P. Bhar et al.

New Astronomy 103 (2023) 102059

where 𝐷2 is an integration constant. where ‘‘timelike geodesics’’ are represented by a vector field called 𝑢𝜇 .
The expression of the shape function of the wormhole in this case Additionally, the Ricci tensor, the expansion parameter, the shear, and
is obtained as, the rotation related to the congruence are represented by 𝑅𝜇𝜈 , 𝜃, 𝜃𝜇𝜈
[ 8𝛾𝑛+𝜅(−1+3𝑛)
] and 𝜔𝜇𝜈 respectively. The Raychaudhuri equation becomes
(4𝛾 + 𝜅)𝑛
𝑏(𝑟) = 𝑟 1 + − 𝐷2 𝑟 𝜅+𝛾(3+𝑛) . (31)
𝜅(1 − 3𝑛) − 8𝛾𝑛 𝑑𝜃 1
= − 𝜃 2 − 𝜎𝜇𝜈 𝜎 𝜇𝜈 + 𝜔𝜇𝜈 𝜔𝜇𝜈 − 𝑅𝜇𝜈 𝑘𝜇 𝑘𝜈 . (40)
𝑑𝜏 2
The pictorial diagram of the shape function 𝑏(𝑟), 𝑏(𝑟) − 𝑟, 𝑏(𝑟)∕𝑟, 𝑏′ (𝑟)for
different values of 𝛾 is shown in Fig. 1. The throat of the wormhole under the scenario of a congruence of ‘‘null geodesics’’ described by the
occurs where 𝑏(𝑟) − 𝑟 cuts the ‘𝑟’ axis. Additionally it can be noted vector field 𝑘𝜇 .
that the redshift function does not approach zero as 𝑟 approaches ∞, It is evident from both Raychaudhuri equations that these energy
and the asymptotic behavior of 𝑏(𝑟)∕𝑟 does not approach zero too, conditions are entirely geometrical and independent of any gravita-
demonstrating that the wormhole spacetime is not asymptotically flat. tional theory. Energy constraints play an important role in both general
From the figure it is clear that 𝑏′ (𝑟0 ) < 1 and hence flaring out condition relativity and modified gravity theories. Each of the four types of
is satisfied. energy constraints is stated using well-known geometrical results. There
The matter density, radial and transverse pressure can be obtained are four types of energy conditions namely, Strong energy condition
as follows: (SEC), Dominant energy condition (DEC), Null energy condition (NEC),
and Weak energy condition (WEC). For an anisotropic distribution it
1
𝜌(𝑟) = can be defined as follows:
(2𝛾 + 𝜅)(𝜅 + 𝛾(3 + 𝑛))(8𝛾𝑛 + 𝜅(−1 + 3𝑛))𝑟2
[
• NEC: 𝜌 + 𝑝𝑟 ≥ 0, 𝜌 + 𝑝𝑡 ≥ 0,
× −3𝛾𝜅 − 𝜅 2 + 24𝛾 2 𝑛 + 13𝛾𝜅𝑛 + 2𝜅 2 𝑛 + 8𝛾 2 𝑛2
• WEC: 𝜌 ≥ 0, 𝜌 + 𝑝𝑟 ≥ 0, 𝜌 + 𝑝𝑡 ≥ 0,
]
3(2𝛾+𝜅)(−1+𝑛)
2+ 𝜅+𝛾(3+𝑛) • SEC: 𝜌 + 𝑝𝑟 ≥ 0, 𝜌 + 𝑝𝑡 ≥ 0, 𝜌 + 𝑝𝑟 + 2𝑝𝑡 ≥ 0,
+ 2𝛾𝜅𝑛2 − 3𝐷2 (2𝛾 + 𝜅)𝑛(8𝛾𝑛 + 𝜅(−1 + 3𝑛))𝑟 , (32)
• DEC: 𝜌 > |𝑝𝑟 |, 𝜌 > |𝑝𝑡 |.
3(2𝛾+𝜅)(−1+𝑛)
𝜅 3𝐷2 𝑟 𝜅+𝛾(3+𝑛) 5.1. Proposed model-I: Anisotropic WH
𝑝𝑟 = + , (33)
(2𝛾 + 𝜅)(8𝛾𝑛 + 𝜅(−1 + 3𝑛))𝑟2 𝜅 + 𝛾(3 + 𝑛)
[ 3(2𝛾+𝜅)(−1+𝑛) ]
3𝐷2 𝑟 𝜅+𝛾(3+𝑛) The following expressions are required to check the following in-
𝜅
𝑝𝑡 = 𝑛 + . (34) equalities:
(2𝛾 + 𝜅)(8𝛾𝑛 + 𝜅(−1 + 3𝑛))𝑟2 𝜅 + 𝛾(3 + 𝑛)
3(2𝛾+𝜅)(−1+𝑛)
2(4𝛾 + 𝜅)𝑛 3𝐷2 (−1 + 𝑛)𝑟 𝜅+𝛾(3+𝑛)
𝜌 + 𝑝𝑟 = −
4.2. Model II: By assuming pressure isotropy (2𝛾 + 𝜅)(8𝛾𝑛 + 𝜅(−1 + 3𝑛))𝑟2 𝜅 + 𝛾(3 + 𝑛)
(41)
To generate the second model, let us assume that the underlying
1
fluid is isotropic in nature, i.e., let us assume that 𝜌 + 𝑝𝑡 = (42)
(2𝛾 + 𝜅)𝑟2
3(2𝛾+𝜅)(−1+𝑛)
𝑝𝑡 = 𝑝𝑟 = 𝑝, (35) 2𝜅(−1 + 𝑛) + 8𝛾𝑛 3𝐷2 (1 + 𝑛)𝑟 𝜅+𝛾(3+𝑛)
𝜌 − 𝑝𝑟 = −
where 𝑝 being the isotropic pressure. (2𝛾 + 𝜅)(8𝛾𝑛 + 𝜅(−1 + 3𝑛))𝑟2 𝜅 + 𝛾(3 + 𝑛)
Using (25)–(26) and using the relation (35) we obtain the expression (43)
of the conformal factor as, 1
𝜌 − 𝑝𝑡 =
𝜓2 2𝑟𝜓𝜓 ′ (2𝛾 + 𝜅)(𝜅 + 𝛾(3 + 𝑛))(8𝛾𝑛 + 𝜅(−1 + 3𝑛))𝑟2
2 − = 1. (36) [
𝐶32 𝐶32 × −3𝛾𝜅 − 𝜅 2 + 24𝛾 2 𝑛 + 10𝛾𝜅𝑛 + 𝜅 2 𝑛
Solving the above equation we get,
+ 8𝛾 2 𝑛2 + 𝛾𝜅𝑛2 − 6𝐷2 (2𝛾 + 𝜅)𝑛(8𝛾𝑛
𝜓2 ]
= (𝐷3 𝑟2 + 1)∕2, (37) 3(2𝛾+𝜅)(−1+𝑛)
𝐶32 + 𝜅(−1 + 3𝑛))𝑟
2+ 𝜅+𝛾(3+𝑛) , (44)
where 𝐷3 is an integration constant.
3(2𝛾+𝜅)(−1+𝑛)
In this model, the shape function of the wormhole is obtained as, 4𝑛 3𝐷2 (1 + 𝑛)𝑟 𝜅+𝛾(3+𝑛)
𝜌 + 𝑝𝑟 + 2𝑝𝑡 = + . (45)
𝑟(1 − 𝐷3 𝑟2 ) (8𝛾𝑛 + 𝜅(−1 + 3𝑛))𝑟2 𝜅 + 𝛾(3 + 𝑛)
𝑏(𝑟) = . (38)
2 Because of the expression’s complexity, we will use the graphical
It is clear that, in this case the shape function does not depend on 𝛾. The representations in Fig. 3 to verify the energy conditions and the 3-D
pictorial representation of 𝑏(𝑟) is shown in Fig. 2 by taking 𝐷3 = 0.002. profiles of the energy conditions are shown in Fig. 4
The throat of the wormhole occurs at the point 𝑟0 = 0. The flaring
out condition is well satisfied since 𝑏′ (𝑟0 ) < 1 (ref. Fig. 2). In this case 5.2. Proposed model-II: Isotropic WH
also, the spacetime is not asymptotically flat as 𝑏(𝑟)∕𝑟 ↛ 0 as 𝑟 → ∞.
However, since the throat radius of the isotropic wormhole is zero, it To check the energy condition for isotropic case we need the fol-
is not a traversable WH. lowing equations:
8𝛾 + 𝜅 − 3𝐷3 (2𝛾 + 𝜅)𝑟2
5. Energy conditions 𝜌 = , (46)
2(2𝛾 + 𝜅)(4𝛾 + 𝜅)𝑟2
𝜅 + 3𝐷3 (2𝛾 + 𝜅)𝑟2
To get a number of findings that are applicable to various situations, 𝑝 = , (47)
2(2𝛾 + 𝜅)(4𝛾 + 𝜅)𝑟2
the energy conditions are used in a variety of circumstances. The
1
Raychaudhuri equation results in the idea of energy conditions, which 𝜌+𝑝 = , (48)
(2𝛾 + 𝜅)𝑟2
are given by, 4𝛾
−3𝐷3 + (2𝛾+𝜅)𝑟2
𝑑𝜃 1
= − 𝜃 2 − 𝜎𝜇𝜈 𝜎 𝜇𝜈 + 𝜔𝜇𝜈 𝜔𝜇𝜈 − 𝑅𝜇𝜈 𝑢𝜇 𝑢𝜈 , (39) 𝜌−𝑝 = , (49)
𝑑𝜏 3 4𝛾 + 𝜅