# Dynamical conditions and causal transport of dissipative spherical collapse
in $f(R,T)$ gravity

Sarbari Guha and Uttaran Ghosh  Department of Physics, St. Xavier’s College
(Autonomous), Kolkata 700016, India

###### Abstract

In this paper, we have investigated the non-adiabatic spherical gravitational
collapse in the framework of the $f(R,T)$ theory of gravity with a locally
anisotropic fluid that undergoes dissipation in the form of heat flux, free-
streaming radiation, and shearing viscosity. The dynamical equations are
analyzed in detail, both in the Newtonian and post-Newtonian regimes. Finally
we couple the dynamical equations to the full causal transport equation in the
context of Israel-Stewart theory of dissipative systems. This yields us a
better understanding of the collapse dynamics and may be connected to various
astrophysical consequences.

##  I Introduction

The final outcome of stellar evolution depends not only on the size of the
object undergoing collapse Chandra  but also on other physical parameters 8 .
On account of the high energy dissipation during collapse, more massive stars
tend to be more unstable because of massive radiation transport and rapid loss
of nuclear fuel 8 ; 9 . The effects of dissipation and slight change in
isotropy alters the subsequent evolution of the collapsing system
considerably. Several researchers 5 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ;
Santos ; OSK ; OPS  have studied extensively the phenomenon of gravitational
collapse of fluids in presence or absence of various conditions like
anisotropy, radiation, dissipation, expansion, shear, along with the
conditions of dynamical instability and causal transport phenomenon occurring
during gravitational collapse, and the list is even longer. The subject of
spherical gravitational collapse have always been at the heart of
astrophysical investigations on account of the symmetry of the matter
distribution, leading to simplification of the field equations, and also due
to the fact that some classes of realistic gravitational collapse can be
modelled as spherical collapse with only small deviations. The analysis is
also simplified due to the absence of gravitational waves. Studies of
radiating spherical collapse in presence of heat flow GGM ; MGG1 ,
relaxational effects in stellar heat transport during collapse GMM  and
temperature profile of such a collapse within the framework of causal
thermodynamics have been conducted MGG2 . Some authors GB ; PHDMS  have also
studied the dissipative collapse of charged configurations.

The limitations of General Relativity (GR) on large scales led to the
investigation of astrophysical processes in modified theories of gravity,
e.g., theories which provide improved models of the accelerating universe.
Among these modified theories (some of the review articles are listed in 17a
), the $f(R)$ gravity presents a very elementary modification by including
higher order curvature terms to incorporate the dark energy components, as
well as the inflationary phase DEinfl . Not only does $f(R)$ gravity reproduce
the $\Lambda$CDM epoch, or is able to mimic the cosmological constant at the
present epoch, but it can also unify the entire evolution history of the
universe evolution . The null dust non-static exact solutions in $f(R)$
gravity constrained by constant curvature describing anti de-Sitter background
evolution, was studied by Ghosh and Maharaj 18 . Cembranos et al. 19  studied
the evolution of gravitating sources in the presence of dust fluid in a
general $f(R)$ model with a view to determine the possible constraints on such
models. Goswami et. al. studied the collapse of massive stars in $f(R)$
gravity and showed that the extra matching conditions arising in modified
gravity impose strong constraints on the stellar structure and thermodynamic
properties Goswami . Chakrabarti and Banerjee investigated the collapse of a
perfect fluid source described by Lemaitre-Tolman-Bondi type geometry CB .
Sharif and Yousaf studied the dynamical instability of charged spherical
collapse in expansion-free condition SY2 , and the stability of dissipative
charged spherical collapse in the CDTT-$f(R)$ model SY3 .

In 2011, Harko et al. 22  introduced the $f(R,T)$ theory of gravity, based on
the non-minimal matter-to-geometry coupling considered by earlier workers 22a
; 22b ; 22c . The action in $f(R,T)$ gravity includes an arbitrary function of
the Ricciscalar $R$ and the trace $T$ of the energy-momentum tensor. The
inclusion of $T$ takes care of quantum effects or the existence of exotic
matter. The higher-curvature terms in this theory can address the flatness
problem which can be seen in the galactic rotation curves. A combination of a
term quadratic in $R$ (as in the Starobinsky model Starobinsky ) along with
linear terms in $R$ and $T$ as used in SZ2 , provides a useful $f(R,T)$
function. It was realized alvarenga2013  that a strong coupling of the
curvature $R$ with the trace $T$ leads to the violation of the conventional
equation of continuity, as a result of which, an extra force arises in the
geodesic equation and test particles do not follow a geodesic motion. However,
this problem can be resolved by choosing a suitable $f(R,T)$ function. These
features of $f(R,T)$ gravity make it _a candidate suitable for_ the
investigation of various astrophysical phenomena within the context of this
theory. The astrophysical, cosmological and thermodynamic implications of
$f(R,T)$ gravity have been widely studied by several groups alvarenga2013 ;
barrientos2014 ; Faraoni2009 ; 23 ; 24 ; SZ1 . Houndjo Houndjo  demonstrated a
possible unification of the accelerated expansion phase with the matter
dominated era in cosmic evolution by choosing a viable $f(R,T)$ function
subject to appropriate constraints, which ensure that either no ghost state
arises where dark energy becomes responsible for the accelerated expansion, or
no tachyon arises. Subsequently Odintsov and Sáez-Gómez OSG  proposed the
$f(R,T,R_{\mu\nu}T^{\mu\nu})$ theory as an extension of $f(R,T)$ gravity. They
derived the general FRLW field equations in the presence of the
$R_{\mu\nu}T^{\mu\nu}$ coupling terms and studied several cosmological
solutions, thereby establishing that the matter sector behaves differently
from that in GR, because the continuity equations are different. In this
theory too, the problem can be dealt with by considering an appropriate
$f(R,T,R_{\mu\nu}T^{\mu\nu})$ function, so that one can assume the usual
evolution for a perfect fluid matter even in presence of the extra degree of
freedom.

Sharif and Yousaf SY  considered the dynamical analysis in $f(R,T)$ gravity
with non-null expansion scalar at Newtonian and post-Newtonian eras and
examined the role played by matter variables on stellar structure. Others
investigated the dynamical instability of spherical stars in locally
anisotropic environment in $f(R,T)$ theory under various situations like
expansion-free, shear-free conditions and from other perspectives NZ1 ; NZ2 ;
NZ3 . Yousaf et al. YBB  studied the factors leading to irregularities in the
case of a spherically symmetric self-gravitating star consisting of
dissipative matter and radiation in $f(R,T)$ gravity. In another work
InflModGrav  they discussed the advantages of working with modified gravity.
If the curvature is low, one can observe accelerated expansion of the
universe, while for high value of curvature, the singularities can be made
smoother. They YBBG  also studied the effect of charge on spherically
symmetric gravitational vacuum stars (gravastars) in $f(R,T)$ gravity. Yousaf
Y2020  considered a charged cylindrically symmetric gravastar with perfect
fluid matter content in $f(R,T)$ gravity. It was found that for an increase in
the amount of charge in the gravastar, the proper length of its middle thin
shell reduces. Also, the energy content in the gravastar reduces for a rise in
the amount of electric charge. The entropy of the system increases if the
charge decreases or with an increase in the thickness of the shell. Bhatti et
al. BYR  considered a spherically symmetric model of a gravastar and studied
some of its characteristics in $f(R,G)$ theory, where $G$ is the Gauss-Bonnet
term.

The physical behavior, stability and validity of energy conditions of compact
objects under the effect of charge was examined in SS , and the collapsing and
expanding solutions of anisotropic charged cylinder was determined in SS1 .
Some authors have studied the higher-dimensional perfect fluid collapse in
$f(R,T)$ theory hdimPfcoll . The effects of charge on the stability range of
anisotropic spherical stellar model in the framework of $f(R,T)$ gravity have
been studied in SW . In the paper AA1  the exact solutions for non-static
anisotropic self-gravitating source in $f(R,T)$ were derived, and the nature
of singularity and trapped surfaces during the collapse of a charged perfect
fluid was determined in AA . They also studied the thermodynamic aspects of a
viscous dissipative collapse in $f(R,T)$ gravity AA2 . Recently the complexity
factor of a self-gravitating system was determined through the orthogonal
splitting of the Riemann tensor and the behaviour of the complexity for a
cylindrically symmetric system in $f(R,T)$ theory of gravity was studied ZA .

A spherically symmetric perfect fluid model for a gravastar in $f(R,T,Q)$
theory was studied in YBA , where $Q=R_{\mu\nu}T^{\mu\nu}$. The paper YBF
studied the instability criteria by the method of dynamical analysis for a
cylindrically symmetric, anisotropic radiating matter fluid in $f(R,T,Q)$
theory. The linear perturbation scheme was invoked under the assumption that
the fluid is initially in hydrostatic equilibrium, with dependence only on the
radial coordinate, but later it begins to undergo radial oscillations. The
instability ranges were calculated in the Newtonian and post-Newtonian
regimes, utilising the Harrison-Wheeler equation of state. The same authors
YBF2  discussed the instability ranges of compact stars by the method of
dynamical analysis in $f(R,T,Q)$ theory and adopting the linear perturbative
scheme. The energy-momentum tensor was that of an anisotropic perfect fluid,
and the interior metric was the most general spherically symmetric with non-
zero expansion scalar. Yousaf, Bamba, Bhatti and Farwa YBBF , studied the
dynamical evolution of a spherical anisotropic fluid with non-vanishing
expansion scalar using perturbative scheme, and its instability ranges
calculated at the Newtonian and post-Newtonian regimes.

Gravitational collapse ensues only when a self-gravitating object is perturbed
from its initial hydrostatic equilibrium condition, when the internal pressure
cannot counteract the gravitational contraction. This leads to the formation
of compact objects like white dwarfs, neutron stars, black holes, etc. The
effect of various factors in the formation of these objects can hardly be
undermined. In this context, Yousaf Yousaf2018  explored the behavior of
scalar variables associated with the shearing viscous dissipative anisotropic
spherical stars in the framework of $f(G,T)$ theory. A correspondence was
drawn between the metric scale factors, tidal forces and structure parameters
of the collapsing configuration. As the technique of orthogonal breaking of
the Riemann tensor helps us to study the reasons behind the emergence of
inhomogeneities in the initially regular spheres as the collapse progresses,
this technique was used to determine the modified structure scalars, and the
role of these invariants in the evolutionary properties of radiating spheres,
was explored. The analysis showed that the evolutionary phases of the
spherical interiors can be well understood in terms of extended versions of
scalar variables. The shear evolution equation (SEE), the expansion evolution
equation (EEE) — also known as the Raychaudhuri equation, and the Weyl
differential equation (WDE) can be expressed with the help of these structure
scalars. For constant $G$ and $T$, in case of a dust ball, the cause of
inhomogeneous energy density in the compact object which is initally regular,
can be expressed through the WDE using one of the structure scalars
Yousaf2019a . Depending on the mass of the collapsing object Yousaf2019b ;
Yousaf2021 , the collapse can result in formation of structures like black
holes, neutron stars or white dwarfs.

It was Misner and Sharp MS  and Misner Misner , who provided a complete
description of the dynamical equations for adiabatic and dissipative collapse.
In fact, all the physical parameters associated with the collapsing matter,
should obey the transport equations of causal thermodynamics. Earlier Tolman
Tolman  had suggested that the heat energy associated with the collapsing
matter should have an inertia associated with it, and this inertial term
appears in the transport equations derived by Eckart and Landau-Lifshitz
Eckart ; Landau . However the Landau-Eckart prescription could only be solved
by a hyperbolic theory involving second-order entropy terms, as done by
Muller, Israel and Stewart Muller ; Israel1 ; Israel2 ; Israel3 . The
dynamical equation can be recasted into the form of Newton’s Second Law of
Motion. The transport equation can be expressed in terms of the acceleration
of the collapsing system, and after we couple the transport equation with the
dynamical equation, both the inertial mass, and the gravitational mass are
found to get reduced by the same factor, in agreement with the principle of
equivalence. The thermal dissipative effects reduce the effective inertial
mass term, and the inertial mass appears to lag behind the collapsing matter
as the collapse progresses HerreraInertia .

Herrera and Santos 12  discussed the importance of dissipative processes in
gravitational collapse and extended the Misner dynamical equations to include
dissipation in the form of radial heat flow along with pure radiation in order
to couple the dynamical equations with the transport equations. Herrera et al.
PHDMS  applied the same prescription to charged, dissipative, spherically
symmetric system with shear viscosity. The paper HPFT2009  examined
spherically symmetric collapse with heat flow, radiation, shear and bulk
viscosity. They applied a full causal treatment to all dissipative variables,
and further included the viscous or heat coupling coefficients, which are
known to influence the evolution of neutron stars Maartens . Plane symmetric
gravitational collapse in the Misner and Sharp formalism was presented for a
fluid undergoing dissipation in the form of heat flow, radiation, shear and
bulk viscosity, and the dynamical equations were coupled with the causal
transport equations srGRG10 . The gravitational collapse of cylindrical
viscous heat conducting anisotropic fluid was investigated in CC , and the
transport equation was derived to determine its effect on the collapsing
system. The effect of dissipation on the dynamics of charged bulk viscous
collapsing cylindrical source with heat flux was studied in SA . Hydrodynamic
and thermodynamic analysis of gravitational collapse of a locally anisotropic
shear-free fluid with dissipative flows was presented in PS . The exact model
satisfied all the energy conditions throughout the interior of the star and
for the entire duration of collapse process, and provided a physically viable
temperature profile in the context of causal thermodynamics.

Spherically symmetric collapse of isotropic non-viscous fluid satisfying the
energy conditions together with matter violating the null energy conditions
was investigated in the framework of $f(R)$ theory and effect of such matter
on the coupling of dynamical and transport equations, was studied in SK .
Dissipative gravitational collapse in non-static cylindrical symmetric
geometry was examined in AbbasNazar2018  by using Misner-Sharp formalism in
the framework of metric $f(R)$ gravity, and the dynamical equation was coupled
with the full causal transportation equations in the context of Israel-Stewart
formalism. Static configurations and radial stability of compact stars have
also been studied within the context of $f(R,T)$ gravity PJRA . Recently, the
dynamics of non-adiabatic charged spherical gravitational collapse of an
anisotropic fluid with heat flux was analyzed in the framework of $f(R,T)$
gravity using the Misner-Sharp approach AA3 . However, the authors did not
consider the effect of shear and free-streaming radiation, nor were the
instability conditions analysed.

In this paper, we intend to explore the dynamical instability and the causal
heat transport of an uncharged anisotropic fluid with shear viscosity in
presence of free-streaming radiation, undergoing spherically symmetric
collapse in $f(R,T)$ gravity. The manuscript is organized as follows: In
Section II, we present the $f(R,T)$ formalism, followed by the choice of the
interior metric and the corresponding field equations in Section III. The
dynamical equations are laid down in Section IV, and the perturbation analysis
along with the Newtonian and post-Newtonian approximation are presented in
Section V. The transport equations are derived in Section VI. Finally, the
summary of our findings and the final outlook are presented in Section VII.

##  II The $f(R,T)$ formalism

The modified Einstein-Hilbert action in $f(R,T)$ gravity is given by 22

$S=\int\leavevmode\nobreak\ d^{4}x\sqrt{-g}\Bigg{(}\frac{f(R,T)}{16\pi
G}+\mathcal{L}_{m}\Bigg{)},$ (1)

where $g$ is the determinant of the 4D metric $g_{\mu\nu}$, $R$ is the
Ricciscalar, $T$ is the trace of the stress-energy tensor $T_{\mu\nu}$, $G$ is
the Newton’s gravitational constant, $\mathcal{L}_{m}$ denotes the Lagrangian
density for matter fields, and $f(R,T)$ is a suitable analytic function of $R$
and $T$ that describes the gravitational interaction. The corresponding field
equations of $f(R,T)$ gravity has the form:

$R_{\mu\nu}f_{R}(R,T)-\frac{1}{2}g_{\mu\nu}f(R,T)+(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu})f_{R}(R,T)=8\pi
GT^{m}_{\mu\nu}-f_{T}(R,T)\left(T^{m}_{\mu\nu}+\Theta_{\mu\nu}\right),$ (2)

where $f_{R}(R,T)$ and $f_{T}(R,T)$ denote the derivatives of $f(R,T)$ with
respect to $R$ and $T$ respectively,
${\Box}=g^{\mu\nu}{\nabla}_{\mu}{\nabla}_{\nu}$ is the d’Alembertian operator,
${\nabla}_{\mu}$ is the covariant derivative associated with the Levi-Civita
connection of the metric tensor, and $\Theta_{{\mu}{\nu}}$ is defined by

$\Theta_{{\mu}{\nu}}=g^{\alpha{\beta}}\frac{{\delta}T_{{\alpha}{\beta}}}{{\delta}g^{\mu{\nu}}}=-2T^{m}_{{\mu}{\nu}}+g_{\mu\nu}\mathcal{L}_{m}-2g^{\alpha\beta}\frac{\partial^{2}\mathcal{L}_{m}}{{\partial}g^{\mu\nu}{\partial}g^{\alpha\beta}}.$
(3)

The covariant divergence of eq. (2) gives barrientos2014

$\displaystyle\nabla^{\mu}T_{\mu\nu}^{m}$ $\displaystyle=$
$\displaystyle\frac{f_{T}(R,T)}{8\pi
G-f_{T}(R,T)}\left[(T_{\mu\nu}^{m}+\Theta_{\mu\nu})\nabla^{\mu}\ln
f_{T}(R,T)+\nabla^{\mu}\Theta_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\nabla^{\mu}T\right].$
(4)

Thus the conventional energy-momentum tensor is not conserved, unlike in GR,
which means that particles do not follow _geodesic paths_ in pure
gravitational fields in the framework of $f(R,T)$ gravity.

While writing out the field equations, various types of $\mathcal{L}_{m}$ can
be chosen to model different types of matter. For this anisotropic fluid, we
choose $\mathcal{L}_{m}=-\rho$, where $\rho$ is the energy density of the
matter distribution, so that (3) reduces to the form

$\Theta_{\mu\nu}=-2T^{m}_{\mu\nu}-\rho g_{\mu\nu}.$

Consequently, (2) becomes

$\displaystyle
R_{{\mu}{\nu}}f_{R}(R,T)-\frac{1}{2}g_{{\mu}{\nu}}f(R,T)+(g_{{\mu}{\nu}}{\Box}-{\nabla}_{\mu}{\nabla}_{\nu})f_{R}(R,T)=8{\pi}GT_{{\mu}{\nu}}^{m}+f_{T}(R,T)\left(T^{m}_{\mu\nu}+\rho
g_{{\mu}{\nu}}\right).$ (5)

Faraoni Faraoni2009  showed that both ${\mathcal{L}}_{m}=p$ and
${\mathcal{L}}_{m}=-\rho$, where $p$ is the isotropic pressure of matter, are
equivalent when the fluid couples minimally to gravity. However, this freedom
of choice is lost when the pressure is not isotropic.

Equation (5) can be written in the form of an effective field equation as
follows:

$G_{\mu\nu}=R_{{\mu}{\nu}}-\frac{1}{2}Rg_{{\mu}{\nu}}=8{\pi}G_{eff}T_{{\mu}{\nu}}^{m}+T^{D}_{{\mu}{\nu}},$
(6)

where

$G_{eff}=\frac{1}{f_{R}(R,T)}\left(G+\frac{f_{T}(R,T)}{8\pi}\right)$

is the effective gravitational coupling in $f(R,T)$ gravity, and

$T^{D}_{{\mu}{\nu}}=\frac{1}{f_{R}(R,T)}\left[\rho
g_{\mu\nu}f_{T}(R,T)+\frac{1}{2}(f(R,T)-Rf_{R}(R,T))g_{\mu\nu}+({\nabla}_{\mu}{\nabla}_{\nu}-g_{{\mu}{\nu}}{\Box})f_{R}(R,T)\right]$
(7)

represents the contribution to the energy-momentum tensor from the matter-
curvature coupling and also includes non-equilibrium description of the field
equations.

The full form of eq. (6) in $f(R,T)$ gravity is then

$G_{\mu\nu}=T_{\mu\nu}^{eff}=\frac{1}{f_{R}}\left[(8\pi
G+f_{T})T^{m}_{\mu\nu}+\rho
g_{\mu\nu}f_{T}+\frac{1}{2}(f-Rf_{R})g_{\mu\nu}+(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box)f_{R}\right],$
(8)

where $T_{\mu\nu}^{eff}$ is the effective energy-momentum tensor incorporating
gravitational effects. Our equation (8) follows the convention adopted by
several authors beginning with Harko and his collaborators (see for example 22
; 22b ; alvarenga2013 ; barrientos2014 ; SY ; NZ1 ; NZ2 ; NZ3 ; SW ). However,
instead of the term $(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box)f_{R}(R,T)$,
Yousaf et. al YBB  considered the term
$\left({\nabla}_{\mu}{\nabla}_{\nu}+g_{\mu\nu}{\Box}\right)f_{R}(R,T)$ in the
$f(R,T)$ field equations (2) in their paper. This led to a difference in the
various components of the field equations obtained by them. Here we have
retained the expression given in 22 .

In the subsequent analysis we will assume $8\pi G=1$ for the sake of
simplicity of presentation.

##  III Interior metric and Field equations for $f(R,T)$ gravity

The physical system consists of a timelike 3D bounding surface, $\Sigma$,
which divides the 4D spacetime into an interior and an exterior portion,
denoted by $V^{-}$ and $V^{+}$ respectively. We assume that the region inside
$\Sigma$ is modeled by the general non-static non-rotating spherically
symmetric metric

$ds^{2}_{-}=-A^{2}dt^{2}+B^{2}dr^{2}+C^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (9)

where $A,B$ and $C$ are in general functions of both $t$ and $r$. The interior
spacetime is filled with shearing viscous, locally anisotropic and radiating
fluid described by the energy-momentum tensor $T^{m}_{\mu\nu}$:

$T^{m}_{\mu\nu}=(\rho+p_{\perp})V_{\mu}V_{\nu}+p_{\perp}g_{\mu\nu}+(p_{r}-p_{\perp})\chi_{\mu}\chi_{\nu}+q_{\mu}V_{\nu}+V_{\mu}q_{\nu}+{\epsilon}l_{\mu}l_{\nu}-2{\eta}{\sigma}_{\mu\nu},$
(10)

where $\rho$ is the energy density, $p_{\perp}$ the tangential pressure,
$p_{r}$ the radial pressure, $q^{\mu}$ the heat flux, $\epsilon$ is the
radiation density, $V^{\mu}$ is the time-like four velocity of the fluid,
$\chi^{\mu}$ is a unit spatial vector along the radial direction, $l^{\mu}$ is
a null 4-vector, and $\eta$ $(>0)$ is the coefficient of shearing viscosity,
respectively, such that

$V^{\mu}V_{\mu}=-1,\;\;V^{\mu}q_{\mu}=0,\;\;\chi^{\mu}\chi_{\mu}=1,\;\;\chi^{\mu}V_{\mu}=0,\;\;l^{\mu}V_{\mu}=-1,\;\;l^{\mu}l_{\mu}=0.$
(11)

In comoving coordinates, we have

$V^{\mu}=A^{-1}\delta^{\mu}_{0},\quad\chi^{\mu}=B^{-1}\delta^{\mu}_{1},\quad
l^{\mu}=A^{-1}\delta^{\mu}_{0}+B^{-1}\delta^{\mu}_{1},\quad
q^{\mu}=B^{-1}q{\delta}^{{\mu}}_{1}=B^{-1}(0,\leavevmode\nobreak\
q,\leavevmode\nobreak\ 0,\leavevmode\nobreak\ 0),$ (12)

where $q=q(r,t)$ and $\epsilon=\epsilon(r,t)$. The acceleration vector
$a^{\mu}$, the expansion scalar $\Theta_{1}$ and the magnitude of the shear
scalar $\sigma$ are given by 11 ; TRM :

$\displaystyle a^{\mu}$ $\displaystyle=$
$\displaystyle\left(0,\frac{A^{\prime}}{AB^{2}},0,0\right),$ (13)
$\displaystyle\Theta_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{A}\left(\frac{\dot{B}}{B}+2\frac{\dot{C}}{C}\right),$
(14) $\displaystyle\sigma$ $\displaystyle=$
$\displaystyle-\frac{1}{3A}\left(\frac{\dot{B}}{B}-\frac{\dot{C}}{C}\right),$
(15)

where primes and dots on the metric functions denote partial derivatives with
respect to $r$ and $t$ respectively.

The components of the Einstein tensor for the interior metric are given by

$\displaystyle G_{00}$ $\displaystyle=$ $\displaystyle
A^{2}\left[\frac{2}{A^{2}}\frac{\dot{B}}{B}\frac{\dot{C}}{C}+\frac{\dot{C}^{2}}{A^{2}C^{2}}+\frac{1}{C^{2}}-\frac{1}{B^{2}}\left(\frac{2C^{\prime\prime}}{C}+\frac{{C^{\prime}}^{2}}{C^{2}}-\frac{2B^{\prime}}{B}\frac{C^{\prime}}{C}\right)\right],$
(16) $\displaystyle G_{11}$ $\displaystyle=$ $\displaystyle
B^{2}\left[\frac{1}{A^{2}}\left(-\frac{2\ddot{C}}{C}-\frac{\dot{C}^{2}}{C^{2}}+\frac{2\dot{A}}{A}\frac{\dot{C}}{C}\right)+\frac{1}{B^{2}}\left(\frac{{C^{\prime}}^{2}}{C^{2}}+\frac{2A^{\prime}}{A}\frac{C^{\prime}}{C}\right)-\frac{1}{C^{2}}\right],$
(17) $\displaystyle G_{22}$ $\displaystyle=$ $\displaystyle
C^{2}\left[-\frac{1}{A^{2}}\left(\frac{\ddot{B}}{B}-\frac{\dot{A}}{A}\frac{\dot{B}}{B}+\frac{\dot{B}}{B}\frac{\dot{C}}{C}-\frac{\dot{A}}{A}\frac{\dot{C}}{C}+\frac{\ddot{C}}{C}\right)+\frac{1}{B^{2}}\left(\frac{A^{\prime\prime}}{A}-\frac{A^{\prime}}{A}\frac{B^{\prime}}{B}+\frac{A^{\prime}}{A}\frac{C^{\prime}}{C}-\frac{B^{\prime}}{B}\frac{C^{\prime}}{C}+\frac{C^{\prime\prime}}{C}\right)\right],$
(18) $\displaystyle G_{01}$ $\displaystyle=$ $\displaystyle
2\left(-\frac{\dot{C}^{\prime}}{C}+\frac{A^{\prime}}{A}\frac{\dot{C}}{C}+\frac{\dot{B}}{B}\frac{C^{\prime}}{C}\right).$
(19)

The Ricciscalar for the interior metric (9) is given by

$R=-\frac{2}{A^{2}}\left(\frac{\dot{A}\dot{B}}{AB}+\frac{2\dot{A}\dot{C}}{AC}-\frac{\ddot{B}}{B}-\frac{2\dot{B}\dot{C}}{BC}-\frac{\dot{C}^{2}}{C^{2}}-\frac{2\ddot{C}}{C}\right)-\frac{2}{B^{2}}\left(\frac{A^{\prime\prime}}{A}-\frac{A^{\prime}B^{\prime}}{AB}+\frac{2A^{\prime}C^{\prime}}{AC}+\frac{C^{\prime
2}}{C^{2}}-\frac{2B^{\prime}C^{\prime}}{BC}+\frac{2C^{\prime\prime}}{C}\right)+\frac{2}{C^{2}}.$
(20)

We find that the Kretschmann scalar for the interior space-time is given by a
lengthy expression and has $\sim 30$ non-zero components Lake . It involves
inverse powers of the field functions $A$, $B$ and $C$. Therefore, both the
Ricci and the Kretschmann scalars will diverge at the time $t=t_{s}$, when
these functions tend to zero, indicating a possible curvature singularity.
However, for an exact analysis we need to find the solution to the $f(R,T)$
field equations in our case, a fairly elaborate program which will be
presented elsewhere.

The $f(R,T)$ field equations (8) for the spherical non-static interior (9) are
found as

$G_{00}=\frac{A^{2}}{f_{R}}\left[{\rho}+{\epsilon}(1+f_{T})-\frac{1}{2}\left(f-Rf_{R}\right)+\frac{\zeta_{00}}{A^{2}}\right],$
(21)

$G_{01}=\frac{AB}{f_{R}}\left[-(1+f_{T})(q+{\epsilon})+\frac{\zeta_{01}}{AB}\right],$
(22)

$\displaystyle
G_{11}=\frac{B^{2}}{f_{R}}\left[(1+f_{T})(p_{r}+\epsilon+4\eta{\sigma})+\rho
f_{T}+\frac{1}{2}\left(f-Rf_{R}\right)+\frac{\zeta_{11}}{B^{2}}\right],$ (23)

$\displaystyle
G_{22}=\frac{C^{2}}{f_{R}}\left[(1+f_{T})({p_{\perp}}-2\eta{\sigma})+\rho
f_{T}+\frac{1}{2}\left(f-Rf_{R}\right)+\frac{\zeta_{22}}{C^{2}}\right],$ (24)

where

$\displaystyle\zeta_{00}$ $\displaystyle=$
$\displaystyle\left[-\left(\frac{\dot{B}}{B}+\frac{2\dot{C}}{C}\right)\dot{f}_{R}+\frac{A^{2}}{B^{2}}\left(\left(\frac{2C^{\prime}}{C}-\frac{B^{\prime}}{B}\right)f^{\prime}_{R}+f^{\prime\prime}_{R}\right)\right],$
(25) $\displaystyle\zeta_{01}$ $\displaystyle=$
$\displaystyle\left[\dot{f}^{\prime}_{R}-\frac{A^{\prime}}{A}\dot{f}_{R}-\frac{\dot{B}}{B}f^{\prime}_{R}\right],$
(26) $\displaystyle\zeta_{11}$ $\displaystyle=$
$\displaystyle\left[\frac{B^{2}}{A^{2}}\left(\ddot{f}_{R}+\left(\frac{2\dot{C}}{C}-\frac{\dot{A}}{A}\right)\dot{f}_{R}\right)-\left(\frac{A^{\prime}}{A}+\frac{2C^{\prime}}{C}\right)f^{\prime}_{R}\right],$
(27) $\displaystyle\zeta_{22}$ $\displaystyle=$
$\displaystyle\left[\frac{C^{2}}{A^{2}}\left(\ddot{f}_{R}+\left(-\frac{\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right)\dot{f}_{R}\right)+\frac{C^{2}}{B^{2}}\left(\left(-\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}-\frac{C^{\prime}}{C}\right)f^{\prime}_{R}-f^{\prime\prime}_{R}\right)\right].$
(28)

##  IV Dynamical Equations for the collapsing system

In his pioneering work, Chandrasekhar showed within the framework of GR C2 ,
that even under conditions of hydrostatic equilibrium, a self-gravitating
gaseous system becomes dynamically unstable (with respect to radial
oscillations) much before they reach the ‘Schwarzschild limit’. In Newtonian
gravity, the condition of dynamical instability is related to the “ratio of
specific heats” of the system, and a difference in the nature of the
gravitational field — general relativistic or other, decides the exact
condition of stability of such a system. In fact, different degrees of
instability will affect the nature of evolution of the collapsing object,
thereby leading to a difference in the end state of collapse HS . The problem
of stability has therefore been widely investigated YBF ; YBBF ; YBF2 , and is
done by examining the dynamical equations of collapse.

The dynamical equations describe the evolution of the parameters of the
collapsing stellar object with time and radius. These equations are derived
from the conservation laws. We consider the conservation of the Einstein
tensor because the matter energy momentum tensor has a non-vanishing
divergence in $f(R,T)$ gravity, as is evident from (4). The divergence of the
effective energy-momentum tensor in (8) provides the following continuity
equations:

$\displaystyle\frac{\dot{f_{R}}}{f_{R}^{2}A}\left[\rho+\epsilon\left(1+f_{T}\right)-\frac{f}{2}\right]+\frac{\left(q+\epsilon\right)\left(1+f_{T}\right)}{Bf_{R}}\left[\frac{f_{R}^{\prime}}{f_{R}}-\frac{4f_{T}}{\left(-1+f_{T}\right)}\left(\frac{A^{\prime}}{A}+\frac{C^{\prime}}{C}\right)\right]+\frac{\dot{f}-\dot{R}f_{R}}{2Af_{R}}$
(29)
$\displaystyle+\frac{f_{T}}{f_{R}A\left(-1+f_{T}\right)}\left[-\frac{2A}{B}\left(1+f_{T}\right)\left(\epsilon^{\prime}+q^{\prime}\right)-2\left(\dot{\rho}+\frac{f_{T}^{\prime}A\left(q+\epsilon\right)}{B}\right)+\frac{\left(1+f_{T}\right)\dot{T}}{2}-2\dot{f_{T}}\epsilon-2\left(1+f_{T}\right)\dot{\epsilon}\right.$
$\displaystyle\left.-2\left(1+f_{T}\right)\left[\frac{\dot{B}}{B}\left(p_{r}+\rho+2\epsilon+4\eta\sigma\right)+\frac{\dot{C}}{C}\left(\rho+p_{\perp}+\epsilon-2\eta\sigma\right)\right]\right]+\mathscr{Z}_{1}(t,r)=0,$
and $\displaystyle\frac{8\eta f_{T}\left(1+f_{T}\right)\left(\sigma
A\right)^{\prime}}{ABf_{R}\left(1-f_{T}\right)}-\frac{4\eta\left(1+f_{T}\right)f_{R}^{\prime}\sigma}{Bf_{R}^{2}}-\frac{f_{R}^{\prime}}{Bf_{R}^{2}}\left[\left(p_{r}+\rho+\epsilon\right)f_{T}+p_{r}+\epsilon+\frac{f}{2}\right]-\frac{\dot{f_{R}}}{Af_{R}^{2}}\left(q+\epsilon\right)\left(1+f_{T}\right)$
$\displaystyle-\frac{4}{AB^{3}C^{2}f_{R}\left(1-f_{T}\right)}\left[f_{T}\left(1+f_{T}\right)\left(6\eta
B^{2}CC^{\prime}A\sigma+\left(q+\epsilon\right)B^{3}C^{2}\left(\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right)+B^{2}C^{\prime}AC\left(p_{r}-p_{\perp}+\epsilon\right)\right)\right.$
$\displaystyle+2A\eta\sigma
f_{T}f_{T}^{\prime}B^{2}C^{2}+\frac{B^{2}C^{2}}{2}\left\\{A^{\prime}f_{T}\left(1+f_{T}\right)\left(p_{r}+\rho+2\epsilon\right)+f_{T}f_{T}^{\prime}A\left(p_{r}+\rho+\epsilon\right)\right.+\left(A\epsilon^{\prime}+B\dot{\epsilon}\right)f_{T}\left(1+f_{T}\right)$
$\displaystyle\left.\left.+\frac{A}{4}f_{T}\left(1+f_{T}\right)\left(T^{\prime}+4p_{r}^{\prime}\right)+Bf_{T}\left\\{\dot{f_{T}}\left(q+\epsilon\right)+\dot{q}\left(1+f_{T}\right)\right\\}\right\\}\right]+\frac{1}{Bf_{R}}\left[\rho^{\prime}f_{T}+\frac{1}{2}\left(f^{\prime}-f_{R}R^{\prime}\right)\right]+\mathscr{Z}_{2}(t,r)=0,$

where the dark source terms are

$\displaystyle\mathscr{Z}_{1}(t,r)$ $\displaystyle=$
$\displaystyle-\frac{\dot{f_{R}}A^{\prime\prime}}{A^{2}B^{2}f_{R}}+\frac{\dot{f_{R}}\ddot{B}}{Bf_{R}A^{3}}-\frac{2f_{R}^{\prime}\dot{C}^{\prime}}{B^{2}f_{R}AC}+\frac{2\dot{f_{R}}\ddot{C}}{Cf_{R}A^{3}}+\frac{\dot{f_{R}}f_{R}^{\prime\prime}}{AB^{2}f_{R}^{2}}-\frac{f_{R}^{\prime}\dot{f_{R}}^{\prime}}{AB^{2}f_{R}^{2}}-\frac{\left(2B\dot{C}+\dot{B}C\right)\dot{f_{R}}^{2}}{A^{3}BCF^{2}}-\frac{\dot{f_{R}}\dot{A}\dot{B}}{BA^{4}f_{R}}$
(31)
$\displaystyle-\frac{2\dot{A}\dot{C}\dot{f_{R}}}{A^{4}f_{R}C}+\frac{\dot{f_{R}}A^{\prime}f_{R}^{\prime}}{A^{2}B^{2}f_{R}^{2}}+\frac{2\dot{C}f_{R}^{\prime}A^{\prime}}{A^{2}B^{2}f_{R}C}+\frac{\left(B^{\prime}C-2BC^{\prime}\right)\left(f_{R}A^{\prime}-Af_{R}^{\prime}\right)\dot{f_{R}}}{A^{2}B^{3}f_{R}^{2}C}+\frac{\left(Cf_{R}^{\prime}+2C^{\prime}f_{R}\right)f_{R}^{\prime}\dot{B}}{AB^{3}f_{R}^{2}C},$
and $\displaystyle\mathscr{Z}_{2}(t,r)$ $\displaystyle=$
$\displaystyle\frac{2\dot{f_{R}}\dot{C}^{\prime}}{A^{2}Bf_{R}C}-\frac{f_{R}^{\prime}A^{\prime\prime}}{AB^{3}f_{R}}+\frac{f_{R}^{\prime}\ddot{B}}{A^{2}B^{2}f_{R}}-\frac{2f_{R}^{\prime}C^{\prime\prime}}{B^{3}f_{R}C}+\frac{\dot{f_{R}}\dot{f_{R}}^{\prime}}{A^{2}Bf_{R}^{2}}-\frac{\ddot{f_{R}}f_{R}^{\prime}}{A^{2}Bf_{R}^{2}}+\frac{f_{R}^{\prime
2}\left(A^{\prime}C+2AC^{\prime}\right)}{AB^{3}Cf_{R}^{2}}-\frac{f_{R}^{\prime}\dot{B}\left(\dot{f_{R}}A+\dot{A}f_{R}\right)}{A^{3}B^{2}f_{R}^{2}}$
(32)
$\displaystyle+\frac{2\dot{B}\dot{C}f_{R}^{\prime}}{A^{2}B^{2}f_{R}C}-\frac{2C^{\prime}\dot{B}\dot{f_{R}}}{A^{2}B^{2}f_{R}C}+\frac{2B^{\prime}C^{\prime}f_{R}^{\prime}}{B^{4}f_{R}C}-\frac{2f_{R}^{\prime}\dot{f_{R}}\dot{C}}{A^{2}BCf_{R}^{2}}+\frac{f_{R}^{\prime}\dot{f_{R}}\dot{A}}{A^{3}Bf_{R}^{2}}+\frac{f_{R}^{\prime}A^{\prime}B^{\prime}}{AB^{4}f_{R}}-\frac{A^{\prime}\dot{f_{R}}^{2}}{A^{3}Bf_{R}^{2}}-\frac{2\dot{C}\dot{f_{R}}A^{\prime}}{A^{3}Bf_{R}C}.$

The parameters $\mathscr{Z}_{1}$ and $\mathscr{Z}_{2}$ are the extra curvature
degrees of freedom representing the dark sources that arise automatically from
the $f(R,T)$ gravitational field. These terms signify the corrections in the
variation of total energy of the self-gravitating relativistic matter across
its boundaries as time evolves. Rearranging the dark source terms, we can
represent them in the following form:

$\displaystyle\mathscr{Z}_{1}$
$\displaystyle=\frac{1}{A}\left[-\frac{1}{A^{2}}\left\\{\left(\frac{A^{2}}{f_{R}}\left[-\frac{f-Rf_{R}}{2}+\frac{D_{00}}{A^{2}}\right]\right)_{,0}-\frac{2\dot{A}A}{f_{R}}\left(-\frac{f-Rf_{R}}{2}+\frac{D_{00}}{A^{2}}\right)-\frac{2A^{\prime}D_{00}}{Bf_{R}}\right\\}\right.$
$\displaystyle\left.+\frac{1}{B^{2}}\left\\{\left(\frac{D_{01}}{f_{R}}\right)_{,1}-\frac{B\dot{B}}{f_{R}}\left(-\frac{f-Rf_{R}}{2}+\frac{D_{00}}{A^{2}}\right)-\left(\frac{B^{\prime}}{B}+\frac{A^{\prime}}{A}\right)\frac{D_{01}}{f_{R}}-\frac{B\dot{B}}{f_{R}}\left(\frac{f-Rf_{R}}{2}+\frac{D_{11}}{B^{2}}\right)\right\\}\right],$

and

$\displaystyle\mathscr{Z}_{2}$
$\displaystyle=\frac{1}{B}\left[-\frac{1}{A^{2}}\left\\{\left(\frac{D_{01}}{f_{R}}\right)_{,0}-\left(\frac{\dot{B}}{B}+\frac{\dot{A}}{A}\right)\frac{D_{01}}{f_{R}}-\frac{AA^{\prime}}{f_{R}}\left(\frac{f-Rf_{R}}{2}+\frac{D_{11}}{B^{2}}\right)-\frac{AA^{\prime}}{f_{R}}\left(-\frac{f-Rf_{R}}{2}+\frac{D_{00}}{A^{2}}\right)\right\\}\right.$
$\displaystyle\left.+\frac{1}{B^{2}}\left\\{\left(\frac{B^{2}}{f_{R}}\left[\frac{f-Rf_{R}}{2}+\frac{D_{11}}{B^{2}}\right]\right)_{,1}-\frac{2B\dot{B}D_{01}}{A^{2}f_{R}}-\frac{2BB^{\prime}}{f_{R}}\left(\frac{f-Rf_{R}}{2}+\frac{D_{11}}{B^{2}}\right)\right\\}\right].$

The Misner-Sharp mass function MS  representing the total gravitational energy
entrapped inside the surface $\Sigma$ bounding the spherical star of radius
‘C’, is given by

$\displaystyle M(v)_{\Sigma}$ $\displaystyle=$
$\displaystyle\left[\frac{C}{2}\left(1+\frac{\dot{C}^{2}}{A^{2}}-\frac{{C^{\prime}}^{2}}{B^{2}}\right)\right]_{\Sigma}.$
(33)

In order to calculate the variation of this mass function through the boundary
surface of the collapsing configuration, we define two well-known operators:
the proper time derivative

$D_{T}=\frac{1}{A}\frac{\partial}{\partial{t}},$ (34)

and the proper radial derivative $D_{C}$ (constructed from the radius of the
sphere inside $\Sigma$)

$D_{C}=\frac{1}{C^{\prime}}\frac{\partial}{\partial{r}}.$ (35)

The relativistic 4-velocity of the fluid for the corresponding collapse is
given by

$U=D_{T}C=\frac{\dot{C}}{A},$ (36)

which must be negative to ensure collapse to occur.

Defining new variable $H(t,r)=\frac{C^{\prime}}{B}$, we obtain from (33) and
(36)

$H=\left[1+U^{2}-\frac{2M}{C}\right]^{1/2}.$ (37)

With the help of the field equations and equations (35)-(37), the mass
variation in the radial direction is found to be

$D_{C}M=\frac{C^{2}}{2f_{R}}\left[\rho+\epsilon(1+f_{T})+\frac{1}{2}\left(f-Rf_{R}\right)+\frac{\zeta_{00}}{A^{2}}+\frac{U}{H}\left\\{(1+f_{T})(q+\epsilon)-\frac{\zeta_{01}}{AB}\right\\}\right],$
(38)

which on integration yields the result

$M=\frac{1}{2}\int^{C}_{0}\frac{C^{2}}{f_{R}}\left[\rho+\epsilon(1+f_{T})+\frac{1}{2}\left(f-Rf_{R}\right)+\frac{\zeta_{00}}{A^{2}}+\frac{U}{H}\left\\{(1+f_{T})(q+\epsilon)-\frac{\zeta_{01}}{AB}\right\\}\right]dC.$
(39)

The temporal variation of the mass function is obtained as

$D_{T}M=-\frac{C^{2}}{2f_{R}}\left[U\left\\{(1+f_{T})(p_{r}+\epsilon+4\eta\sigma)+\rho
f_{T}+\frac{1}{2}\left(f-Rf_{R}\right)+\frac{\zeta_{11}}{B^{2}}\right\\}+H\left\\{(1+f_{T})(q+\epsilon)-\frac{\zeta_{01}}{AB}\right\\}\right].$
(40)

In order to determine the variation of the physical parameters of the self-
gravitating system in course of time, we now consider a suitable perturbation
scheme.

##  V $f(R,T)$ function and Perturbation Scheme

To find the approximate solution of a differential equation, very often one
uses the so-called ‘perturbation method’. This technique has been successfully
applied to analyze the dynamics of gravitational collapse by several authors
10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; Santos ; 24 ; SZ1 ; SY ; NZ1 ; NZ2 ; NZ3 ;
SW ; SS . We use this technique to analyze the evolution of the stellar object
of our study under the effect of $f(R,T)$ model of gravity. To apply the
perturbation theory we assume that initially the matter is in a state of
hydrostatic equilibrium, so that the initial values of the parameters depend
only on the radial coordinate. But in course of time the perturbed quantities
gather both radial and time dependence. The radial heat flow $q$ is of the
order of $\varepsilon$ ($0<\varepsilon\ll 1$) 10 , (where $\epsilon$ and
$\varepsilon$ are different quantities). We consider a combination of the
Starobinsky model Starobinsky  and a linear term of the trace $T$ as the
$f(R,T)$ model SZ2 , which can be written as

$\displaystyle f(R,T)=R+\alpha R^{2}+\lambda T,$ (41)

where $\alpha$ has positive real values, while $\lambda$ is a coupling
parameter, and $\lambda T$ represents the extent of modification to the
Starobinsky $f(R)$ gravity. Thus the metric functions and the physical
parameters may be written as follows:

$\displaystyle A(t,r)$ $\displaystyle=$ $\displaystyle A_{0}(r)+\varepsilon
D(t)a(r),$ (42) $\displaystyle B(t,r)$ $\displaystyle=$ $\displaystyle
B_{0}(r)+\varepsilon D(t)b(r),$ (43) $\displaystyle C(t,r)$ $\displaystyle=$
$\displaystyle C_{0}(r)+\varepsilon D(t)\bar{c}(r),$ (44)
$\displaystyle\rho(t,r)$ $\displaystyle=$
$\displaystyle\rho_{0}(r)+\varepsilon{\bar{\rho}}(t,r),$ (45) $\displaystyle
p_{r}(t,r)$ $\displaystyle=$ $\displaystyle
p_{r0}(r)+\varepsilon{\bar{p}_{r}}(t,r),$ (46) $\displaystyle p_{\bot}(t,r)$
$\displaystyle=$ $\displaystyle p_{\bot
0}(r)+\varepsilon{\bar{p}_{\bot}}(t,r)$ (47) $\displaystyle\epsilon(t,r)$
$\displaystyle=$ $\displaystyle\varepsilon{\bar{\epsilon}}(t,r)$ (48)
$\displaystyle m(t,r)$ $\displaystyle=$ $\displaystyle
m_{0}(r)+\varepsilon{\bar{m}}(t,r),$ (49) $\displaystyle R(t,r)$
$\displaystyle=$ $\displaystyle R_{0}(r)+\varepsilon D_{1}(t)e_{1}(r),$ (50)
$\displaystyle T(t,r)$ $\displaystyle=$ $\displaystyle T_{0}(r)+\varepsilon
D_{2}(t)e_{2}(r),$ (51) $\displaystyle f(R,T)$ $\displaystyle=$
$\displaystyle[R_{0}(r)+\alpha R_{0}^{2}(r)+\lambda T_{0}]+\varepsilon
D_{1}(t)e_{1}(r)$ (52) $\displaystyle\times[1+2\alpha
R_{0}(r)]+\varepsilon\lambda D_{2}(t)e_{2}(r),$ $\displaystyle f_{R}$
$\displaystyle=$ $\displaystyle[1+2\alpha R_{0}(r)]+2\alpha\varepsilon
D_{1}(t)e_{1}(r),$ (53) $\displaystyle f_{T}$ $\displaystyle=$
$\displaystyle\lambda,$ (54) $\displaystyle\Theta_{1}(t,r)$ $\displaystyle=$
$\displaystyle\varepsilon\bar{\Theta}_{1},$ (55) $\displaystyle\sigma(t,r)$
$\displaystyle=$ $\displaystyle\varepsilon{\bar{\sigma}}(t,r),$ (56)
$\displaystyle q(t,r)$ $\displaystyle=$
$\displaystyle\varepsilon\bar{q}(t,r).$ (57)

For $\bar{c}(r)=0,$ we get the shear-free case. Without loss of generality, we
consider $C_{0}(r)=r$ as the Schwarzschild coordinate. In the above equations,
$R_{0}$ represents the static part of the Ricci scalar which has the following
form

$\displaystyle
R_{0}(r)=-\frac{2}{rB_{0}^{2}}\left(\frac{1}{r}-\frac{2B^{\prime}_{0}}{B_{0}}+\frac{2A^{\prime}_{0}}{A_{0}}\right)-\frac{2}{B_{0}^{2}}\left(\frac{A^{\prime\prime}_{0}}{A_{0}}-\frac{A^{\prime}_{0}B^{\prime}_{0}}{A_{0}B_{0}}\right)+\frac{2}{r^{2}},$
(58)

while the value of perturbed part of Ricci scalar is given by

$De=\frac{2\ddot{D}}{A_{0}^{2}}\left(\frac{b}{B_{0}}+\frac{\bar{c}}{r}\right)+D\left[-\frac{2}{B_{0}^{2}}\left\\{\frac{A_{0}^{\prime}}{A_{0}}\left(\frac{2\bar{c}^{\prime}}{r}-\frac{b^{\prime}}{B_{0}}\right)-\frac{B_{0}^{\prime}}{B_{0}}\left(\frac{2\bar{c}^{\prime}}{r}+\frac{a^{\prime}}{A_{0}}\right)+\frac{2}{r}\left(\frac{a^{\prime}}{A_{0}}-\frac{b^{\prime}}{B_{0}}+\frac{\bar{c}}{r}\right)+\frac{a^{\prime\prime}}{A_{0}}+\frac{\bar{c}^{\prime\prime}}{r}\right\\}\right],$
(59)

where we have assumed that the perturbed quantities $D_{1}=D_{2}=D$, and
$e_{1}=e_{2}=e$. The matter content of the spherical collapsing configuration
under static conditions as well as non-static equilibrium phases, are given by

$\displaystyle m_{0}$ $\displaystyle=$
$\displaystyle\frac{r}{2}\left(1-\frac{1}{B_{0}^{2}}\right),$ (60)
$\displaystyle{\bar{m}}(t,r)$ $\displaystyle=$
$\displaystyle\frac{D(t)}{B_{0}^{2}}\left[r\left(\frac{b}{B_{0}}-\bar{c}^{\prime}\right)+\frac{\bar{c}}{2}\left(B_{0}^{2}-1\right)\right].$
(61)

Let $Z=1+2\alpha R_{0}$, $Z_{R_{0}}=\frac{\partial Z}{\partial R_{0}}$, and
$Z_{R_{0}R_{0}}=\frac{\partial^{2}Z}{\partial R_{0}^{2}}$. Using these
symbols, the static configuration of $f(R,T)$ field equations is obtained in
the form

$\displaystyle\frac{1}{r^{2}}\left(1-\frac{1}{B_{0}^{2}}\right)+\frac{2B_{0}^{\prime}}{rB_{0}^{3}}$
$\displaystyle=$ $\displaystyle\frac{1}{Z}\left[\rho_{0}+\frac{\alpha
R_{0}^{2}-\lambda T_{0}}{2}+\frac{D_{00}^{(S)}}{A_{0}^{2}}\right],$ (62)
$\displaystyle\frac{1}{r^{2}}\left(\frac{1}{B_{0}^{2}}-1\right)+\frac{2A_{0}^{\prime}}{A_{0}B_{0}^{2}r}$
$\displaystyle=$
$\displaystyle\frac{1}{Z}\left[p_{r_{0}}+\lambda\left(\rho_{0}+p_{r_{0}}\right)-\frac{\alpha
R_{0}^{2}-\lambda T_{0}}{2}+\frac{D_{11}^{(S)}}{B_{0}^{2}}\right],$ (63)
$\displaystyle\frac{1}{B_{0}^{2}}\left[\frac{A_{0}^{\prime\prime}}{A_{0}}+\frac{A_{0}^{\prime}}{A_{0}r}-\frac{A_{0}^{\prime}B_{0}^{\prime}}{A_{0}B_{0}}-\frac{B_{0}^{\prime}}{B_{0}r}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{Z}\left[p_{\perp_{0}}+\lambda\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{\alpha
R_{0}^{2}-\lambda T_{0}}{2}+\frac{D_{22}^{(S)}}{r^{2}}\right],$ (64)

where

$\displaystyle D_{00}^{(S)}$ $\displaystyle=$
$\displaystyle\frac{A_{0}^{2}}{B_{0}^{2}}\left[Z_{R_{0}}\left\\{R_{0}^{\prime\prime}-R_{0}^{\prime}\left(\frac{B_{0}^{\prime}}{B_{0}}-\frac{2}{r}\right)\right\\}+Z_{R_{0}R_{0}}\left(R_{0}^{\prime}\right)^{2}\right],$
(65) $\displaystyle D_{11}^{(S)}$ $\displaystyle=$ $\displaystyle-
Z_{R_{0}}R_{0}^{\prime}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{2}{r}\right),$
(66) $\displaystyle D_{22}^{(S)}$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{B_{0}^{2}}\left[Z_{R_{0}}\left\\{R_{0}^{\prime}\left(\frac{B_{0}^{\prime}}{B_{0}}-\frac{A_{0}^{\prime}}{A_{0}}-\frac{1}{r}\right)-R_{0}^{\prime\prime}\right\\}-Z_{R_{0}R_{0}}\left(R_{0}^{\prime}\right)^{2}\right].$
(67)

In the static configuration, the first dynamical equation is identically
satisfied, while the second equation is given by

$\displaystyle-\frac{Z_{R_{0}}R_{0}^{\prime}}{B_{0}Z^{2}}\left\\{p_{r_{0}}\left(1+\lambda\right)+\rho_{0}\lambda+\frac{R_{0}+\alpha
R_{0}^{2}+\lambda
T_{0}}{2}\right\\}+\frac{4\lambda\left(1+\lambda\right)}{B_{0}Zr\left(-1+\lambda\right)}\left(p_{r_{0}}-p_{\perp_{0}}\right)+\frac{\lambda\left(1+\lambda\right)T_{0}^{\prime}}{2\left(-1+\lambda\right)B_{0}Z}$
$\displaystyle+\frac{2\lambda}{\left(-1+\lambda\right)}\frac{1}{B_{0}Z}\left\\{\frac{A_{0}^{\prime}}{A_{0}}\left(p_{r_{0}}+\rho_{0}\right)\left(1+\lambda\right)+p_{r_{0}}^{\prime}\left(1+\lambda\right)+\rho_{0}^{\prime}\lambda\right\\}+\mathscr{Z}_{2}^{(S)}=0,$
(68)

where $\mathscr{Z}_{2}^{(S)}$ represents the static part of
$\mathscr{Z}_{2}(t,r)$ and has the following form:

$\displaystyle\mathscr{Z}_{2}^{(S)}=\frac{Z_{R_{0}}R_{0}^{\prime}}{B_{0}^{3}Z}\left[-\frac{A_{0}^{\prime\prime}}{A_{0}}+\frac{Z_{R_{0}}R_{0}^{\prime}\left(A_{0}^{\prime}r+2A_{0}\right)}{A_{0}rZ}+\frac{2B_{0}^{\prime}}{rB_{0}}+\frac{A_{0}^{\prime}B_{0}^{\prime}}{A_{0}B_{0}}\right].$
(69)

After the application of the perturbation scheme, the perturbed form of the
first dynamical equation is

$\displaystyle-\frac{2\lambda\dot{\bar{\rho}}}{\left(-1+\lambda\right)A_{0}Z}+\frac{\left(\bar{q}+\bar{\epsilon}\right)\left(1+\lambda\right)}{B_{0}Z}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-4\frac{\lambda}{\left(-1+\lambda\right)}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{1}{r}\right)\right]-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)Z}\left[\frac{\bar{\epsilon}^{\prime}+\bar{q}^{\prime}}{B_{0}}+\frac{\dot{\bar{\epsilon}}}{A_{0}}\right]\qquad\qquad\qquad$
$\displaystyle+\dot{D}\left[\frac{2\alpha
e}{Z^{2}A_{0}}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)ZA_{0}}\left\\{\frac{b}{B_{0}}\left(p_{r_{0}}+\rho_{0}\right)+\frac{2\bar{c}}{r}\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{e}{4}\right\\}+\frac{\lambda
e}{2A_{0}Z}+\mathscr{Z}_{1}^{(P)}\right]$ $\displaystyle=0,$ (70)

where $\mathscr{Z}_{1}^{(P)}$ is given in the Appendix.

The perturbed form of the second dynamical equation is

$\displaystyle\frac{4\eta(1+\lambda)}{B_{0}Z}\left[\frac{2\lambda}{\left(-1+\lambda\right)}\left(\bar{\sigma}^{\prime}+\frac{\bar{\sigma}A_{0}^{\prime}}{A_{0}}+\frac{3\bar{\sigma}}{r}\right)-\frac{\bar{\sigma}Z_{R_{0}}R_{0}^{\prime}}{Z}\right]-\frac{Z_{R_{0}}R_{0}^{\prime}}{B_{0}Z^{2}}\left[\left(\bar{p_{r}}+\bar{\epsilon}\right)\left(1+\lambda\right)+\lambda\bar{\rho}\right]$
$\displaystyle+\frac{\lambda e^{\prime}}{2B_{0}Z}-\frac{\alpha
eR_{0}^{\prime}}{B_{0}Z}+\frac{eZ_{R_{0}}R_{0}^{\prime}}{2B_{0}Z}+\frac{2\lambda}{\left(-1+\lambda\right)}\left[\frac{2\left(1+\lambda\right)}{B_{0}Zr}\left(\bar{p_{r}}-\bar{p_{\perp}}+\bar{\epsilon}\right)+\frac{A_{0}^{\prime}}{A_{0}B_{0}Z}\left(\bar{p_{r}}+\bar{\rho}+2\bar{\epsilon}\right)\right.$
$\displaystyle\left.+\frac{\left(1+\lambda\right)}{A_{0}B_{0}Z}\left\\{A_{0}\left(\bar{p_{r}}^{\prime}+\bar{\epsilon}^{\prime}\right)+B_{0}\left(\dot{\bar{\epsilon}}+\dot{\bar{q}}\right)\right\\}+\frac{\bar{\rho}^{\prime}\lambda}{B_{0}Z}\right]+\frac{\ddot{D}bZ_{R_{0}}R_{0}^{\prime}}{A_{0}^{2}B_{0}^{2}Z}$
$\displaystyle+D\left[-\frac{Z_{R_{0}}R_{0}^{\prime}}{B_{0}Z^{2}}\frac{e}{2}\left(Z+\lambda\right)-\frac{2\alpha
e^{\prime}}{B_{0}Z^{2}}\left(\rho_{0}\lambda+p_{r_{0}}\left(1+\lambda\right)+\frac{R_{0}+\alpha
R_{0}^{2}+\lambda T_{0}}{2}\right)\right.$
$\displaystyle+\frac{2\lambda}{\left(-1+\lambda\right)}\left[\frac{2\bar{c}^{\prime}}{B_{0}Zr}\left(1+\lambda\right)\left(p_{r_{0}}-p_{\perp_{0}}\right)+\frac{\left(1+\lambda\right)}{A_{0}B_{0}Z}\left[a^{\prime}\left(p_{r_{0}}+\rho_{0}\right)+ap_{r_{0}}^{\prime}\right]\right.$
$\displaystyle\left.+\frac{a}{A_{0}B_{0}Z}\left\\{\rho_{0}^{\prime}\lambda-\frac{ZR_{0}^{\prime}}{2}-\frac{\left(R_{0}^{\prime}+2\alpha
R_{0}R_{0}^{\prime}+\lambda
T_{0}^{\prime}\right)}{2}\right\\}\right]\left.+\frac{e^{\prime}\lambda\left(1+\lambda\right)}{2B_{0}Z\left(-1+\lambda\right)}+\mathscr{Z}_{2}^{(P)}\right]=0,$
(71)

where $\mathscr{Z}_{2}^{(P)}$ is given in the Appendix. From $(\ref{59a})$,
integrating with respect to $t$, we obtain,

$\displaystyle\bar{\rho}$
$\displaystyle=\frac{\left(-1+\lambda\right)}{2\lambda}\frac{A_{0}}{B_{0}}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\frac{4\lambda}{\left(-1+\lambda\right)}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{1}{r}\right)\right]\left(1+\lambda\right)\int\left(\bar{q}+\bar{\epsilon}\right)dt-\left(1+\lambda\right)\frac{A_{0}}{B_{0}}\int\left(\bar{\epsilon}^{\prime}+\bar{q}^{\prime}\right)dt-\left(1+\lambda\right)\bar{\epsilon}$
$\displaystyle+D\frac{\left(-1+\lambda\right)}{2\lambda}\left[\frac{2\alpha
e}{Z}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)}\left\\{\frac{b}{B_{0}}\left(p_{r_{0}}+\rho_{0}\right)+\frac{2\bar{c}}{r}\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{e}{4}\right\\}+\frac{\lambda
e}{2}+A_{0}Z\mathscr{Z}_{1}^{(P)}\right].$ (72)

The perturbed quantities $\bar{\rho}$ and $\bar{p}_{r}$ are related through
the ratio of specific heats, if we consider the second law of thermodynamics
and assume a Harrison-Wheeler type equation of state HW , expressed in the
following form:

$\displaystyle\bar{p}_{r}$ $\displaystyle=$ $\displaystyle\frac{\Gamma
p_{r0}}{\rho_{0}+p_{r_{0}}}\bar{\rho},$ (73)

where $\Gamma$ is the adiabatic index which determines the change of pressure
for a given change in density. We assume that $\Gamma$ remains constant while
the collapse progresses. Now substituting $\bar{\rho}$ from (V) in (73), we
obtain

$\displaystyle\bar{p}_{r}$ $\displaystyle=\frac{\Gamma
p_{r_{0}}}{\rho_{0}+p_{r_{0}}}\left[\frac{\left(-1+\lambda\right)}{2\lambda}\frac{A_{0}}{B_{0}}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\frac{4\lambda}{\left(-1+\lambda\right)}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{1}{r}\right)\right]\left(1+\lambda\right)\int\left(\bar{q}+\bar{\epsilon}\right)dt-\frac{A_{0}}{B_{0}}\left(1+\lambda\right)\int\left(\bar{\epsilon}^{\prime}+\bar{q}^{\prime}\right)dt-\left(1+\lambda\right)\bar{\epsilon}\right.$
$\displaystyle\left.+D\frac{\left(-1+\lambda\right)}{2\lambda}\left[\frac{2\alpha
e}{Z}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)}\left\\{\frac{b}{B_{0}}\left(p_{r_{0}}+\rho_{0}\right)+\frac{2\bar{c}}{r}\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{e}{4}\right\\}+\frac{\lambda
e}{2}+A_{0}Z\mathscr{Z}_{1}^{(P)}\right]\right].$ (74)

Considering the perturbed variant of the field equation (24), we obtain the
following form of the perturbed term $\bar{p}_{\perp}$:

$\bar{p}_{\perp}=-\frac{\ddot{D}Z\left(\bar{c}+\frac{br}{B_{0}}\right)}{A_{0}^{2}r\left(1+\lambda\right)}-\frac{\bar{\rho}\lambda}{1+\lambda}+2\eta\bar{\sigma}+DH_{\perp},$
(75)

where $H_{\perp}$ is given in the Appendix. Substituting (V), (74), and (75),
in (V), we get an equation of the form

$\ddot{D}-Q(r)D=G(r,t),$ (76)

where $Q(r)$ and $G(r,t)$ are given in the Appendix. The terms in (76) are
chosen to ensure that $Q(r)$ remains positive throughout the evolution. The
solution of this differential equation is given by

$D=\frac{1}{2\sqrt{Q}}\left[e^{\sqrt{Q}t}\int
Ge^{-\sqrt{Q}t}dt-e^{-\sqrt{Q}t}\int Ge^{\sqrt{Q}t}dt\right].$ (77)

The explicit form of equation (76) is given in equation (Appendix) in the
appendix.

###  V.1 Conditions of instability

We now examine the instability conditions of the collapsing system by
analysing the Newtonian and post Newtonian regimes of the collapse equations.
This analysis also reveals the significance of the adiabatic index $\Gamma$ in
the collapse dynamics.

####  V.1.1 Newtonian Approximation

To deal with the Newtonian Approximation of the collapsing configuration, let
us consider $A_{0}=1$, $B_{0}=1$. Along with this we assume that $\rho_{0}\gg
p_{r_{0}}$, and $\rho_{0}\gg p_{\perp_{0}}$. Putting these assumptions in the
explicit form (Appendix) of equation (76), we find that the range of stability
of the collapsing matter is specified by the following inequality:

$\Gamma<\frac{ZH_{1}\left(-1+\lambda\right)}{2\lambda
p_{r_{0}}\left(\frac{\bar{\rho}_{N}}{\rho_{0}}\right)^{\prime}\left(1+\lambda\right)},$
(78)

where $H_{1}$ is given in the appendix. The collapse will become unstable
unless this condition is satisfied. We then have

$\displaystyle\bar{\rho}_{N}$
$\displaystyle=\frac{\left(-1+\lambda\right)}{2\lambda}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\frac{4\lambda}{\left(-1+\lambda\right)}\frac{1}{r}\right]\left(1+\lambda\right)\int\left(\bar{q}+\bar{\epsilon}\right)dt-\left(1+\lambda\right)\int\left(\bar{\epsilon}^{\prime}+\bar{q}^{\prime}\right)dt-\left(1+\lambda\right)\bar{\epsilon}$
$\displaystyle+D\frac{\left(-1+\lambda\right)}{2\lambda}\left[\frac{2\alpha
e}{Z}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)}\left[\rho_{0}\left(b+\frac{2\bar{c}}{r}\right)-\frac{e}{4}\right]+\frac{\lambda
e}{2}+Z\mathscr{Z}_{1_{N}}^{(P)}\right].$ (79)

To ensure the positivity on the right hand side of the inequality (78), we
need the following constraints:

$\displaystyle\frac{2\lambda}{\left(-1+\lambda\right)}\left(\bar{\sigma}^{\prime}+\frac{2\bar{\sigma}}{r}\right)<\frac{\bar{\sigma}Z_{R_{0}}R_{0}^{\prime}}{Z},$
(80)
$\displaystyle\bar{\epsilon}\left(1+\lambda\right)+\lambda\bar{\rho}_{N}>0,$
(81)
$\displaystyle\frac{Z_{R_{0}}R_{0}^{\prime}}{Z^{2}}>\frac{2\lambda}{\left(-1+\lambda\right)}\frac{2}{Zr},$
(82) $\displaystyle e^{\sqrt{Q}t}\int Ge^{-\sqrt{Q}t}dt>e^{-\sqrt{Q}t}\int
Ge^{\sqrt{Q}t}dt,$ (83) $\displaystyle
G+\frac{\sqrt{Q}}{2}\left[e^{\sqrt{Q}t}\int
Ge^{-\sqrt{Q}t}dt-e^{-\sqrt{Q}t}\int Ge^{\sqrt{Q}t}dt\right]<0,$ (84)
$\displaystyle\frac{2\left(\bar{c}+br\right)}{r^{2}}+\frac{\left(-1+\lambda\right)}{2\lambda}\frac{bZ_{R_{0}}R_{0}^{\prime}}{Z}>0,$
(85) $\displaystyle 0<p_{r_{0}}<p_{\perp_{0}},$ (86)
$\displaystyle\bar{\epsilon}^{\prime}+\dot{\bar{\epsilon}}+\dot{\bar{q}}<0,$
(87) $\displaystyle\frac{\lambda\bar{\rho}_{N}^{\prime}}{Z}<0,$ (88)
$\displaystyle\lambda\rho_{0}+\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}>0,$ (89)
$\displaystyle\frac{Z_{R_{0}}R_{0}^{\prime}e\left(Z+\lambda\right)}{2Z^{2}}>0,$
(90)
$\displaystyle\frac{a^{\prime}\left(1+\lambda\right)\rho_{0}}{Z}+\frac{\lambda
e^{\prime}}{2Z}-\frac{\alpha
eR_{0}^{\prime}}{Z}+\frac{eZ_{R_{0}}R_{0}^{\prime}}{2Z}<0,$ (91)
$\displaystyle\rho_{0}^{\prime}\lambda-\frac{ZR_{0}^{\prime}}{2}-\frac{\left(R_{0}^{\prime}+2\alpha
R_{0}R_{0}^{\prime}+\lambda T_{0}^{\prime}\right)}{2}<0,$ (92)
$\displaystyle\mathscr{Z}_{2_{N}}^{(P)}<0,$ (93)
$\displaystyle\left(\frac{\bar{\rho}_{N}}{\rho_{0}}\right)^{\prime}>0.$ (94)

####  V.1.2 Post-Newtonian Approximation

For the Post-Newtonian Approximation, we consider $A_{0}=1-\frac{m_{0}}{r}$,
and $B_{0}=1+\frac{m_{0}}{r}$. So, we have

$\frac{A_{0}}{B_{0}}=\frac{r-m_{0}}{r+m_{0}},\qquad\frac{A_{0}^{\prime}}{A_{0}}=\frac{m_{0}}{r\left(r-m_{0}\right)},\qquad\textrm{and}\qquad\frac{B_{0}^{\prime}}{B_{0}}=-\frac{m_{0}}{r\left(r+m_{0}\right)}.$

Applying these conditions to equation (Appendix), we arrive at the stability
criterion

$\Gamma<\frac{\left(-1+\lambda\right)Z\left(r+m_{0}\right)\left(\rho_{0}+p_{r_{0}}\right)H_{2}}{2\lambda
p_{r_{0}}\left\\{\left(1+\lambda\right)r\left(\bar{\rho}_{pN}^{\prime}-\frac{\bar{\rho}_{pN}\rho_{0}^{\prime}}{\left(\rho_{0}+p_{r_{0}}\right)}\right)-\bar{\rho}_{pN}\left(\frac{\left(-1+\lambda\right)}{2\lambda}\frac{Z_{R_{0}}R_{0}^{\prime}r\left(1+\lambda\right)}{Z}-2\left(1+\lambda\right)-\frac{m_{0}}{\left(r-m_{0}\right)}\right)\right\\}},$
(95)

where $H_{2}$ is given in the appendix. We also have

$\displaystyle\bar{\rho}_{pN}$
$\displaystyle=\frac{\left(-1+\lambda\right)}{2\lambda}\frac{\left(r-m_{0}\right)}{\left(r+m_{0}\right)}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\frac{4\lambda}{\left(-1+\lambda\right)}\frac{1}{\left(r-m_{0}\right)}\right]\left(1+\lambda\right)\int\left(\bar{q}+\bar{\epsilon}\right)dt$
$\displaystyle-\frac{\left(r-m_{0}\right)}{\left(r+m_{0}\right)}\left(1+\lambda\right)\int\left(\bar{\epsilon}^{\prime}+\bar{q}^{\prime}\right)dt-(1+\lambda)\bar{\epsilon}+\frac{D\left(\lambda-1\right)}{2\lambda}\times\left[\frac{2\alpha
e}{Z}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)\right.$
$\displaystyle\left.\qquad\qquad-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)}\left\\{\frac{br}{r+m_{0}}\left(p_{r_{0}}+\rho_{0}\right)+\frac{2\bar{c}}{r}\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{e}{4}\right\\}+\frac{\lambda
e}{2}+\left(1-\frac{m_{0}}{r}\right)Z\mathscr{Z}_{1_{pN}}^{(P)}\right].$ (96)

To ensure that the right hand side of the inequality (95) remains positive, we
impose at the following constraints:

$\displaystyle\left(\bar{\rho}_{pN}^{\prime}-\frac{\bar{\rho}_{pN}\rho_{0}^{\prime}}{\left(\rho_{0}+p_{r_{0}}\right)}\right)>\bar{\rho}_{pN}\left(\frac{\left(-1+\lambda\right)}{2\lambda}\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\frac{2}{r}-\frac{m_{0}}{r\left(r-m_{0}\right)\left(1+\lambda\right)}\right),$
(97) $\displaystyle r+m_{0}>0,$ (98) $\displaystyle\rho_{0}+p_{r_{0}}>0,$ (99)
$\displaystyle\frac{2\lambda}{\left(-1+\lambda\right)}\left(\bar{\sigma}^{\prime}+\frac{m_{0}\bar{\sigma}}{r\left(r-m_{0}\right)}+\frac{2\bar{\sigma}}{r}\right)<\frac{\bar{\sigma}Z_{R_{0}}R_{0}^{\prime}}{Z},$
(100)
$\displaystyle\bar{\epsilon}\left(1+\lambda\right)+\lambda\bar{\rho}_{pN}>0,$
(101)
$\displaystyle\frac{Z_{R_{0}}R_{0}^{\prime}r}{Z^{2}}>\frac{2\lambda}{\left(-1+\lambda\right)}\frac{2}{Z},$
(102) $\displaystyle e^{\sqrt{Q}t}\int Ge^{-\sqrt{Q}t}dt>e^{-\sqrt{Q}t}\int
Ge^{\sqrt{Q}t}dt,$ (103) $\displaystyle
G+\frac{\sqrt{Q}}{2}\left[e^{\sqrt{Q}t}\int
Ge^{-\sqrt{Q}t}dt-e^{-\sqrt{Q}t}\int Ge^{\sqrt{Q}t}dt\right]<0,$ (104)
$\displaystyle\frac{2r\left(\bar{c}+\frac{br^{2}}{r+m_{0}}\right)}{\left(r+m_{0}\right)\left(r-m_{0}\right)^{2}}+\frac{\left(-1+\lambda\right)}{2\lambda}\frac{bZ_{R_{0}}R_{0}^{\prime}r^{4}}{\left(r^{2}-m_{0}^{2}\right)^{2}Z}>0,$
(105)
$\displaystyle\frac{\bar{\epsilon}^{\prime}}{\left(r+m_{0}\right)}+\frac{\dot{\bar{\epsilon}}+\dot{\bar{q}}}{\left(r-m_{0}\right)}<0,$
(106)
$\displaystyle\frac{m_{0}}{\left(r^{2}-m_{0}^{2}\right)Z}\left(2\bar{\epsilon}+\bar{\rho}_{pN}\right)<0,$
(107) $\displaystyle\frac{\lambda
r\bar{\rho}_{pN}^{\prime}}{\left(r+m_{0}\right)Z}<0,$ (108)
$\displaystyle\frac{Z_{R_{0}}R_{0}^{\prime}re\left(Z+\lambda\right)}{2Z^{2}\left(r+m_{0}\right)}>0,$
(109)
$\displaystyle\lambda\rho_{0}+p_{r_{0}}\left(1+\lambda\right)+\frac{R_{0}+\alpha
R_{0}^{2}+\lambda T_{0}}{2}>0,$ (110) $\displaystyle
0<p_{r_{0}}<p_{\perp_{0}},$ (111)
$\displaystyle\frac{a^{\prime}r^{2}\left(1+\lambda\right)\left(p_{r_{0}}+\rho_{0}\right)}{\left(r^{2}-m_{0}^{2}\right)Z}+\frac{\lambda
e^{\prime}r}{2Z\left(r+m_{0}\right)}-\frac{\alpha
erR_{0}^{\prime}}{Z\left(r+m_{0}\right)}+\frac{erZ_{R_{0}}R_{0}^{\prime}}{2\left(r+m_{0}\right)Z}<0,$
(112)
$\displaystyle\rho_{0}^{\prime}\lambda-\frac{ZR_{0}^{\prime}}{2}-\frac{\left(R_{0}^{\prime}+2\alpha
R_{0}R_{0}^{\prime}+\lambda T_{0}^{\prime}\right)}{2}<0,$ (113)
$\displaystyle\frac{e^{\prime}\lambda\left(1+\lambda\right)r}{2\left(-1+\lambda\right)\left(r+m_{0}\right)Z}<0,$
(114) $\displaystyle\mathscr{Z}_{2_{pN}}^{(P)}<0.$ (115)

##  VI Transport Equations

To derive the transport equation for an arbitrary fluid, we have to keep in
mind that both the energy-momentum tensor $T^{ab}$ and the total particle flux
$N^{a}$ are conserved, and the second law of thermodynamics is obeyed by the
entropy flux $S^{a}$. Mathematically, we have

$\displaystyle T_{;b}^{ab}=0,\qquad N_{;a}^{a}=0,\qquad S_{;a}^{a}\geq 0.$

There exists a unique timelike eigenvector $u_{E}^{a}$ to $T^{ab}$, and
another timelike vector $u_{N}^{a}$ parallel to $N^{a}$. In equilibrium, a
rest-frame can be defined in which these eigenvectors, and $S^{a}$ are
parallel to each other. However they are not so when the system deviates from
the equilibrium state Israel1 .

Utilising the above three requirements for the energy-momentum tensor, the
particle flux and the entropy flux, the transport equation is obtained as

$\tau
h^{ab}u^{c}q_{b;c}+q^{a}=-Kh^{ab}(T_{1,b}+T_{1}a_{b})-\frac{1}{2}KT_{1}^{2}\left(\frac{\tau
u^{b}}{KT_{1}^{2}}\right)_{;b}q^{a}.$ (116)

where, $\tau$ is the relaxation time (within which the system reverts back to
the equilibrium state after being disturbed), $K$ is the thermal conductivity,
$T_{1}$ is the temperature, $h^{ab}$ is the projection tensor, $a_{b}$ is the
acceleration vector, $u^{a}$ is the four-velocity and $q^{a}$ is the heat-flux
vector. This is the causal heat transport equation derived from the Müller-
Israel-Stewart theory Muller ; Israel1 ; Israel2 ; Israel3  for dissipative
fluids, and used in 12 ; PHDMS . Using the specified definition of the four-
velocity in the comoving frame, and radial heat flow, the transport equation
can be written as

$\dot{q}=-\frac{Aq}{\tau}-\frac{K(T_{1}A)^{\prime}}{\tau
B}-\frac{qA\Theta_{1}}{2}-\frac{qKT_{1}^{2}}{2\tau}\dot{\left(\frac{\tau}{KT_{1}^{2}}\right)}.$
(117)

Rearranging this equation we get

$\tau\dot{q}=-\frac{qKT_{1}^{2}}{2}\dot{\left(\frac{\tau}{KT_{1}^{2}}\right)}-\tau
q\left(\frac{\dot{B}}{2B}+\frac{\dot{C}}{C}\right)-\frac{K(T_{1}A)^{\prime}}{B}-qA.$
(118)

which is equation (47) of PHDMS . This equation can be further simplified as

$\dot{q}=-\frac{K(T_{1}A)^{\prime}}{\tau
B}-qA\left[\frac{1}{\tau}+\frac{\Theta_{1}}{2}+\frac{1}{2}D_{T}\left[\ln\left(\frac{\tau}{KT_{1}^{2}}\right)\right]\right],$
(119)

where $D_{T}=\frac{1}{A}\frac{\partial}{\partial t}$. Finally we can write the
transport equation in the form

$D_{T}q=-\frac{K(T_{1}A)^{\prime}}{A\tau
B}-q\left[\frac{1}{\tau}+\frac{\Theta_{1}}{2}+\frac{1}{2}D_{T}\left[\ln\left(\frac{\tau}{KT_{1}^{2}}\right)\right]\right].$
(120)

The collapse velocity is given by $U=D_{T}C=\frac{\dot{C}}{A}$, which must be
negative in order to ensure collapse. Considering $D_{T}U$ as the collapse
acceleration, and the expression for the Misner-Sharp mass-energy $M$
contained in a sphere of radius $C$, we obtain from our field equation (23),

$D_{T}U=-\frac{M}{C^{2}}-\frac{C}{2f_{R}}\left[\left(1+f_{T}\right)\left(p_{r}+\epsilon+4\eta\sigma\right)+\rho
f_{T}+\frac{1}{2}\left(f-RF\right)+\frac{D_{11}}{B^{2}}\right]+\frac{A^{\prime}H}{AB},$
(121)

where $H$ is given in (37).

To understand the effects of dissipation on the collapse dynamics, we now
combine the transport equation with the dynamical equation, remembering that,
$D_{C}=\frac{1}{C^{\prime}}\frac{\partial}{\partial r}$. Substituting for
$\frac{A^{\prime}}{A}$ and coupling $D_{T}q$ with the 2nd dynamical equation
(LABEL:cont_2), and utilising all the above relations, we get

$\displaystyle\left(p_{r}+\rho+2\epsilon-4\eta\sigma\right)\left(1-\Lambda\right)D_{T}U=\left(1-\Lambda\right)F_{grav}+F_{hyd}$
$\displaystyle-H\left(q+\epsilon\right)\left[\frac{\left(1-f_{T}\right)}{2f_{R}f_{T}}D_{T}f_{R}+\frac{D_{T}f_{T}}{1+f_{T}}+2\frac{D_{T}B}{B}\right]-\frac{H^{2}\left(D_{C}f_{R}\right)\left(1-f_{T}\right)}{f_{R}f_{T}\left(1+f_{T}\right)}\left[\left(1+f_{T}\right)\left(p_{r}+\epsilon+4\eta\sigma\right)+\rho
f_{T}+\frac{f}{2}\right]$
$\displaystyle-2H\left(q+\epsilon\right)\left[\frac{U}{C}\right]+\frac{H^{2}KD_{C}T_{1}}{\tau}+Hq\left[\frac{1}{\tau}+\frac{\Theta_{1}}{2}\right]$
$\displaystyle+H\left[\frac{q}{2}D_{T}\left[\ln\left(\frac{\tau}{KT_{1}^{2}}\right)\right]-D_{T}\epsilon\right]+\frac{H^{2}f_{R}\left(1-f_{T}\right)}{2f_{T}\left(1+f_{T}\right)}\left[\frac{f_{T}D_{C}\rho}{f_{R}}+\frac{D_{C}f}{2f_{R}}-D_{C}R+\frac{\mathscr{Z}_{2}\left(t,r\right)}{H}\right]$

where

$\Lambda=\frac{KT_{1}}{\tau}\left(p_{r}+\rho+2\epsilon-4\eta\sigma\right)^{-1},$
(123)
$F_{grav}=-\left(p_{r}+\rho+2\epsilon-4\eta\sigma\right)\left[\frac{M}{C^{2}}+\frac{C}{2f_{R}}\left[\left(1+f_{T}\right)\left(p_{r}+\epsilon+4\eta\sigma\right)+\rho
f_{T}+\frac{1}{2}\left(f-RF\right)+\frac{D_{11}}{B^{2}}\right]\right],$ (124)

and

$F_{hyd}=-H^{2}\left[D_{C}\left(p_{r}+\epsilon-4\eta\sigma+\frac{T}{4}\right)+\frac{2}{C}\left(p_{r}-p_{\perp}+\epsilon+6\eta\sigma\right)\right].$
(125)

A close inspection of (VI) reveals that the effective inertial mass on the
left, and the passive gravitational mass on the right (which is associated
with $F_{grav}$), are both reduced by a factor of $\Lambda$, which arises as a
result of heat conduction. Hence the ratio of the gravitational mass and the
inertial mass remains unchanged, which is precisely what the weak equivalence
principle is. This result is in agreement with HPFT2009 . The hydrodynamic
force term $F_{hyd}$ remains unaffected by the thermal dissipative effects, as
its coefficient is unity. It represents the gradient of the effective pressure
which involves radiation pressure and also the shear viscosity. It also has a
term involving the pressure anistropy, radiation and shear. Since $U$ must be
negative for collapse to occur, the term
$-2H\left(q+\epsilon\right)\left[\frac{U}{C}\right]$ has a positive
contribution, and denotes the matter energy which is escaping in the form of
heat-flux and radiation in the free-streaming approximation, which reduces the
collapse rate. The terms involving the temperature gradient, the expansion
scalar, the thermal conductivity and the relaxation time all arise from the
$D_{T}q$ term coming from the transport equation. For $\Lambda=1$, the left
side of (VI) vanishes, and the coefficient of $F_{grav}$ reduces to zero.
Hence the passive gravitational mass density will no longer affect the
collapse. Also, for values of $\Lambda>1$, the inertial mass density on the
left hand side becomes negative, and the sign of $F_{grav}$ also becomes
negative. Hence at the critical value of $\Lambda=1$, we expect a bounce in
the system: values of $\Lambda$ less than 1 indicate increase in the inertial
mass, which implies collapse of the system, while those greater than 1 implies
that the system is undergoing expansion, due to a decrease in the inertial
mass AA3 . The terms
$\frac{H^{2}f_{R}\left(1-f_{T}\right)}{2f_{T}\left(1+f_{T}\right)}\left[\frac{D_{C}f}{2f_{R}}-D_{C}R+\frac{\mathscr{Z}_{2}\left(t,r\right)}{H}\right]$
on the right-hand side of equation (VI) arise purely from the geometry of the
space-time metric and represent the effective dark energy contribution of
$f(R,T)$ gravity to the collapsing process.

##  VII Summary and Outlook

In this paper, we have considered the most general non-rotating spherically
symmetric metric for the interior of the collapsing matter cloud, and have
studied the dynamical instability of the dissipative collapse in the $f(R,T)$
theory of gravity. An effective measure of the instability is available from
an examination of the limit for the ratio of specific heat, both in the
Newtonian approximation as well as the post-Newtonian approximation. Further,
we have coupled the dynamical equations with the transport equation in order
to examine the effects of dissipation on the evolution of the collapsing
system.

Our findings from the analysis of dynamical instability are listed below:

  * •

We see that in case of a non-dissipative, non-radiative fluid with no shear,
the differential equation describing the collapse becomes a homogeneous
equation and the solution is a negative exponential in $t$ similar to the
result obtained in 10 . The entire expression for $G(r,t)$ in (76) vanishes in
that case.

  * •

In the Newtonian regime, the term $H_{1}$ will be greater for a shear-free
fluid, similar to the result in 16 . It also increases if the heat flux and
radiation coefficients are zero. This is because the term involving time
derivatives of the heat flux and radiation coefficients contribute negatively
to the expression of $H_{1}$. It further increases in the case of isotropic
pressure.

  * •

In the case of Post-Newtonian Approximation, we find that the instability
range is greater than that in the Newtonian regime, since the denominator on
the right hand side of the inequality involving the ratio of specific heats
decreases, and the numerator of the inequality increases. This result is also
similar to that obtained in 10 . This conclusion matches with that of
Chandrasekhar C2  where it was shown that relativistic effects increase the
instability limit for the specific heat ratio.

  * •

Further, in the post-Newtonian regime, we also see that the term $H_{2}$
increases if all the following quantities vanishes, viz. the shear, radiation,
heat flux, and isotropic pressure.

  * •

However it must be mentioned that in order to ensure the positivity of all
terms in the expressions for $H_{1}$ and $H_{2}$, we have assumed that
$p_{r_{0}}<p_{\perp_{0}}$, which indeed increases the instability. However the
reverse condition, i.e., $p_{r_{0}}>p_{\perp_{0}}$ will lead to a reduction in
the instability compared to the case $p_{r_{0}}<p_{\perp_{0}}$ HS .

  * •

We also see, from the expression for the total energy trapped inside the fluid
surface, that the heat flux and the radiation coefficient contribute
negatively towards the total energy, since their coefficient involves
$\dot{C}$ which must be negative to ensure collapse of the fluid HS ; 14 .
This is consistent with the result that both $H_{1}$ and $H_{2}$ would have
increased if the heat flux and radiation coefficients were put to zero.

Hence it can be concluded that the simultaneous presence of heat flux,
radiation and shear decreases the instability range for the adiabatic index,
in both Newtonian and relativistic regimes. In other words, the adiabatic
index becomes more constrained in presence of dissipative, radiative and
viscous effects. Further, the pressure anisotropy also reduces the instability
bound. Dynamical instability for a spherical matter distribution with pressure
anisotropy, but without any shear, radiation or heat flux was studied in
$f(R,T)$ theory NZ1 . These authors also studied dynamical analysis for
collapse of bodies with axial symmetry and with pressure anisotropy in NZ2 ,
and the effect of the shear-free condition on the adiabatic index in NZ3  for
the same type of matter distribution with anisotropic pressure. Although the
dynamics of non-adiabatic charged spherical gravitational collapse of an
anisotropic fluid with heat flux have been studied recently in the framework
of $f(R,T)$ gravity AA3 , but the authors did not consider the effect of shear
and free-streaming radiation, nor analysed the instability conditions. For the
first time, we have studied the dynamical instability for the collapse of a
more generalised fluid involving shear, radiation and heat flux and in
$f(R,T)$ gravity, considering a linear coupling of the trace of the energy-
momentum tensor with the Starobinsky model of $f(R)$, and showed that the
instability range for the adiabatic index becomes more restricted in presence
of heat flux, shear viscosity and radiation.

We have also derived the transport equation for the given matter in the free
streaming approximation. By coupling it to the dynamical equation, we have
arrived at the following results:

  * •

Both the active inertial mass and the passive gravitational mass get reduced
by the same factor, in agreement with the equivalence principle, and with the
works by 12 ; HPFT2009 ; AbbasNazar2018 .

  * •

The hydrodynamic force term remains unaffected by the thermal dissipation
terms. Also, there is a reduction of mass-energy by the outgoing heat-flux and
radiation, which slows down the collapse, as is evident from the negative sign
before the term $-2H\left(q+\epsilon\right)\left[\frac{U}{C}\right]$, and the
fact that the collapse velocity $U$ must be negative.

  * •

Depending upon whether the value of $\Lambda$ in equation (VI) is greater than
1 or less than 1, the active inertial mass gets reduced or increased
accordingly, and the system undergoes expansion or collapse. The value
$\Lambda=1$ represents the critical point where the system undergoes a bounce
between expansion and collapse. At this point, the passive gravitational mass
does not affect the collapsing process AA3 . The hydrodynamic force term does
not get influenced by changing the values of $\Lambda$ .

There is scope for further investigation of dynamical instability and end
results of collapse of the same type of matter distribution in presence of an
electromagnetic field, which will be presented in a future work. We are also
investigating the thermodynamic behavior of such type of distribution.

## Acknowledgments

The initial part of this work was done earlier in IUCAA, India under the
associateship programme. SG gratefully acknowledges the warm hospitality and
the facilities of work at IUCAA. We also thank Dr. Subenoy Chakraborty and Dr.
Narayan Banerjee for their suggestions and comments. We thank the anonymous
reviewer for the useful suggestions.

## References

  * (1) S. Chandrasekhar, Mon. Not. R. Astr. Soc. 91, 456 (1931); ibid. 95, 207 (1935); Astrophys. J. 74, 81 (1931); Observatory 57, 373 (1934).
  * (2) C. Hansen and S. Kawaler, Stellar Interiors: Physical Principles, Structure and Evolution (Springer Verlag, 1994).
  * (3) R. Kippenhahn and A. Weigert, Stellar Structure and Evolution (Springer Verlag, 1990).
  * (4) E.N. Glass, Phys. Lett. A 86, 351 (1981).
  * (5) N. O. Santos, Mon. Not. R. Astron. Soc. 216, 403 (1985).
  * (6) A. K. G. De Oliveira, N. O. Santtos and C. A. Kolassis, Mon. Not. R. Astron. Soc. 216, 1001 (1985).
  * (7) A. K. G. de Oliveira, J. A. de F. Pacheco, and N. O. Santos, Mon. Not. R. Astron. Soc. 220, 405 (1986).
  * (8) L. Herrera, G. Le Denmat and N.O. Santos, Mon. Not. R. Astron. Soc. 237, 257 (1989).
  * (9) R. Chan, S. Kichenassamy, G. Le Denmat and N.O. Santos, Mon. Not. R. Astron. Soc. 239, 91 (1989).
  * (10) R. Chan, L. Herrera and N.O. Santos, Mon. Not. R. Astron. Soc. 265, 533 (1993).
  * (11) R. Chan, L. Herrera and N.O. Santos, Mon. Not. R. Astron. Soc. 267, 637 (1994).
  * (12) R. Chan, Mon. Not. R. Astron. Soc. 316, 588 (2000).
  * (13) L. Herrera and N.O. Santos, Phys. Rev. D 70, 084004 (2004).
  * (14) L. Herrera, N.O. Santos and G. Le Denmat, Gen. Relativ. Gravit. 44, 1143 (2012).
  * (15) M. Govender, K. S. Govinder and S. D. Maharaj, Int. Jour. Mod. Phys. D 12, 667 (2003).
  * (16) S. D. Maharaj, G. Govender and M. Govender, Gen. Relativ. Gravit. 44, 1089 (2012).
  * (17) M. Govender, R. Maartens and S. D. Maharaj, Mon. Not. R. Astron. Soc. 310, 557 (1999).
  * (18) S. D. Maharaj, G. Govender and M. Govender, PRAMANA 77, 469 (2011).
  * (19) S. Guha and R. Banerji, Int. J. Theor. Phys. 53, 2332 (2014).
  * (20) A. Di Prisco, L. Herrera, G. Le Denmat, M. A. H. MacCallum and N. O. Santos, Phys. Rev. D 76, 064017 (2007).
  * (21) S. Nojiri and S. D. Odintsov, Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007); Phys. Rept. 505, 59 (2011); S. Capozziello and M. De Laurentis, Phys. Rept. 509, 167 (2011); T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Phys. Rept. 513, 1 (2012); O. Bertolami and M. C. Sequeira, Phys. Rev. D 79, 104010 (2009); T. Harko, F. S. N. Lobo, G. Otalora and E. N. Saridakis, Phys. Rev. D 89, 124036 (2014); T. Harko, F. S. N. Lobo, G. Otalorac and E.N. Saridakis, JCAP 12, 021 (2014); M. Sharif and M. Zubair, J. High. Energy Phys. 12, 079 (2013).
  * (22) S. Nojiri and S. Odintsov, Phys. Rev. D 74, 086005 (2006); Gen. Rel. Grav. 36, 1765 (2004); S. Capozziello, Int. J. Mod. Phys. D 11, 483 (2002); S. M. Carroll, et al., Phys. Rev. D 71, 063513 (2005); S. Carloni, P. K. S. Dunsby, S. Capozziello and A. Troisi, Class. Quant. Grav. 22, 4839 (2005); J. A. R. Cembranos, Phys. Rev. D 73, 064029 (2006); T. Clifton and J. D. Barrow, Phys. Rev. D 72, 103005 (2005); K. Bamba, C. Q. Geng and C. C. Lee, JCAP 08, 021 (2010).
  * (23) A. de la Cruz-Dombriz and A. Dobado, Phys. Rev. D 74, 087501 (2006); P. K. S. Dunsby, E. Elizalde, R. Goswami, S. Odintsov and D. Sáez-Gómez, Phys. Rev. D 82, 023519 (2010); S. Nojiri and S. D. Odintsov, Phys. Rev. D 68, 123512 (2003).
  * (24) S. G. Ghosh and S.D. Maharaj, Phys. Rev. D 85, 124064 (2012).
  * (25) J. A. R. Cembranos, A. de la Cruz-Dombriz and B. M. Nunez, JCAP 04, 021 (2012).
  * (26) R. Goswami, A. M. Nzioki, S. D. Maharaj and S. G. Ghosh, Phys. Rev. D 90, 084011 (2014).
  * (27) S. Chakrabarti and N. Banerjee, Gen. Relativ. Gravit., 48, 57 (2016).
  * (28) M. Sharif and Z. Yousaf, Phys. Rev. D 88, 024020 (2013).
  * (29) M. Sharif and Z. Yousaf, Mon. Not. R. Astron. Soc. 434, 2529-2538 (2013).
  * (30) T. Harko, F. S. N. Lobo, S. Nojiri and S.D. Odinstov, Phys. Rev. D 84, 024020 (2011).
  * (31) T. Harko and F. S. N. Lobo, Galaxies 2, 410 (2014).
  * (32) T. Harko and F. S. N. Lobo, Eur. Phys. J. C 70, 373 (2010).
  * (33) T. Harko, Phys. Rev. D 81, 044021 (2010).
  * (34) A. A. Starobinsky, Phys. Lett. B 91, 99 (1980).
  * (35) M. Sharif and M. Zubair, J. Phys. Soc. Jpn. 81, 114005 (2012).
  * (36) F. G. Alvarenga, A. de la Cruz-Dombriz, M. J. S. Houndjo, M. E. Rodrigues and D. Sáez-Gómez, Phys. Rev. D 87, 103526 (2013).
  * (37) J. O. Barrientos and G.F. Rubilar, Phys. Rev. D 90, 028501 (2014).
  * (38) V. Faraoni, Phys. Rev. D 80, 124040 (2009).
  * (39) H. Shabani and M. Farhoudi, Phys. Rev. D. 90, 044031 (2014).
  * (40) M. Sharif and M. Zubair, JCAP 03, 028 (2012); M. Jamil, D. Momeni and R. Myrzakulov, Chin. Phys. Lett. 29, 109801 (2012).
  * (41) M. Sharif and M. Zubair, Gen. Relativ. Gravit., 46, 1723 (2014).
  * (42) M. J. S. Houndjo, Int. J. Mod. Phys. D 21, 1250003 (2012).
  * (43) S. D. Odintsov and D. Sáez-Gómez, Phys. Lett. B 725, 437 (2013).
  * (44) M. Sharif and Z. Yousaf, Astrophys. Space Sci. 354, 471 (2014).
  * (45) Ifra Noureen and M. Zubair, Astrophys. Space Sci. 356, 103 (2015).
  * (46) Ifra Noureen and M. Zubair, Eur. Phys. J. C 75, 62 (2015).
  * (47) Ifra Noureen, M. Zubair, A. A. Bhatti and G. Abbas, Eur. Phys. J. C 75, 323 (2015).
  * (48) Z. Yousaf, K. Bamba and M. Z. Bhatti, Phys. Rev. D 93, 124048 (2016).
  * (49) Z. Yousaf, K. Bamba and M. Z. Bhatti, Phys. Rev. D 93, 064059 (2016).
  * (50) Z. Yousaf, K. Bamba, M. Z. Bhatti and U. Ghafoor, Phys. Rev. D 100, 024062 (2019).
  * (51) Z. Yousaf, Phys. Dark Univ. 28, 100509 (2020).
  * (52) M. Z. Bhatti, Z. Yousaf and A. Rehman, Phys. Dark Univ. 29, 100561 (2020).
  * (53) M. Sharif and A. Siddiqa, Eur. Phys. J. Plus 132, 529 (2017).
  * (54) M. Sharif and A. Siddiqa, AHEP (2019), 8702795 (2019).
  * (55) S. Khan, M. S. Khan and A. Ali, Modern Physics Letters A 33, 1850065 (2018); M. Sharif and A. Anwar, Astrophys. Space Sci. 363, 123 (2018).
  * (56) M. Sharif and A. Waseem, Gen Relativ Gravit 50, 78 (2018).
  * (57) G. Abbas and R. Ahmed, Eur. Phys. J. C 77, 441 (2017).
  * (58) G. Abbas and R. Ahmed, Mod. Phys. Lett. A 34, 1950153 (2019).
  * (59) R. Ahmed and G. Abbas, Can. J. Phys. 97, 994 (2019).
  * (60) M. Zubair and H. Azmat, Phys. Dark Univ. 28, 100531 (2020).
  * (61) Z. Yousaf, M. Z. Bhatti and H. Asad, Phys. Dark Univ. 28, 100527 (2020).
  * (62) Z. Yousaf, M. Z. Bhatti and U. Farwa, Class. Quant. Grav. 34, 145002 (2017).
  * (63) Z. Yousaf, M. Z. Bhatti and U. Farwa, Mon. Not. R. Astron. Soc. 464, 45094519 (2017).
  * (64) Z. Yousaf, K. Bamba, M. Z. Bhatti and U. Farwa, Eur. Phys. J. A 54, 122 (2018).
  * (65) Z. Yousaf, Astrophys. Space Sci. 363, 226 (2018).
  * (66) Z. Yousaf, Eur. Phys. J. Plus 134, 245 (2019).
  * (67) Z. Yousaf, Mod. Phys. Lett. A 34, 1950333 (2019).
  * (68) Z. Yousaf, Eur. Phys. J. Plus 136, 281 (2021).
  * (69) C. W. Misner and D. Sharp, Phys. Rev. 136 B571 (1964).
  * (70) C.W. Misner, Phys. Rev. 137, B1360, (1965).
  * (71) R. Tolman, Phys. Rev. 35, 904 (1939).
  * (72) C. Eckart, Phys. Rev. 58, 919 (1940).
  * (73) L. Landau and E. Lifshitz, Fluid Mechanics (Pergamon Press, London, 1959).
  * (74) I. Muller, Z. Physik 198, 329 (1967).
  * (75) W. Israel, Ann. Phys. (N.Y.) 100, 310 (1976).
  * (76) W. Israel and J. Stewart, Phys. Lett. A 58, 213 (1976).
  * (77) W. Israel and J. Stewart, Ann. Phys. (N.Y.) 118, 341 (1979).
  * (78) L. Herrera, Int. J. Mod. Phys. D 15, 2197 (2006).
  * (79) L. Herrera and J. Martinez, Astrophys. Space Sci. 259, 235 (1998).
  * (80) L. Herrera, G. Le. Denmat, N. O. Santos and A. Wang, Int. J. Mod. Phys. D 13, 583 (2004).
  * (81) L. Herrera, A. D. Prisco, E. Fuenmayor and O. Troconis, Int. Jour. of Mod. Phys. D 18, 129, (2009).
  * (82) R. Maartens, Causal Thermodynamics in Relativity. Lectures given at the Hanno Rund Workshop on Relativity and Thermodynamics, University of Natal, June 1996. astro-ph/9609119.
  * (83) M. Sharif and Z. Rehmat, Gen Relativ Gravit 42, 1795 (2010).
  * (84) S. Chakraborty and S. Chakraborty, AHEP (2017), 8786791 (2017).
  * (85) S. M. Shah and G. Abbas, Eur. Phys. J. C 77, 251 (2017).
  * (86) J. M. Z. Pretel and M. F. A. da Silva, Gen. Relativ. Gravit. 51, 3 (2019).
  * (87) M. Sharif and H. R. Kausar, Journal of Physics: Conference Series 354, 012020 (2012).
  * (88) G. Abbas and H. Nazar, Adv. High Energy Phys. 2018, 9250786 (2018).
  * (89) J. M. Z. Pretel, S. E. Jorás, R. R. R. Reis and J. D. V. Arbañil, arXiv:2012.03342v1 [gr-qc].
  * (90) R. Ahmed and G. Abbas, Chinese Journal of Physics 65, 177 (2020).
  * (91) S. Thirukkanesh, S. S. Rajah and S. D. Maharaj, J. Math. Phys. 53, 032506 (2012).
  * (92) These calculations were done with the help of Maple program and GRTensor package: K. Lake and P. J. Musgrave, GRTensor (Queen’s University, Kingston, 2003).
  * (93) S. Chandrasekhar, Astrophys. J. 140, 417 (1964).
  * (94) L. Herrera and N.O. Santos, Phys. Rept. 286, 53 (1997).
  * (95) B.K. Harrison, K.S. Thorne, J.A. Wheeler, _Gravitation Theory and Gravitational Collapse_(University of Chicago Press, Chicago, 1965).

## Appendix

$\displaystyle\mathscr{Z}_{1}^{(P)}$ $\displaystyle=$
$\displaystyle\frac{1}{A_{0}B_{0}^{2}Z}\left[-\frac{2\alpha
eA_{0}^{\prime\prime}}{A_{0}}-\frac{2\bar{c}^{\prime}Z_{R_{0}}R_{0}^{\prime}}{r}+\frac{2\alpha
e\left\\{Z_{R_{0}R_{0}}\left(R_{0}^{\prime}\right)^{2}+Z_{R_{0}}R_{0}^{\prime\prime}\right\\}}{Z}-\frac{2\alpha
e^{\prime}Z_{R_{0}}R_{0}^{\prime}}{Z}+\frac{2\alpha
eA_{0}^{\prime}Z_{R_{0}}R_{0}^{\prime}}{A_{0}Z}\right]$ (126)
$\displaystyle+\frac{1}{A_{0}B_{0}^{2}Z}\left[\frac{2\alpha
e\left(B_{0}^{\prime}r-2B_{0}\right)\left(ZA_{0}^{\prime}-A_{0}Z_{R_{0}}R_{0}^{\prime}\right)}{A_{0}B_{0}Zr}+\frac{2\bar{c}A_{0}^{\prime}Z_{R_{0}}R_{0}^{\prime}}{A_{0}r}+\frac{bZ_{R_{0}}R_{0}^{\prime}\left[rZ_{R_{0}}R_{0}^{\prime}+2Z\right]}{B_{0}Zr}\right],$
$\displaystyle\mathscr{Z}_{2}^{(P)}$ $\displaystyle=$
$\displaystyle\frac{2\alpha
e^{\prime}A_{0}^{\prime\prime}+a^{\prime\prime}Z_{R_{0}}R_{0}^{\prime}}{A_{0}B_{0}^{3}Z}-\frac{2\bar{c}^{\prime\prime}Z_{R_{0}}R_{0}^{\prime}}{B_{0}^{3}Zr}+\frac{\left(Z_{R_{0}}R_{0}^{\prime}\right)^{2}}{A_{0}B_{0}^{3}Z^{2}r}\left(a^{\prime}r+\bar{c}^{\prime}A_{0}^{\prime}+\bar{c}^{\prime}A_{0}+2a\right)+\frac{2\alpha
e^{\prime}A_{0}^{\prime}B_{0}^{\prime}}{A_{0}B_{0}^{4}Z}$ (127)
$\displaystyle+\frac{4\alpha
e^{\prime}Z_{R_{0}}R_{0}^{\prime}}{A_{0}B_{0}^{3}Z^{2}r}\left(A_{0}^{\prime}r+2A_{0}\right)+\frac{2\left(b^{\prime}+\bar{c}^{\prime}B_{0}^{\prime}\right)Z_{R_{0}}R_{0}^{\prime}}{B_{0}^{4}Zr}+\frac{4\alpha
e^{\prime}B_{0}^{\prime}}{B_{0}^{4}Zr}+\frac{Z_{R_{0}}R_{0}^{\prime}\left(a^{\prime}B_{0}^{\prime}+b^{\prime}A_{0}^{\prime}\right)}{A_{0}B_{0}^{4}Z}.$

$\displaystyle H_{\perp}$
$\displaystyle=\frac{1}{1+\lambda}\left[\frac{Z}{r^{2}}\left(\frac{\left(a^{\prime\prime}r^{2}+2A_{0}^{\prime\prime}\bar{c}r+a^{\prime}r+\bar{c}^{\prime}A_{0}^{\prime}r+\bar{c}A_{0}^{\prime}\right)}{A_{0}B_{0}^{2}}+\frac{\bar{c}^{\prime\prime}r}{B_{0}}\right.\right.-\frac{\left(a^{\prime}B_{0}^{\prime}r^{2}+b^{\prime}A_{0}^{\prime}r^{2}+2\bar{c}rA_{0}^{\prime}B_{0}^{\prime}\right)}{A_{0}B_{0}^{3}}$
$\displaystyle\left.-\frac{\left(b^{\prime}r+\bar{c}^{\prime}B_{0}^{\prime}r+\bar{c}B_{0}^{\prime}\right)}{B_{0}^{3}}-\frac{2\bar{c}r}{Z}\left\\{\left(1+\lambda\right)p_{\perp_{0}}+\rho_{0}\lambda+\frac{R_{0}+\alpha
R_{0}^{2}+\lambda
T_{0}-R_{0}Z}{2}+\frac{D_{22}^{(P)}}{r^{2}}\right\\}\right)\left.-\frac{e\lambda}{2}+\alpha
R_{0}e\right],$ (128)

where

$D_{22}^{(P)}=\frac{2\alpha
R_{0}^{\prime}B_{0}^{\prime}r^{2}}{B_{0}^{3}}-\frac{2\alpha
r}{B_{0}^{2}}\left(R_{0}^{\prime}+R_{0}^{\prime\prime}r+\frac{R_{0}^{\prime}A_{0}^{\prime}r}{A_{0}}\right).$
(129)

$\displaystyle Q(r)$
$\displaystyle=\frac{1}{\left[\frac{2\lambda}{\left(-1+\lambda\right)}\frac{2\left(\bar{c}+\frac{br}{B_{0}}\right)}{B_{0}A_{0}^{2}r^{2}}+\frac{bZ_{R_{0}}R_{0}^{\prime}}{A_{0}^{2}B_{0}^{2}Z}\right]}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{B_{0}Z^{2}}\left(\lambda+\left(1+\lambda\right)\Gamma\frac{p_{r_{0}}}{\rho_{0}+p_{r_{0}}}\right)\frac{\left(-1+\lambda\right)}{2\lambda}\left[\frac{2\alpha
e}{Z}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)\right.\right.$
$\displaystyle-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)}\left\\{\frac{b}{B_{0}}\left(p_{r_{0}}+\rho_{0}\right)+\frac{2\bar{c}}{r}\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{e}{4}\right\\}\left.+\frac{\lambda
e}{2}+A_{0}Z\mathscr{Z}_{1}^{(P)}\right]$
$\displaystyle-\left[\frac{2\left(1+\lambda\right)}{B_{0}Zr}\left[\Gamma\frac{p_{r_{0}}}{\rho_{0}+p_{r_{0}}}+\frac{\lambda}{1+\lambda}\right]\frac{\left(-1+\lambda\right)}{2\lambda}\left[\frac{2\alpha
e}{Z}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)\right.\right.$
$\displaystyle-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)}\left\\{\frac{b}{B_{0}}\left(p_{r_{0}}+\rho_{0}\right)+\frac{2\bar{c}}{r}\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{e}{4}\right\\}$
$\displaystyle\left.+\frac{\lambda
e}{2}+A_{0}Z\mathscr{Z}_{1}^{(P)}\right]+\frac{2\left(1+\lambda\right)H_{\perp}}{B_{0}Zr}-\frac{A_{0}^{\prime}}{A_{0}B_{0}Z}\left(\Gamma\frac{p_{r_{0}}}{\rho_{0}+p_{r_{0}}}+1\right)\frac{\left(-1+\lambda\right)}{2\lambda}\left[\frac{2\alpha
e}{Z}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)\right.$
$\displaystyle-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)}\left\\{\frac{b}{B_{0}}\left(p_{r_{0}}+\rho_{0}\right)+\frac{2\bar{c}}{r}\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{e}{4}\right\\}\left.+\frac{\lambda
e}{2}+A_{0}Z\mathscr{Z}_{1}^{(P)}\right]$
$\displaystyle-\frac{\left(1+\lambda\right)}{B_{0}Z}\left[\Gamma\frac{p_{r_{0}}}{\rho_{0}+p_{r_{0}}}\frac{\left(-1+\lambda\right)}{2\lambda}\left[\frac{2\alpha
e}{Z}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)}\left\\{\frac{b}{B_{0}}\left(p_{r_{0}}+\rho_{0}\right)+\frac{2\bar{c}}{r}\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{e}{4}\right\\}\right.\right.$
$\displaystyle\left.\left.+\frac{\lambda
e}{2}+A_{0}Z\mathscr{Z}_{1}^{(P)}\right]\right]^{\prime}-\frac{\lambda}{B_{0}Z}\frac{\left(-1+\lambda\right)}{2\lambda}\left[\frac{2\alpha
e}{Z}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)\right.$
$\displaystyle\left.\left.-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)}\left\\{\frac{b}{B_{0}}\left(p_{r_{0}}+\rho_{0}\right)+\frac{2\bar{c}}{r}\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{e}{4}\right\\}+\frac{\lambda
e}{2}+A_{0}Z\mathscr{Z}_{1}^{(P)}\right]^{\prime}\right]\frac{2\lambda}{\left(-1+\lambda\right)}-\left[-\frac{Z_{R_{0}}R_{0}^{\prime}}{B_{0}Z^{2}}\frac{e}{2}\left(Z+\lambda\right)\right.$
$\displaystyle-\frac{2\alpha
e^{\prime}}{B_{0}Z^{2}}\left(\rho_{0}\lambda+p_{r_{0}}\left(1+\lambda\right)+\frac{R_{0}+\alpha
R_{0}^{2}+\lambda
T_{0}}{2}\right)+\frac{2\lambda}{\left(-1+\lambda\right)}\left[\frac{2\bar{c}^{\prime}}{B_{0}Zr}\left(1+\lambda\right)\left(p_{r_{0}}-p_{\perp_{0}}\right)\right.$
$\displaystyle+\frac{\left(1+\lambda\right)}{A_{0}B_{0}Z}\left[a^{\prime}\left(p_{r_{0}}+\rho_{0}\right)+ap_{r_{0}}^{\prime}\right]+\frac{\lambda
e^{\prime}}{2B_{0}Z}-\frac{\alpha
eR_{0}^{\prime}}{B_{0}Z}+\frac{eZ_{R_{0}}R_{0}^{\prime}}{2B_{0}Z}$
$\displaystyle\left.\left.\left.+\frac{a}{A_{0}B_{0}Z}\left\\{\rho_{0}^{\prime}\lambda-\frac{ZR_{0}^{\prime}}{2}-\frac{\left(R_{0}^{\prime}+2\alpha
R_{0}R_{0}^{\prime}+\lambda
T_{0}^{\prime}\right)}{2}\right\\}\right]+\frac{e^{\prime}\lambda\left(1+\lambda\right)}{2B_{0}Z\left(-1+\lambda\right)}+\mathscr{Z}_{2}^{(P)}\right]\right].$
(130)

$\displaystyle G(r,t)$
$\displaystyle=\frac{1}{\left[\frac{2\lambda}{\left(-1+\lambda\right)}\frac{2\left(\bar{c}+\frac{br}{B_{0}}\right)}{B_{0}A_{0}^{2}r^{2}}+\frac{bZ_{R_{0}}R_{0}^{\prime}}{A_{0}^{2}B_{0}^{2}Z}\right]}\left[-\frac{4\eta\left(1+\lambda\right)}{B_{0}Z}\left\\{\frac{2\lambda}{\left(-1+\lambda\right)}\left(\bar{\sigma}^{\prime}+\frac{\bar{\sigma}A_{0}^{\prime}}{A_{0}}+\frac{2\bar{\sigma}}{r}\right)-\frac{\bar{\sigma}Z_{R_{0}}R_{0}^{\prime}}{Z}\right\\}+\frac{Z_{R_{0}}R_{0}^{\prime}}{B_{0}Z^{2}}\left[\bar{\epsilon}\left(1+\lambda\right)\right.\right.$
$\displaystyle+\left(\lambda+\left(1+\lambda\right)\Gamma\frac{p_{r_{0}}}{\rho_{0}+p_{r_{0}}}\right)\left\\{\frac{\left(-1+\lambda\right)}{2\lambda}\frac{A_{0}}{B_{0}}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\frac{4\lambda}{\left(-1+\lambda\right)}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{1}{r}\right)\right]\left(1+\lambda\right)\int\left(\bar{q}+\bar{\epsilon}\right)dt-\left(1+\lambda\right)\bar{\epsilon}\right.$
$\displaystyle\left.\left.-\frac{A_{0}}{B_{0}}\left(1+\lambda\right)\int\left(\bar{\epsilon}^{\prime}+\bar{q}^{\prime}\right)dt\right\\}\right]$
$\displaystyle-\left[\frac{2\left(1+\lambda\right)}{B_{0}Zr}\left[\Gamma\frac{p_{r_{0}}}{\rho_{0}+p_{r_{0}}}+\frac{\lambda}{1+\lambda}\right]\left[\frac{\left(-1+\lambda\right)}{2\lambda}\frac{A_{0}}{B_{0}}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\frac{4\lambda}{\left(-1+\lambda\right)}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{1}{r}\right)\right]\left(1+\lambda\right)\int\left(\bar{q}+\bar{\epsilon}\right)dt\right.\right.$
$\displaystyle\left.-\frac{A_{0}}{B_{0}}\left(1+\lambda\right)\int\left(\bar{\epsilon}^{\prime}+\bar{q}^{\prime}\right)dt-\left(1+\lambda\right)\bar{\epsilon}\right]-\frac{\left(1+\lambda\right)\bar{\epsilon}^{\prime}}{B_{0}Z}-\frac{\left(1+\lambda\right)\left(\dot{\bar{\epsilon}}+\dot{\bar{q}}\right)}{A_{0}Z}-\frac{2\left(1+\lambda\right)\bar{\epsilon}}{B_{0}Zr}-\frac{2\bar{\epsilon}A_{0}^{\prime}}{A_{0}B_{0}Z}$
$\displaystyle-\frac{A_{0}^{\prime}}{A_{0}B_{0}Z}\left(\Gamma\frac{p_{r_{0}}}{\rho_{0}+p_{r_{0}}}+1\right)\left[\frac{\left(-1+\lambda\right)}{2\lambda}\frac{A_{0}}{B_{0}}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\frac{4\lambda}{\left(-1+\lambda\right)}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{1}{r}\right)\right]\left(1+\lambda\right)\int\left(\bar{q}+\bar{\epsilon}\right)dt\right.$
$\displaystyle\left.-\frac{A_{0}}{B_{0}}\left(1+\lambda\right)\int\left(\bar{\epsilon}^{\prime}+\bar{q}^{\prime}\right)dt-\left(1+\lambda\right)\bar{\epsilon}\right]$
$\displaystyle-\frac{\left(1+\lambda\right)}{B_{0}Z}\left[\Gamma\frac{p_{r_{0}}}{\rho_{0}+p_{r_{0}}}\left[\frac{\left(-1+\lambda\right)}{2\lambda}\frac{A_{0}}{B_{0}}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\frac{4\lambda}{\left(-1+\lambda\right)}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{1}{r}\right)\right]\left(1+\lambda\right)\int\left(\bar{q}+\bar{\epsilon}\right)dt\right.\right.$
$\displaystyle\left.\left.-\left(1+\lambda\right)\bar{\epsilon}-\frac{A_{0}}{B_{0}}\left(1+\lambda\right)\int\left(\bar{\epsilon}^{\prime}+\bar{q}^{\prime}\right)dt\right]\right]^{\prime}$
$\displaystyle-\frac{\lambda}{B_{0}Z}\left[\frac{\left(-1+\lambda\right)}{2\lambda}\frac{A_{0}}{B_{0}}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\frac{4\lambda}{\left(-1+\lambda\right)}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{1}{r}\right)\right]\left(1+\lambda\right)\int\left(\bar{q}+\bar{\epsilon}\right)dt\right.$
$\displaystyle\left.\left.\left.-\left(1+\lambda\right)\bar{\epsilon}-\frac{A_{0}}{B_{0}}\left(1+\lambda\right)\int\left(\bar{\epsilon}^{\prime}+\bar{q}^{\prime}\right)dt\right]^{\prime}\right]\frac{2\lambda}{\left(-1+\lambda\right)}\right].$
(131)

The explicit form of equation (76) is obtained in the form

$\displaystyle\frac{4\eta\left(1+\lambda\right)}{B_{0}Z}\left[\frac{2\lambda}{\left(-1+\lambda\right)}\left(\bar{\sigma}^{\prime}+\frac{\bar{\sigma}A_{0}^{\prime}}{A_{0}}+\frac{2\bar{\sigma}}{r}\right)-\frac{\bar{\sigma}Z_{R_{0}}R_{0}^{\prime}}{Z}\right]-\frac{Z_{R_{0}}R_{0}^{\prime}}{B_{0}Z^{2}}\left[\bar{\epsilon}\left(1+\lambda\right)+\right.$
$\displaystyle\left(\lambda+\left(1+\lambda\right)\Gamma\frac{p_{r_{0}}}{\rho_{0}+p_{r_{0}}}\right)\left[\frac{\left(-1+\lambda\right)}{2\lambda}\frac{A_{0}}{B_{0}}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\frac{4\lambda}{\left(-1+\lambda\right)}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{1}{r}\right)\right]\left(1+\lambda\right)\int\left(\bar{q}+\bar{\epsilon}\right)dt\right.$
$\displaystyle-\left(1+\lambda\right)\bar{\epsilon}-\frac{A_{0}}{B_{0}}\left(1+\lambda\right)\int\left(\bar{\epsilon}^{\prime}+\bar{q}^{\prime}\right)dt+D\frac{\left(-1+\lambda\right)}{2\lambda}\left[\frac{2\alpha
e}{Z}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)\right.$
$\displaystyle\left.\left.\left.-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)}\left\\{\frac{b}{B_{0}}\left(p_{r_{0}}+\rho_{0}\right)+\frac{2\bar{c}}{r}\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{e}{4}\right\\}+\frac{\lambda
e}{2}+A_{0}Z\mathscr{Z}_{1}^{(P)}\right]\right]\right]$
$\displaystyle+\frac{2\lambda}{\left(-1+\lambda\right)}\left[\frac{2\left(1+\lambda\right)\bar{\epsilon}}{B_{0}Zr}+\frac{2\left(1+\lambda\right)}{B_{0}Zr}\left[\Gamma\frac{p_{r_{0}}}{\rho_{0}+p_{r_{0}}}+\frac{\lambda}{1+\lambda}\right]\left[\frac{\left(-1+\lambda\right)}{2\lambda}\frac{A_{0}}{B_{0}}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\right.\right.\right.$
$\displaystyle\left.\frac{4\lambda}{\left(-1+\lambda\right)}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{1}{r}\right)\right]\left(1+\lambda\right)\int\left(\bar{q}+\bar{\epsilon}\right)dt-\frac{A_{0}}{B_{0}}\left(1+\lambda\right)\int\left(\bar{\epsilon}^{\prime}+\bar{q}^{\prime}\right)dt-\left(1+\lambda\right)\bar{\epsilon}$
$\displaystyle+D\frac{\left(-1+\lambda\right)}{2\lambda}\left[\frac{2\alpha
e}{Z}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)}\left\\{\frac{b}{B_{0}}\left(p_{r_{0}}+\rho_{0}\right)+\frac{2\bar{c}}{r}\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{e}{4}\right\\}\right.$
$\displaystyle\left.\left.+\frac{\lambda
e}{2}+A_{0}Z\mathscr{Z}_{1}^{(P)}\right]\right]-\frac{2\left(1+\lambda\right)DH_{\perp}}{B_{0}Zr}+\frac{2\ddot{D}\left(\bar{c}+\frac{br}{B_{0}}\right)}{B_{0}A_{0}^{2}r^{2}}+\frac{2\bar{\epsilon}A_{0}^{\prime}}{A_{0}B_{0}Z}$
$\displaystyle+\frac{A_{0}^{\prime}}{A_{0}B_{0}Z}\left(\Gamma\frac{p_{r_{0}}}{\rho_{0}+p_{r_{0}}}+1\right)\left[\frac{\left(-1+\lambda\right)}{2\lambda}\frac{A_{0}}{B_{0}}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\frac{4\lambda}{\left(-1+\lambda\right)}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{1}{r}\right)\right]\left(1+\lambda\right)\int\left(\bar{q}+\bar{\epsilon}\right)dt\right.$
$\displaystyle-\frac{A_{0}}{B_{0}}\left(1+\lambda\right)\int\left(\bar{\epsilon}^{\prime}+\bar{q}^{\prime}\right)dt-\left(1+\lambda\right)\bar{\epsilon}$
$\displaystyle+D\frac{\left(-1+\lambda\right)}{2\lambda}\left[\frac{2\alpha
e}{Z}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)}\left\\{\frac{b}{B_{0}}\left(p_{r_{0}}+\rho_{0}\right)+\frac{2\bar{c}}{r}\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{e}{4}\right\\}\right.$
$\displaystyle\left.\left.+\frac{\lambda
e}{2}+A_{0}Z\mathscr{Z}_{1}^{(P)}\right]\right]+\frac{\left(1+\lambda\right)\bar{\epsilon}^{\prime}}{B_{0}Z}+\frac{\left(1+\lambda\right)\left(\dot{\bar{\epsilon}}+\dot{\bar{q}}\right)}{A_{0}Z}$
$\displaystyle+\frac{\left(1+\lambda\right)}{B_{0}Z}\left[\Gamma\frac{p_{r_{0}}}{\rho_{0}+p_{r_{0}}}\left[\frac{\left(-1+\lambda\right)}{2\lambda}\frac{A_{0}}{B_{0}}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\frac{4\lambda}{\left(-1+\lambda\right)}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{1}{r}\right)\right]\left(1+\lambda\right)\int\left(\bar{q}+\bar{\epsilon}\right)dt\right.\right.$
$\displaystyle-\left(1+\lambda\right)\bar{\epsilon}-\frac{A_{0}}{B_{0}}\left(1+\lambda\right)\int\left(\bar{\epsilon}^{\prime}+\bar{q}^{\prime}\right)dt$
$\displaystyle+D\frac{\left(-1+\lambda\right)}{2\lambda}\left[\frac{2\alpha
e}{Z}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)}\left\\{\frac{b}{B_{0}}\left(p_{r_{0}}+\rho_{0}\right)+\frac{2\bar{c}}{r}\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{e}{4}\right\\}\right.$
$\displaystyle\left.\left.\left.+\frac{\lambda
e}{2}+A_{0}Z\mathscr{Z}_{1}^{(P)}\right]\right]\right]^{\prime}+\frac{\lambda}{B_{0}Z}\left[\frac{\left(-1+\lambda\right)}{2\lambda}\frac{A_{0}}{B_{0}}\left[\frac{Z_{R_{0}}R_{0}^{\prime}}{Z}-\frac{4\lambda}{\left(-1+\lambda\right)}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{1}{r}\right)\right]\left(1+\lambda\right)\int\left(\bar{q}+\bar{\epsilon}\right)dt\right.$
$\displaystyle-\left(1+\lambda\right)\bar{\epsilon}-\frac{A_{0}}{B_{0}}\left(1+\lambda\right)\int\left(\bar{\epsilon}^{\prime}+\bar{q}^{\prime}\right)dt$
$\displaystyle+D\frac{\left(-1+\lambda\right)}{2\lambda}\left[\frac{2\alpha
e}{Z}\left(\rho_{0}-\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)-\frac{2\lambda\left(1+\lambda\right)}{\left(-1+\lambda\right)}\left\\{\frac{b}{B_{0}}\left(p_{r_{0}}+\rho_{0}\right)+\frac{2\bar{c}}{r}\left(\rho_{0}+p_{\perp_{0}}\right)-\frac{e}{4}\right\\}\right.$
$\displaystyle\left.\left.\left.+\frac{\lambda
e}{2}+A_{0}Z\mathscr{Z}_{1}^{(P)}\right]\right]^{\prime}\right]+\frac{\ddot{D}bZ_{R_{0}}R_{0}^{\prime}}{A_{0}^{2}B_{0}^{2}Z}+D\left[-\frac{Z_{R_{0}}R_{0}^{\prime}}{B_{0}Z^{2}}\frac{e}{2}\left(Z+\lambda\right)-\frac{2\alpha
e^{\prime}}{B_{0}Z^{2}}\left(\rho_{0}\lambda+p_{r_{0}}\left(1+\lambda\right)+\frac{R_{0}+\alpha
R_{0}^{2}+\lambda T_{0}}{2}\right)\right.$
$\displaystyle+\frac{2\lambda}{\left(-1+\lambda\right)}\left[\frac{2\bar{c}^{\prime}}{B_{0}Zr}\left(1+\lambda\right)\left(p_{r_{0}}-p_{\perp_{0}}\right)\right.+\frac{\left(1+\lambda\right)}{A_{0}B_{0}Z}\left[a^{\prime}\left(p_{r_{0}}+\rho_{0}\right)+ap_{r_{0}}^{\prime}\right]+\frac{\lambda
e^{\prime}}{2B_{0}Z}-\frac{\alpha
eR_{0}^{\prime}}{B_{0}Z}+\frac{eZ_{R_{0}}R_{0}^{\prime}}{2B_{0}Z}$
$\displaystyle\left.\left.+\frac{a}{A_{0}B_{0}Z}\left\\{\rho_{0}^{\prime}\lambda-\frac{ZR_{0}^{\prime}}{2}-\frac{\left(R_{0}^{\prime}+2\alpha
R_{0}R_{0}^{\prime}+\lambda
T_{0}^{\prime}\right)}{2}\right\\}\right]+\frac{e^{\prime}\lambda\left(1+\lambda\right)}{2B_{0}Z\left(-1+\lambda\right)}+\mathscr{Z}_{2}^{(P)}\right]=0.$
(132)

$\displaystyle H_{1}$
$\displaystyle=-\frac{4\eta\left(1+\lambda\right)}{Z}\left[\frac{2\lambda}{\left(-1+\lambda\right)}\left(\bar{\sigma}^{\prime}+\frac{2\bar{\sigma}}{r}\right)-\frac{\bar{\sigma}Z_{R_{0}}R_{0}^{\prime}}{Z}\right]+\frac{Z_{R_{0}}R_{0}^{\prime}}{Z^{2}}\left[\bar{\epsilon}\left(1+\lambda\right)+\lambda\bar{\rho}_{N}\right]$
$\displaystyle+\frac{2\lambda}{\left(-1+\lambda\right)}\left[-\frac{2\bar{\epsilon}\left(1+\lambda\right)}{Zr}\right.-\frac{2\lambda\bar{\rho}_{N}}{Zr}+\frac{2\left(1+\lambda\right)H_{\perp_{N}}}{Zr}\frac{1}{2\sqrt{Q}}\left[e^{\sqrt{Q}t}\int
Ge^{-\sqrt{Q}t}dt-e^{-\sqrt{Q}t}\int Ge^{\sqrt{Q}t}dt\right]$
$\displaystyle-\left[G+\frac{\sqrt{Q}}{2}\left[e^{\sqrt{Q}t}\int
Ge^{-\sqrt{Q}t}dt-e^{-\sqrt{Q}t}\int
Ge^{\sqrt{Q}t}dt\right]\right]\left[\frac{2\left(\bar{c}+br\right)}{r^{2}}+\frac{\left(-1+\lambda\right)}{2\lambda}\frac{bZ_{R_{0}}R_{0}^{\prime}}{Z}\right]$
$\displaystyle\left.-\frac{\left(1+\lambda\right)}{Z}\left[\bar{\epsilon}^{\prime}+\dot{\bar{\epsilon}}+\dot{\bar{q}}\right]-\frac{\lambda\bar{\rho}_{N}^{\prime}}{Z}\right]$
$\displaystyle-\frac{1}{2\sqrt{Q}}\left[e^{\sqrt{Q}t}\int
Ge^{-\sqrt{Q}t}dt-e^{-\sqrt{Q}t}\int
Ge^{\sqrt{Q}t}dt\right]\left[-\frac{Z_{R_{0}}R_{0}^{\prime}e\left(Z+\lambda\right)}{2Z^{2}}\right.$
$\displaystyle-\frac{2\alpha
e^{\prime}}{Z^{2}}\left(\lambda\rho_{0}+\frac{R_{0}+\alpha R_{0}^{2}+\lambda
T_{0}}{2}\right)$
$\displaystyle+\frac{2\lambda}{\left(-1+\lambda\right)}\left[\frac{2\bar{c}^{\prime}}{Zr}\left(1+\lambda\right)\left(p_{r_{0}}-p_{\perp_{0}}\right)+\frac{a^{\prime}\left(1+\lambda\right)\rho_{0}}{Z}+\frac{\lambda
e^{\prime}}{2Z}-\frac{\alpha eR_{0}^{\prime}}{Z}\right.$
$\displaystyle\left.\left.+\frac{eZ_{R_{0}}R_{0}^{\prime}}{2Z}+\frac{a}{Z}\left\\{\rho_{0}^{\prime}\lambda-\frac{ZR_{0}^{\prime}}{2}-\frac{\left(R_{0}^{\prime}+2\alpha
R_{0}R_{0}^{\prime}+\lambda
T_{0}^{\prime}\right)}{2}\right\\}+\mathscr{Z}_{2_{N}}^{(P)}\right]\right].$
(133)

$\displaystyle H_{2}$ $\displaystyle=-\frac{4\eta
r\left(1+\lambda\right)}{\left(r+m_{0}\right)Z}\left[\frac{2\lambda}{\left(-1+\lambda\right)}\left(\bar{\sigma}^{\prime}+\frac{m_{0}\bar{\sigma}}{r\left(r-m_{0}\right)}+\frac{2\bar{\sigma}}{r}\right)-\frac{\bar{\sigma}Z_{R_{0}}R_{0}^{\prime}}{Z}\right]$
$\displaystyle+\frac{Z_{R_{0}}R_{0}^{\prime}r}{\left(r+m_{0}\right)Z^{2}}\left[\bar{\epsilon}\left(1+\lambda\right)+\lambda\bar{\rho}_{pN}\right]+\frac{2\lambda}{\left(-1+\lambda\right)}\left[-\frac{2\bar{\epsilon}\left(1+\lambda\right)}{\left(r+m_{0}\right)Z}\right.$
$\displaystyle-\frac{2\lambda\bar{\rho}_{pN}}{\left(r+m_{0}\right)Z}+\frac{2\left(1+\lambda\right)H_{\perp_{pN}}}{\left(r+m_{0}\right)Z}\frac{1}{2\sqrt{Q}}\left[e^{\sqrt{Q}t}\int
Ge^{-\sqrt{Q}t}dt-e^{-\sqrt{Q}t}\int Ge^{\sqrt{Q}t}dt\right]$
$\displaystyle-\left[G+\frac{\sqrt{Q}}{2}\left[e^{\sqrt{Q}t}\int
Ge^{-\sqrt{Q}t}dt-e^{-\sqrt{Q}t}\int
Ge^{\sqrt{Q}t}dt\right]\right]\left(\frac{2r\left(\bar{c}+\frac{br^{2}}{r+m_{0}}\right)}{\left(r+m_{0}\right)\left(r-m_{0}\right)^{2}}+\frac{\left(-1+\lambda\right)}{2\lambda}\frac{bZ_{R_{0}}R_{0}^{\prime}r^{4}}{\left(r^{2}-m_{0}^{2}\right)^{2}Z}\right)$
$\displaystyle-\frac{\left(1+\lambda\right)r}{Z}\left[\frac{\bar{\epsilon}^{\prime}}{\left(r+m_{0}\right)}+\frac{\dot{\bar{\epsilon}}+\dot{\bar{q}}}{\left(r-m_{0}\right)}\right]-\frac{m_{0}}{\left(r^{2}-m_{0}^{2}\right)Z}\left(2\bar{\epsilon}+\bar{\rho}_{pN}\right)-\frac{\lambda
r\bar{\rho}_{pN}^{\prime}}{\left(r+m_{0}\right)Z}$
$\displaystyle-\frac{1}{2\sqrt{Q}}\left[e^{\sqrt{Q}t}\int
Ge^{-\sqrt{Q}t}dt-e^{-\sqrt{Q}t}\int
Ge^{\sqrt{Q}t}dt\right]\left[-\frac{Z_{R_{0}}R_{0}^{\prime}re\left(Z+\lambda\right)}{2Z^{2}\left(r+m_{0}\right)}\right.$
$\displaystyle-\frac{2\alpha
e^{\prime}r}{\left(r+m_{0}\right)Z^{2}}\left(\lambda\rho_{0}+p_{r_{0}}\left(1+\lambda\right)+\frac{R_{0}+\alpha
R_{0}^{2}+\lambda
T_{0}}{2}\right)+\frac{2\lambda}{\left(-1+\lambda\right)}\left[\frac{2\bar{c}^{\prime}}{Z\left(r+m_{0}\right)}\left(1+\lambda\right)\left(p_{r_{0}}-p_{\perp_{0}}\right)\right.$
$\displaystyle+\frac{a^{\prime}r^{2}\left(1+\lambda\right)\left(p_{r_{0}}+\rho_{0}\right)}{\left(r^{2}-m_{0}^{2}\right)Z}+\frac{\lambda
e^{\prime}r}{2Z\left(r+m_{0}\right)}-\frac{\alpha
erR_{0}^{\prime}}{Z\left(r+m_{0}\right)}+\frac{erZ_{R_{0}}R_{0}^{\prime}}{2\left(r+m_{0}\right)Z}$
$\displaystyle\left.\left.+\frac{ar^{2}}{Z\left(r^{2}-m_{0}^{2}\right)}\left\\{\rho_{0}^{\prime}\lambda-\frac{ZR_{0}^{\prime}}{2}-\frac{\left(R_{0}^{\prime}+2\alpha
R_{0}R_{0}^{\prime}+\lambda
T_{0}^{\prime}\right)}{2}\right\\}\right]+\frac{e^{\prime}\lambda\left(1+\lambda\right)r}{2\left(-1+\lambda\right)\left(r+m_{0}\right)Z}+\mathscr{Z}_{2_{pN}}^{(P)}\right].$
(134)