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
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
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 .
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.
. The solution contains the exterior Schwarzschild so-
lution and provides a potential explanation for observed
∗Electronic address: firstname.lastname@example.org
†Electronic address: email@example.com
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 . 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 . The causal structure, using Penrose
diagrams, of the static spherically symmetric vacuum so-
lution to conformal Weyl gravity was also investigated
. 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 . Exact topological black hole solu-
tions of conformal Weyl gravity, with negative, zero or
positive scalar curvature at infinity were also found ,
the former generalizing the well-known topological black
holes in anti-de Sitter gravity.
. 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 , 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 . In Ref.  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. , 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-
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 ; wormhole
solutions in semi-classical gravity (see Ref.  and ref-
erences therein); exact wormhole solutions using a more
systematic geometric approach were found ; and solu-
tions supported by equations of state responsible for the
cosmic acceleration , etc (see Refs. [25, 26] for more
details and  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-
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
IW = −2α
Varying the action with respect to the metric gµνpro-
vides the following relationship
µν and W(2)
µν given by
µν= 2gµνR;β;β− 2R;µν− 2RRµν+1
2gµνR;β;β+ Rµν;β;β− Rµβ;νβ− Rνβ;µβ
The stress energy tensor is defined as
where Lm is the Lagrangian density corresponding to
The final gravitational field equation is given by
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
µν. Both sides are symmetric,
metric (3), the term Wrrmay be deduced from 
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
From the Bianchi identity,
one obtains the following relationship
where we have defined
for notational simplicity.
From the trace identity, Wµµ = 0, one obtains the
− B Wtt+ AWrr+ 2r2Wθθ= 0.
Finally, using Eqs. (13) and (15), the gravitationalten-
sor components Wttand Wθθare related to Wrrthrough
the following expressions
B − B′r/2(1 + rD) Wrr,
4r(B − B′r/2)(B′+ 2B D) Wrr,
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:
??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)
−4B3B′′(8AA′′− 19A′2) − 49A2B′4− 58AA′BB′3+ (24AA′′− 57A′2)B2B′2
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
64A2B3B′′′− 104AB2(AB′+ A′B)B′′
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).
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-
ogous form to the Einstein field equation, given by
where the effective stress energy tensor is given by Teff
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
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
where R is the curvature scalar.
The second term T(W)
ture Weyl stress energy tensor, and is provided by
may be denoted as the curva-
with the tensor Wµνdefined as
Wµν = −1
6gµνR;β;β+ Rµν;β;β− Rµβ;νβ− Rνβ;µβ
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
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. . 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
1 − b(r)/r+ r2(dθ2+ sin2θ dφ2),
where Φ(r) and b(r) are arbitrary functions of the radial
coordinate, r, denoted as the redshift function and the
form function, respectively . 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)
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 .
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
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
2r2?b′′′r2− 2b′′r + 2b′??
2(b − b′r)
(b − b′r) + 2b2
2r2(−b′′r + 2b′)
(b − b′r) +b
2(3b + b′r)
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
2r3(b′′′r − b′′)
+?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
ρ = −3r2
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
which shows that the NEC is violated throughout the
2.Form function: b(r) = r2
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
,pr= −4(r2− r2
The energy density negative in the range r0 ≤ r <
?5/3r0. This example also differs from its general rela-
The NEC is provided by
tivistic counterpart in that the radial pressure is zero at
ρ + pr= −8(r2− 2r2
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
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
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
1.2 1.41.6 1.822.2 2.4
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
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.
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
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-
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-
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.
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
. 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 . Another particularly interesting example is in
the context of the D-dimensional Einstein-Gauss-Bonnet
theory of gravitation , 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.
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
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