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arX
iv:2
003.
0029
8v1
[ph
ysic
s.ge
n-ph
] 1
7 Fe
b 20
20
A new shape function for wormholes in f(R) gravity and General
Relativity
Ambuj Kumar Mishra1 Umesh Kumar Sharma2
1,2Department of Mathematics, Institute of Applied Sciences and Humanities, GLAUniversity, Mathura-281 406, Uttar Pradesh, India
1E-mail: ambuj [email protected]
2E-mail:[email protected]
Abstract
In the present work, a new shape function is proposed inside modified f(R) grav-ity and General Relativity in wormhole (WH) geometry. The shape function obeyedall the desired conditions of WH geometry. The equation of state (EoS) parameter,anisotropy parameter and the energy conditions are computed. The tangential nullenergy conditions and the weak energy condition are validated, as well as the radialenergy conditions, which demonstrates the nonappearance of exotic matter due to mod-ified gravity allied with such a new proposal.
Keywords: f(R) gravity, Wormholes, Shape function, Energy conditions
PACS number: 04.50.kd
1 Introduction
The geometrical structures as throat like, called wormholes, which interface two isolated anddifferent areas of space-times and have no singularity or horizon. As a bridge model of aparticle, the wormholes solutions were firstly found in 1935 by Einstein and Rosen [1]. Theinvestigation of Lorentzian wormholes with regards to General Relativity (GR) was invigo-rated by the remarkable work of Morris and Thorne in 1988 [2, 3]. The flaring-out state ofthe throat is the basic component in wormhole physics, which in general relativity involvesthe infringement of the null energy condition (NEC). The violation of null energy conditionrepresents the presence of exotic matter. A significant issue in wormhole solutions is thesetting up of energy conditions. In such manner, different techniques have been suggested inthe previous works that manage the scenarios of energy conditions with in wormhole back-ground. Along this line, work on thin-shell wormholes has been done in [4] and dynamicalwormholes [5, 6, 7], where the supporting matter is focused on the wormholes throat.
Different autonomous high-precision observational information have acknowledged withalarming proof that the Universe is experiencing a period of accelerating expansion [8]. Manyproposals have been suggested in the published work to clarify this idea, varying from modi-fied gravity theory to dark energy models. In modified gravity theory, one may suppose that
1
the Einstein’s hypothesis of general relativity (GR) does not work at large scales, and a moregeneralized action depicts the gravitational field. Hilbert derived the Einstein field equationof GR from the action principle, by embracing the scalar curvature as a linear function, R,in the Lagrangian gravitational density. Although, there are no presumptive basis to limitthe gravitational Lagrangian to this structure, and naturally a few speculations have beenproposed. In the context of modified theories of gravity, the presence of higher order termsin curvature would allow for building thin-shell wormholes supported by ordinary matter [9].Recently, many people try to build and study wormhole solutions within the framework ofmodified theories of gravity in various context [10].
The modified f(R) gravity, a popular gravity theory, which modifies the Einstein GRgravitational action R by an arbitrary function f(R), where R is a well- known Ricci scalar.The accelerated expansion of the Universe may be demonstrated by these models [11]. It isobserved that, in f(R) gravity, the energy conditions can be satisfy by the higher-order cur-vature invariants [12]. The wormhole solutions in modified f(R) gravity with non constantRicci scalar are obtained in [13]. Lately, the junction conditions applied to the developmentof pure double layer bubbles and thin-shell wormholes in f(R) gravity [14]. Using the non-commutative geometry, the new static wormholes are built in f(R) gravity [15] and the worm-hole solutions cosmological evolution is explored in [16]. Wormhole solutions (Lorentzian),which are consistent with the cosmological evolution and solar system observations were alsoinvestigated in viable modified f(R) theories of gravity [17]. These wormholes were observedin explicit f(R) models which are considered to repeat sensible situations of cosmologicalbehavior dependent on the accelerated expansion of the Universe. It was demonstrated thatthe weak energy conditions (WEC) holds at the region of the throat for specific scopes ofthe free parameters of the hypothesis. The weak energy condition is fulfilled at a particularpoint in space with the simple choices of shape function for the asymptotically flat wormholesolutions. A hybrid and exponential shape function in modified gravity are introduced in[18, 19]. Recently, two different shape function for wormhole structure in f(R) gravity wereproposed in [20, 21].
Because of the absence of wormholes observations at present, in spite of the consider-able number of endeavors and proposition [22], some material and geometrical highlights ofwormholes, for example, the equation of state (EoS) and the shape function, are as yet notexactly known. Especially, a few structures for the shape function b(r) have been suggestedand investigated so far, as one may check, for example, [23]. As the shape function b(r) haveto obey several conditions and not arbitrary. We propose a new shape function for modifiedf(R) gravity and GR in this work, which must be, normally as per the shape function con-ditions.
This paper is composed in the accompanying way: In Sec. II, the field equations and thewormhole metric is presented in reference to f(R) gravity. In Sec. III, energy conditions aredefined. In Sec. IV, wormholes models are described. In Sec. V, results and discussions areexplored. Finally, conclusions are given in Sec. VI.
2
2 Field Equations and Wormhole Metric
The theory of f(R) gravity starts from the generalization of the Einstein’s theory of relativity.The action in f(R) theory is defined as
S =1
2κ
∫ √−g [f(R) + Lm] d4x, (1)
where κ = 2πG, g is the metric determinant and Lm is matter Lagrangian density of themetric gυµ. For mathematical simplicity κ is considered to be unity.
Varying Eq. (1) with respect to gυµ, the fourth-order non-linear field equations are ob-tained as
FRRυµ −1
2fgυµ −▽υ ▽µ FR +✷FR gυµ = Tmυµ, (2)
where ✷ = ▽υ▽υ, ▽υ denotes, covariant derivative operator. R and Rυµ denote curvaturescalar and Ricci tensor respectively. Tmυµ denotes energy momentum tensor and FR = df
dR.
The effective field equation can be obtained by using Eq. (2) as
Gυµ = T effυµ = Rυµ −1
2Rgυµ = T gυµ +
8π
FRTmυµ, (3)
where Gυµ denotes the Einstein tensor, T effυµ denotes effective energy momentum tensor and
T gυµ = 1FR
[
▽υ ▽µ FR −(
✷FR + 12RFR − 1
2f)
gυµ]
. The energy momentum tensor for the
matter source of the wormholes is Tυµ = ∂Lm
∂gυµ, which can be written as
Tυµ = (ρ+ pl) uυuµ − plgυµ + (pr − pl)XυXµ, (4)
such that uυuυ = −1 and XυXυ = 1. Here rho, pr and pl denote energy density, radialpressure and tangential pressure respectively.
We explore static and spherically symmetric wormhole geometries in f(R) in gravity, sowe take metric describing wormhole geometry as
ds2 = −e2χ(r)dt2 + eψ(r)dr2 + r2(
dθ2 + sin2θ2φ2)
, (5)
where χ(r) is arbitrary function of r, indicates gravitational redshift (called redshift function).
ψ(r) =(
1− b(r)r
)−1
and r0 < r < ∞. The minimum value r0 denotes throat radius of
wormhole, b(r) describes geometry of the wormhole and is known as the shape function.This shape function is responsible for shape of the wormhole and it must have the followingproperties: (i) b(r0) = r0, (ii) b
′(r0) < 1, (iii) b(r)−b′(r)r
b(r)2> 0, (iv) b(r)
r→ 0 as r → ∞, (v) b(r)
r<
1 for r > r0. Moreover, to avoid event horizon (non-traversable condition) χ(r) shouldbe every where finite. However, In this article, we consider constant redshift function formathematical simplicity.
Here R (Ricci scalar tensor) is given by R = 2b′(r)r2
. This yields Einstein’s fields equationsfor the action Eq.(1) in f(R) gravity as:
ρ =b′(r)FRr2
−H, (6)
pr = −b(r)FRr3
−(
1− b(r)
r
)
F ′′
R +F ′
R (rb′(r)− b(r))
2r2(
1− b(r)r
)
+H, (7)
3
pl =(b(r)− rb′(r))FR
2r3− F ′
R
r
(
1− b(r)
r
)
+H, (8)
whereH = 14(RFR +✷FR + T ) and prime upon the function indicates derivative with respect
to radial coordinate r.The radial state parameter i.e EOS parameter in terms of radial pressure is defined as
ω =pr
ρ. (9)
The anisotropy parameter ∆ in terms of radial pressure pr and tangential pressure pl isdefined as
∆ = pl − pr. (10)
The negative and positive sign of δ indicate that geometry is attractive or repulsive, δ = 0indicates geometry has an isotropic pressure.
3 Energy Conditions
The energy conditions play important roll to analyze physical existence of wormhole ge-ometry. The important energy conditions are NEC (null energy condition), WEC (weakenergy condition), SEC (strong energy condition), and DEC (dominant energy condition).NEC for any null vector is defined as Tυµk
υkµ ≥ 0 ⇔ NEC. Alternatively, NEC canalso be defined in terms of principal pressures as ∀i, ρ + pi ≥ 0 ⇔ NEC. WEC for atimelike vector is defined as TυµV
υV µ ≥ 0 ⇔ WEC. This WEC in terms of principalpressures can also be defined as ρ ≥ 0 and ∀i, ρ + pi ≥ 0 ⇔ WEC. For a time-like vec-tor, SEC is defined as (Tυµ − T
2gυµ)V
υV µ ≥ 0 ⇔ SEC, where T denotes trace of energymomentum tensor. SEC in terms of principal pressures is defined as T = −ρ +
∑
j pjand ∀j, ρ + pj ≥ 0, ρ +
∑
j pj ≥ 0 ⇔ SEC. DEC for a time-like vector is defined asDEC ⇔ TυµV
υV µ ≥ 0 and Tυµ is not space like. Alternately, DEC is also defined in termsof principal pressures ρ ≥ 0 and ∀i, pi ∈ [−ρ,+ρ] ⇔ DEC.
In the present article, These energy conditions are investigated in terms of radial pressurepr and tangential pressure pl as follows:
• ρ+ pr ≥ 0, ρ+ pl ≥ 0 (NEC),
• ρ ≥ 0, ρ+ pr ≥ 0, ρ+ pl ≥ 0 (WEC),
• ρ+ pr ≥ 0, ρ+ pl ≥ 0, ρ+ pr + 2pl ≥ 0 (SEC),
• ρ ≥ 0, ρ− |pr| ≥ 0, ρ− |pl| ≥ 0 (DEC).
4 Models of Wormholes
In the present paper, f(R) model is taken, which is defined by Nojiri and Odintsov [24] as
f(R) = R + αRm − βR−n, (11)
4
(a)
Figure 1: Nature of b(r), throat condition b(r) < r, flaring out condition b′(r) < 1 and
asymptotically flatness limr→∞
b(r)r
= 0 for r0 = 1.
where α, β,m and n are taken as positive constants. The term Rm dominates when R−n
vanishes in the case of curvature increases and it represent evolution of inflation at earlytimes. Similarly when curvature decrease the term R−n dominates and Rm vanishes thatshows the production of present acceleration. In the framework of FRW universe such formsof f(R) are considered by Cao et al. [25] for exploration of the probabilities of late timeacceleration. Moradpour [26] investigated traversable wormholes in the background of Lyramanifold and general relativity. He studies energy conditions by using hyperbolic shapefunction. Godani and Samanta [27] investigated non violation of energy conditions in f(R)gravity by using the same hyperbolic shape function.
In this section, we will obtain wormhole models that for their matter content are con-structed from different hypotheses in viable f(R) gravity. Here, a new shape function isdefined as
b(r) = r0
(
ar
ar0
)
, (12)
where a is base and 0 < a < 1. We can see directly in Fig.1 that the proposed shape functionsatisfies all the conditions i.e. basic requirements of shape functions, such as throat condition,flaring out condition and typical asymptotically flatness condition for WH geometry.
This new shape function is taken into account to investigate the wormhole solutions. Theenergy density (ρ), radial pressure (pr), tangential pressure (pl) can be obtained by usingfield equations (6)- (8) as
ρ =1
r2
[
r0ar−r0 ln (a) + 2−1+mα
(
r0ar−r0 ln (a)
r2
)m
mr2
+ 2−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
nr2
]
, (13)
5
pr = − 1
4r3r0 (ln (a))
[
4 r02ar−r0 ln (a) + 4α 2m
(
r0ar−r0 ln (a)
r2
)m
mar0−rr3
− 12αm2
(
r0ar−r0 ln (a)
r2
)m
2mar0−rr3 + 12 β n2
(
r0ar−r0 ln (a)
r2
)−n
2−nar0−rr3
+ 4 β 2−n(
r0ar−r0 ln (a)
r2
)−n
nar0−rr3 + 4 21+mα
(
r0ar−r0 ln (a)
r2
)m
m3ar0−rr3
+ 4 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3ar0−rr3 − 4 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3r2r0
− 4 21+mα
(
r0ar−r0 ln (a)
r2
)m
m3r2r0 − 13αm2
(
r0ar−r0 ln (a)
r2
)m
2mr3r0 ln (a)
+ 13 β n2
(
r0ar−r0 ln (a)
r2
)−n
2−nr3r0 ln (a) + 4 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3r3r0 ln (a)
+ 4 21+mα
(
r0ar−r0 ln (a)
r2
)m
m3r3r0 ln (a)− 4 2−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3 (ln (a))2 r4r0
− 3 β n2r4 (ln (a))2(
r0ar−r0 ln (a)
r2
)−n
2−nr0 − 4 2−1+mα
(
r0ar−r0 ln (a)
r2
)m
m3 (ln (a))2 r4r0
+ 3αm2r4 (ln (a))2(
r0ar−r0 ln (a)
r2
)m
2mr0 − 4 21+mα
(
r0ar−r0 ln (a)
r2
)m
mar0−rr4 ln (a)
− 4 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
nar0−rr4 ln (a)− 4 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3ar0−rr4 ln (a)
− 4 21+mα
(
r0ar−r0 ln (a)
r2
)m
m3ar0−rr4 ln (a) + 16αm2r4(
r0ar−r0 ln (a)
r2
)m
2mar0−r ln (a)
− 16 β n2r4(
r0ar−r0 ln (a)
r2
)−n
2−nar0−r ln (a)− 4αm2r5 (ln (a))2(
r0ar−r0 ln (a)
r2
)m
2mar0−r
+ 4 2−1+mα
(
r0ar−r0 ln (a)
r2
)m
mar0−rr5 (ln (a))2 + 4 2−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
nar0−rr5 (ln (a))2
+ 4 2−1+mα
(
r0ar−r0 ln (a)
r2
)m
m3ar0−r (ln (a))2 r5 + 4 2−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3ar0−r (ln (a))2 r5
+ 4 β n2r5 (ln (a))2(
r0ar−r0 ln (a)
r2
)−n
2−nar0−r − 20 2−1−nβ n2
(
r0ar−r0 ln (a)
r2
)−n
r2r0
+ 20 2−1+mαm2
(
r0ar−r0 ln (a)
r2
)m
r2r0 + 5 2mr3r0α
(
r0ar−r0 ln (a)
r2
)m
m ln (a)
− 2mα
(
r0ar−r0 ln (a)
r2
)m
mr4 (ln (a))2 r0 + 5 2−nr3r0β
(
r0ar−r0 ln (a)
r2
)−n
n ln (a)
− 2−nβ
(
r0ar−r0 ln (a)
r2
)−n
nr4 (ln (a))2 r0
]
, (14)
6
pl = − 1
4r3r0 (ln (a))
[
4 r42−1+mαm2
(
r0ar−r0 ln (a)
r2
)m
ar0−r ln (a)
− 4 r3r02−1+mαm2
(
r0ar−r0 ln (a)
r2
)m
ln (a)− 4αm2
(
r0ar−r0 ln (a)
r2
)m
2mar0−rr3
+ 4αm2
(
r0ar−r0 ln (a)
r2
)m
2mr2r0 − 4 r42−1+mα
(
r0ar−r0 ln (a)
r2
)m
mar0−r ln (a)
+ 3 2mr3r0α
(
r0ar−r0 ln (a)
r2
)m
m ln (a) + 4α 2m(
r0ar−r0 ln (a)
r2
)m
mar0−rr3
− 5α 2m(
r0ar−r0 ln (a)
r2
)m
mr2r0 − 4 r42−1−nβ n2
(
r0ar−r0 ln (a)
r2
)−n
ar0−r ln (a)
+ 4 r3r02−1−nβ n2
(
r0ar−r0 ln (a)
r2
)−n
ln (a) + 4 β n2
(
r0ar−r0 ln (a)
r2
)−n
2−nar0−rr3
− 4 β n2
(
r0ar−r0 ln (a)
r2
)−n
2−nr2r0 − 4 r42−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
nar0−r ln (a)
+ 3 2−nr3r0β
(
r0ar−r0 ln (a)
r2
)−n
n ln (a) + 4 β 2−n(
r0ar−r0 ln (a)
r2
)−n
nar0−rr3
− 5 β 2−n(
r0ar−r0 ln (a)
r2
)−n
nr2r0 − 2 r02ar−r0 ln (a) + 2 r0
2 (ln (a))2 rar−r0
]
, (15)
7
ρ+ pr =1
4r3r0 (ln (a))
[
4 r02 (ln (a))2 rar−r0 − 4 r0
2ar−r0 ln (a)
− 4α 2m(
r0ar−r0 ln (a)
r2
)m
mar0−rr3 + 12αm2
(
r0ar−r0 ln (a)
r2
)m
2mar0−rr3
− 12 β n2
(
r0ar−r0 ln (a)
r2
)−n
2−nar0−rr3 − 4 β 2−n(
r0ar−r0 ln (a)
r2
)−n
nar0−rr3
− 4 21+mα
(
r0ar−r0 ln (a)
r2
)m
m3ar0−rr3 − 4 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3ar0−rr3
+ 4 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3r2r0 + 4 21+mα
(
r0ar−r0 ln (a)
r2
)m
m3r2r0
+ 13αm2
(
r0ar−r0 ln (a)
r2
)m
2mr3r0 ln (a)− 13 β n2
(
r0ar−r0 ln (a)
r2
)−n
2−nr3r0 ln (a)
− 4 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3r3r0 ln (a)− 4 21+mα
(
r0ar−r0 ln (a)
r2
)m
m3r3r0 ln (a)
+ 4 2−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3 (ln (a))2 r4r0 + 3 β n2r4 (ln (a))2(
r0ar−r0 ln (a)
r2
)−n
2−nr0
+ 4 2−1+mα
(
r0ar−r0 ln (a)
r2
)m
m3 (ln (a))2 r4r0 − 3αm2r4 (ln (a))2(
r0ar−r0 ln (a)
r2
)m
2mr0
+ 4 21+mα
(
r0ar−r0 ln (a)
r2
)m
mar0−rr4 ln (a) + 4 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
nar0−rr4 ln (a)
+ 4 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3ar0−rr4 ln (a) + 4 21+mα
(
r0ar−r0 ln (a)
r2
)m
m3ar0−rr4 ln (a)
− 16αm2r4(
r0ar−r0 ln (a)
r2
)m
2mar0−r ln (a) + 16 β n2r4(
r0ar−r0 ln (a)
r2
)−n
2−nar0−r ln (a)
+ 4αm2r5 (ln (a))2(
r0ar−r0 ln (a)
r2
)m
2mar0−r − 4 2−1+mα
(
r0ar−r0 ln (a)
r2
)m
× mar0−rr5 (ln (a))2 − 4 2−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
nar0−rr5 (ln (a))2
− 4 2−1+mα
(
r0ar−r0 ln (a)
r2
)m
m3ar0−r (ln (a))2 r5
− 4 2−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3ar0−r (ln (a))2 r5
− 4 β n2r5 (ln (a))2(
r0ar−r0 ln (a)
r2
)−n
2−nar0−r + 20 2−1−nβ n2
(
r0ar−r0 ln (a)
r2
)−n
r2r0
− 20 2−1+mαm2
(
r0ar−r0 ln (a)
r2
)m
r2r0 − 3 2mr3r0α
(
r0ar−r0 ln (a)
r2
)m
m ln (a)
+ 2mα
(
r0ar−r0 ln (a)
r2
)m
mr4 (ln (a))2 r0 − 3 2−nr3r0β
(
r0ar−r0 ln (a)
r2
)−n
n ln (a)
+ 2−nβ
(
r0ar−r0 ln (a)
r2
)−n
nr4 (ln (a))2 r0
]
, (16)
8
ρ+ pl =1
4r3r0 (ln (a))
[
2 r02 (ln (a))2 rar−r0 + 4 r3r02
−1+mαm2
(
r0ar−r0 ln (a)
r2
)m
ln (a)
− 2mr3r0α
(
r0ar−r0 ln (a)
r2
)m
m ln (a)− 4 r3r02−1−nβ n2
(
r0ar−r0 ln (a)
r2
)−n
ln (a)
− 2−nr3r0β
(
r0ar−r0 ln (a)
r2
)−n
n ln (a) + 2 r02ar−r0 ln (a)
+ 5 β 2−n(
r0ar−r0 ln (a)
r2
)−n
nr2r0 + 5α 2m(
r0ar−r0 ln (a)
r2
)m
mr2r0
− 4α 2m(
r0ar−r0 ln (a)
r2
)m
mar0−rr3 + 4αm2
(
r0ar−r0 ln (a)
r2
)m
2mar0−rr3
− 4 β n2
(
r0ar−r0 ln (a)
r2
)−n
2−nar0−rr3 − 4 β 2−n(
r0ar−r0 ln (a)
r2
)−n
nar0−rr3
+ 4 β n2
(
r0ar−r0 ln (a)
r2
)−n
2−nr2r0 − 4αm2
(
r0ar−r0 ln (a)
r2
)m
2mr2r0
− 4 r42−1+mαm2
(
r0ar−r0 ln (a)
r2
)m
ar0−r ln (a)
+ 4 r42−1+mα
(
r0ar−r0 ln (a)
r2
)m
mar0−r ln (a)
+ 4 r42−1−nβ n2
(
r0ar−r0 ln (a)
r2
)−n
ar0−r ln (a)
+ 4 r42−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
nar0−r ln (a)
]
, (17)
ρ+ pr + 2 pl = − 1
4rr0 (ln (a))
[
−20αm2r
(
r0ar−r0 ln (a)
r2
)m
2mar0−r
+ 4 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3ar0−rr + 20 β n2r
(
r0ar−r0 ln (a)
r2
)−n
2−nar0−r
+ 4 21+mα
(
r0ar−r0 ln (a)
r2
)m
m3ar0−rr + 12α 2m(
r0ar−r0 ln (a)
r2
)m
mar0−rr
+ 12 β 2−n(
r0ar−r0 ln (a)
r2
)−n
nar0−rr − 4 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3r0
− 4 21+mα
(
r0ar−r0 ln (a)
r2
)m
m3r0 + 36 2−1+mαm2
(
r0ar−r0 ln (a)
r2
)m
r0
− 20 2−1+mα
(
r0ar−r0 ln (a)
r2
)m
mr0 − 36 2−1−nβ n2
(
r0ar−r0 ln (a)
r2
)−n
r0
9
− 20 2−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
nr0 − 2mα
(
r0ar−r0 ln (a)
r2
)m
mr2 (ln (a))2 r0
− 2−nβ
(
r0ar−r0 ln (a)
r2
)−n
nr2 (ln (a))2 r0 + 20αm2r2 ln (a)
(
r0ar−r0 ln (a)
r2
)m
2mar0−r
− 20 β n2r2 ln (a)
(
r0ar−r0 ln (a)
r2
)−n
2−nar0−r − 17αm2
(
r0ar−r0 ln (a)
r2
)m
2mrr0 ln (a)
+ 17 β n2
(
r0ar−r0 ln (a)
r2
)−n
2−nrr0 ln (a) + 4 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3rr0 ln (a)
+ 4 21+mα
(
r0ar−r0 ln (a)
r2
)m
m3rr0 ln (a)− 4 2−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
× n3 (ln (a))2 r2r0 − 3 β n2r2 (ln (a))2(
r0ar−r0 ln (a)
r2
)−n
2−nr0
− 4 2−1+mα
(
r0ar−r0 ln (a)
r2
)m
m3 (ln (a))2 r2r0
+ 3αm2r2 (ln (a))2(
r0ar−r0 ln (a)
r2
)m
2mr0 − 4 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
× n3ar0−rr2 ln (a)− 4 21+mα
(
r0ar−r0 ln (a)
r2
)m
m3ar0−rr2 ln (a)
− 4αm2r3 (ln (a))2(
r0ar−r0 ln (a)
r2
)m
2mar0−r + 4 2−1+mα
(
r0ar−r0 ln (a)
r2
)m
× mar0−rr3 (ln (a))2 + 2−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
nar0−rr3 (ln (a))2
+ 4 2−1+mα
(
r0ar−r0 ln (a)
r2
)m
m3ar0−r (ln (a))2 r3 + 2−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
× n3ar0−r (ln (a))2 r3 + 4 β n2r3 (ln (a))2(
r0ar−r0 ln (a)
r2
)−n
2−nar0−r
− 12 2mα
(
r0ar−r0 ln (a)
r2
)m
mar0−rr2 ln (a)− 12 2−nβ
(
r0ar−r0 ln (a)
r2
)−n
nar0−rr2 ln (a)
+ 9 2mα
(
r0ar−r0 ln (a)
r2
)m
mrr0 ln (a) + 9 2−nβ
(
r0ar−r0 ln (a)
r2
)−n
nrr0 ln (a)
]
, (18)
10
ρ− |pr| =1
4r2
[
4 r0ar−r0 ln (a) + 4 2−1+mα
(
r0ar−r0 ln (a)
r2
)m
mr2
+ 4 2−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
nr2 −∣
∣
∣
∣
(
−2mα
(
r0ar−r0 ln (a)
r2
)m
mr4 (ln (a))2 r0
− 2−nβ
(
r0ar−r0 ln (a)
r2
)−n
nr4 (ln (a))2 r0 − 21+mα
(
r0ar−r0 ln (a)
r2
)m
m3 (ln (a))2 r4r0
− 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3 (ln (a))2 r4r0
+ 21+mαmar0−rr5 (ln (a))2(
r0ar−r0 ln (a)
r2
)m
+ 21+mαm3ar0−r (ln (a))2 r5(
r0ar−r0 ln (a)
r2
)m
+ 21−nβ nar0−rr5 (ln (a))2(
r0ar−r0 ln (a)
r2
)−n
+ 21−nβ n3ar0−r (ln (a))2 r5(
r0ar−r0 ln (a)
r2
)−n
− 13αm2
(
r0ar−r0 ln (a)
r2
)m
2mr3r0 ln (a) + 13 β n2
(
r0ar−r0 ln (a)
r2
)−n
2−nr3r0 ln (a)
− 3 β n2r4 (ln (a))2(
r0ar−r0 ln (a)
r2
)−n
2−nr0 + 3αm2r4 (ln (a))2(
r0ar−r0 ln (a)
r2
)m
2mr0
+ 16αm2r4(
r0ar−r0 ln (a)
r2
)m
2mar0−r ln (a)− 16 β n2r4(
r0ar−r0 ln (a)
r2
)−n
2−nar0−r ln (a)
− 4αm2r5 (ln (a))2(
r0ar−r0 ln (a)
r2
)m
2mar0−r + 4 β n2r5 (ln (a))2(
r0ar−r0 ln (a)
r2
)−n
2−nar0−r
+ 8 2mα
(
r0ar−r0 ln (a)
r2
)m
m3r3r0 ln (a)− 8 2mα
(
r0ar−r0 ln (a)
r2
)m
mar0−rr4 ln (a)
− 8 2mα
(
r0ar−r0 ln (a)
r2
)m
m3ar0−rr4 ln (a) + 8 2−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3r3r0 ln (a)
− 8 2−nβ
(
r0ar−r0 ln (a)
r2
)−n
nar0−rr4 ln (a)− 8 2−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3ar0−rr4 ln (a)
+ 8 2mα
(
r0ar−r0 ln (a)
r2
)m
m3ar0−rr3 − 8 2mα
(
r0ar−r0 ln (a)
r2
)m
m3r2r0
+ 8 2−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3ar0−rr3 − 8 2−nβ
(
r0ar−r0 ln (a)
r2
)−n
n3r2r0
+ 5 2−nr3r0β
(
r0ar−r0 ln (a)
r2
)−n
n ln (a) + 5 2mr3r0α
(
r0ar−r0 ln (a)
r2
)m
m ln (a)
+ 4α 2m(
r0ar−r0 ln (a)
r2
)m
mar0−rr3 − 12αm2
(
r0ar−r0 ln (a)
r2
)m
2mar0−rr3
+ 12 β n2
(
r0ar−r0 ln (a)
r2
)−n
2−nar0−rr3 + 4 β 2−n(
r0ar−r0 ln (a)
r2
)−n
nar0−rr3
− 10 β n2
(
r0ar−r0 ln (a)
r2
)−n
2−nr2r0 + 10αm2
(
r0ar−r0 ln (a)
r2
)m
2mr2r0
+ 4 r02ar−r0 ln (a)
)
r−3r0−1 (ln (a))−1
∣
∣ r2]
, (19)
11
(a) (b) (c)
Figure 2: (a) NEC, ρ + pr for α = 0, β = 0, (b) NEC, ρ + pl for α = 0, β = 0,(c) SEC,ρ+ pr + 2pl for for α = 0, β = 0 .
ρ− |pl| =1
4r2
[
4 r0ar−r0 ln (a) + 4 2−1+mα
(
r0ar−r0 ln (a)
r2
)m
mr2
+ 4 2−1−nβ
(
r0ar−r0 ln (a)
r2
)−n
nr2 −∣
∣
∣
∣
(
21+mαm2r4ar0−r ln (a)
(
r0ar−r0 ln (a)
r2
)m
− 21+mαm2
(
r0ar−r0 ln (a)
r2
)m
r3r0 ln (a)− 4αm2
(
r0ar−r0 ln (a)
r2
)m
2mar0−rr3
+ 4αm2
(
r0ar−r0 ln (a)
r2
)m
2mr2r0 − 21+mα
(
r0ar−r0 ln (a)
r2
)m
mar0−rr4 ln (a)
+ 3 2mr3r0α
(
r0ar−r0 ln (a)
r2
)m
m ln (a) + 4α 2m(
r0ar−r0 ln (a)
r2
)m
mar0−rr3
− 5α 2m(
r0ar−r0 ln (a)
r2
)m
mr2r0 − 21−nβ n2r4ar0−r ln (a)
(
r0ar−r0 ln (a)
r2
)−n
+ 21−nβ n2
(
r0ar−r0 ln (a)
r2
)−n
r3r0 ln (a) + 4 β n2
(
r0ar−r0 ln (a)
r2
)−n
2−nar0−rr3
− 4 β n2
(
r0ar−r0 ln (a)
r2
)−n
2−nr2r0 − 21−nβ
(
r0ar−r0 ln (a)
r2
)−n
nar0−rr4 ln (a)
+ 3 2−nr3r0β
(
r0ar−r0 ln (a)
r2
)−n
n ln (a) + 4 β 2−n(
r0ar−r0 ln (a)
r2
)−n
nar0−rr3
− 5 β 2−n(
r0ar−r0 ln (a)
r2
)−n
nr2r0 − 2 r02ar−r0 ln (a)
+ 2 r02 (ln (a))2 rar−r0
)
r−3r0−1 (ln (a))−1
∣
∣ r2]
. (20)
5 Results and Discussion
The wormhole geometries are very much affected on shape function b(r), so in literature vari-ous shape functions are defined and wormhole solutions are investigated. Several researchersexplored violation or non-violation of energy conditions in the context of wormhole solutions.In the present investigation, the shape function b(r) = r0
(
ar
ar0
)
is taken into account and ex-plored the wormhole solutions in the framework of f(R) gravity with f(R) = R+αRm−βR−n,where α, β,m and n are positive real numbers. Here different cases are analyzed for various
12
(a) (b) (c) (d)
Figure 3: (a) DEC, ρ − pr for α = 0, β = 0, (b) DEC, ρ − pl for α = 0, β = 0, (c) EOS,forα = 0, β = 0 (d)△ = pl − pr for α = 0, β = 0 .
(a) (b) (c) (d)
Figure 4: (a) WEC, ρ for β = 0, a = 0.5, m = 2 (b) NEC, ρ + pr for β = 0, a = 0.5, m = 2,(c) NEC, ρ+ pl forβ = 0, a = 0.5, m = 2,(d) SEC, ρ+ pr + 2pl forβ = 0, a = 0.5, m = 2.
(a) (b) (c) (d)
Figure 5: (a) DEC, ρ−pr for β = 0, a = 0.5, m = 2, (b) DEC, ρ−pl forβ = 0, a = 0.5, m = 2,(c) EOS,for β = 0, a = 0.5, m = 2 (d)△ = pl − pr or β = 0, a = 0.5, m = 2.
(a) (b) (c) (d)
Figure 6: (a) WEC, ρ for α = 0, a = 0.5, n = 4 (b) NEC, ρ+ pr for α = 0, a = 0.5, n = 4, (c)NEC, ρ+ pl forα = 0, a = 0.5, n = 4,(d) SEC, ρ+ pr + 2pl forα = 0, a = 0.5, n = 4.
13
(a) (b) (c) (d)
Figure 7: (a) DEC, ρ− pr for α = 0, a = 0.5, n = 4, (b) DEC, ρ− pl forβ = 0, a = 0.5, n = 4,(c) EOS,for α = 0, a = 0.5, n = 4 (d)△ = pl − pr for α = 0, a = 0.5, n = 4.
(a) (b) (c) (d)
Figure 8: (a) WEC, ρ for α = 5, a = 0.5, m = 2, n = 4 (b) NEC, ρ + pr for α = 5, a =0.5, m = 2, n = 4, (c) NEC, ρ + pl forα = 5, a = 0.5, m = 2, n = 4,(d) SEC, ρ + pr + 2plforα = 5, a = 0.5, m = 2, n = 4.
(a) (b) (c) (d)
Figure 9: (a) DEC, ρ − pr for α = 5, a = 0.5, m = 2, n = 4, (b) DEC, ρ − pl forα =5, a = 0.5, m = 2, n = 4, (c) EOS,for α = 5, a = 0.5, m = 2, n = 4 (d)△ = pl − pr forα = 5, a = 0.5, m = 2, n = 4.
14
(a) (b) (c) (d)
Figure 10: (a) WEC, ρ for β = 5, a = 0.5, m = 2, n = 4 (b) NEC, ρ + pr for β = 5, a =0.5, m = 2, n = 4, (c) NEC, ρ + pl forβ = 5, a = 0.5, m = 2, n = 4,(d) SEC, ρ + pr + 2plforβ = 5, a = 0.5, m = 2, n = 4.
(a) (b) (c) (d)
Figure 11: (a) DEC, ρ − pr for β = 5, a = 0.5, m = 2, n = 4, (b) DEC, ρ − pl forβ =5, a = 0.5, m = 2, n = 4, (c) EOS,for β = 5, a = 0.5, m = 2, n = 4 (d)△ = pl − pr forβ = 5, a = 0.5, m = 2, n = 4.
15
values of parameters in function f(R) and NEC, WEC, SEC, DEC, EOS (equation of state)and anisotropy parameter are examined.
5.1 Case 1: General Relativity Case
For general relativity case the parameters α and β is considered to be zero i.e. f(R) = R.The energy density ρ is found to be a negative function of radial coordinate, which indicatethe presence of exotic matter in wormhole geometry. In Fig. 2(a) and 2(b), NEC is plottedin terms of radial coordinate and tangential coordinate. ρ + pr is found to be negative andρ + pl is found to be positive. In Fig.2(c), SEC is plotted. It is observed that SEC is alsoviolating in this case. DEC in terms of ρ−|pr| and ρ−|pl| is also plotted in Fig.3(a) and 3(b).Both are obtained to be negative. Therefore, for present model NEC, SEC, and DEC areviolated. Hence, in general relativity this model strongly support that the wormhole is filledwith exotic matter. In Fig.3(c), EOS parameter ω is plotted, which comes out to be positive.This shows the role of matter in the universe. From Fig.3(d), it is found that anisotropyparameter △ is positive, that supports the repulsive geometry of universe.
5.2 Case 2: f(R) = R + αRm
For this case energy density is plotted in Fig.4(a), which comes out to be positive function ofr. We also found NEC in term of ρ+ pr is violating at the throat. The first NEC is plottedin Fig.4(b). The second NEC in term of ρ + pl is plotted in Fig.4(c). It is observed thatsecond NEC is violating for α > 125. In Fig.4(d), it is analyzed that SEC is also violating forr > r0 = 1. The DEC in terms of ρ− |pr| and ρ− |pl| is plotted in Figs.5(a) and 5(b), whichare observed to be negative. Therefore, in this model violation of energy conditions confirmedthe presence of exotic matter in wormhole geometry. EOS parameter ω is plotted in Fig.5(c).It is analyzed that phantom fluid is dominant in wormhole. The anisotropy parameter △ isshown in Fig.5(d), which is found to be positive at throat r0 = 1. Also anisotropy parameteris observed positive for r > 2, that indicate the repulsive nature of the geometry of Universe.
5.3 Case 3: f(R) = R − βR−n
In this model β and n are positive real numbers. In Fig.6(a), energy density is plotted fora = 0.5, n = 4 and β > 20, which is found to be positive. For the same values of parametersthe first and second NEC are plotted in Figs.6(b)and 6(c), which are observed to be positive.The SEC is also analyzed as positive function of r, which can be seen in Fig.6(d). However,first and second DEC are violating as both are observed to be negative in Figs.7(a)and7(b). In Fig.7(c), EOS parameter is plotted, which is found to be positive and anisotropyparameter is found to be negative in Fig.7(d). Therefore, in the present model validation ofNEC and SEC indicates presence of non-exotic matter and w > 0 shows role of non-phantomfluid, △ < 0 indicated attractive geometry. Thus in this model we conclude that wormholesolution with attractive geometry could exists without violating NEC and SEC with presenceof non-phantom fluid.
16
5.4 Case 4: f(R) = R + αRm − βR−n
In this model α 6= 0, β 6= 0, m and n are positive real numbers. In the first sub-case energydensity is plotted in Fig.8(a) for a = 0.5, α = 5, m = 2, n = 4 and β > 10. Energy density isfound to be positive. For the same values of parameters first and second NEC are analyzed,which are observed positive in Figs.8(b) and 8(c). The SEC is also found to be positive inFig.8(d). However, both DEC are observed to be negative in Figs.9(a) and 9(b). The EOSparameter is found to be positive function of radial coordinate r in Fig.9(c). However, △ isobserved to be negative.
In the second sub-case the parameters are taken as a = 0.5, β = 5, m = 2, n = 4 andα > 1. The energy density is plotted in Fig.10(a), which is found to be positive function ofradial coordinate. The first and second NEC are observed to be positive in Figs.10(b) and10(c). The SEC is also found to be positive in Fig.10(d). However, first and second DECsare analyzed to be negative in Figs. 11(a)and 11(b). The EOS parameter is observed aspositive and anisotropy parameter is negative in Figs.11(c) and 11(d). Therefore, in boththe sub-cases this model indicated the existence of wormhole without violating NECs andSEC. However, DECs are violating. The anisotropy parameter and EOS parameter indicatesattractive geometry with non-phantom fluid.
Thus, we observed that results obtained in viable modified theories of gravity in cases(2-4) are completely different from general relativity case.
6 Conclusion
Modified gravitational theories have received considerable attention over the last few decadesas a possible alternative to GR. Obviously, this incorporates two stages of universe expansion:rapid expansion (early inflation) and present cosmic acceleration. The present paper isfocused on wormhole models using the framework of f(R) gravity by considering a newshape function b(r) = r0
(
ar
ar0
)
, 0 < a < 1. The lower bound of radial coordinate r = r0 = 1defines the throat radius, also there are no black holes/horizons i.e wormhole regions arevalid everywhere for r > r0. The shape function satisfies all desired condition for wormholestructure like b′(r0) < 1 etc. Here, the function f(R) is taken in well known form as f(R) =R+ αRm − βR−n (see in [24]) and exact solutions are obtained. On varying the parameters(α, β,m and n), energy conditions (NEC, SEC, DEC), equation of state parameter andanisotropic parameter are analyzed in four different cases. It is observed that in the caseof GR, wormhole is filled with exotic matter, geometry is repulsive, while in other cases off(R) = R + Rm, wormhole is found filled with exotic matter with phantom fluid whosegeometry is repulsive. In other two case of f(R), it is analyzed that wormhole solutions mayexists without violation of NEC and SEC, moreover role of non-phantom fluid is observed inthese cases with attractive geometry. In this case wormhole satisfies energy conditions (NECand SEC) due to matter threading and gravitational fluid is also interpreted by its higherorder curvature derivative terms. Therefore, framework of modified gravity found significantwhile discussing the existence of wormhole solutions. Thus, f(R) gravity can be a usefulframework in wormhole physics in addition to its applications in cosmology.
17
Acknowledgments
The authors are thankful to GLA University, Mathura, India for providing help and supportto do this research work. The authors are also appreciative to A. Pradhan for his fruitfulsuggestions which improved the paper in present form.
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