4 A. Das et al. / Nuclear Physics B 954 (2020) 114986
in the geometry of the space-time. We have extensively studied the shell region which contains
the stiff fluid and the total mass of the gravastar. We have explored different physical features,
viz., the pressure-density, energy, entropy, proper length of the shell region analytically along
with graphical representation unlike any other previous
work in this field. From the graphical
analysis, it is clear that the pressure or density of the shell increases gradually with the thickness
of the shell. This eventually suggests that the shell becomes denser at the exterior than the interior
boundary. We have found out the surface energy density and
surface pressure at the boundary of
the gravastar and arrived at the equation of state for boundary region. Interestingly, unlike the
previous work in gravastar we have found out the exterior solution in f(T ) gravity which is
turned out to be well-known Schwarzchild-de Sitter solution.
The outline of the present investigation is as follows: In Section 2 we provide the basic math-
ematical formalism of the f(T ) theory of gravity. Thereafter the field equations in f(T ) gravity
have been written assuming a specific form of gravitational Lagrangian, i.e., f(T ) in Section 3
whereas in Section 4, the solutions
of the field equations have been provided for different re-
gions, i.e., interior, exterior and the shell of the gravastar. In Section 5 we discuss the junction
conditions which are very important in connection to the three regions of the gravastar. Several
physical features of the model, viz., the energy, entropy, proper
length and equation of state have
been discussed in Section 6. Finally, in Sections 7 and 8 we respectively discuss on the status of
gravastar and pass some concluding remarks.
2. Basic mathematical formalism of the f
(
T
)
theory
The action of f(T ) theory [40,50,53,55]is taken as (with geometrized units G =c = 1)
S[e
i
μ
,φ
A
]=
d
4
xe
1
16π
f(T ) +L
matter
(φ
A
)
, (1)
where φ
A
represents the matter fields and f(T ) is an arbitrary analytic function of the torsion
scalar T . The torsion scalar is usually constructed from the torsion and contorsion tensor as
follows:
T = S
μν
σ
T
σ
μν
, (2)
where
T
σ
μν
=
˜
σ
μν
−
˜
σ
νμ
= e
i
σ
∂
μ
e
i
ν
−∂
ν
e
i
μ
, (3)
K
μν
σ
=−
1
2
T
μν
σ
−T
νμ
σ
−T
σ
μν
(4)
are torsion and contorsion tensor respectively and new components of tensor S
σ
μν
can be written
as
S
σ
μν
=
1
2
K
μν
σ
+δ
μ
σ
T
βν
β
−δ
ν
σ
T
βμ
β
. (5)
Here e
i
μ
are the tetrad fields by which we can define any metric as g
μν
= η
ij
e
i
μ
e
j
ν
with
η
ij
= diag(1, −1, −1, −1) and e
i
μ
e
i
ν
= δ
μ
ν
, e =
√
−g = det[e
i
μ
].
Variation of the action (1) with respect to the tetrad, yield the field equations of f(T ) gravity
[40,50,53,55]as
S
i
μν
f
TT
∂
μ
T + e
−1
∂
μ
(eS
i
μν
)f
T
−T
σ
μi
S
νμ
σ
f
T
+
1
4
e
i
ν
f = 4πT
ν
i
, (6)