#### Models of collapsing and expanding anisotropic gravitating source in f(R, T) theory of gravity

Eur. Phys. J. C
Models of collapsing and expanding anisotropic gravitating source in f ( R, T ) theory of gravity
G. Abbas 1
Riaz Ahmed 0 1
0 Department of Mathematics, University of the Central Punjab , Bahawalpur Campus, Lahore , Pakistan
1 Department of Mathematics, The Islamia University of Bahawalpur , Bahawalpur , Pakistan
In this paper, we have formulated the exact solutions of the non-static anisotropic gravitating source in f (R, T ) gravity which may lead to expansion and collapse. By assuming there to be no thermal conduction in gravitating source, we have determined parametric solutions in f (R, T ) gravity with a non-static spherical geometry filled using an anisotropic fluid. We have examined the ranges of the parameters for which the expansion scalar becomes negative and positive, leading to collapse and expansion, respectively. Further, using the definition of the mass function, the conditions for the trapped surface have been explored, and it has been investigated that there exists a single horizon in this case. The impact of the coupling parameter λ has been discussed in detail in both cases. For the various values of the coupling parameter λ, we have plotted the energy density, anisotropic pressure and anisotropy parameter in the cases of collapse and expansion. The physical significance of the graphs has been explained in detail.
1 Introduction
Gravity is fundamentally and obviously present in our daily
life, yet it still remains the force most difficult to understand
and interpret as an interaction compared to all the other
interactions. The gravitational force is most easily studied,
without any sophisticated and deep knowledge, and was the first
one to be tested experimentally due to its nature and the
apparatus needed [
1
].
A lot of work has been done [
2–4
] to explore the
dynamical aspects of stars without finding the exact solutions in
f (R, T ) gravity. The f (R, T ) theory of gravity has been
introduced by Harko et al. [
5
]; for the formulation of this
theory they have modified the Lagrangian of general
relativity as a general function of R (Ricci scalar) and T (trace of the
stress-energy tensor). They have formulated the equations of
motion by using the metric approach instead of the Palatini
approach. It has been investigated that the importance of T in
the theory may be prominently observed by the exotic form of
matter or phenomenological aspects of quantum gravity. The
f (R, T ) theory is an explicit generalization of f (R) theory,
in which many cosmological and astrophysical results have
been discussed so far [
7
]. But, there is a still room to study
some cosmological and astronomical processes in f (R, T )
theory which have not yet been studied. In our present work
the f (R, T ) model will be selected in the following form:
f (R, T ) = f1(R) + f2(T ).
(1)
Here, we take f1(R) = R and f2(T ) = 2λT where R is the
Ricci scalar, λ is some positive constant and T is the trace of
the stress-energy tensor, as already mentioned.
Recently, Zubair et al. [
6
] analyzed the dynamical
stability of a cylindrically symmetric collapsing object with
locally anisotropic fluid in f (R, T ) theory. Alves et al.
[
8
] investigated the existence of spacetime fluctuations in
f (R, T ) and f (R, T φ ) theories of gravity. The study of
collapse and dynamics of collisionless self-gravitating
systems has been carried out by the coupled collisions using
the Boltzmann and Poisson equations in f (R, T ) gravity
[
9,10
]. Chakraborty [11] has proved that the unknown
generalized function f (R, T ) can be evaluated in closed form
if this theory obeys the conservation of the stress-energy
tensor. Sharif and Zubair [
12
] derived the energy
conditions in f (R, T ) gravity, which correspond to the results
of f (R) gravity. Houndjo et al. [
13
] investigated some little
rip model in f (R, T ) gravity using the standard
reconstruction approach. Also, they remarked that the second law of
thermodynamics remains valid for the little rip model if the
temperature inside the horizon is the same that of the apparent
horizon.
Oppenheimer and Snyder [
14
] investigated the process of
collapse in 1939; they observed the contraction of an
inhomogeneous spherically symmetric dust cloud. This study
involves the exterior and interior regions as Schwarzschild
and Friedman like solutions, respectively. It has been
investigated in [
15
] that when massive stars collapse by the force of
their own gravity, the final fate of such gravitational collapse
is a white dwarf, neutron star, a black hole or a naked
singularity. Misner and Sharp [
16,17
] studied a perfect fluid in
spherically symmetric collapse and also some authors [
18–
32
] have discussed the phenomenon of gravitational collapse
using the dissipative and viscous fluid in general relativity.
It has been shown that one can go beyond the general
relativity; the more one has chances of admitting an
uncovered singularity [33]. In fact most modified gravity theories
go out of their way to avoid making any changes to
gravity near massive objects. The reason for this is that there is
no evidence of any odd gravitational behavior near massive
objects, so all of the modified gravity theories are designed
to match the standard gravity at short non-galactic distances.
There are also a large number of modified gravity theories
that attempt to deal with dark energy, i.e. they explain the
observations that lead us to think dark energy exists as a
modified gravity in a similar fashion to how modified
Newtonian dynamics is supposed to do away with the need for
dark matter. As far as it has proposed a modified gravity
theory that consistently accounts for all observational data,
it might work really well for galaxies as well as for
cosmology. That being said, the current dark energy and dark
matter model for the universe is not completely without
its own problems, so modified gravity remains a
possibility, albeit less likely on current evidence than the current
model.
2 f ( R, T ) theory of gravity
Harko et al. [
5,33
] proposed a generalization of the f (R)
theories, namely f (R, T ) gravity. It depends on a general
function of R (Ricci scalar) and T (the trace of the tensor
Tμν ) but in f (R) theories the action depends on just the Ricci
scalar R. According to the authors, the dependence of the
theory on T arises from quantum mechanical aspects which
are usually neglected in f (R) or GR theories, for instance.
The full action of f (R, T ) gravity is in [5] as follows:
S =
d4x √−g( f (R, T ) + Lm),
where Lm is the matter Lagrangian and g is the determinant
of the metric gab. Here, we choose Lm = ρ , and the above
action yields
1
Gμν = f R
( fT + 1)Tμ(mν) − ρgμν fT
+
f − R f R
2
gμν + (∇μ∇ν − gμν ) f R ,
where Tμ(mν) is the matter stress-energy tensor. The spacetime
in this case has the following form:
ds2
= W 2(t, r )dt 2 − X 2(t, r )dr 2
−Y 2(t, r )(dθ 2 + si n2θ dφ2).
The stress-energy tensor for the anisotropic source is
Tuv(m) = (ρ + P⊥)Vu Vv − P⊥guv + ( Pr − P⊥)Xu Xv. (5)
Here ρ, Vu , Xu , Pr , P⊥ are the energy density of matter,
comoving four-velocity of the source fluid, radial four-vector,
and radial and tangential pressures, respectively. Also, for the
given line element the quantities appearing in Tuv(m) must
satisfy
V u = W −1δ0u , V u Vu = 1,
X u = X −1δu ,
1
X u Xu = −1.
The volumetric rate of expansion
is
1
= W X Y
X˙ Y + 2X Y˙ ,
where · = ∂t and
= ∂r . The dimensionless anisotropy is
a =
Pr − P⊥ .
Pr
For the given line element and stress-energy tensor the set of
field equations is
(3)
(4)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(2)
G22 = f R
f
p⊥ + (ρ + P⊥) fT − 2 +
R f R
2
f¨R f R
+ W 2 − X 2
G00 = f R
f˙R
G01 = f R
G11 = f R
X 2
W 2
− f˙R
ρ +
f − R f R
2
X˙ 2Y˙
X − Y
f R
+ X 2
1 f R
W 2 − X 2
W f˙R X˙ f R
1 − W f˙R − X f˙R
Pr + (ρ + Pr ) fT −
f˙R
− Y W X 2 Y W˙ − 2W Y˙
f R
− Y W X 2
Y 2
W Y + 2Y W
f˙R
− W 2
f R
− X 2
W˙ X˙ Y˙
W − X − Y
W X Y
W − X − Y
X 2Y
X − Y
f − f R
2
The Misner and Sharp mass is [
17
]
Y
m(t, r ) = 2X 2W 2
,
1
H (t ) + (r − 2αr ) 1−2α
−2(1+α)
(4αλ2 − 18λ2 + αλ − 7λ − 1)
(1 + 2λ)(12λ2 + 6λ + 1)
1 4α
(1 + 2α) H (t ) + (r − 2αr ) 1−2α
,
Pr =
1
H (t ) + (r − 2αr ) 1−2α
2(−1+α)
(1 + 2α)(12λ2 − 2αλ + 6λ + 1)
(1 + 2λ)(12λ2 + 6λ + 1)
1 −2(1+α)
H (t ) + (r − 2αr ) 1−2α
2
(r − 2αr ) 1−2α (6λ2 + 5λ + 1)
(r − 2αr ) −11−+24αα
(1 + 2λ)(12λ2 + 6λ + 1)r 2(−1 + 2α)
1 −2(1+α)
H (t ) + (r − 2αr ) 1−2α
(1 + 2λ)(12λ2 + 6λ + 1)
(r − 2αr ) −11−+24αα
The auxiliary solution of Eq. (15) is
Y
W = Y˙ ,
X = Y α,
where α is arbitrary constant. Now using Eq. (7), we have
the following form for the expansion scalar:
= (2 + α)Y (α−1).
For α > −2 and α < −2, we have expanding and collapsing
solutions. With the help of Eq. (18) the mass function is given
by
2m(t, r )
Y
Y 2
− 1 = Y 2α − Y 2α .
When Y = Y 2α, there exists a trapped surface at Y = 2m,
hence Y = Y 2α is the trapping condition. The trapping
condition Y = Y 2α has the integral
Yt(r1ap−2α) = r (1 − 2α) + H (t ),
where H (t ) appears as the integration function. Using Eqs.
(18) and (21), we get the following explicit form of the source
variables:
(18)
(19)
(20)
(21)
(22)
(23)
1
× −4(α2λ + 6αλ2) − (r − 2αr ) 1−2α (12αλ + 6λ + 1) +
1
H (t ) + (r − 2αr ) 1−2α
(1 + 2λ)
−2
P⊥ =
3 Generating solutions
For various possible values of α, we now discuss the nature
of solutions.
5
3.1 Collapse solution α = − 2
When the value of the expansion scalar is negative, we have
gravitational collapse, so Eq. (19) implies that < 0, if
α < −2, for convenience we take α = − 25 and the condition
Y = Y 2α leads to Y = Y −5, which further is integrated to
1
Ytrap = (6r + h1(t )) 6 .
Here h1(t ) is the integration function. Using Eqs. (18), (25)
in Eqs. (14)–(17), with some tedious algebra, we obtain the
following explicit form of the matter variables:
ρ = −
+
5λ(1 + 4λ)
7
(6r + h1(t )) 6 (1 + 2λ)(12λ2 + 6λ + 1)
1
1
(6r + h1(t )) 3 (1 + 2λ)
5λ
Pr = − 24λ3 + 24λ2 + 8λ + 1 (6r + h1(t ))7/6
1
P⊥ = −
− (2λ + 1) √36r + h1(t )
5(1 + 2λ)
7
2 (6r + h1(t )) 6 (12λ2 + 6λ + 1)
Fig. 1 The variation of ρ with
respect to r for the various
values of λ and h1(t) = 1
(25)
(26)
(27)
(28)
(24)
(29)
(30)
The mass function given in Eq. (19) becomes
m1(r, t ) = 21 6 6r + h1(t ).
The dimensionless parameter a from Eq. (8) takes the form
a =
5
−5 − 10λ(1 + 2λ)+(6r + h1(t )) 6 (2 + 6λ + 12λ2)
5
10λ + 2 (6r + h1(t )) 6 (1 + 6λ + 12λ2)
.
In the above expressions h1(t ) is an arbitrary function of
time t ; by taking h1(t ) = 1, we have analyzed the results.
By choosing α = − 25 , we get < 0 and the energy density
remains a positive and decreasing function of r . The
graphical behavior of ρ with various values of λ is shown in Fig.
1. The radial pressure increases first and then decreases
continuously with respect to radius at different values of λ as
shown in Fig. 2, but the transverse pressure is decreasing
with respect to radius as shown in Fig. 3. To account for this
we can say the pressure is minimum when the value of λ is
minimum, as shown in Figs. 2 and 3. The maximum value
of the anisotropy occurs near the center of the sphere, so the
anisotropy parameter a attains a maximum value near the
center and its value decreases when r increases with various
values of λ; see Fig. 4.
Fig. 2 The variation of Pr with
respect to r for the various
values of λ and h1(t) = 1
Fig. 3 The variation of P
with respect to r for the va⊥rious
values of λ and h1(t) = 1
3
3.2 Expansion with α = 2
When the expansion scalar attains positive values, we have
an expanding solution, so Eq. (19), implies that > 0, if
α > −2, for convenience we take α = 23 and assume that
Y = (r 2 + r02)−1 + h2(t ),
(31)
where h2(t ) is an integration function and r0 > 0. For
simF
plicity, we take F (t, r ) = 1+h2(t )(r 2+r02) and Y = (r2+r02) ,
then using Eqs. (18), (31) with Eqs. (14)–(17), with
simplification, we obtain the following explicit form of the matter
variables:
14λ2 + 7λ + 1
ρ = (1 + 2λ)(12λ2 + 6λ + 1) −
2
8r 2(r 2 + r0 )
F 5
+
2
4(3r 2 − r0 )
F 4
4F
− (r 2 + r02) −
,
2
8r 2(r 2 + r0 )
F 5
λ(1 + 6λ)
Pr = (1 + 2λ)(12λ2 + 6λ + 1)
2
8r 2(r 2 + r0 )
F 5
+
4(3r 2 − r02)(r 2 + r0 )
2
F 4
+
4λ2 8r2(rF2+r02) − 4(3r 2 − r02)(r 2 + r02) + (r 2 + r02)F 2 + 4F 5
(1 + 2λ) 4(3r 2 − r02)(r 2 + r02)λ + 6((1r+2+2λr0)2F)5
+
r2(r2+r02)(2r02−9)(1+2λ)
F
+
+
(r 2 + r02)(4λ2 + 2λ + 1)
3r4−r02(1+r0 )
2
F
6F5
+ (r2+r02)2 +
r2(2r02)−9
F
(1 + 2λ) 4(3r 2 − r02)(r 2 + r02)λ + 6((1r+2+2λr0)2F)5
+
r2(r2+r02)(2r02−9)(1+2λ)
F
(8λ2 + 6λ + 1)
8r2(rF2+r02) + (r 2 + r02)F 2 + 4F 5
(1 + 2λ) 4(3r 2 − r02)(r 2 + r02)λ + 6((1r+2+2λr0)2F)5
+
r2(r2+r02)(2r02−9)(1+2λ)
F
1
m2(t, r ) = 2
F
In this case, when α = 23 we get > 0 and the energy
density remains finitely positive for the selected arbitrary time
profile t ; by fixing the values of α, the variation is given by
λ and under such restrictions the density ρ can be observed
to be a decreasing function, as shown in Fig. 5. The radial
pressure decreases continuously with respect to radius at
different values of λ, but the transverse pressure is increasing
with respect to radius as shown in Figs. 6 and 7, respectively.
To account for this we can say the pressure has reverse effects
to the effects observed in the previous case. It is observed that
the anisotropy is an increasing function of r but with respect
to the value of λ the anisotropy is going to be maximum as
the value of λ is being increased, as shown in Fig. 8.
(35)
Fig. 5 The variation of ρ with
respect to r for the various
values of λ and h2(t ) = 1
Fig. 6 The variation of Pr with
respect to r for the various
values of λ and h2(t ) = 1
Fig. 7 The variation of P⊥
with respect to r for the various
values of λ and h2(t ) = 1
4 Conclusion
Motivated by the f (R, T ) theory of gravity formulated by
Harko et al. [
5
], a lot of work related to cosmology and
stability of the dynamics of a collapsing stellar system has been
done in recent years [
34–48
]. This theory has a wide range
of cosmological and astrophysical applications in modern
physics. According to the available observation data, our
universe is in the phase of accelerating expansion, and to
explain the physical significance of this phenomenon a
number of modifications to GR have been proposed. The f (R, T )
theory of gravity is one that has been at the center of
attention of researchers in the current era; this type of theories
seems to provide a capacity of working successfully for dark
matter. The conformal relation of f (R, T ) to GR with a
selfinteracting scalar field has been discussed by Zubair et al.
[
49
].
Here, we have developed the generating solutions to
collapse and expansion of fluid sphere in f(R,T) theory of
gravity. For the interior matter distribution, the collapse solution
yields a unique trapped surface. It has been investigated that
during gravitational contraction, the phase transition would
occur for the massive stellar system, for instance most of
the condensed matter configuration transits to a π -meson
condensed state. The gravitational collapse is a highly
dissipative process and a great amount of heat energy is released
during gravitational collapse according to Herrera et al. [
50
].
In order to model the inhomogeneous cosmological solutions
Collins [
51
] has explored the non-static expanding solutions.
Also, the inclusion of anisotropic stress in the fluid source is
very important, and the influence of the non-zero anisotropy
parameter δa on the late time evaluation of the universe with
non-homogeneous background has been explored by
Barrow et al. [
52
]. Due to the valid selection of R(t, r ) and
α, one can obtain an anisotropy interconnection which
collapses or expands as studied by Glass [
53
]. In this paper,
we extend the work of Glass [
53
] to the f (R, T ) theory of
gravity.
In this paper interior solutions for anisotropic fluids have
been discussed in detail, which are being used in
modeling of anisotropic stars in the context of the modified theory
of gravity based on f (R, T ). Using the auxiliary form of
the metric functions, we have determined the trapping
conditions for a fluid sphere in f (R, T ) gravity. The resulting
solutions have been classified as collapsing and expanding,
depending on the nature of the scalar expansion. The matter
density, radial and transverse pressures, anisotropy
parameter and mass function have been calculated in the context of
the f (R, T ) theory of gravity. For the collapse solution when
α = − 25 the density decreases as the value of λ increases,
as shown in Fig. 1. The radial pressure Pr and the matter
density ρ have maximum values at center and they decrease
from center to the surface of star. It has been observed that
the anisotropy will be directed outward when Pr > P⊥; this
gives a > 0 as observed graphically in Figs. 4 and 8. It is
found in Fig. 4 that the anisotropy decreases with the increase
in radius.
Further, the expansion of the gravitating source would
occur when α = 23 , and , the expansion scalar, is positive.
In this case the matter density decreases as shown in Fig. 5.
The radial/transverse pressures and the anisotropy parameter
with various values of λ have a reverse behavior as compared
to the case of gravitational collapse.
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