Physics Letters A 373 (2008) 1–4
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Physics Letters A
www.elsevier.com/locate/pla
Thin-shell wormholes in Brans–Dicke gravity
Ernesto F. Eiroa a,b,∗ , Martín G. Richarte a , Claudio Simeone a
a
b
Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria Pab. I, 1428 Buenos Aires, Argentina
Instituto de Astronomía y Física del Espacio, C.C. 67, Suc. 28, 1428 Buenos Aires, Argentina
a r t i c l e
i n f o
Article history:
Received 9 September 2008
Received in revised form 27 October 2008
Accepted 28 October 2008
Available online 31 October 2008
Communicated by P.R. Holland
a b s t r a c t
Spherically symmetric thin-shell wormholes are constructed within the framework of Brans–Dicke
gravity. It is shown that, for appropriate values of the Brans–Dicke constant, these wormholes can be
supported by matter satisfying the energy conditions.
© 2008 Elsevier B.V. All rights reserved.
PACS:
04.20.Gz
04.50.Kd
04.40.Nr
Keywords:
Lorentzian wormholes
Exotic matter
Brans–Dicke gravity
1. Introduction
After the leading work by Morris and Thorne [1], traversable
Lorentzian wormholes [2] have received considerable attention.
Such geometries connect two regions of the same universe – or
of two universes – by a traversable throat. A central objection
against the existence of wormholes is that, within the framework
of general relativity, the flare-out condition [3] to be satisfied at
the throat requires the presence of exotic matter, that is, matter which violates the energy conditions [1–4]. However, it was
shown in Ref. [5] that the amount of exotic matter necessary for
the existence of a wormhole can be made infinitesimally small by
suitably choosing the geometry, though it may be at the expense of
large stresses at the throat [6,7]. On the other hand, it was demonstrated that in some alternative theories of gravity the requirement
of exotic matter can be avoided [8,9]; in particular, some years
ago, Anchordoqui et al. [10] showed that, in Brans–Dicke gravity,
Lorentzian wormholes of the Morris–Thorne type are compatible
with matter which, apart from the Brans–Dicke scalar field, satisfies the energy conditions. Other related aspects of wormholes
*
Corresponding author at: Instituto de Astronomía y Física del Espacio, C.C. 67,
Suc. 28, 1428 Buenos Aires, Argentina.
E-mail addresses: [email protected] (E.F. Eiroa), [email protected] (M.G. Richarte),
[email protected] (C. Simeone).
0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2008.10.065
in Brans–Dicke or in scalar–tensor theories were also discussed in
Refs. [11,12].
Thin-shell wormholes [13] are mathematically constructed by
cutting and pasting two manifolds to yield another one with a
throat placed at the joining surface. The mechanical stability and
the matter content of spherically symmetric wormholes of this
kind have been studied by several authors, both within general relativity [14] and in alternative theories of gravity [6,8,15]. Other geometries were also recently explored [16]. In the present work we
apply the Darmois–Israel formalism [17,18] generalized to Brans–
Dicke gravity [19] for the construction of spherically symmetric
thin-shell wormholes. We calculate the energy density and the
pressure on the shell. We find that for certain values of the Brans–
Dicke constant ω , the wormhole radius can be chosen so that the
matter on the shell satisfies the energy conditions, being the existence of the throat made possible by the presence of the Brans–
Dicke field. Throughout the Letter units such that c = G = 1 are
used.
2. Thin-shells in Brans–Dicke gravity
In the framework of present unified theories, a scalar field
should exist besides the metric of the spacetime. Scalar–tensor theories of gravitation would be important when studying the early
Universe, where it is supposed that the coupling of the matter
to the scalar field could be non-negligible. In Brans–Dicke theory,
matter and non gravitational fields generate a long-range scalar
2
E.F. Eiroa et al. / Physics Letters A 373 (2008) 1–4
field φ which, together with them, acts as a source of gravitational
field. The metric equations generalizing those of general relativity
are
R μν −
1
g μν R =
2
8π
T μν +
φ
+
1
φ
ω
φ2
φ;μ;ν −
φ,μ φ,ν −
1
φ
ω
2φ 2
g μν φ,α φ ,α
g μν φ:;αα ,
1
=√
∂
√
− g ∂ xμ
∂φ
− g g μν ν
∂x
=
8π T
3 + 2ω
(1)
(2)
,
μ
where T is the trace of T ν . In the limit
ω → ∞ the Einstein equations are recovered, provided that φ(ω → ∞) = 1/G = 1. In Brans–
Dicke theory, the junction conditions across a smooth timelike hypersurface Σ in the four-dimensional manifold, can be obtained by
projecting on Σ the equations above. The extrinsic curvature associated with the two sides of Σ in terms of the unit normals n±
γ
(nγ nγ = 1) is given by
K i±j = −n±
γ
μ
ν γ ∂X ∂X
,
+
Γ
μν
∂ξ i ∂ξ j
∂ξ i ∂ξ j ∂ Xγ
2
Σ
8π
S
− K ij + K δ ij =
S ij −
δ ij ,
φ
3 + 2ω
8π S
3 + 2ω
(4)
(5)
where the notation · stands for the jump of a given quantity
across the hypersurface Σ , N labels the coordinate normal to this
surface and S ij is the energy–momentum tensor of matter and
fields (except the field φ ) on the shell located at Σ . The quantities
K and S are the traces of K ij and S ij , respectively. The components
of the metric and the Brans–Dicke field are continuous across the
shell ( g μν = 0, φ = 0). Note that in the general relativity limit
ω → ∞ the Lanczos equations [17,18] are recovered.
3. Spherically symmetric wormholes
Now let us apply the formalism introduced above to the construction of spherically symmetric thin-shell wormholes. We start
from the metric
ds2 = − f (r ) dt 2 + g (r ) dr 2 + h(r ) dθ 2 + sin2 θ dϕ 2 ,
(6)
so that X = (t , r , θ, ϕ ). From this geometry we choose a radius
a and take two copies M+ and M− of the region r a, and
paste them at the hypersurface Σ defined by r = a, obtaining a
new manifold M = M+ ∪ M− . The radius a is chosen so that
there are no horizons and singularities in M. If h (a) is positive
the flare-out condition is satisfied, and the resulting geometry describes a wormhole having a throat of radius a connecting the two
regions M+ and M− . Introducing the coordinates ξ = (τ , θ, ϕ )
on Σ (with τ the proper time on the shell), the jump of the components of the extrinsic curvature is given by
τ
Kτ =
f (a)
√
f (a)
ϕ
K θθ = K ϕ =
g (a)
(7)
,
h (a)
√
h(a)
g (a)
.
σ +p=
f (a)
h (a)
φ(a)
−
.
√
h(a)
8π g (a) f (a)
(8)
(10)
(11)
Non-exotic matter should satisfy the weak energy condition,1 i.e.
σ 0 and σ + p 0; this condition would be violated if the flareout condition is to be fulfilled within pure general relativity, which
is easy to see from Eq. (9) taking the limit ω → ∞.
As a particular case, we consider the spherically symmetric vacuum solution of Brans–Dicke equations. Thus the functions f , g
and h are given by [11]
f (r ) = 1 −
g (r ) = 1 −
h(r ) = 1 −
2η
r
2η
r
2η
A
,
(12)
,
(13)
B
1+ B
r
r2,
(14)
and the field φ takes the form
−( A + B )/2
2η
φ(r ) = A 2 1 −
,
(15)
r
,
(9)
where φ(a) is the value of the Brans–Dicke field on the surface Σ .
Consequently, we have
(3)
where X γ are the coordinates of the four-dimensional manifold,
γ
ξ i are the coordinates on the hypersurface, and Γμν are the components of the connection associated with the metric g μν . Then,
with this definition, the junction conditions in Brans–Dicke theory
(generalized Darmois–Israel conditions) have the form [19]
φ, N =
2h (a)
1 f (a)
2h (a)
φ(a)
+
+
,
√
ω f (a)
h(a)
8π g (a) h(a)
f (a)
h (a)
1 f (a)
2h (a)
φ(a)
p=
+
+
+
,
√
h(a)
ω f (a)
h(a)
8π g (a) f (a)
σ =−
where R μν is the Ricci tensor, T μν is the energy–momentum tensor of matter and fields – not including the Brans–Dicke field –
and ω is a dimensionless constant. The field φ is a solution of the
equation
;μ
φ;μ
ϕ
Then the energy density σ = − S ττ and the pressures p = S θθ = S ϕ
of the matter and fields in the shell (apart from the field φ ) are
given by
where
A=
4 + 2ω
3 + 2ω
η=M
B =−
,
3 + 2ω
4 + 2ω
ω + 1 4 + 2ω
,
ω + 2 3 + 2ω
(16)
,
with M the mass of the spherically symmetric source. For finite ω , when ω > −3/2 this geometry represents a naked singularity [11] with radius r s = 2η , and when ω < −2 the metric
corresponds to a wormhole of the Morris–Thorne type [11], while
for −2 ω −3/2 the metric is not defined. If |ω| is infinite, we
recover the Einstein gravity solution, which is the Schwarzschild
black hole. To construct the wormhole, we take ω > −3/2 and a
throat radius larger than r s = 2η . It is easy to see that in this case
the flare out condition h (a) > 0 is satisfied. For ω slightly below
zero, the bracket in the right-hand side of Eq. (9) is negative, because f (a) > 0. Thus in this case the condition σ 0 could be
satisfied by the matter on the shell. Instead, when σ + p is positive cannot be easily guessed from Eq. (11). Replacing the explicit
form of the metric and field, we have
σ =−
A2
4π a2
a
1+ B + A / 2
a − 2η
× 2η( B − 1) + 2a +
1
ω
η A + 2η( B − 1) + 2a ,
(17)
1
The weak energy condition (WEC) states that for any timelike vector
T μν V μ V ν 0, which means that the local energy density as measured by any
timelike observer is positive. In terms of the principal pressures it takes the form
ρ 0, ρ + p j 0 ∀ j. The WEC implies the null energy condition (NEC) T μν kμ kν 0 for any null vector, which in terms of the principal pressures takes the form
ρ + p j 0 ∀ j.
E.F. Eiroa et al. / Physics Letters A 373 (2008) 1–4
Fig. 1. Values of the Brans–Dicke constant ω for which wormholes with throat radius a and mass M exist such that σ 0 is satisfied by the matter on the shell
(grey zone).
p=
A2
4π a2
a − 2η
A2
a
Then the condition
1
ω
η A + 2η( B − 1) + 2a ,
1+ B + A /2
a − 2η
4π a2
Fig. 2. Values of the Brans–Dicke constant ω for which wormholes with throat radius a and mass M exist such that the condition σ + p 0 is satisfied by the matter
on the shell (grey zone).
1+ B + A / 2
a
× η A + η( B − 1) + a +
σ +p=
3
η A + η(1 − B ) − a .
σ 0 is fulfilled when
and the condition
(19)
η
A
2η < a and
( B − 1)(ω + 1) +
0,
(ω + 1) +
ω
ω
2
a
(18)
(20)
σ + p 0 is satisfied if
2η < a η( A + 1 − B ).
(21)
Summarizing, we conclude that if the inequalities above are satisfied the weak energy condition is not violated, and then the matter
on the shell is not exotic. In Figs. 1 and 2 values of the throat radius and the Brans–Dicke constant such that conditions (20) and
(21) are fulfilled are respectively shown. Fig. 3 corresponds to the
region where both conditions are simultaneously satisfied, so that
configurations defined by such values contain non-exotic matter
in the shell placed at the wormhole throat. It can be seen that
this happens for negative values of ω (up to values slightly smaller
than −1), and radius a slightly larger than r s .
with
S̃ ij = S ij −
4. Discussion
We have applied the generalized Darmois–Israel formalism to
the construction of spherically symmetric thin-shell wormholes in
the framework of Brans–Dicke gravity. We have obtained the energy density and the pressure of matter and fields other than φ ,
in the shell placed at the throat of the wormhole. We have shown
that, for certain negative values of the Brans–Dicke constant, this
matter satisfies the energy conditions if the throat radius is suitably chosen. Such values of the constant ω seem to be unphysical
in the present Universe, but they could make sense in another
scenario (i.e. far in the past). If the right-hand side of Eq. (4) is
understood as an effective source, the junction conditions present
the same form as in general relativity:
8π i
− K ij + K δ ij =
S̃ ,
φ j
Fig. 3. Values of the Brans–Dicke constant ω for which wormholes with throat
radius a and mass M exist such that the weak energy condition (σ 0 and
σ + p 0) is satisfied by the matter on the shell (grey zone).
(22)
S
3 + 2ω
δ ij
(23)
the effective energy–momentum tensor on the surface. With this
definition, we would have that
σ̃ = − S̃ ττ = −
φ(a) h (a)
√
4π g (a) h(a)
(24)
is negative because of the flare-out condition. So, in this picture,
the energy conditions are not satisfied by this effective surface tensor. The violation of these conditions comes from the Brans–Dicke
field, even in the presence of non-exotic matter and other fields.
This result is analogous to what was obtained by Anchordoqui et
al. [10] for wormholes of the Morris–Thorne type.
4
E.F. Eiroa et al. / Physics Letters A 373 (2008) 1–4
Acknowledgements
This work has been supported by Universidad de Buenos Aires
(UBA) and CONICET. Some calculations in this Letter were done
with the help of the package GRTensorII [20].
[13]
[14]
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