞÞ, such that theequation of state (27) holds. Note that a boundary valuekmin depends on !, and �max and �max depend on both !and k. Note also that p > 0 because of kminð!Þ> 0, and somatter filling the thin shell is not the dust with zeropressure.Finally, we may conclude that the thin shell dividing two

static spherically symmetric regions with different Brans-Dicke vacua can be made from ordinary matter. In particu-lar, it can be the perfect fluid with the barotropic equationof state.

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IV. COMPOSITE CONFIGURATIONWITH AWORMHOLE

In previous sections we have constructed the staticspherically symmetric configuration composed of twoBrans-Dicke vacua and demonstrated that the thin shelldividing the regions with different vacua can be made fromordinary matter. In this section we will discuss the follow-ing problem: Under which conditions does the compositeconfiguration represent a wormhole?

The exterior region of the composite configuration isdescribed by the Schwarzschild metric and does not con-tain any wormholes. Let us consider the interior region.The interior metric (10) has an explicit singular behavior atr ¼ B. To determine whether it is a real or fictitious(coordinate) singularity, we should explore the behaviorof curvature invariants. For example, the scalar curvaturecalculated in the metric (10) reads

R ¼ � 8e�2�0r4B2ðA2 � C2 � C� 1ÞA2ðr� BÞ2ð2A�C�1Þ=Aðrþ BÞ2ð2AþCþ1Þ=A : (28)

It is obvious that R becomes singular at points where thedenominator of Eq. (28) is equal to zero. In particular, if the

power of the term ðr� BÞ2ð2A�C�1Þ=A is positive, thenr ¼ B is a naked singularity. And vice versa, the scalarcurvature R remains regular at r ¼ B provided the power

of ðr� BÞ2ð2A�C�1Þ=A is negative. Substituting Cð!Þand Að!Þ given by Eqs. (8) and (25) into the inequality2ð2A� C� 1Þ=A < 0, we obtain

2�!þ 1

!þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2!þ 4

2!þ 3

s< 0: (29)

The last inequality is fulfilled in a narrow interval

� 2�ffiffiffi3

p3

<!<�2: (30)

Hence the scalar curvature R is regular at r ¼ B if and onlyif ! takes its value within the interval (30). Moreover,

since R / ðr� BÞ2j2A�C�1j=A, it is equal to zero at r ¼ B.

Note that in this case the metric function e2�intðrÞ given byEq. (10b) tends to infinity as r ! B, and hence r ¼ B is aflat spatial infinity.

Finally, the composite vacuum configuration with�2�ffiffi3

p3 <!<�2 is regular in the range r 2 ðB;1Þ, does notcontain horizons in this range, and is asymptotically flatboth as r ! B and r ! 1. Therefore, we can conclude thatsuch a configuration is nothing but a wormhole.

Let us determine the position of the wormhole throat. Itcorresponds to a sphere r ¼ rth with the radius rth provid-

ing a global minimum of the function r2e2�intðrÞ (this guar-antees the minimality of area of the sphere). The value rthis called a throat radius. Taking into account Eq. (10b), wefind

rth ¼ B

24Cþ 1

Aþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Cþ 1

A

�2 � 1

s 35: (31)

Substituting Eqs. (8) and (25) for Cð!Þ and Að!Þ into thelast expression yields

rth ¼ Bð!Þ; (32)

where

ð!Þ ¼ !þ 1

!þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2!þ 4

2!þ 3

sþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�3!� 4

ð2!þ 3Þð!þ 2Þ

s:

One can easily check that ð!Þ> 1 for any !<�2, andhence rth > B.Since B ¼ a�, we have rth ¼ a�ð!Þ. As was shown in

the previous section, if the thin shell is made from theperfect fluid with the barotropic equation of state p ¼ k�,then �<�maxð!; kÞ. A numerical analysis shows that�maxð!; kÞð!Þ< 1 for any !<�2 and k > kminð!Þ,and so rth < a. Therefore, we can conclude that the worm-hole throat r ¼ rth is situated within the interior region r 2ðB; aÞ of a composite vacuum configuration.

V. SUMMARYAND DISCUSSION

In this paper we have constructed a new static spheri-cally symmetric configuration composed of interior andexterior Brans-Dicke vacua divided by a thin matter shell.Both vacua correspond to the same Brans-Dicke couplingparameter !; however, they are described by the Bransclass I solution (3) with different sets of parameters ofintegration. In particular, the exterior vacuum solution hasCextð!Þ � 0. In this case the Brans class I solution with any! just reduces to the Schwarzschild one being consistentwith restrictions on the post-Newtonian parameters follow-ing from recent Cassini data. The interior region possessesa strong gravitational field, and so, generally, Cintð!Þ � 0.In particular, we have used a specific choice Cð!Þ ¼�1=ð!þ 2Þ. In this case the Brans class I solution de-scribes a wormhole, provided ! lies in the narrow interval

�2�ffiffi3

p3 <!<�2. The interior and exterior regions are

matched at a thin shell made from ordinary matter withpositive energy density and pressure. We have studied indetail the shell made from a perfect fluid with the baro-tropic equation of state p ¼ k� with 0 � k � 1. Theresulting configuration represents a composite wormhole,i.e. the thin matter shell with the Schwarzschild-likeexterior region and the interior region containing thewormhole throat.Note that the restriction obtained for the Brans-Dicke

coupling parameter !, namely, �2�ffiffi3

p3 <!<�2, cru-

cially depends on the form of Cð!Þ. As was discussed inRefs. [10,11], an appropriate choice of Cð!Þ providesdifferent viability regions and less restrictive intervals for!; in particular, one is able to construct a more general

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class of vacuum Brans-Dicke wormholes that include thevalue of ! ¼ 0, proving the consistency of the solutionsconstructed in fðRÞ gravity [10]. Analogously, the freedomin the choice of Cð!Þ would provide more general modelsof composite Brans-Dicke wormholes which need furtherinvestigation.

An interesting feature of composite wormholes is thatthe strong-field interior region containing all exotic ghost-like matter is hidden behind the matching surface, whereasthe exterior weak-field region possesses the usualSchwarzschild vacuum. Such a configuration is similar tothe model of trapped-ghost wormholes [31]. Note that inboth models wormholes have two asymptotically flat re-gions. However, in the trapped-ghost wormhole model theghost is hidden in some restricted region around the throat,

whereas in the composite wormhole model the ghostlikeBrans-Dicke scalar occupies the ‘‘half’’ of the wormholespacetime behind the matching surface. Anyway, in thecomposite wormhole configuration a ghost is hidden inthe strong-field interior region, which may, in principle,explain why no ghosts are observed under the usualconditions.

ACKNOWLEDGMENTS

The work was supported in part by the RussianFoundation for Basic Research Grant No. 11-02-01162.Also, S. S. acknowledges Douglas Singleton andCalifornia State University Fresno for hospitality duringthe Fulbright visit.

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