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arXiv:0801.4401v3 [gr-qc] 4 Jul 2008

General class of wormhole geometries in conformal Weyl gravity

Francisco S. N. Lobo∗

Institute of Gravitation & Cosmology, University of Portsmouth, Portsmouth PO1 2EG, UK†and

Centro de Astronomia e Astrof´ ısica da Universidade de Lisboa,

Campo Grande, Ed. C8 1749-016 Lisboa, Portugal

(Dated: July 4, 2008)

In this work, a general class of wormhole geometries in conformal Weyl gravity is analyzed. A

wide variety of exact solutions of asymptotically flat spacetimes is found, in which the stress energy

tensor profile differs radically from its general relativistic counterpart. In particular, a class of

geometries is constructed that satisfies the energy conditions in the throat neighborhood, which is

in clear contrast to the general relativistic solutions.

PACS numbers: 04.20.Gz, 04.20.Jb, 04.50.Kd

I. INTRODUCTION

The Einstein field equation reflects the dynamics of

general relativity, and is formally obtained from the

Einstein-Hilbert action, IEH =

is the curvature scalar. However, the latter action can

be generalized to include other scalar invariants. An in-

triguing example is conformal Weyl gravity, involving the

following purely gravitational sector of the action

?d4x√−gR, where R

IW = −α

?

d4x√−g CµναβCµναβ, (1)

where Cµναβ is the Weyl tensor, and α is a dimension-

less gravitational coupling constant. It was argued in

Ref. [1, 2] that in analogy to the principle of local gauge

invariance that severely restricts the structure of pos-

sible Lorentz invariant actions in flat spacetimes, then

the principle of local conformal invariance is a requi-

site invariance principle in curved spacetimes. The lat-

ter principle requires that the gravitational action to

remain invariant under the conformal transformations

gµν(x) → Ω2(x)gµν(x). The conformal Weyl tensor

Cµναβ= Rµναβ−gµ[αRβ]ν+gν[αRβ]µ+1

3Rgµ[αgβ]ν, (2)

also transforms as Cµναβ→ Ω2(x)Cµναβ. The action IW

is an interesting theoretical construct, for instance, being

a strictly conformally invariant theory, particle masses

may possibly arise through the spontaneous symmetry

breaking of the action [1].

Being a fourth order gravity theory, with respect to

the derivatives of the metric, finding exact solutions of

the gravitational field equations yields a formidable en-

deavor. Nevertheless, the exact vacuum exterior solution

for a static and spherically symmetric spacetime in lo-

cally conformal invariant Weyl gravity was found in Ref.

[1]. The solution contains the exterior Schwarzschild so-

lution and provides a potential explanation for observed

∗Electronic address: francisco.lobo@port.ac.uk

†Electronic address: flobo@cosmo.fis.fc.ul.pt

galactic rotation curves without the need for dark mat-

ter [3, 4].The time-dependent spherically symmetric

solution was further explored in Ref.

solutions to the Reissner-Nordstrm problem associated

with a static and spherically symmetric point electric

and/or magnetic charge coupled to fourth-order confor-

mal Weyl gravity were found [5]. In addition to this, ex-

act solutions associated with the fourth-order Kerr and

Kerr-Newman problems in which a stationary and axi-

ally symmetric rotating system with or without electric

and/or magnetic charge is coupled to gravity, were fur-

ther explored [5]. The causal structure, using Penrose

diagrams, of the static spherically symmetric vacuum so-

lution to conformal Weyl gravity was also investigated

[6]. New vacuum solutions were found using a covariant

(2+2)-decomposition of the field equation, which covers

the spherically and the plane symmetric space-times as

special subcases [7]. Exact topological black hole solu-

tions of conformal Weyl gravity, with negative, zero or

positive scalar curvature at infinity were also found [8],

the former generalizing the well-known topological black

holes in anti-de Sitter gravity.

[2].The exact

The weak-field limit of conformal Weyl gravity for

an arbitrary spherically symmetric static distribution of

matter in the physical gauge with a constant scalar field

was also analyzed [9], and it was argued that the con-

formal theory of gravity is inconsistent with the Solar

System observational data. In a cosmological context,

exact analytical solutions to conformal Weyl gravity for

the matter and radiation dominated eras, and the primor-

dial nucleosynthesis process were exhaustively analyzed.

It was found that the cosmological models are unlikely

to reproduce the observational properties of our Uni-

verse, as they fail to fulfill the observational constraints

on present cosmological parameters and on primordial

light element abundances [10]. In Ref. [11] it was also

argued that in the limit of weak fields and non-relativistic

velocities the theory does not agree with the predictions

of general relativity, and is therefore ruled out by Solar

System observations. Nevertheless, in Ref. [12], it was

counter-argued in the presence of macroscopic long range

scalar fields, the standard Schwarzschild phenomenology

is still recovered. To check the viability of Weyl grav-

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ity, two additional classical tests of the theory, namely,

the deflection of light and time delay in the exterior of

a static spherically symmetric source were analyzed, and

it was shown that the parameters fit the experimental

constraints [13, 14].

An interesting application of conformal Weyl gravity

would be to analyze traversable wormhole solutions in

the theory. We emphasize that an important and intrigu-

ing challenge in wormhole physics is the quest to find

a realistic matter source that will support these exotic

spacetimes. In classical general relativity, wormholes are

supported by exotic matter, which involves a stress en-

ergy tensor that violates the null energy condition (NEC)

[15, 16]. Note that the NEC is given by Tµνkµkν≥ 0,

where kµis any null vector. Several candidates have been

proposed in the literature, amongst which we refer to

solutions in higher dimensions, for instance in Einstein-

Gauss-Bonnet theory [17, 18], wormholes on the brane

[19, 20]; solutions in Brans-Dicke theory [21]; wormhole

solutions in semi-classical gravity (see Ref. [22] and ref-

erences therein); exact wormhole solutions using a more

systematic geometric approach were found [23]; and solu-

tions supported by equations of state responsible for the

cosmic acceleration [24], etc (see Refs. [25, 26] for more

details and [26] for a recent review). In conformal Weyl

gravity, as the gravitational field equations differ radi-

cally from the Einstein field equation, one would expect

a wider class of solutions. This is indeed the case, and

the solutions found contain interesting physical proper-

ties and characteristics, amongst which we refer to a zero

or positive radial pressure at the throat, or more impor-

tant the non-violation of the energy conditions in the

throat neighborhood, contrary to their general relativis-

tic counterparts.

This paper is outlined in the following manner: In Sec-

tion II, we outline the general formalism and the gravi-

tational field equations governing static and spherically

symmetric spacetimes in conformal Weyl gravity. In Sec-

tion III, we further explore specific wormhole solutions,

and finally, in Section IV, we conclude.

II. GENERAL FORMALISM

A.Gravitational field equations

The metric used throughout this work, in curvature

coordinates, is given by

ds2= −B(r)dt2+A(r)dr2+r2?dθ2+ sin2θdφ2?. (3)

The Weyl action, Eq. (1), may be simplified by noting

that the quantity

√−g?RµναβRµναβ− 4RµνRµν+ R2?,

is a total divergence, and thus IW may be rewritten as

(4)

IW = −2α

?

d4x√−g

?

RµνRµν−1

3R2

?

. (5)

Varying the action with respect to the metric gµνpro-

vides the following relationship

(−g)−1/2gµαgνβδIW

δgαβ

= −2α

?

W(2)

µν−1

3W(1)

µν

?

, (6)

with W(1)

µν and W(2)

µν given by

W(1)

µν= 2gµνR;β;β− 2R;µν− 2RRµν+1

2gµνR2, (7)

and

W(2)

µν

=

1

2gµνR;β;β+ Rµν;β;β− Rµβ;νβ− Rνβ;µβ

−2RµβRνβ+1

2gµνRαβRαβ, (8)

respectively.

The stress energy tensor is defined as

Tµν= −

2

√−g

δ(√−gLm)

δ(gµν)

, (9)

where Lm is the Lagrangian density corresponding to

matter.

The final gravitational field equation is given by

4αWµν= Tµν, (10)

with Wµν = W(2)

traceless and covariantly conserved. Note that the intrin-

sic Newtonian constant that arises in the Einstein-Hilbert

action is absent.

Determining Wµν from Eqs. (7) and (8) presents a

formidable endeavor. However, one may use the fact that

for an arbitrary action I =? √−gd4xL, and using the

√−gWrr= −1

2α δA∂A

+∂2

∂r2

µν −1

3W(1)

µν. Both sides are symmetric,

metric (3), the term Wrrmay be deduced from [1]

δI

=

∂

?√−gL?−∂

?√−g

∂r

?√−g∂L

?

∂A′

?

∂L

∂A′′

, (11)

where the prime denotes a derivative with respect to the

radial coordinate, r. Likewise for Wttfrom δI/δB, etc.

However, rather than use this method, which for calcu-

lational purposes is rather intractable, the gravitational

tensor components Wttand Wθθmay be determined

from the Bianchi and trace identities, and given in terms

of Wrr[1].

From the Bianchi identity,

Wµν;µ= (−g)−1/2?

one obtains the following relationship

(−g)1/2Wµν?

,µ+ Γν

µλWµλ= 0,

(12)

DWrr+B′

2AWtt−2r

AWθθ= 0, (13)

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where we have defined

D =

∂

∂r+2

r+A′

A+B′

2B,

(14)

for notational simplicity.

From the trace identity, Wµµ = 0, one obtains the

following relationship

− B Wtt+ AWrr+ 2r2Wθθ= 0.

Finally, using Eqs. (13) and (15), the gravitationalten-

sor components Wttand Wθθare related to Wrrthrough

the following expressions

(15)

Wtt=

A

B − B′r/2(1 + rD) Wrr,

A

4r(B − B′r/2)(B′+ 2B D) Wrr,

(16)

Wθθ=

(17)

so that all the information is contained in the Wrrterm.

The stress-energy tensor components, through the

gravitational field equation, are given by

ρ = −4αWtt,pr= 4αWrr,pt= 4αWθθ, (18)

in which ρ(r) is the energy density, pr(r) is the radial

pressure, and pt(r) is the lateral pressure measured in

the orthogonal direction to the radial direction. Note

that in conformal Weyl gravity, the stress energy tensor

components are constrained through the trace identity,

i.e., −ρ + pr+ 2pt= 0.

Although extremely lengthy, we present the relevant

gravitational terms, namely, Wrr and Wtt, which will

be used extensively throughout this work:

Wrr =

??4A2B2(2B′B′′′− B′′2) − 4ABB′′?3AB′2+ 2A′BB′?+ 7A2B′4+ 6AA′BB′3+ B2B′2?7A′2− 4AA′′??r4

+?− 16A2B3B′′′+ 16AB2B′′(3AB′+ A′B) − 20A2BB′3− 16AA′B2B′2+ 4B3B′(4AA′′− 7A′2)?r3

+?− 4A2B2(8BB′′+ B′2) + 8AA′B3B′+ 4B4(7A′2− 4AA′′)?r2

+32A2B3B′r + 16A2B4(A2− 1)

?

/(48A4B4r4), (19)

and

Wtt =

??

−4B3B′′(8AA′′− 19A′2) − 49A2B′4− 58AA′BB′3+ (24AA′′− 57A′2)B2B′2

+4

A

4ABB′2(11AB′+ 12A′B) − 4(6AA′′− 13A′2)B3B′+ 16AA′′′B4− 104AA′′B4+ 112A′3

20AB2B′(AB′+ 2A′B) + 16AA′′B4− 28A′2A′′B4?

16A2B3B′′′′− 48AB2(AB′+ A′B)B′′′− 36A2B2(B′′)2+ 4ABB′(29AB′+ 27A′B)B′′

?

−2AA′′′+ 13A′A′′− 14A′2

?

B3B′?

r4+

?

64A2B3B′′′− 104AB2(AB′+ A′B)B′′

AB4?

/(48A4B4r4), (20)

r3

?

r2+ 32A2B3B′r + 16A2B4(A2− 1)

?

respectively. The term Wθθmay be given by Eq. (17),

or simply using the trace identity, i.e., Wθθ= −(Wtt+

Wrr)/2, through Eqs. (19) and (20).

B.Energy conditions

In this work we are interested in deducing exact solu-

tions of traversable wormholes in conformal Weyl gravity

and, therefore, a fundamental point is the energy con-

dition violations. However, a subtle issue needs to be

pointed out in this respect. Note that the energy con-

ditions arise when one refers back to the Raychaudhuri

equation for the expansion where a term Rµνkµkνap-

pears, and kµis a null vector. The positivity of this

quantity ensures that geodesic congruences focus within

a finite value of the parameter labelling points on the

geodesics. However, in general relativity, through the

Einstein field equation one can write the above condi-

tion in terms of the stress energy tensor Tµν, and conse-

quently one ends up with the null energy condition given

by Tµνkµkν≥ 0. In any other theory of gravity, one

would require to know how one can replace Rµν using

the corresponding field equations and hence using mat-

ter stresses. In particular, in a theory where we still have

an Einstein-Hilbert term, the task of evaluating Rµνkµkν

is trivial. However, in the conformal Weyl gravity under

consideration, things are not so straightforward.

To this effect, one may rewrite the gravitational field

equation (10) in terms of the Einstein tensor, in an anal-

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4

ogous form to the Einstein field equation, given by

Gµν≡ Rµν−1

2gµνR =

1

4αTeff

µν, (21)

where the effective stress energy tensor is given by Teff

T(m)

µν

+ T(W)

µν . Note that this relationship differs funda-

mentally from the Einstein field equation, as one is con-

sidering a dimensionless gravitational coupling constant

α, contrary to the Newtonian gravitational constant G.

Nevertheless, the gravitational field equation written in

this form proves extremely useful in deducing a defini-

tion of the null energy condition, in terms of the effective

stress energy tensor, from the Raychaudhuri expansion

term Rµνkµkν.

The first term, i.e., T(m)

µν , in the effective stress energy

tensor, is defined in terms of the matter stress energy

tensor, Eq. (9), and is given by

µν=

T(m)

µν

≡

3

2RTµν, (22)

where R is the curvature scalar.

The second term T(W)

ture Weyl stress energy tensor, and is provided by

µν

may be denoted as the curva-

T(W)

µν

≡ −6α

RWµν, (23)

with the tensor Wµνdefined as

Wµν = −1

−2RµβRνβ+1

6gµνR;β;β+ Rµν;β;β− Rµβ;νβ− Rνβ;µβ

2gµνRαβRαβ+2

3R;µν+1

6gµνR2.(24)

Note that the gravitational field equation (21) im-

poses interesting conservation equations. Through the

the Bianchi identities, Gµν;ν = 0 and the conservation

of the stress energy tensor Tµν;ν= 0, which can also be

verified from the diffeomorphism invariance of the matter

part of the action, one verifies the following conservation

law

T(W)µν;ν=

3

2R2TµνR,ν.(25)

Now the positivity condition, Rµνkµkν≥ 0, in the

Raychaudhuri equation provides the following form for

the null energy condition Teff

ified gravitational field equation (21). For this case, in

principle, one may impose that the matter stress energy

tensor satisfies the energy conditions and the respective

violations arise from the Weyl curvature term T(W)

analogy to the case carried out in Ref. [20]. Although

this analysis is an interesting avenue to study, we con-

sider an alternative approach which is described below.

Another approach to the energy conditions considers

in taking the condition Tµνkµkν≥ 0 at face value. Note

that this is useful as using local Lorentz transformations

it is possible to show that the above condition implies

µνkµkν≥ 0, through the mod-

µν , in

that the energy density is positive in all local frames

of reference. However, if the theory of gravity is cho-

sen to be non-Einsteinian, then the assumption of the

above condition does not necessarily imply focusing of

geodesics. The focusing criterion is different and will fol-

low from the nature of Rµνkµkν. In the next section, we

consider this latter approach to the energy conditions,

which provides interesting results.

III.TRAVERSABLE WORMHOLES IN

CONFORMAL WEYL GRAVITY

In this section, we consider the equations of structure

for traversable wormholes in conformal Weyl gravity. For

this, it is convenient to express the metric in a more

familiar form [15, 16], given by

ds2= −e2Φ(r)dt2+

dr2

1 − b(r)/r+ r2(dθ2+ sin2θ dφ2),

(26)

where Φ(r) and b(r) are arbitrary functions of the radial

coordinate, r, denoted as the redshift function and the

form function, respectively [15]. The radial coordinate

has a range that increases from a minimum value at r0,

corresponding to the wormhole throat, to ∞.

To avoid the presence of event horizons, Φ(r) is im-

posed to be finite throughout the coordinate range. At

the throat r0, one has b(r0) = r0, which implies that

A(r0) → ∞. A fundamental condition is the flaring-out

condition given by (b′r−b)/b2< 0, which is provided by

the mathematics of embedding [15, 16].

In analogy to their general relativistic counterparts,

one may consider asymptotically flat spacetimes. How-

ever, it is also possible to match the interior wormhole

solution to the unique vacuum solution given by

B(r) = A−1(r) = 1−β(2 − 3βγ)

r

−3βγ+γr−kr2, (27)

where β, γ and k are constants of integration [1, 2]. Note

that the general relativistic Schwarzschild solution is pa-

rameterized by β. The constant k characterizes a back-

ground de Sitter spacetime, although the metric fields

(27) in Weyl gravity correspond to a vacuum solution.

The integration constant γ measures departures from the

respective solution in classical general relativity. There-

fore, it is possible to have a cosmology that admits a

de Sitter solution without a cosmological constant [27].

This latter term vanishes identically due to the conformal

invariance of the theory. Thus, conformal Weyl gravity

naturally avoids the theoretical–observational value dis-

crepancy of the cosmological constant.

In the analysis that follows, we consider that the factor

that appears in the gravitational field equation be equal

to unity, i.e., 4α = 1, for notational and computational

simplicity.

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A. Specific case: constant redshift function

A particularly interesting case are the solutions with

a constant redshift function, Φ′= 0. Without a loss of

generality one may impose Φ = 0, which is equivalent to

considering B = 1. This specific case simplifies the field

equations significantly, and provide particularly intrigu-

ing solutions, which differ from their general relativistic

counterparts. This is due to the fact that the fourth order

gravitationalfield equation in conformal Weyl gravity dif-

fers from the general relativistic Einstein field equation.

The energy density and radial pressure, taking into

account Eqs. (19) and (20), reduce to

ρ = −1

6r6

?

2r2?b′′′r2− 2b′′r + 2b′??

b′′r2+5

2(b − b′r)

1 −b

r

?

+

?

?

(b − b′r) + 2b2

?

, (28)

pr =

1

6r6

?

2r2(−b′′r + 2b′)

?

1 −b

r

?

+b′r

2

(b − b′r) +b

2(3b + b′r)

?

, (29)

respectively. The NEC is given by Tµνkµkν≥ 0, as men-

tioned in the Introduction, and for a diagonal stress en-

ergy tensor takes the form ρ + pr≥ 0. For the present

case, the NEC is given by

ρ + pr = −1

6r6

?

2r3(b′′′r − b′′)

?

1 −b

r

?

?

+?b′′r2+ 3(b − b′r)?(b − b′r). (30)

To verify the non-violation of the NEC at the throat,

Eq. (30) imposes the following inequality

b′′r0≤ 3(b′− 1), (31)

where the flaring-out condition evaluated at the throat

has been taken into account, i.e., b′(r0) < 1. We consider

next specific choices for the form function.

1. Form function: b(r) = r0

For this case, the stress energy tensor components are

given by

ρ = −3r2

0

4r6,pr=

r2

4r6.

0

(32)

Note that in this simple case, one already obtains a solu-

tion that deviates from the general relativistic counter-

part, in that the radial pressure is positive at the throat.

Recall that in general relativity the radial pressure is al-

ways negative at the throat, implying the necessity of a

radial tension to maintain the throat open. In addition

to this, we recall that for the specific case of b(r) = r0,

the energy density in general relativity is zero, whilst in

conformal Weyl gravity it is negative.

The NEC is provided by

ρ + pr= −r2

0

2r6,(33)

which shows that the NEC is violated throughout the

spacetime.

2. Form function: b(r) = r2

0/r

The specific case of b(r) = r2

tive energy density in general relativity. Equations (28)-

(29) provide the following stress energy tensor scenario

0/r corresponds to a nega-

ρ =4(3r2− 5r2

0)r2

0

3r8

,pr= −4(r2− r2

0)r2

0

3r8

. (34)

The energy density negative in the range r0 ≤ r <

?5/3r0. This example also differs from its general rela-

the throat.

The NEC is provided by

tivistic counterpart in that the radial pressure is zero at

ρ + pr= −8(r2− 2r2

0)r2

0

3r8

.(35)

which shows that the NEC is violated for r0≤ r <√2r0.

3.Form function: b(r) = r0+ γr0(1 − r0/r)

The specific choice of

b(r) = r0+ γr0

?

1 −r0

r

?

, (36)

where 0 < γ < 1, is particularly interesting. The stress

energy tensor components are somewhat lengthy, so that

the respective profile of the energy density, radial pres-

sure and the NEC are depicted in Fig. 1.

It is interesting to note that for this specific case the

NEC evaluated at the throat is given by

(ρ + pr)??

r0= −5γ2− 8γ + 3

6r4

0

.(37)

This choice does indeed eliminate the need for the vio-

lation of the NEC, in the interval 0.6 ≤ γ < 1. Note

that this is consistent with the general condition given

by inequality (31). Nevertheless, the energy density is

negative throughout the spacetime, which violates the

weak energy condition (WEC). The WEC, TµνUµUν≥

0, where Uµis a timelike vector, implies ρ ≥ 0 and

ρ + pr ≥ 0. Note that the radial pressure is positive

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NEC

ρ

r p

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

1.21.41.6 1.82 2.22.4

r/ro

FIG. 1: The energy density, radial pressure and NEC profile

for the specific case of Φ′(r) = 0 and b(r) = r0+γr0(1−r0/r)

for γ = 0.9. The energy density is negative, the radial pressure

positive; and the NEC is satisfied at the throat neighborhood.

In particular, at the throat the NEC is satisfied in the range

of 0.6 ≤ γ < 1. See the text for details.

as depicted in Fig. 1. The specific case of γ = 0.9 has

been used in the figure, which may be considered as a

representative for this specific case.

The qualitative behavior of the NEC is depicted in

Fig. 2. Note that the NEC is satisfied for high values of

γ and low values of r. In particular, the NEC is satisfied

for increasing values of r, as γ tends to its limiting value

of 1.

0.6

0.7

0.8

γ

0.9

1

1

1.04

α.

1.08

1.12

0.01

0.02

0.03

0.04

FIG. 2: The NEC profile, with ρ + pr ≥ 0, for the specific

case of Φ′(r) = 0 and b(r) = r0 + γr0(1 − r0/r). We have

defined α = r/r0. The NEC is satisfied at the throat in the

range of 0.6 ≤ γ < 1. One verifies, qualitatively, that the

NEC is satisfied for high values of γ and low values of r, i.e.,

as r increases, then γ tends to its limiting value of 1.

B.Specific case: Φ(r) = r0/r

For this case the stress energy tensor components are

extremely lengthy, so that they are also depicted in the

respective plots for the specific choices of the form func-

tion, considered below.

1.Form function: b(r) = r0

The energy density, radial pressure and NEC are de-

picted in Fig. 3. Note that the radial pressure is zero

at the throat, and then remains negative throughout the

coordinate range. The energy density and the NEC are

negative in the throat neighborhood.

NEC

ρ

r p

–2

–1.5

–1

–0.5

0

1.2 1.41.6

r/ro

1.82 2.2

FIG. 3: The energy density, radial pressure and NEC profile

for the specific case of Φ(r) = r0/r and b(r) = r0. The radial

pressure is zero at the throat; the energy density is negative

and the NEC is violated in the throat’s neighborhood.

2.Form function: b(r) = r2

0/r

The energy density, radial pressure and NEC are de-

picted in Fig. 4. This choice is qualitatively analogous

to the previous case, except that the radial pressure is

negative at the throat.

C. Specific case: Φ(r) = −r0/r and b(r) = r0

This specific example is a considerable improvement

to the solutions considered above. The energy density,

radial pressure and NEC profile are depicted in Fig. 5.

Note that the pressure is always positive, and the energy

density and NEC are also positive in the neighborhood

of the throat, thus satisfying all of the energy conditions.

The profile for the specific case of b(r) = r2

itatively analogous to this case. One may then match

these solutions to the exterior vacuum given by Eq. (27),

at a junction surface a0, in which the energy conditions

are satisfied in the interval r0≤ r ≤ a0. This shows that

one may, in principle, construct a class of traversable

0/r is qual-

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NEC

ρ

r p

–10

–8

–6

–4

–2

0

2

1.21.4 1.6

r/ro

1.82 2.2

FIG. 4: The energy density, radial pressure and NEC profile

for the specific case of Φ(r) = r0/r and b(r) = r2

dial pressure is negative throughout the spacetime; the energy

density is negative and the NEC is violated in the neighbor-

hood of the throat.

0/r. The ra-

NEC

ρ

r p

–0.2

–0.1

0

0.1

0.2

0.3

0.4

0.5

1.2 1.41.6

r/ro

1.82 2.2

FIG. 5: The energy density, radial pressure and NEC profile

for the specific case of Φ(r) = −r0/r and b(r) = r0. The ra-

dial pressure is positive throughout. The energy density and

the NEC are positive in the throat’s neighborhood. Conse-

quently, this example shows that one may, in principle, con-

struct a class of traversable wormholes, within the context of

conformal Weyl gravity, that satisfies all of the energy condi-

tions, in the vicinity of the throat.

wormholes, within the context of conformal Weyl grav-

ity, that satisfies all of the energy conditions, contrary to

their general relativistic counterparts.

IV. CONCLUSION

In general relativity, the null energy condition vio-

lation is a fundamental ingredient of static traversable

wormholes. Despite this fact, it was shown that for time-

dependent wormhole solutions the null energy condition

and the weak energy condition can be avoided in certain

regions and for specific intervals of time at the throat

[28]. Nevertheless, in certain alternative theories to gen-

eral relativity, taking into account the modified Einstein

field equation, one may impose in principle that the stress

energy tensor threading the wormhole satisfies the NEC.

However, we emphasize that the latter is necessarily vi-

olated by an effective total stress energy tensor. This is

the case, for instance, in braneworld wormhole solutions,

where the matter confined on the brane satisfies the en-

ergy conditions, and it is the local high-energy bulk ef-

fects and nonlocal corrections from the Weyl curvature

in the bulk that induce a NEC violating signature on the

brane [20]. Another particularly interesting example is in

the context of the D-dimensional Einstein-Gauss-Bonnet

theory of gravitation [17], where it was shown that the

weak energy condition can be satisfied depending on the

parameters of the theory.

In this work, a general class of wormhole geometries

in conformal Weyl gravity was analyzed. In conformal

Weyl gravity, as the fourth order gravitational field equa-

tions differ radically from the Einstein field equation, one

would expect a wider class of solutions. This is indeed

the case, in which the stress energy tensor profile dif-

fers radically from its general relativistic counterpart,

amongst which we may refer to a zero or positive ra-

dial pressure at the throat, or at a more fundamental

level, the non-violation of the energy conditions in the

throat neighborhood, which is in clear contrast to the

classical general relativistic static wormhole solutions.

Note that as for their general relativistic counterparts,

these Weyl variations have far-reaching physical impli-

cations, namely apart from being used for interstellar

shortcuts, and being multiply-connected spacetimes an

absurdly advanced civilization may convert them into

time-machines [16, 29, 30], probably implying the vio-

lation of causality.

Acknowledgements

I thank Demos Kazanas for extremely stimulating dis-

cussions. This work was funded by Funda¸ c˜ ao para

a Ciˆ encia e a Tecnologia (FCT)–Portugal through the

grant SFRH/BPD/26269/2006.

[1] P. D. Mannheim and D. Kazanas, “Exact Vacuum Solu-

tion To Conformal Weyl Gravity And Galactic Rotation

Curves,” Astrophys. J. 342, 635 (1989).

[2] D. Kazanas and P. D. Mannheim, “General structure of

Page 8

8

the gravitational equations of motion in conformal weyl

gravity,” Astrophys. J. Suppl. 76, 431 (1991).

[3] P. D. Mannheim, “Attractive and Repulsive Gravity,”

Found. Phys. 30, 709 (2000) [arXiv:gr-qc/0001011].

[4] A. Edery, A. A. Methot and M. B. Paranjape, “Gauge

choice and geodetic deflection in conformal gravity,” Gen.

Rel. Grav. 33, 2075 (2001) [arXiv:astro-ph/0006173].

[5] P. D. Mannheim and D. Kazanas, “Solutions to the Kerr

and Kerr-Newman problems in fourth order conformal

Weyl gravity,” Phys. Rev. D 44, 417 (1991).

[6] A. Edery and M. B. Paranjape, “Causal structure of vac-

uum solutions to conformal (Weyl) gravity,” Gen. Rel.

Grav. 31, 1031 (1999) [arXiv:astro-ph/9808345].

[7] V. D. Dzhunushaliev and H. J. Schmidt, “New vacuum

solutions of conformal Weyl gravity,” J. Math. Phys. 41,

3007 (2000) [arXiv:gr-qc/9908049].

[8] D. Klemm, “Topological black holes in Weyl confor-

mal gravity,” Class. Quant. Grav. 15, 3195 (1998)

[arXiv:gr-qc/9808051].

[9] O. V. Barabash and H. P. Pyatkovska, “Weak-field limit

of conformal Weyl gravity,” arXiv:0709.1044 [astro-ph].

[10] D. Elizondo and G. Yepes, “Can conformal Weyl gravity

be considered a viable cosmological theory?,” Astrophys.

J. 428, 17 (1994) [arXiv:astro-ph/9312064].

[11] E. E. Flanagan, “Fourth order Weyl gravity,” Phys. Rev.

D 74, 023002 (2006) [arXiv:astro-ph/0605504].

[12] P. D. Mannheim, “Schwarzschild limit of conformal grav-

ity in the presence of macroscopic scalar fields,” Phys.

Rev. D 75, 124006 (2007) [arXiv:gr-qc/0703037].

[13] A. Edery and M. B. Paranjape, “Classical tests for Weyl

gravity: deflection of light and radar echo delay,” Phys.

Rev. D 58, 024011 (1998) [arXiv:astro-ph/9708233].

[14] S. Pireaux, “Light deflection in Weyl gravity: critical dis-

tances for photon paths,” Class. Quant. Grav. 21, 1897

(2004) [arXiv:gr-qc/0403071];

S. Pireaux, “Light deflection in Weyl gravity: constraints

on the linear parameter,” Class. Quant. Grav. 21, 4317

(2004) [arXiv:gr-qc/0408024].

[15] M. S. Morris and K. S. Thorne, “Wormholes in space-

time and their use for interstellar travel: A tool for teach-

ing general relativity,” Am. J. Phys. 56, 395 (1988).

[16] M. Visser, Loretzian wormholes: from Einstein to Hawk-

ing AIP Press (1995).

[17] B. Bhawal and S. Kar, “Lorentzian wormholes in

Einstein-Gauss-Bonnet theory,” Phys. Rev. D 46, 2464-

2468 (1992).

[18] G. Dotti, J. Oliva, and R. Troncoso, “Static wormhole so-

lution for higher-dimensional gravity in vacuum,” Phys.

Rev. D 75, 024002 (2007) [arXiv:hep-th/0607062].

[19] L. A. Anchordoqui and S. E. P Bergliaffa, “Worm-

hole surgery and cosmology on the brane: The world

is not enough,” Phys. Rev. D 62, 067502 (2000)

[arXiv:gr-qc/0001019];

K. A. Bronnikov and S.-W. Kim, “Possible wormholes

in a brane world,” Phys. Rev. D 67, 064027 (2003)

[arXiv:gr-qc/0212112];

M. La Camera, “Wormhole solutions in the Randall-

Sundrum scenario,” Phys. Lett. B573, 27-32 (2003)

[arXiv:gr-qc/0306017].

[20] F. S. N. Lobo, “General class of braneworld wormholes,”

Phys. Rev. D75, 064027 (2007) [arXiv:gr-qc/0701133].

[21] K. K. Nandi, B. Bhattacharjee, S. M. K. Alam and

J. Evans, “Brans-Dicke wormholes in the Jordan and Ein-

stein frames,” Phys. Rev. D 57, 823 (1998).

[22] R. Garattini and F. S. N. Lobo, “Self sustained phantom

wormholes in semi-classical gravity,” Class. Quant. Grav.

24, 2401 (2007) [arXiv:gr-qc/0701020].

[23] C. G. Boehmer, T. Harko and F. S. N. Lobo, “Confor-

mally symmetric traversable wormholes,” Phys. Rev. D

76, 084014 (2007) [arXiv:0708.1537 [gr-qc]];

C. G. Boehmer, T. Harko and F. S. N. Lobo, “Wormhole

geometries with conformal motions,” arXiv:0711.2424

[gr-qc].

[24] S.Sushkov, “Wormholes

tomenergy,”Phys. Rev.

[arXiv:gr-qc/0502084];

F. S. N. Lobo, “Phantom energy traversable wormholes,”

Phys. Rev. D71, 084011 (2005) [arXiv:gr-qc/0502099];

F. S. N. Lobo, “Stability of phantom wormholes,” Phys.

Rev. D71, 124022 (2005) [arXiv:gr-qc/0506001];

F. S. N. Lobo, “Chaplygin traversable wormholes,” Phys.

Rev. D73, 064028 (2006) [arXiv:gr-qc/0511003];

F. S. N. Lobo, “Van der Waals quintessence stars,” Phys.

Rev. D 75, 024023 (2007) [arXiv:gr-qc/0610118].

[25] J. P. S. Lemos, F. S. N. Lobo and S. Quinet de

Oliveira, “Morris-Thorne wormholes with a cosmo-

logical constant,” Phys. Rev. D 68, 064004 (2003)

[arXiv:gr-qc/0302049].

[26] F. S. N. Lobo, “Exotic solutions in General Relativ-

ity: Traversable wormholes and ’warp drive’ spacetimes,”

arXiv:0710.4474 [gr-qc].

[27] P. D. Mannheim, “Conformal cosmology with no cosmo-

logical constant,” Gen. Rel. Grav. 22, 289 (1990).

[28] D. Hochberg and M. Visser, “The null energy condition

in dynamic wormholes,” Phys. Rev. Lett. 81, 746 (1998)

[arXiv:gr-qc/9802048];

D. Hochberg and M. Visser, “Dynamic wormholes, anti-

trapped surfaces, and energy conditions,” Phys. Rev. D

58, 044021 (1998) [arXiv:gr-qc/9802046];

S. Kar, “Evolving wormholes and the weak energy con-

dition,” Phys. Rev. D 49, 862 (1994);

S. Kar and D. Sahdev, “Evolving Lorentzian wormholes,”

Phys. Rev. D 53, 722 (1996) [arXiv:gr-qc/9506094];

S. W. Kim, “The Cosmological model with traversable

wormhole,” Phys. Rev. D 53, 6889 (1996);

A. V. B. Arellano and F. S. N. Lobo, “Evolving wormhole

geometries within nonlinear electrodynamics,” Class.

Quant. Grav. 23, 5811 (2006) [arXiv:gr-qc/0608003].

[29] M. S. Morris, K. S. Thorne and U. Yurtsever, “Worm-

holes, Time Machines and the Weak Energy Condition,”

Phy. Rev. Lett. 61, 1446 (1988).

[30] For review articles, see for example:

F. Lobo and P. Crawford, “Time, closed timelike curves

and causality,” NATO Sci. Ser. II 95, 289 (2003)

[arXiv:gr-qc/0206078];

F. S. N. Lobo, “Nature of time and causality in Physics,”

arXiv:0710.0428 [gr-qc].

supported

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71,

bya phan-

(2005)043520