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30 Page 4 of 19 Eur. Phys. J. C (2020) 80 :30
d
2
r
dz
2
=
b(r) − b
(r)r
2b(r)
2
> 0. (12)
This flaring-out condition is an important constituent of
wormhole physics and plays a major role in the analysis
of the violation of the energy conditions. In the light of the
above discussion, for a traversable wormhole, the shape func-
tion should satisfy the following properties: (i)
b(r)
r
< 1for
r > r
0
, (ii) b(r
0
) = r
0
at r = r
0
, (iii)
b(r)
r
→ 0asr →∞,
(iv)
b(r)−b
(r)r
b(r)
2
> 0forr > r
0
and (v) b
(r)−1 < 0atr = r
0
.
The energy momentum tensor for the matter source of the
wormhole is defined as
T
μν
=
∂ L
m
∂g
μν
,
= (ρ + p
t
)u
μ
u
ν
− p
t
g
μν
+ ( p
r
− p
t
)X
μ
X
ν
, (13)
where ρ is the energy density, p
t
and p
r
are tangential and
radial pressures respectively and u
μ
& X
μ
denote the four
velocity and radial vectors respectively such that
u
μ
u
μ
=−1 and X
μ
X
μ
= 1. (14)
The effective field equations for the metric (7) can be
expressed as follows:
ρ =
Fb
(r)
r
2
−
1 −
b(r)
r
F
(r) − H (15)
p
r
=−
b(r)
r
3
− 2
1 −
b(r)
r
(r)
r
F
−
1 −
b(r)
r
F
+
F
(rb
(r) − b(r))
2r
2
1 −
b(r)
r
+ H (16)
p
t
= F
1 −
b(r)
r
(r) −
(b
(r)r − b(r))
2r(r − b(r))
(r) +
2
+
(r)
r
−
(b
(r)r − b(r))
2r
2
(r − b(r))
−
F
r
1 −
b(r)
r
+ H,
(17)
where F ≡
df
dR
, R =
4
(r)
r
+ 2
(r) + 2
2
(r)
1 −
b(r)
r
−
(b
(r)r−b(r))
r
2
(r)−
2b
(r)
r
2
, H (r) =
1
4
(FR+F+T ),
F =
1 −
b(r)
r
F
+
rb
(r)−b(r )
2r
2
1−
b(r)
r
F
+ 2
F
r
+ F
(r)
and T =−ρ + p
r
+ 2 p
t
.
These are the standard terminologies of the matter thread-
ing the wormhole, as a function of the shape function b(r),
redshift function (r ) and function F (r ). One can compre-
hend the matter content of the wormhole by specifying the
above functions. Thus, one can consider a specific choice
of shape function to obtain a wormhole solution. There-
fore, in this paper a specific form of the shape function
b(r) =
r
exp(r−r
0
)
is considered.
Morris and Thorne [3] defined the dimensionless function
ξ =
τ −ρ
|ρ|
. However, in this paper, we define the dimension-
less function
ξ
ef f
=
−p
ef f
r
− ρ
ef f
|ρ
ef f
|
. (18)
where τ =−p
ef f
r
and ρ = ρ
ef f
.Now,theEqs. (15) and
(16) yields
ξ
ef f
=
b(r) − rb
(r)
r|b
(r)|
(19)
Let us combine the flaring-out condition, given in Eq. (12),
with the Eq. (19), the effective exotic function takes the form
ξ
ef f
=
2b(r)
r|b
(r)|
d
2
r
dz
2
(20)
At the throat, we have the following condition
ξ(r
0
) =
−p
ef f
r
(r
0
) − ρ
ef f
(r
0
)
|ρ
ef f
(r
0
)|
> 0 (21)
Thus, from the above condition it is observed that the radial
tension should exceed the total density of mass energy i.e.
−p
ef f
r
(r
0
)>ρ
ef f
(r
0
). We shall call matter with this prop-
erty, −p
ef f
r
(r
0
)>ρ
ef f
(r
0
)>0, exotic [3]. The presence of
exotic matter at the throat of the wormhole indicates that the
observer who moves through the throat with a radial velocity
close to the speed of light will see a negative energy density.
Overall, we can say that the field equations plus the absence
of a horizon at throat indicates −p
ef f
r
(r
0
)>ρ
ef f
(r
0
) at the
throat. This implies traveler moving through the throat with
speed very close to the speed of light can see the negative
energy density. This implies the violation of the null, weak,
strong and dominant energy conditions at the throat.
3 Wormhole solutions
In literature, the cosmologists have studied several mod-
els with quantum corrections of Einstein’s field equations
[87,88]. Starobinsky followed the same idea and studied the
cosmology of a model which in its simplified form is known
as Starobinsky model [18]. He replaced Einstein’s action R
with R + α R
2
and presented a first compatible model of
inflation. In this action, term R
2
is a responsible factor for
acceleration at high energies in the early stage of the uni-
verse. The results obtained from the Planck satellite [89]are
also consistent with the Starobinsky model. This model has
been studied in several aspects [90–94]. In this paper, a gen-
eral form of this model f (R) = R + αR
n
, where α and
n are arbitrary constants, is considered. Further, the shape
function b(r) =
r
exp(r−r
0
)
defined by Samanta et al. [70] and
the redshift function introduced by Pavlovic and Sossich [76]
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