#### A note on thin-shell wormholes with charge in F(R)-gravity

Eur. Phys. J. C
A note on thin-shell wormholes with charge in F( R)-gravity
S. Habib Mazharimousavi 0
0 Department of Physics, Faculty of Arts and Sciences, Eastern Mediterranean University , Via Mersin 10, Famagusta, North Cyprus , Turkey
In their recent work (Eiroa and Aguirre in Eur Phys J C 76:132, 2016), Eiroa and Aguirre introduced thinshell wormholes in F (R) = R + α R2-gravity coupled with the Maxwell electromagnetic field. Here in this note we shall address an interesting feature of their results which has been missed. It will be shown that thin-shell wormhole can not be formed in the black hole spacetime solution of this theory but instead there are rooms for making stable thin-shell wormholes in non-black hole bulk spacetime as was noted in Eiroa and Aguirre (2016). This study is not a comment on very correct results of Eiroa and Aguirre (2016) but instead it is a complementary result to their paper.
1 Introduction
Thin-shell wormhole in modified theory of gravity, namely
F (R)-gravity, seems to be more restrictive than its former
version in R-gravity due to the modified junction conditions
introduced in [
2
]. In [
1
], Eiroa and Aguirre constructed
thinshell wormholes in F (R) = R + α R2-gravity coupled with
the Maxwell’s electrodynamic field in the framework of
constant curvature i.e., R = R0 = const. spherically
symmetric bulk spacetime. Apparently their stability analysis results
in stable thin-shell wormhole against a radial perturbation.
Due to the modified junction conditions, one should
consider additional constraint on the radius of the throat which
is assumed to be larger than the radius of the event horizon. In
other words, unlike R-gravity where the standard Israel
junction conditions [
3–5
] are applicable and there is no restriction
on choosing the radius of the throat except it has to be larger
than the possible event horizon, in F (R) -gravity it has to
satisfy Eq. (12) as well. This is because of the continuity of
the trace of the extrinsic curvature in F (R)-gravity [
2
] i.e.,
[Kii ] = 0. We shall show that upon this condition thin-shell
d4x √−g F (R) − Fμν F μν
F (R) = R + α R2
and
1
F = 2 Fμν d x μ ∧ d x ν
is the Maxwell’s electromagnetic field. With constant
extrinsic curvature R = R0 the solution for the F (R)-Maxwell’s
field equations in spherically symmetric spacetime is found
to be [
6,7
]
ds2 = gμν d x μd x ν = − A(r )dt 2 + Ad(rr2) + r 2d 2
where
A(r ) = 1 −
2M
r
Q2
+ F (R0) r 2 −
Applying the standard method of cut and paste one constructs
thin-shell wormhole whose throat is located at r = a (τ ) in
which τ is the proper time on the throat. Hence, the extrinsic
curvature tensor on the sides of the throat are found to be
(1)
(2)
(3)
(4)
(5)
K ij± = ±diag
in which a prime and a dot stand for the derivative with respect
to r and τ respectively. As it was introduced in [
2
] and
properly applied in [
1
] the following junction conditions have to
be satisfied. First the metric tensor, the Ricci scalar and the
trace of the extrinsic curvature should be continuous across
the shell i.e., [hi j ] = 0, [R] = 0 and [K ] = 0 respectively.
Second, due to the identical constant curvature bulks in both
sides of the thin-shell, the jump of the extrinsic curvature
tensor gives the surface energy-momentum tensor as
in which σ is the energy density and p is the angular pressure.
[K ] = 0 implies
A(a) + a˙ 2 = 0
κ Sij = −F (R0) Kij
where
Sij = di ag (−σ, p, p)
A (a) + 2a¨
2 A(a) + a˙ 2 + a
2
and (7) yields to
F 2a¨ + A
κ√
A + a˙ 2
,
σ =
and
2F
p = − κa
A + a˙ 2.
F A (a0) ,
σ0 = κ√ A (a0)
and
2F
p0 = − κa0
A (a0).
Introducing the equilibrium radius a = a0 [
1
], where a˙ =
a¨ = 0 one finds from (9) that a0 has to satisfy
a0 A (a0) + 4 A (a0) = 0
which is the additional constraint on the radius of the
thinshell wormhole and the main concern of this note. Also (10)
and (11) give
Furthermore, the trace of Eq. (7) implies that the trace of
Sij vanishes i.e., Sii = 0 which gives directly the equation
A (a) + 2a¨ , 1
2 A(a) + a˙ 2 a
A(a) + a˙ 2, 1
a
A(a) + a˙ 2
(6)
of state p = σ2 for the perfect fluid presented on the shell.
The stability analysis of the thin-shell wormhole ends up to a
one-dimensional equation of motion for the throat given by
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
a˙ 2 + V (a) = 0
Hence, at the radius of the equilibrium a = a0, satisfying
(12), if V (a0) > 0 the thin-shell wormhole is stable against
the radial perturbation. In Fig. 2 of [
1
] it is clearly shown that
the solid curve outside the shaded region for each individual
case is the stable region.
3 The new observation
Let’s recall that a = a0 is the equilibrium radius of the
thinshell wormhole satisfying two critical conditions: (i) a0 >
rh , in which rh is the possible event horizon of the bulk
spacetime and (ii) a0 should satisfy Eq. (12). Now in this
section we show that any possible event horizon is larger
than a0 and therefore the stable thin-shell is not possible for
the black hole spacetime solution to the R + α R2-Maxwell
gravity. This however, does not contradict the results in [
1
]
because in Fig. 2 of [
1
] the bulk does not need to be a black
hole. We note that from Fig. 1 in [
1
], |Q| < Qc corresponds
to an inner and an event horizon for the bulk while |Q| >
Qc presents a naked singularity for the bulk spacetime. At
|Q| = Qc the two horizons coincide.
First let’s look at the cases mentioned in [
1
] more closely as
is shown in Fig. 1. In this figure the region where V (a0) > 0
is shown (as shaded region) in terms of the horizontal axes
x = aM0 and vertical axes y = M|√Q|F , together with other two
curves. The dot-dashed (blue) curve represents the points
on the plane of x y which satisfy Eq. (12) which implicitly
gives the radius of the throat (up to a coefficient M1 ). Finally
the long dashed (brown) curve stands for the point of the
x y plane satisfying A (r ) = 0 where the horizontal axes
x = rMh and therefore this curves implicitly reveals the event
and Cauchy horizons of the spacetime. One observes that in
all cases for the region where the bulk spacetime is a black
hole the curve of a0 lies under the curve of rh showing that
a0 < rh . A portion of a0 curve remains inside the shaded
tively. The value of ξ = R0 M2 are given on each individual plot. The
points I and J marked on the curves are the intersections of the curves
qa vs xa with qh vs xh and qv vs xa respectively. Definitions of these
quantities are given in Sect. 3
region where the spacetime is not black hole is the stable
thin-shell wormhole reported in [
1
].
To complete our note let’s find above observation
analytically. The two equations i.e., A (rh ) = 0, and a0 A (a0) +
4 A (a0) = 0 after change of variable as xa = M rh
a0 , xh = M
|Q| become
and q = M√F
and
qa =
xa −8xa + 12 + ξ xa3
4
(21)
1 − 1ξ2 xh2 − x2h + x
q2
2 = 0
h
and
4 − ξ2 xa2 − x6a + xa2
2q2
= 0,
respectively, in which ξ = R0 M 2. Solving both equations
(18) and (19) for q, reveals
qh =
xh −12xh + 24 + ξ xh3
12
(18)
(19)
(20)
in which a sub h/a stands for the horizon/throat and refers
to the solutions of the Eq. (18)/(19). In Fig. 1 the brown
long-dashed and the blue dot-dashed curves are qh (= y)
versus xh (= x ) and qa (= y) versus xa (= x ), respectively,
for different values of ξ . Next we find the intersection point
between two curves. From Fig. 1 we see that there are two
points of intersections between two curves, one at the origin
which is trivially seen from (20) and (21) and the second point
is the maximum point of the curve qh versus xh i.e., point
I shown on the curve, which is not trivial. Here we show
that irrespective of the value of ξ the second intersection
point is actually the maximum of qh . To do so we find the
extremum/maximum of qh by finding its first derivative and
equate it to zero i.e., ddqxhh = 0 which gives
1
3 ξ x˜h3 − 2x˜h + 2 = 0
in which x˜h is the horizontal location of the extremum point
of qh . Solving (22) for ξ and inserting it in (20) one finds the
extremum of qh = q˜h which is given by
q˜h =
x˜h (3 − x˜h ) .
2
Next, we find the value of qa at xa = x˜h with the same value
of ξ as
qa (xa = x˜h ) =
x˜h (3 − x˜h )
2
which is equal to q˜h . This is the end of the proof. Hence we
see that only at the location of the extremal black hole when
the event horizon and the Cauchy horizon coincide (point I )
the value of a0 can be equal to the radius of the horizon while
for any other black hole case rc < a0 < rh .
A similar calculation shows that, the boundary curve of the
region V (a0) > 0 which is given by V (a0) = 0 reduces
to
3ξ 20 36
2 − x 2 + x 3 −
a a
14q2
x 4
a
= 0.
The solution of (25) for q is found to be
qv =
xa 3ξ xa3 − 40xa + 72
28
in which a sub v refers to the solution of Eq. (25). In Fig. 1 qv
versus xa is shown with Green-Solid curve. This curve
intersects the curve of qa versus xa at the origin and its extremum
point J . The proof is similar to the case of point I which
we have worked out earlier. Hence one concludes that the
possible stable thin-shell wormhole is located on the curve
of qa versus xa between the two points I and J marked on
Fig. 1.
4 Conclusion
In this note the results of the recent work of Eiroa and Aguirre
on stability of thin-shell wormhole in F (R) = R + α
R2(22)
(24)
(25)
(26)
Maxwell theory of gravity have been reconsidered. It was
shown that for the black hole bulk solution in this theory
there is no possible stable thin-shell wormhole. No need to
mention that, the non-black hole solution to the R + α
R2Maxwell theory with constant curvature is naked singular and
the radius of the throat of the thin-shell wormhole is located
to the left of the local minimum of A (r ) where A (r ) = 0.
Also to keep the bulk spacetime a non-black hole solution
one must consider |Q| > Qc in which Qc is a minimum
value for the charge (see Fig. 1 in Ref. [
1
]). Restriction on
|Q| affects the physical properties of the constructed TSW,
for instance the amount of the exotic matter. Therefore this
study is not trivial. Once more we would like to add that
our results are in agreement with the original work of Eiroa
and Aguirre [
1
]. As our final remark we would like to look
at the condition (12) once more. In this equation A (a0) =
0 and therefore A (a0) = 0, in other words the point of
the throat and the point where derivative of A (r ) is zero
do not coincide. In [
8
] it is shown that wormhole solutions
with this property are asymmetric wormhole. In our case
although the condition used in [
8
] is satisfied but our original
bulk metric is not a wormhole solution. Hence, the thin-shell
wormhole considered in this study remains symmetric
thinshell wormhole.
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Funded by SCOAP3.
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