Nonminimal curvaturematter coupled wormholes with
matter satisfying the null energy condition
Nadiezhda Montelongo Garcia, Francisco S N Lobo
To cite this version:
Nadiezhda Montelongo Garcia, Francisco S N Lobo. Nonminimal curvaturematter coupled
wormholes with matter satisfying the null energy condition. Classical and Quantum Gravity,
IOP Publishing, 2011, 28 (8), pp.85018. .
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Confidential: not for distribution. Submitted to IOP Publishing for peer review 7 January 2011
Nonminimal curvature-matter coupled wormholes with matter
satisfying the null energy condition
Nadiezhda Montelongo Garcia∗ and Francisco S. N. Lobo†
Centro de Astronomia e Astrofı́sica da Universidade de Lisboa,
Campo Grande, Edifı́cio C8 1749-016 Lisboa, Portugal
(Dated: January 7, 2011)
Recently, exact solutions of wormhole geometries supported by a nonminimal curvature-matter
coupling were found, where the nonminimal coupling minimizes the violation of the null energy
condition of normal matter at the throat. In this brief report, we present a solution where normal
matter satisfies the energy conditions at the throat and it is the higher order curvature derivatives
of the nonminimal coupling that is responsible for the null energy condition violation, and consequently for supporting the respective wormhole geometries. For simplicity, we consider a linear
R nonmiminal curvature-matter coupling and an explicit monotonically increasing function for the
energy density. Thus, the solution found is not asymptotically flat, but may in principle be matched
to an exterior vacuum solution.
PACS numbers: 04.50.-h, 04.50.Kd, 04.20.Jb, 04.20.Cv
Introduction: Recently, f (R) modified theories of gravity have received a great deal of attention [1], in particular, to account for the late-time cosmic acceleration [2].
An interesting generalization of f (R) gravity involves the
inclusion of an explicit coupling between the curvature
scalar and matter [3]. The latter coupling imposes the
nonconservation of the stress-energy tensor, and consequently implies nongeodesic motion [3].
The action for a nonminimal curvature-matter coupling in f (R) modified theories of gravity, considered in
this work [3], is given by
Z √
1
f1 (R) + [1 + λf2 (R)] Lm
−g d4 x , (1)
S=
2
where fi (R) (with i = 1, 2) are arbitrary functions of
the Ricci scalar R and Lm is the Lagrangian density corresponding to matter. Note that the coupling constant
λ characterizes the strength of the interaction between
f2 (R) and the matter Lagrangian.
Varying the action with respect to the metric gµν yields
the field equations, given by
1
F1 (R)Rµν − f1 (R)gµν − ∇µ ∇ν F1 (R) + gµν F1 (R)
2
= −2λF2 (R)Lm Rµν + 2λ(∇µ ∇ν − gµν )Lm F2 (R)
(m)
+ [1 + λf2 (R)]Tµν
, (2)
with the definition Fi (R) = dfi (R)/dR.
f (R) gravity with the nonminimal curvature-matter
coupling has been explored in different astrophysical and
cosmological contexts [4], and in particular has also been
applied in the context of wormhole physics [5]. In the latter case, it was shown that the nonminimal coupling minimizes the violation of the null energy condition (NEC)
of normal matter at the throat. In the present paper,
we further extend this analysis and find an exact wormhole solution, where normal matter satisfies the NEC at
the throat, and it is the higher order curvature derivatives of the nonminimal coupling that are responsible for
the NEC violation, and consequently for supporting the
respective wormhole geometries.
Nonminimal curvature-matter coupled wormholes:
Consider the following wormhole metric [6]
ds2 = −e2Φ(r) dt2 +
dr2
+r2 (dθ2 +sin2 θdφ2 ) . (3)
1 − b(r)/r
The redshift function Φ(r) must be finite everywhere to
avoid the presence of event horizons [6]. In the analysis
outlined below, we consider Φ0 = 0, which simplifies the
calculations considerably, and provides interesting exact
wormhole solutions. The shape function b(r) obeys several conditions, namely, the flaring out condition of the
throat, given by (b − b0 r)/b2 > 0 [6], and at the throat
b(r0 ) = r = r0 , the condition b0 (r0 ) < 1 is imposed to
have wormhole solutions. It is these restrictions that impose the NEC violation in classical general relativity. Another condition that needs to be satisfied is 1−b(r)/r > 0.
For simplicity, throughout this paper, we consider the
specific case of f1 (R) = f2 (R) = R, and we choose
the Lagrangian form of Lm = −ρ(r). The gravitational
field equation (2) can be expressed in the following form
eff
, where the effective stressGµν ≡ Rµν − 12 R gµν = Tµν
energy tensor is given by
eff
(m)
= (1 + λR)Tµν
+ 2λ [ρRµν − (∇µ ∇ν − gµν )ρ] .
Tµν
(4)
The curvature scalar, R, for the wormhole metric (3),
taking into account Φ0 = 0, is given by R = 2b0 /r2 .
Thus, the gravitational field equations are given by the
following relationships
2λρ00 r(b−r)+λρ0 (rb0 +3b−4r)+ρ(r2 +2λb0 )−b0 = 0 , (5)
∗ Electronic
† Electronic
address: [email protected]
address: [email protected]
4λrρ0 (b − r) + 2λρ(b − b0r) − rpr (r2 + 2λb0 ) − b = 0 , (6)
2
4λr2 ρ00 (b − r) + 2λrρ0 (rb0 + b − 2r) − 2λρ(rb0 + b)
−2rpt r2 + 2λb0 + b − rb0 = 0 . (7)
Relative to the energy conditions, considering a radial
eff µ ν
null vector, the violation of the NEC, i.e., Tµν
k k < 0,
takes the following form
1h
b
eff
eff
2 00
ρ + pr = 2 − 2λr ρ 1 −
r
r
0
br−b i
2
0
0
+(ρ + pr ) r + 2λb + λ (rρ + 2ρ)
< 0. (8)
r
At the throat, taking into account the finiteness of the
factor ρ00 at the throat, one has the following general
condition (ρ0 + pr0) r02 + 2λb00 < λ (r0 ρ00 + 2ρ0 ) (1 − b00 ).
Obtaining explicit solutions to the gravitational field
equations is extremely difficult due to the nonlinearity of
the equations, although the problem is mathematically
well-defined. One has three equations, with four functions, namely, the field equations (5)-(7), with four unknown functions of r, i.e., ρ(r), pr (r), pt (r) and b(r). It
is possible to adopt different strategies to construct solutions with the properties and characteristics of wormholes
(see [5]) for more details).
The approach followed in this paper is to specify a
physically reasonable form for the energy density, and
thus solve Eq. (5) to find the shape function. The radial
pressure and the lateral pressure are consequently given
by Eqs. (6) and (7).
Specific solution: ρ(r) = ρ0 (r/r0 ): Adopting this latter
approach, consider the specific energy density given by
the following monotonically increasing function
r
,
ρ(r) = ρ0
r0
(9)
where ρ0 > 0 is the energy density at the throat. As the
stress-energy profile is monotonically increasing, one may
in principle match the interior wormhole solution to an
exterior vacuum solution [7]. Thus, taking into account
Eq. (9) and solving Eq. (5), the shape function is given
by
−ρ0 r2 14 r2 − 2λ + 14 r02 (ρ0 r02 + 4λρ0 − 4)
. (10)
b(r) =
3λρ0 r − r0
Note that the asymptotically flatness condition, b/r →
0, as r → ∞, is not verified. However, as mentioned
above, one may in principle match the interior wormhole
geometry at a junction interface r0 < a0 < a, to an
exterior vacuum solution. The flaring-out condition at
the throat also entails the restriction b0 (r0 ) < 1. Taking
the radial derivative of Eq. (10), and evaluating at the
throat, one arrives at the following relationship
b0 (r0 ) = −
(r02 − λ)ρ0
,
3λρ0 − 1
(11)
where 3λρ0 − 1 6= 0. The condition b0 (r0 ) < 1 imposes
λ > (1 − r02 ρ0 )/(2ρ0 ).
Using Eqs. (9) and (10), the radial pressure and tangential pressure are given by
pr = [λρ20 r0 r5 − (r02 + 48ρ20 λ3 )ρ0 r4 + 56λ2 ρ20 r0 r3 + 8(12ρ20 λ2 + 3ρ0 (ρ0 r02 − 4)λ − 1)λr02 ρ0 r2
+9(−4λρ0 − ρ0 r02 + 4)λρ0 r03 r + (4λρ0 + r02 ρ0 − 4)r04 ]/
/2[9λ2 ρ20 r4 − 8λρ0 r0 r3 + 2(r02 + 12λ3 ρ20 )r2 − 16λ2 ρ0 r0 r + 3(−4λρ0 − ρ0 r02 + 4)λ2 ρ0 r02 ]r0 r,
(12)
pt = 2[6λρ20 r0 r5 − ((12)2 λ3 ρ20 + 3r02 )ρ0 r4 + 96λ2 ρ20 r0 r3 − 8λρ0 r02 r2 + 6(ρ0 r02 + 4λρ0 − 4)r03 ρ0 λr +
+r04 (−4λρ0 − r02 ρ0 + 4)](3λρ0 r − r0 )2 /[9λ2 ρ20 r4 − 8λρ0 r0 r3 + 2(r02 + 12λ3 ρ20 )r2 − 16λ2 ρ0 r0 r −
−3(ρ0 r02 + 4λρ0 − 4)r02 λ2 ρ0 ](9λ2 ρ20 r2 − 6λρ0 r0 r + r02 )r0 r.
(13)
The NEC at the throat is given by
(ρ + pr )|r0 =
6λ2 ρ20 + (3ρ0 r02 − 5)ρ0 λ − ρ0 r02 + 1
, (14)
2λ2 ρ0 + λr02 ρ0 − r02
which becomes singular when
λ=
−r02 ρ0 ± [r02 ρ0 (8 + r02 ρ0 )]1/2
.
4ρ0
(15)
Thus, the nonsingular regimes lie outside the regions
defined by Eq. (15) and the condition imposed by b00 < 1,
i.e., λ = 1/(3ρ0 ), which are depicted in Figure 1.
The regions where the condition (1 − b/r) ≥ 0 (which
can be expressed as (r − b) ≥ 0) is satisfied are plotted in Fig. 2, for the values ρ0 = 1, r0 = 1. The
left plot depicted is for the region in [−r02 ρ0 + [r02 ρ0 (8 +
r02 ρ0 )]1/2 ]/(4ρ0 ) < λ < 2. The right plot depicted is for
the region −2 < λ < [−r02 ρ0 ± [r02 ρ0 (8 + r02 ρ0 )]1/2 ]/(4ρ0 ).
Note that for these regions, one cannot obtain the general
relativistic limit, as one necessarily crosses the singular
region depicted by the curves in Fig. 1.
We analyze now the regions where the flaring-out condition, b0 (r0 ) < 1, and the NEC at the throat, (ρ +
3
FIG. 1: The nonsingular regions lie outside the curves λ =
1/(3ρ0 ), depicted by the dashed curve, and λ = [−r02 ρ0 ±
[r02 ρ0 (8+r02 ρ0 )]1/2 ]/(4ρ0 ), where the upper (lower) solid curve
depicts the positive (negative) sign, respectively.
pr )|r0 > 0, are satisfied. Figure 3 depicts these regions for
the domain [−r02 ρ0 + [r02 ρ0 (8 + r02 ρ0 )]1/2 ]/(4ρ0 ) < λ < 2
and 0.3 < ρ0 < 1. Figure 4 depicts these regions for the
domain −2 < λ < [−r02 ρ0 +[r02 ρ0 (8+r02 ρ0 )]1/2 ]/(4ρ0 ) and
0.2 < ρ0 < 2.
It is important to emphasize that for the regions
[−r02 ρ0 − [r02 ρ0 (8 + r02 ρ0 )]1/2 ]/(4ρ0 ) < λ < [−r02 ρ0 +
[r02 ρ0 (8 + r02 ρ0 )]1/2 ]/(4ρ0 ) and 0 < ρ0 < 1/3, and
1/(3ρ0 ) < λ < [−r02 ρ0 + [r02 ρ0 (8 + r02 ρ0 )]1/2 ]/(4ρ0 ) and
ρ0 > 1/3, where λ can smoothly run to the GR limit,
i.e., λ → 0, the NEC is always violated. However, for
large values of λ, one verifies that the NEC violation of
normal matter can be minimized as in Ref. [5].
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Conclusion: In this work we have considered wormhole
geometries supported by a nonminimal curvature-matter
coupling in a generalized f (R) modified theory of gravity. For simplicity, we considered a linear R nonminimal curvature-matter coupling and an explicit monotonically increasing function for the energy density. The
wormhole solution obtained is not asymptotically flat,
but may in principle be matched to an exterior vacuum
solution. This brief report extends previous work [5], in
that normal matter satisfies the null energy condition at
the throat, and it is the higher order curvature derivatives that is responsible for the null energy condition violation, and consequently for supporting the respective
wormhole geometries. A drawback for the solution obtained is that one cannot regain the general relativistic regime, i.e., λ → 0, due to the presence of singular regions. We emphasize that it is extremely difficult
to obtain exact solutions due to the nonlinearity of the
field equations. However, in principle it is possible to
find solutions of asymptotically flat wormhole geometries
for a general f (R) function, which tend continuously to
the general relativistic regime through a reconstruction
method, where one imposes a specific wormhole geometry and finally finds the form of f (R), much in the spirit
of [8]. Work along these lines is presently underway.
Acknowledgments: NMG acknowledges financial support from CONACYT-Mexico. FSNL acknowledges financial support of the Fundação para a Ciência e Tecnologia through Grants PTDC/FIS/102742/2008 and
CERN/FP/109381/2009.
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4
FIG. 2: The shape function restriction r − b(r) > 0 is plotted for the values ρ0 = 1, r0 = 1. The left plot is given for the range
[−r02 ρ0 + [r02 ρ0 (8 + r02 ρ0 )]1/2 ]/(4ρ0 ) < λ < 2. The right plot is given for the range −2 < λ < [−r02 ρ0 ± [r02 ρ0 (8 + r02 ρ0 )]1/2 ]/(4ρ0 ).
FIG. 3: Plots for the regions [−r02 ρ0 + [r02 ρ0 (8 + r02 ρ0 )]1/2 ]/(4ρ0 ) < λ < 2 and 0.3 < ρ0 < 1: The left plot depicts the regions of
b0 (r0 ) < 1; the right plot depicts the region satisfying the NEC at the throat (ρ + pr )|r0 > 0.
FIG. 4: Plots for the regions −2 < λ < [−r02 ρ0 + [r02 ρ0 (8 + r02 ρ0 )]1/2 ]/(4ρ0 ) and 0.2 < ρ0 < 2: The left plot depicts the regions
of b0 (r0 ) < 1; the right plot depicts the region satisfying the NEC at the throat, i.e., (ρ + pr )|r0 > 0.