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Eur. Phys. J. C (2020) 80:175 Page 3 of 15 175
surface redshift, has been discussed in Sect. 5.4. Finally,
in Sect. 6, we have provided an overall conclusion of
our detailed study from different aspect of strange star
model.
2 Mathematics behind f ( R, T ) gravity
Following Harko [47], we can represent action in f (R, T )
theory as
S =
d
4
x£
m
√
−g +
1
16π
d
4
xf(R, T )
√
−g. (1)
Here the arbitrary function f (R, T ) contains the Ricci
scalar R as well as T , the trace of the energy momentum
tensor, £
m
is the matter Lagrangian density which represents
the possibility of non-minimal coupling between matter and
geometry, g is the determinant of g
μν
metric (c = G = 1).
Varying the action (1) w.r.t. g
μν
metric, we can derive
modified EFEs for f (R, T ) gravity as
f
R
(R, T )R
μν
−
1
2
g
μν
f (R, T )
+ f
R
(R, T )(g
μν
−∇
μ
∇
ν
)
= 8πT
μν
− T
μν
f
T
(R, T ) − Θ
μν
f
T
(R, T ), (2)
where f
T
(R, T ) =
∂ f (R,T )
∂T
, f
R
(R, T ) =
∂ f (R,T )
∂ R
, ≡
∂
μ
(
√
−gg
μν
∂
ν
)
√
−g
, R
μν
represents the Ricci tensor, ∇
μ
is the
covariant derivative w.r.t. the symmetry connected to g
μν
,
Θ
μν
= g
αβ
δT
αβ
/δg
μν
and T
μν
= g
μν
£
m
− 2
∂£
m
∂g
μν
repre-
sents the stress-energy tensor.
From Eq. (2), the covariant divergence becomes [108]
∇
μ
T
μν
=
f
T
(R, T )
8π − f
T
(R, T )
[(Θ
μν
+ T
μν
)∇
μ
ln f
T
(R, T )
−
1
2
g
μν
∇
μ
T +∇
μ
Θ
μν
]. (3)
Equation (3) states that in f (R, T ) theory of grav-
ity, energy-momentum tensor is not conserved whereas it
remains conserved in general relativity.
For a perfect anisotropic fluid, the energy-momentum ten-
sor takes the following form
T
μν
= (ρ + p
t
)u
μ
u
ν
− p
t
g
μν
+ ( p
r
− p
t
)v
μ
v
ν
, (4)
with u
μ
∇
ν
u
μ
= 0 and u
μ
u
μ
= 1. Here ρ(r ), p
r
(r), p
t
(r),
u
μ
, v
μ
stand for the energy density, radial pressure and tan-
gential pressure, four-velocity and radial four-vector respec-
tively for a static fluid source. Besides these, we have another
condition Θ
μν
=−2T
μν
− pg
μν
.
Following Harko et al. [47], f (R, T ) takes the form
f (R, T ) = R + 2χ T. (5)
Here χ is a constant arises due to the coupling between
matter and geometry in modified gravity.
Substituting Eq. (5)inEq.(2), we get
G
μν
= 8πT
μν
+ χ Tg
μν
+ 2χ(T
μν
+ pg
μν
). (6)
where G
μν
is known as the Einstein tensor. Now, substituting
χ = 0intheEq.(6), we get different results which represent
the results of GR.
Combination of Eqs. (3) and (5) gives the following result
(8π + 2χ)∇
μ
T
μν
=−2χ
∇
μ
( pg
μν
) +
1
2
g
μν
∇
μ
T
. (7)
Incorporation of χ = 0inEq.(7) shows the invariance of
energy-momentum tensor in general relativity.
3 Solution of Einstein’s field equations
To study a strange star, we consider the static, spherically
symmetric stellar configuration, for which the line element
of the interior space-time, can be described by the following
metric:
ds
2
= e
ν(r )
dt
2
− e
λ(r)
dr
2
−r
2
(dθ
2
+ si n
2
θdφ
2
). (8)
Here the metric potentials λ(r ) and ν(r) are chosen as
λ(r) = ln(1 + ar
2
+ br
4
) and ν(r) = Br
2
+ 2lnC, known
as Tolman-Kuchowicz metric potentials [1,2]. The constants
a, b, B and C can be determined using matching conditions.
The non-zero components of energy-momentum tensor can
be written as T
0
0
= ρ(r), T
1
1
=−p
r
(r) and T
2
2
= T
3
3
=
−p
t
(r).
For an anisotropic spherically symmetric uncharged stel-
lar system, combining Eqs. (4), (6) and (8), we get the EFE
as follows
e
−λ
r
2
(−1 + e
λ
+r λ
)
= 8πρ + 2χ
2ρ −
p
r
+ 2 p
t
3
= 8πρ
ef f
, (9)
e
−λ
r
2
(1 − e
λ
+r ν
)
= 8π p
r
− 2χ
ρ −
4 p
r
+ 2 p
t
3
= 8π p
ef f
r
, (10)
e
−λ
4r
2(ν
− λ
) + (2ν
+ ν
2
− λ
ν
)r
= 8π p
t
− 2χ
ρ −
p
r
3
−
5 p
t
3
= 8π p
ef f
t
. (11)
Here expresses the differentiation of the respective param-
eters w.r.t. r.
123