F(T)引力中Finch-Skea几何下的各向异性紧凑星解与EoS研究
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更新于2024-09-05
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本文探讨了在F(T)引力框架下，具有Finch-Skea (FS) 几何结构的各向异性紧凑天体的相对论解。F(T)理论是TEGR（TEGR, Teleparallel Equivalent of General Relativity）的一种扩展，它将引力与宇宙学常数联系起来，提供了对引力的新理解。FS几何是研究此类物体的重要工具，因为它能够描述天体内部的不均匀性。
作者们研究了不同质量和半径的已知恒星，通过修改FS几何，试图推导出这些天体的状态方程（EoS）。EoS决定了物质在极端条件下如何相互作用，对于理解恒星的内部结构和性质至关重要。然而，目前对恒星内部的EoS仍知之甚少，这使得使用诸如修正的Chaplygin气体（MCG）作为备选EoS来推测恒星特性成为必要。
MCG是一种特殊类型的流体模型，其压力具有各向异性，即$$ p_t > p_r $$，表示在不同方向上的压力差异。在F(T)引力的背景下，这种各向异性能够在星体中心达到最大，有助于形成紧凑的星体结构。尽管如此，文章指出，如果MCG在F(T)理论中缺失，这可能会对紧凑恒星的形成和稳定性产生影响。
研究结果对于理解已知恒星的物理特性具有重要意义，尽管这些特性在观测上仍然是个谜。作者们通过分析已知恒星的物理特性，提出了一个假设，即在某些情况下，排斥核力量可能在恒星核心起着关键作用，促进了紧凑恒星的形成。
总结来说，本文不仅提供了F(T)引力框架下各向异性紧凑恒星的理论模型，还探讨了EoS在确定恒星性质中的作用以及可能的新型EoS如MCG的应用。这一工作有助于拓展我们对宇宙中天体行为的理解，并可能为未来更深入的观测和实验提供理论指导。
Eur. Phys. J. C (2019) 79 :502 Page 3 of 12 502
as
α
μν
= e
α
i
∂
ν
e
i
μ
=−e
μ
i
∂
ν
e
α
i
. (3)
The torsion and con-torsion tensors are:
T
α
μν
=
α
νμ
−
α
μν
= e
α
i
∂
μ
e
i
ν
− ∂
ν
e
i
μ
. (4)
K
μν
α
=−
1
2
T
μν
α
− T
νμ
α
− T
μν
α
. (5)
The above two tensors given by Eqs. (3) and (4) are used to
construct a new tensor given by,
S
μν
α
=
1
2
K
μν
α
+ δ
μ
α
T
βν
β
− δ
ν
α
T
βμ
β
. (6)
The torsion scalar now can be expressed as:
T = T
α
μν
S
μν
α
. (7)
Analogous to modified gravitational action in f (R) theory
one can obtain another modified gravity f (T ) replacing R
by T ,as,
S =
d
4
x
1
16π
f (T ) + L
matter
e (8)
where, G = c = 1 and L
matter
is the matter Lagrangian. Vari-
ation of Eq. (8) with respect to tetrad field yield the following
set of equations of motion:
e
α
i
S
μν
α
∂
μ
Tf
TT
+ e
−1
∂
μ
(ee
α
i
S
μν
α
) f
T
+ e
μ
i
T
λ
μk
S
νk
λ
f
T
−
1
4
e
ν
i
f =−4π e
λ
i
T
ν
λ
, (9)
where T
μν
is the energy–momentum tensor of a particular
matter, whereas f
T
and f
TT
represent first and second deriva-
tives of f (T ) with respect to the torsion scalar T respec-
tively. For an anisotropic fluid the energy–momentum tensor
is described by,
T
ν
μ
= (ρ + p
t
)u
μ
u
ν
− p
t
δ
ν
μ
+ ( p
r
− p
t
)v
μ
v
ν
(10)
where u
μ
and v
μ
are the four velocity and radial four vec-
tors, p
r
and p
t
represents pressures in radial and transverse
directions.
In covariant derivative formalism, f (T ) field equations
can be recasted as,
ϒ
μν
f
TT
+
T
2
f
T
− f
T
g
μν
+ G
μν
f
T
=
1
16π
T
(m)
μν
(11)
where, ϒ
μν
= S
α
νμ
∇
α
T , while G
μν
is the Einstein tensor. In
the stream of GR as well as f (R) field equations, Eq.(9) can
be rewritten as,
G
μν
=
1
16π f
T
T
(T )
μν
+ T
(m)
μν
, (12)
where,
T
(T )
μν
=
1
16π
− ϒ
μν
f
TT
−
1
4
(T − ϒ f
TT
+ Rf
T
)g
μν
(13)
is a tensor incorporating corrections coming from torsion
scalar. It is evident that for a linear f (T ) i.e., f (T ) = T ,
Eq. (11) leads to GR equations.
3 Model of anisotropic compact stars in f (T ) gravity
with Finch–Skea geometry
We consider a spherically symmetric static space-time metric
given by,
ds
2
=−e
2ν(r )
dt
2
+ e
2λ(r)
dr
2
+r
2
d
2
2
(14)
where, ν(r) and λ(r) are the two unknown metric poten-
tials, and d
2
2
= dθ
2
+ si n
2
θdφ
2
represents the metric on
the 2-sphere in polar co-ordinates. We consider the energy
momentum tensor for an anisotropic compact object in 4-
dimensions as,
T
ab
= diag(−ρ, p
r
, p
t
, p
t
) (15)
where, ρ is the energy density, p
r
is the radial pressure and p
t
is the tangential pressure.The pressure anisotropy is defined
by Δ = p
t
− p
r
. The value of the anisotropy depends on
the metric potentials λ(r) and ν(r ). The tetrad matrix for the
metric in Eq. (10) is given by,
[e
i
μ
]=diag[e
ν
(r), e
λ
(r), r, rsin(θ )]. (16)
The determinant of the tetrad matrix is given by e =
det[e
i
μ
]=e
ν+λ
r
2
si n(θ), the torsion scalar and its derivative
in terms of r is given by,
T (r) =
2e
−2λ
r
2ν
+
1
r
(17)
T
(r) =
2e
−2λ
r
2ν
−
1
r
2
−
2ν
+
1
r
2λ
+
1
r
(18)
where the prime ()
denotes the derivative with respect to the
radial coordinates.
Now substituting the above tetrad field (16) and inserting
the values of torsion scalar and its derivative in Eq. (9), one
can find out the Einstein field equations explicitly in f (T )
gravity for an anisotropic fluid as,
4πρ =
f
4
−
f
T
2
T −
1
r
2
− 2
e
−2λ
r
ν
+ λ
, (19)
4π p
r
=
f
T
2
T −
1
r
2
−
f
4
, (20)
123
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