# Role of Structure Scalars on the evolution of Compact Objects in Palatini
$f(R)$ Gravity

M. Z. Bhatti , Z. Yousaf , and Z. Tariq
Department of Mathematics, University of the Punjab,
Quaid-i-Azam Campus, Lahore-54590, Pakistan
mzaeem.math@pu.edu.pkzeeshan.math<EMAIL_ADDRESS>
###### Abstract

The utmost concern of this article is the construction of modified scalar
functions (structure scalars) by taking Palatini $f(R)$ gravitational theory
into account. At first, a general formalism is established in which we assess
gravitational stellar equations by putting into use the Palatini’s technique.
Later, from the perspective of tilted observer, we Lorentz boosted the
components of energy-momentum tensor using relative velocity $\omega$. To
examine the physical as well as mathematical aspects of the fluid source, we
carry out a detailed analysis of kinematical variables by evaluating shear
tensor and scalar, four-acceleration and expansion scalar. For the fluid
content inside our spherical star, we inferred the mass function (geometric
mass) and the active gravitational mass. Raychaudhuri equation, Bianchi
identities in addition to few other equations are worked out to discern the
structure formation and analyze the object’s evolutionary stages. The Riemann
tensor is then broken up orthogonally to set up few scalar functions connected
with fundamental physical characteristics of the fluid source like energy
density, effects of tidal forces and anisotropic stresses etc.

Keywords: Structure Scalars, Palatini Formalism, Anisotropic fluids; self-
gravitating systems.
PACS: 04.40.Nr, 04.20.Fy, 04.20.Jb, 04.40.Dg.

##  1 Introduction

Pursuant to the big bang theory, our universe was once an infinitely dense and
extraordinarily hot small point resembling a mathematical infinity. Since, the
mathematical infinities are irrational when it comes to physical equations, so
the comprehension of universe at that infinite scale is laborious. Due to some
unidentified reasons, the space started inflating, i.e., it endured an
exponential expansion phase driving apart all the constituents of the
universe. Owing to such expansion of the space, the temperature dropped down
which resulted in the collision of protons and neutrons present in the space.
This phenomenon produced Deuterium (an isotope of hydrogen gas) which in turn
combined with each other to produce Helium (elementary ingredient for Star
formation). Observations from NASA demonstrated that minor fluctuations in the
matter density generated a gravitational pull that engendered an enormous
cosmic web-like configuration of stars and space. Pulling in more and more
material from space via this gravitational pull, the dense regions became more
massive leading to the formation of stellar structures e.g., stars, galaxies,
clusters and super-clusters of galaxies. Low density regions, on the other
hand, showed no growth rather they evolved into emptiness known as void in
space. The gravitational pull boosting the structure formation process was so
intense that it decelerated the expansion phenomenon. Couple of years later,
it was observed that universe is actually accelerating inexplicably, the root
cause of which is an unidentified driving force termed as dark energy (DE).
Its only physical characteristic identified till now is that it is a repulsive
force exerting negative pressure on the objects.

Another oddity that prevails the space is the ’missing mass’ required to
maintain the stars orbiting around the galaxy’s center in addition to the
baryonic mass. Such mass (termed as dark matter (DM)) shows no interaction
with electromagnetic radiation, light or any other detective source which
makes it complicated to detect. General relativity (GR) forecasted DE and DM
but it cannot explicate both of them physically and mathematically. Many
theoretical frameworks have been constructed [43, 45] to account for them but
no theory till now is able to entirely explicate them. One of the credible
approach is the introduction of substitutive gravitational theories hinged on
the amendment of Einstein-Hilbert action integral by substituting a generic
function $f(R)$ of Ricci scalar instead of Ricci scalar itself. Performing the
variation of such action with the connection symbol and the metric tensor
assuming their independence relation imparts the modified gravitational field
equations which potentially explicate DM. Such theories, based upon the
generic model employed, elucidates distinct eras of evolutionary phases of our
universe.

Seifert [1] investigated the stability of static solutions of a spherically
symmetric matter content by making use of generalized variational principle in
three substitutive gravitational theories. They concluded that $f(R)$
gravitational theory behaves like a highly unstable theory when the matter
content is present. Lobo [2] reviewed certain alternative gravitational
theories that cope with two current cosmological problems, DM and DE. They
proposed that the galactic kinematics can be explicated using alternative
gravitational theories without the introduction of DM terms. Multamäki and
Vijla [3] studied the empty space solutions for static spherically symmetric
matter content by taking into consideration the $f(R)$ theories of gravity. By
reducing the number of modified field equations to one, they demonstrated that
exact solutions can be rendered corresponding to distinct $f(R)$ gravity
models. Yousaf et al. [4] analyzed the dynamics of compact objects in modified
theory via structure scalars. Taking into consideration a heat emitting
shearing matter content, they evaluated structure scalars by breaking up the
Riemann tensor orthogonally in $f(R,T)$ gravity. It is inferred that by making
use of the modified scalar functions, the evolutionary behavior of realistic
astrophysical objects can be discussed. Olmo [5] over-viewed the Newtonian
framework and devoted their attention towards Nordström’s scalar theory and
its impact on Einstein’s GR. They also presented the cosmic speed-up issue and
inferred that several quantum gravity issues can be well addressed by the
Palatini $f(R)$ gravity and $f(R,Q)$ theories [6].

In order to establish consistent and viable models of relativistic bodies, the
interior region is matched with the exterior region at a boundary known as
hypersurface. Such matching leads to certain conditions termed as junction
conditions. Olmo and Garcia [7] evaluated the matching conditions in $f(R)$
gravity using a tensor distributional technique and inferred that few of such
restrictions deviate from standard ones. They also illustrated the
significance of these restrictions by taking into consideration various
characteristics of stellar objects in polytropes. Their work can be used to
safely model Neutron stars and White dwarfs in Palatini $f(R)$ gravity. Yousaf
et al. described the role of modified gravity terms including Palatini $f(R)$
[8, 9, 10] and $f(G,T)$ [11, 12, 13] corrections on the stability of
homogeneous and inhomogeneous relativistic stars. Senovilla [14] inferred
matching conditions in modified gravity i.e. $f(R)$ to explicate matter shells
and braneworld. For this purpose, author acquired the modified field equations
for the stress-energy tensor on the shell and deduced that it was similar to
that found for GR. Huber [15] proposed a framework based on the geometric
deformations to address the issue of joining two spacetime manifolds having
distinct physical characteristics and geometries. The author exhibited that
such a geometric technique assists in the depiction of spacetimes in GR.
Deruelle et al. [16] generalized the matching conditions for $f(R)$ theories
by directly integrating the Einstein’s equations. This is possible because of
the equivalent relation between GR coupled to scalar field and $f(R)$ theory.
Bonnor and Vickers [17] analyzed certain junction conditions in GR and showed
the existence of equivalence relation between few of them. Out of all the
conditions, it was discovered that the Darmois conditions are the most
trustworthy and easy to handle.

An algebraic expression that assists in assessment of total energy of the
gravitating body is called the active gravitational mass. Herrera et al. [18]
acquired an algebraic expression for the Tolman mass of a collapsing matter
content. This expression brings into light the part played by the energy
density inhomogeneity and anisotropy in the collapsing process of the
spherical object. It was deduced that the energy density inhomogeneity
increases in the Tolman mass expression during the collapsing phenomena and
local anisotropy assists in the development of singularities. Bonnor [19]
calculated the active and passive gravitational masses of a static spherically
symmetric fluid content having uniform density. The results exhibit that the
active gravitational mass is much smaller than the passive one. The dynamics
of such a sphere are discussed in detail. Devitt and Florides [20] worked out
an expression for modified active gravitational mass of a spherically
symmetric fluid content having no dependence on temporal coordinate. They
enlisted certain characteristics of the modified formula thus obtained
including the fact that such an expression provides the correct mass of the
overall system. Herrera et al. [21] attained an algebraic expression for the
Tolman mass of a source just after its withdrawal from equilibrium phase. For
the case of compact sources, it is noticed that small withdrawal from
sphericity results in a substantial change in Tolman mass of the considered
system as compared to the value in equilibrium phase. Misner and Putnam [22]
scrutinized a paradox based on the active gravitational mass of a spherical
source for the case when interior gravitational influences are insignificant.

The changes induced in a star’s interior with the passage of time is termed as
stellar evolution process. Schaab et al [23] applied the theoretical equation
of state to analyze the cooling characteristics of compact astronomical
objects and tested a number of uncertainties including the dynamics of
coupling strength of matter and the super fluidity. Althaus et al. [24]
reviewed the physical characteristics of white dwarfs with focus on the
processes involved in their formation. The pulsational characteristics of
white dwarfs and their implications in modern cosmology are also investigated.
Chabrier and Baraffe [25] analyzed the structural formation and evolutionary
behavior of low-mass stars with special focus on the thermal and mechanical
characteristics of such objects. They also demonstrated mass-radius luminosity
relationship via numerical tabulation for different ages of these stars and
observed that the minimum masses are much smaller than previous
approximations. Sakashita and Hayashi [26] computed few models of highly
massive starts and concluded that such stars evolve in a same manner as low-
mass stars after their formation.

A persistent battle between the force of gravity and pressure helps a stellar
object in maintaining its physical state. When one of these forces supersedes
the other one, stellar collapse occurs. Astashenok et al. [27] carried out a
comparative analysis of the collapsing phenomena for perfect homogeneous
matter content via distinct equations of state in GR and $R$-squared
gravitational theory. Considering the cases of radiating and stiff matter,
they also found the possibility of presence of vacuum energy along with its
impact on the collapsing phenomena. Chiba [28] explored the dust collapse for
a cylindrical symmetry and presented a novel theoretical approach to analyze
the singularity formation in such symmetry. They concluded that negligible
gravitational waves are emitted within the free fall time. Taking into account
the quantum mechanical influences, Balakrishna et al [29] modeled a collapsing
star into a black hole. They derived certain solutions for the path of the
practical on the collapsing star’s surface in crucial and Schwarzschild
coordinators. Fryer and New [30] over-viewed all the possible collapsing
sources of gravitational waves including the collapsing phenomena of white
dwarf. They also investigated the collapse of Black holes and neutron stars in
addition with other super massive stars.

For a detailed examination of fluid characteristics, few kinematical variables
are ascertained. These include the four-acceleration, the vorticity vector,
shear tensor and expansion scalar etc. Keeping in view the quasistatic
approximation, Herrera and Santos [31] inquired the shear-free restriction for
the case of dissipative self-gravitating matter content. They inferred that
the expansion rate in addition to the shear-free restriction are equivalent to
the homology restriction. They also worked out the implications of such models
to some astronomical scenarios. Yousaf et al. [32] examined the irregularity
factors for a spherically symmetric star by taking into consideration an
imperfect fluid content. They further investigated two differential equations
utilizing the conservation laws and the Weyl tensor and concluded that the
complexity of object increases with the increase in anisotropic stresses.
Herrera et al. [33] revisited the notion of titled congruence for Lemaitre-
Tolman bondi spacetime with an imperfect fluid distribution. They evaluated
the factors responsible for the presence of energy density inhomogeneities and
carried out a comparison with the outcomes obtained for the case of non tilted
congruence. Herrera et al. [34] demonstrated the instability of spherical
systems undergoing non-adiabatic collapse with radial heat dissipation. They
presented that the Newtonian correction causes of heat flow maximizes the
fluid instability whereas the relativistic correction lessens the instability
of the fluid. Herrera et al. [35] explored on the expansion-free spherically
symmetric stars by evaluating the field equations along with the junction
conditions, for an anisotropic fluid with heat flux to comprehend the
expansion-free motion. They defined radial velocity of matter in two distinct
ways and analyzed each in detail.

Few of the factors that describe the physical state of fluid source create
complications in the analysis of gravitating relativistic bodies. These
factors are called complexity factors. Herrera [36] proposed a novel concept
of complexity for the case of time independent spherically symmetric objects.
They determined certain exact solutions to the Einstein’s stellar equations
fulfilling the zero complexity factor criterion and also found implications of
this novel concept of the analysis of structure development of compact object.
Herrera et al. [37] propounded a systematic study of complexity for time
independent system with spherical symmetry along with considering the
complexity factor of fluid content. They also obtained the minimal complexity
condition analyzing the dissipative and non-dissipative cases in detail and
also explored the stability of zero complexity condition. Yousaf [38]
revisited the complexity factor for relativistic spheres having non-dynamical
equation in Palatini $f(R)$ gravitational theory. Manipulation of the field
equations along with the TOV equation assists in constructing the complexity
factor of the system. The part played by the $f(R)$ terms and fluid variables
is also analyzed in the discussion of evolution of such systems. Sañudo and
Pacheo [39] studied the density profiles of white dwarfs in order to evaluate
its complexity. They inferred that the complexity of such a structure
increased with the value of atomic number. Chatziavvas et al. [40] applied the
notion of measuring complexity statistically to a Neutron star model in
continuation to the work of [39] from the analysis of gravity and short range
force. They observed that under their theoretical framework. The Neutron stars
less complex objects. Bhatti and Tariq [41, 42] carried out a study of
spherically symmetric astronomical object with anisotropy in fluid having heat
dissipation in the presence and absence of electromagnetic field in both GR
and $f(R)$ modified gravitational theories. They manipulated certain evolution
equation to demonstrate the physical processes involved in the evolutionary
phases of such systems via structure scalars.

The structure of this research article is organized in the following pattern:
Section 1 encompasses Palatini $f(R)$ field equations, the dynamical
quantities expressing fluid properties and the Junction conditions. Misner-
Sharp mass, electric part of the Weyl tensor and the active gravitational mass
are evaluated in Section 2. In order to comprehend how the fluid evolves with
time, certain evolution equations are also worked out. The Riemann tensor is
divided into three tensor quantities $X_{\varrho\beta},Y_{\varrho\beta}$ and
$Z_{\varrho\beta}$ in Section 3 to acquire few scalars connected with the
fluid dynamics. Such scalars are then interlinked with few equations of motion
to get an idea about how the fluid evolves with time. Section 4 enlists three
locally anisotropic static spheres. Section 5 incorporates outlines of the
acquired results.

##  2 Palatini Field Equations along with Kinematical Variables

Amendment in the gravitational constituent of the Einstein-Hilbert action
renders the $f(R)$ gravitational theory as follows

$S_{f(R)}=\frac{1}{2\kappa}\int f(R)\sqrt{-g}d^{4}x+S_{M},$ (1)

where $f(R)$ represents a non-linear generic Ricci function and $\kappa$ and
$S_{M}$ are referred to as the coupling constant and the matter action
respectively. In order to examine a dissipative fluid content having spherical
symmetry and enduring the collapsing phenomenon under anisotropic stresses, we
specify the following metric

$ds^{2}=e^{\nu(t,r)}dt^{2}-e^{\lambda(t,r)}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}).$ (2)

Exploiting the Palatini’s technique, the following composition of Palatini
gravitational equations is attained

$\displaystyle\frac{1}{f_{R}}\left(\nabla_{\mu}\nabla_{\delta}f_{R}-g_{\mu\delta}\Box
f_{R}\right)+\frac{\kappa
T_{\mu\delta}}{f_{R}}-\frac{1}{2}g_{\mu\delta}\left(R-\frac{f}{f_{R}}\right)+\frac{3}{2}f_{R}^{2}\left(\frac{1}{2}g_{\mu\delta}(\nabla
f_{R})^{2}-\nabla_{\mu}f_{R}\nabla_{\delta}f_{R}\right)$
$\displaystyle=R_{\mu\delta}-\frac{1}{2}g_{\mu\delta}R.$

Compiling this equation in an Einstein-like structure of field equations, we
infer

$G^{\mu}_{\delta}=\frac{\kappa}{f_{R}}\left[T^{\mu}_{\delta}+\mathcal{T}^{\mu}_{\delta}\right],$
(3)

where $\mathcal{T}_{\mu\delta}$ symbolizes the effective energy-momentum
tensor composed of pure geometric elements and is provided below

$\mathcal{T}^{\mu}_{\delta}=\frac{1}{\kappa}\left(\nabla^{\mu}\nabla_{\delta}-\delta^{\mu}_{\delta}\Box\right)f_{R}+\frac{\delta^{\mu}_{\delta}f_{R}}{2\kappa}+\frac{3}{2\kappa
f_{R}}\left[\frac{1}{2}\delta^{\mu}_{\delta}(\nabla
f_{R})^{2}-\nabla^{\mu}f_{R}\nabla_{\delta}f_{R}\right]\left(\frac{f}{f_{R}}-R\right).$

Equation (3) leads us to the following Palatini gravitational stellar
equations

$\displaystyle\kappa T^{0}_{0}$
$\displaystyle=f_{R}\left[\frac{1}{r^{2}}-\frac{e^{-}\lambda}{r^{2}}+\frac{\lambda^{\prime}e^{-\lambda}}{r}\right]+\bar{\tau}_{0},$
(4) $\displaystyle\kappa T^{1}_{1}$
$\displaystyle=f_{R}\left[\frac{1}{r^{2}}-\frac{e^{-\lambda}}{r^{2}}-\frac{\nu^{\prime}e^{-\lambda}}{r}\right]+\bar{\tau_{1}},$
(5) $\displaystyle\kappa T^{2}_{2}$
$\displaystyle=f_{R}\left[\frac{\ddot{\lambda}e^{-\nu}}{2}+\frac{\dot{\lambda^{2}}e^{-\nu}}{4}-\frac{\dot{\lambda}\dot{\nu}e^{-\nu}}{4}+\frac{\lambda^{\prime}e^{-\lambda}}{2r}-\frac{\nu^{\prime\prime}e^{-\lambda}}{2}-\frac{\nu^{{}^{\prime}2}e^{-\lambda}}{4}-\frac{\nu^{\prime}e^{-\lambda}}{2r}+\frac{\lambda^{\prime}\nu^{\prime}e^{-\lambda}}{4}\right]+\bar{\tau_{2}},$
$\displaystyle\kappa T_{10}$
$\displaystyle=\frac{\dot{\lambda}f_{R}}{r}+\bar{\tau_{3}}.$ (6)

The dark source terms arising due to Palatini $f(R)$ emendations are

$\displaystyle\bar{\tau}_{0}$
$\displaystyle=\frac{9e^{-\nu}\dot{f_{R}}^{2}}{4f_{R}}+\frac{e^{-\lambda}f_{R}^{\prime
2}}{4f_{R}}-e^{-\lambda}f_{R}^{\prime\prime}+\frac{\dot{\lambda}e^{-\nu}\dot{f_{R}}}{2}-\frac{\lambda^{\prime}e^{-\lambda}f_{R}^{\prime}}{2}-\frac{2e^{-\lambda
f_{R}^{\prime}}}{r}+\frac{f_{R}}{2}\left(R-\frac{f}{f_{R}}\right),$
$\displaystyle\bar{\tau}_{1}$
$\displaystyle=\frac{-e^{-\nu}\dot{f_{R}}^{2}}{4f_{R}}-\frac{9e^{-\lambda}f_{R}^{\prime
2}}{4f_{R}}+\ddot{f_{R}}e^{-\nu}-\frac{\dot{\nu}\dot{f_{R}}e^{-\nu}}{2}-\frac{\nu^{\prime}e^{-\lambda}f_{R}^{\prime}}{2}-\frac{2e^{-\lambda}f_{R}^{\prime}}{r}-\frac{f_{R}}{2}(R-\frac{f}{R}),$
$\displaystyle\bar{\tau}_{2}$
$\displaystyle=-\frac{e^{-\nu}\dot{f_{R}}^{2}}{4f_{R}}+\frac{e^{-\nu}\dot{f_{R}}^{2}}{4f_{R}}-\frac{e^{-\lambda}f_{R}^{\prime}}{r}+e^{-\nu}\ddot{f_{R}}-\frac{\dot{\nu}\dot{f_{R}}e^{-\nu}}{2}-\frac{\nu^{\prime}e^{-\lambda}f_{R}^{\prime}}{2}-e^{-\lambda}f_{R}^{\prime\prime}+\frac{\dot{\lambda}e^{-\lambda}\dot{f_{R}}}{2}$
$\displaystyle+\frac{\lambda^{\prime}e^{-\lambda}f_{R}^{\prime}}{2}+\frac{f_{R}}{2}(R-\frac{f}{f_{R}}),$
$\displaystyle\bar{\tau}_{3}$
$\displaystyle=-\dot{f_{R}^{\prime}}+\frac{\nu^{\prime}\dot{f_{R}}}{2}+\frac{\dot{\lambda}f_{R}^{\prime}}{2}+\frac{5\dot{f_{R}}f_{R}^{\prime}}{2f_{R}}.$

The dotted and primes values of the metric coefficients signify the temporal
and radial differentiation respectively.
The gravitational influences from the fluid sources decides the geometry of
the astronomical bodies. Four-velocities of such fluid sources are the
elementary constituents in the composition of energy-momentum tensor. If two
distinct relativistic explications of a spacetime manifold are interconnected
with the boost of one observer relative to the other, then the kinematics of
the fluid sources can be different. Keeping in view the work of Bondi [46], we
write the Minkowskian coordinates as

$\displaystyle d\tilde{\eta}=e^{\nu/2}dt;\quad d\tilde{y}=rd\theta;\quad
d\tilde{x}=e^{\lambda/2}dr;\quad d\tilde{z}=rsin\theta d\phi.$

To exploit the tilted congruence notion, it is supposed that the fluid source
has radial velocity $\omega$ in accordance with the new reference frame. We
employ Lorentz boost from Minkowskian frame to new frame. So, we write
Minkowskian energy-momentum tensor components as

$\displaystyle\check{T}_{0}^{0}=T_{0}^{0};\quad\check{T}_{1}^{1}=T_{1}^{1};\quad\check{T}_{2}^{2}=T_{2}^{2};\quad\check{T}_{3}^{3}=T_{3}^{3};\quad\check{T}_{01}=e^{-(\nu+\lambda)/2}T_{01}.$

Thus, a comoving observer observes the following covariant energy-momentum
tensor components

$\begin{pmatrix}\rho+\varepsilon&-q-\varepsilon&0&0\\\
-q-\varepsilon&P_{r}+\varepsilon&0&0\\\ 0&0&P_{\bot}&0\\\ 0&0&0&P_{\bot}\\\
\end{pmatrix},$

with $\varepsilon,\rho,q,P_{r}$ and $P_{\bot}$ signify radiation and energy
density, the heat flux vector and the radial and tangential stresses
respectively. Putting the idea of Lorentz transformation in use, we render

$\displaystyle\check{T}_{0}^{0}$
$\displaystyle=T_{0}^{0}=\frac{\rho+P_{r}\omega^{2}}{1-\omega^{2}}+\frac{2\omega
q}{1-\omega^{2}}+\frac{\varepsilon(1+\omega)}{1-\omega},$ (7)
$\displaystyle\check{T}_{1}^{1}$
$\displaystyle=T_{1}^{1}=-\left(\frac{\rho\omega^{2}+P_{r}}{1-\omega^{2}}+\frac{2\omega
q}{1-\omega^{2}}+\frac{\varepsilon(1+\omega)}{1-\omega}\right),$ (8)
$\displaystyle\check{T}_{01}=e^{-(\nu+\lambda)/2}T_{01}$
$\displaystyle=-\left(\frac{\omega
e^{(\nu+\lambda)/2}(\rho+P_{r})}{1-\omega^{2}}+\frac{qe^{(\nu+\lambda)/2}(1+\omega^{2})}{1-\omega^{2}}\right.$
$\displaystyle\left.+\frac{\varepsilon(1+\omega)e^{(\nu+\lambda)/2}}{1-\omega}\right),$
(9) $\displaystyle\check{T}_{2}^{2}$
$\displaystyle=T_{2}^{2}=-P_{\bot};\quad\check{T}_{3}^{3}=T_{3}^{3}=-P_{\bot}.$
(10)

For the fluid source velocity $\omega$, we have

$\omega=\left(\frac{dr}{dt}\right)e^{\frac{(\lambda-\nu)}{2}}.$

Substituting Eqs. (7-10) into Eqs. (3-6), we attain

$\displaystyle\frac{\rho+P_{r}\omega^{2}}{1-\omega^{2}}+\frac{2\omega
q}{1-\omega^{2}}+\frac{\varepsilon(1+\omega)}{1-\omega}=\frac{f_{R}}{\kappa}\left(\left[\frac{\lambda^{\prime}e^{-\lambda}}{r}+\frac{1}{r^{2}}-\frac{e^{-\lambda}}{r^{2}}\right]+\frac{\bar{\tau}_{0}}{f_{R}}\right),$
(11)
$\displaystyle\left(\frac{\rho\omega^{2}+P_{r}}{1-\omega^{2}}+\frac{2\omega
q}{1-\omega^{2}}+\frac{\varepsilon(1+\omega)}{1-\omega}\right)=-\frac{f_{R}}{\kappa}\left(\left[-\frac{\nu^{\prime}e^{-\lambda}}{r}+\frac{1}{r^{2}}-\frac{e^{-\lambda}}{r^{2}}\right]+\frac{\bar{\tau}_{1}}{f_{R}}\right),$
(12) $\displaystyle
P_{\bot}=-\frac{f_{R}}{\kappa}\left(\left[e^{-\nu}\left(\frac{\ddot{\lambda}}{2}+\frac{\dot{\lambda}^{2}}{4}-\frac{\dot{\lambda}\dot{\nu}}{4}\right)+e^{-\lambda}\left(\frac{{\lambda}^{\prime}}{2r}-\frac{{\nu}^{\prime\prime}}{2}-\frac{{\nu}^{\prime
2}}{4}-\frac{{\nu}^{\prime}}{2r}+\frac{{\lambda}^{\prime}{\nu}^{\prime}}{4}\right)\right]+\frac{\bar{\tau}_{2}}{f_{R}}\right),$
(13) $\displaystyle\left(\frac{\omega
e^{(\nu+\lambda)/2}(\rho+P_{r})}{1-\omega^{2}}+\frac{qe^{(\nu+\lambda)/2}(1+\omega^{2})}{1-\omega^{2}}+\frac{\varepsilon(1+\omega)e^{(\nu+\lambda)/2}}{1-\omega}\right)=-\frac{f_{R}}{\kappa}\left(\frac{\dot{\lambda}}{r}+\frac{\bar{\tau}_{3}}{f_{R}}\right).$
(14)

The notion of tilted congruence supports the use of following four-velocity
vector

$u^{\nu}=\left(\frac{e^{-\nu/2}}{(1-\omega^{2})^{1/2}},\frac{\omega
e^{-\lambda/2}}{(1-\omega^{2})^{1/2}},0,0\right)$

Kinematical Variables

Being already familiar with the expression for four-acceleration i.e.
$a^{\eta}=u^{\eta}_{;\lambda}u^{\lambda}$, we evaluate its surviving
components as

$\displaystyle a_{0}$ $\displaystyle=\frac{\nu^{\prime}\omega
e^{\frac{\nu-\lambda}{2}}}{2(1-\omega^{2})}+\frac{\omega\dot{\omega}}{(1-\omega^{2})^{2}}+\frac{\omega^{2}\omega^{\prime}e^{\frac{\nu-\lambda}{2}}}{(1-\omega^{2})^{2}}+\frac{\dot{\lambda}\omega^{2}}{2(1-\omega^{2})}+\frac{wf_{R}^{\prime}e^{\frac{\nu-\lambda}{2}}}{2f_{R}(1-\omega^{2})}+\frac{\dot{f_{R}}(1+\omega^{2})}{2f_{R}(1-\omega^{2})},$
$\displaystyle a_{1}$
$\displaystyle=-\frac{\nu^{\prime}}{2(1-\omega^{2})}-\frac{\dot{\omega}e^{\frac{\lambda-\nu}{2}}}{(1-\omega^{2})^{2}}-\frac{\dot{\lambda}\omega
e^{\frac{\lambda-\nu}{2}}}{2(1-\omega^{2})}-\frac{\omega\omega^{\prime}}{(1-\omega^{2})^{2}}-\frac{f_{R}^{\prime}(1+\omega^{2})}{2f_{R}(1-\omega^{2})}-\frac{\dot{f_{R}}\omega
e^{\frac{\lambda-\nu}{2}}}{f_{R}(1-\omega^{2})}.$

To explicate the physical characteristics of the fluid content, we manipulate
the shear tensor components
$(\sigma_{\beta\eta}=u_{\beta;\eta}+u_{\eta;\beta}-u_{\beta}a_{\eta}-u_{\eta}a_{\beta}-\frac{2\Theta
h_{\beta\eta}}{3})$ along with the evaluation of expansion scalar
$(\Theta=u^{\eta}_{;\eta})$ as follows

$\displaystyle\Theta$
$\displaystyle=\frac{\dot{\lambda}e^{-\nu/2}}{2\sqrt{(1-\omega^{2})}}+\frac{\omega\dot{\omega}e^{-\nu/2}}{(1-\omega^{2})^{3/2}}+\frac{\omega\nu^{\prime}e^{-\lambda/2}}{2\sqrt{(1-\omega^{2})}}+\frac{\omega^{\prime}e^{-\lambda/2}}{(1-\omega^{2})^{3/2}}+\frac{2\omega
e^{-\lambda/2}}{r\sqrt{(1-\omega^{2})}}$
$\displaystyle+\frac{2f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}+\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}\sqrt{(1-\omega^{2})}},$

and

$\displaystyle\sigma_{00}$
$\displaystyle=-\frac{4\omega^{3}\dot{\omega}e^{\nu/2}}{3(1-\omega^{2})^{5/2}}-\frac{2\nu^{\prime}\omega^{3}e^{\nu-\lambda/2}}{3(1-\omega^{2})^{3/2}}-\frac{4\omega^{2}\omega^{\prime}e^{\nu-\lambda/2}}{3(1-\omega^{2})^{5/2}}-\frac{2\dot{\lambda}\omega^{2}e^{\nu/2}}{3(1-\omega^{2})^{3/2}}+\frac{4\omega^{3}e^{\nu-\lambda/2}}{3r(1-\omega^{2})^{3/2}}$
$\displaystyle-\frac{2\dot{f_{R}}e^{\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}-\frac{2\omega^{2}\dot{f_{R}}e^{\nu/2}}{3f_{R}(1-\omega^{2})^{3/2}}+\frac{4\omega^{3}f_{R}^{\prime}}{3f_{R}(1-\omega^{2})^{3/2}},$
$\displaystyle\sigma_{11}$
$\displaystyle=-\frac{4\omega^{\prime}e^{\lambda/2}}{3(1-\omega^{2})^{5/2}}-\frac{2\dot{\lambda}e^{\frac{\lambda-\nu}{2}}}{3(1-\omega^{2})^{3/2}}+\frac{\dot{f_{R}}e^{\frac{\lambda-\nu}{2}}}{3f_{R}(1-\omega^{2})^{5/2}}-\frac{2\omega
f_{R}^{\prime}e^{\lambda/2}}{3f_{R}(1-\omega^{2})^{3/2}}-\frac{4\omega\dot{\omega}e^{\frac{\lambda-\nu}{2}}}{3(1-\omega^{2})^{5/2}}$
$\displaystyle-\frac{2\nu^{\prime}\omega
e^{\lambda/2}}{3(1-\omega^{2})^{5/2}}+\frac{4\omega
e^{\lambda/2}}{3r(1-\omega^{2})^{3/2}},$ $\displaystyle\sigma_{22}$
$\displaystyle=\frac{r^{2}e^{-\nu/2}\dot{f_{R}}}{3f_{R}(1-\omega^{2})^{1/2}}+\frac{\omega
r^{2}e^{-\lambda/2}f_{R}^{\prime}}{3f_{R}(1-\omega^{2})^{1/2}}-\frac{2\omega
re^{-\lambda/2}}{3(1-\omega^{2})^{1/2}}+\frac{2r^{2}\omega\dot{\omega}e^{-\nu/2}}{3(1-\omega^{2})^{3/2}}+\frac{2r^{2}\omega^{\prime}e^{-\lambda/2}}{3(1-\omega^{2})^{3/2}}$
$\displaystyle+\frac{r^{2}\nu^{\prime}\omega
e^{-\lambda/2}}{3(1-\omega^{2})^{1/2}}+\frac{\dot{\lambda}r^{2}e^{-\nu/2}}{3(1-\omega^{2})^{1/2}},$
$\displaystyle\sigma_{33}$ $\displaystyle=\sigma_{22}Sin^{2}\theta,$
$\displaystyle\sigma_{01}$ $\displaystyle=\frac{2\omega\lambda
e^{\frac{\lambda-\nu}{2}}}{3(1-\omega^{2})^{3/2}}+\frac{4\omega^{2}\dot{\omega}e^{\frac{\lambda-\nu}{2}}}{3(1-\omega^{2})^{5/2}}+\frac{2\omega^{2}\nu^{\prime}e^{\nu/2}}{3(1-\omega^{2})^{3/2}}+\frac{4\omega\omega^{\prime}e^{\nu/2}}{3(1-\omega^{2})^{5/2}}+\frac{4\omega
e^{\lambda/2}}{3_{r}(1-\omega^{2})^{3/2}}$
$\displaystyle+\frac{7\omega^{2}f_{R}^{\prime}e^{\nu/2}}{6f_{R}(1-\omega^{2})^{3/2}}+\frac{7\dot{f_{R}}\omega
e^{\lambda/2}}{6f_{R}(1-\omega^{2})^{3/2}}-\frac{\dot{f_{R}}\omega^{3}e^{\lambda/2}}{2f_{R}(1-\omega^{2})^{3/2}},$

where $h_{\beta\eta}=g_{\beta\eta}-u_{\beta}u_{\eta}$. The shear scalar using
the Palatini’s technique renders the following outcome

$\displaystyle\sigma$ $\displaystyle=\frac{2\omega
e^{-\lambda/2}}{r(1-\omega^{2})^{1/2}}-\frac{\dot{\lambda}e^{-\nu/2}}{(1-\omega^{2})^{1/2}}\frac{-2\omega^{\prime}e^{\lambda/2}}{(1-\omega^{2})^{3/2}}-\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{f_{R}\omega^{2}}+\frac{2\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}$
$\displaystyle-\frac{2\omega\dot{\omega}e^{-\nu/2}}{(1-\omega^{2})^{3/2}}-\frac{\nu^{\prime}\omega
e^{-\lambda/2}}{(1-\omega^{2})^{1/2}}.$

Another four-vector field with the following definition is presumed

$s^{\nu}=\left(\frac{\omega
e^{-(\nu/2)}}{(1-\omega^{2})^{1/2}},\frac{e^{-\lambda/2}}{(1-\omega^{2})^{1/2}},0,0\right).$
(15)

Pressure anisotropy has crucial contribution in the study of collapsing
astronomical objects as it increases the complexity in them. For our work, we
choose anisotropic fluid source that can be indicated by the following energy-
momentum tensor

$T^{\beta}_{\eta}=\tilde{\rho}u^{\beta}u_{\eta}-\check{P}h^{\beta}_{\eta}+\Pi^{\beta}_{\eta}+\tilde{q}(s^{\beta}u_{\eta}+s_{\eta}u^{\beta}),$

with

$\displaystyle\Pi^{\beta}_{\eta}$
$\displaystyle=\Pi\left(s^{\beta}s_{\eta}+\frac{h^{\beta}_{\eta}}{3}\right);\quad\check{P}=\frac{2P_{\bot}+\tilde{P}_{r}}{3};\quad\tilde{q}^{\gamma}=\tilde{q}s^{\gamma};\quad\tilde{\rho}=\rho+\varepsilon;$
$\displaystyle\tilde{P_{r}}$
$\displaystyle=P_{r}+\varepsilon;\quad\tilde{q}=q+\varepsilon;\quad\Pi=\tilde{P}_{r}-P_{\bot}.$

In order to discern the singularity areas of curvature and matter contents at
hypersurface $\Sigma$, we exploit the junction conditions and determine the
permitted values for discontinuity of certain quantities across $\Sigma$. The
radiating Schwarzschild metric i.e. the Vaidya’s metric given below is
considered for this purpose

$ds^{2}=\left(1-\frac{2M(\bar{\eta})}{\Re}\right)d\bar{\eta}^{2}+2d\bar{\eta}d\Re-\Re^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}).$

For our spherically symmetric object, the Darmois junction conditions are
determined as

$\displaystyle
e^{\nu_{\Sigma}}=\left(1-\frac{2M(\bar{\eta})}{\Re_{\Sigma}}\right),e^{-\lambda_{\Sigma}}=\left(1-\frac{2M(\bar{\eta})}{\Re_{\Sigma}}\right),\quad[(P_{r})_{eff}=(q)_{eff}]_{\Sigma}.$

###  2.1 The Curvature Tensors (Weyl and Riemann tensors)

It is a formerly known fact that the fluid content engenders a distortion
(curvature) in spacetime. We can measure the extent of such curvature with the
help of curvature tensors i.e the Riemann and the Weyl tensor. For a spherical
star, the Weyl tensor comprises only the electric part as the magnetic part
becomes zero due to symmetric property. From the expression used to explicate
the Weyl tensor, we can extract the definition of Riemann tensor too. This is
provided as follows

$R^{\nu}_{\eta\rho\mu}=C^{\nu}_{\eta\rho\mu}+\frac{1}{2}R^{\nu}_{\rho}g_{\eta\mu}+\frac{1}{2}R_{\eta\rho}\delta^{\nu}_{\mu}+\frac{1}{2}R_{\eta\mu}\delta^{\nu}_{\rho}-\frac{1}{2}R^{\nu}_{\mu}g_{\eta\rho}-\frac{1}{6}R\left(\delta^{\nu}_{\rho}g_{\eta\mu}-g_{\eta\rho}\delta^{\nu}_{\mu}\right).$
(16)

The Weyl tensor in terms of its electric part is defined as

$C_{\xi\nu\pi\lambda}=E^{\beta\delta}u^{\rho}u^{\gamma}(g_{\xi\nu\rho\beta}g_{\pi\lambda\gamma\delta}-\eta_{\xi\nu\rho\beta}\eta_{\pi\lambda\gamma\delta}),$

where $g_{\xi\nu\rho\beta}$ is equal to
$g_{\xi\nu\alpha\beta}=g_{\xi\rho}g_{\nu\beta}-g_{\xi\beta}g_{\nu\rho}$ and
$\eta_{\pi\lambda\gamma\delta}$ indicates Levi-Civita tensor.
$E_{\varrho\beta}$ has the following alternative form

$E_{\varrho\beta}=E\left(s_{\varrho}s_{\beta}+\frac{1}{3}h_{\varrho\beta}\right).$
(17)

The Weyl scalar is determined to be as follows

$\displaystyle E$
$\displaystyle=\frac{\ddot{\lambda}e^{-\nu}}{4}+\frac{\dot{\lambda}^{2}e^{-\nu}}{8}-\frac{\dot{\lambda}\dot{\nu}e^{-\nu}}{8}-\frac{{\nu}^{\prime\prime}e^{-\lambda}}{4}-\frac{\nu^{\prime
2}e^{-\lambda}}{8}+\frac{\lambda^{\prime}\nu^{\prime}e^{-\lambda}}{8}+\frac{\nu^{\prime}e^{-\lambda}}{4r}-\frac{\lambda^{\prime}e^{-\lambda}}{4r}-\frac{e^{-\lambda}}{2r^{2}}+\frac{1}{2r^{2}},$

with $E$ fulfilling the following criteria

$E^{\eta}_{\eta}=0;\quad\quad E_{\eta\sigma}=E_{(\eta\sigma)};\quad\quad
E_{\eta\sigma}u^{\sigma}=0.$

###  2.2 Mass Function for the Spherical Interior

The interior geometric mass is inferred as

$R^{3}_{232}=-e^{-\lambda}+1=\frac{2m}{r}.$ (18)

Putting into use the Weyl scalar, the field equations attained from Platini’s
variational technique and the definition of Reimann tensor provided in Eq.
(16),we acquire

$\frac{3m}{r^{3}}=E+\frac{\kappa}{2f_{R}}(\tilde{\rho}-\tilde{P}_{r}+P_{\bot})-\frac{1}{2f_{R}}(\bar{\tau}_{0}+\bar{\tau_{1}}).$
(19)

Eqs. (4) and (18) lead us to the following expression for the interior
geometric mass

$m=\frac{1}{2}\int^{r}_{0}\frac{\kappa
r^{2}}{f_{R}}\left(T^{0}_{0}-\frac{\bar{\tau}_{0}}{\kappa}\right).$ (20)

This result when compared with Eq. (43) in [48] demonstrates that the
appearance of the dark source term $\bar{\tau}_{0}$ emerging due to Palatini’s
technique affects the interior mass of the astronomical body. Using Eqs. (16)
and (17) in addition with Eq. (20) and field equations, we infer

$m=\frac{r^{3}}{6f_{R}}\kappa(T^{1}_{1}+T^{0}_{0}-T^{2}_{2})+\frac{Er^{3}}{3}-\frac{(\bar{\tau_{0}}+\bar{\tau_{1}})r^{3}}{6f_{R}}.$
(21)

The radial differentiation of this equation in combination with Eq. (20)
provides

$\displaystyle\left(\frac{r^{3}}{3}E\right)^{\prime}$
$\displaystyle=-\frac{r^{3}}{6f_{R}}\left[\kappa(T^{0}_{0})^{\prime}+\frac{3\bar{\tau}_{0}}{r}\right]+\left[\frac{r^{3}}{6f_{R}}\kappa(T^{2}_{2}-T^{1}_{1})\right]^{\prime}+\left[\frac{(\bar{\tau}_{0}+\bar{\tau}_{1})r^{3}}{6f_{R}}\right]^{\prime}.$
(22)

Carrying out the integration of above equation, we acquire

$\displaystyle E$
$\displaystyle=-\frac{1}{2r^{3}}\int^{r}_{0}\frac{r^{3}}{f_{R}}\left[\kappa(T^{0}_{0})^{\prime}+\frac{3\bar{\tau}_{0}}{r}\right]dr+\frac{1}{2f_{R}}\kappa(T^{2}_{2}-T^{1}_{1})+\frac{\bar{\tau}_{0}+\bar{\tau}_{1}}{2f_{R}}.$
(23)

This outcome has correspondence with Eq. (46) in [48] with a distinction that
the tidal force for our theory, in addition to the energy density
inhomogeneity and stress anisotropy, is also dependent on the dark source
quantities $\bar{\tau}_{0}$ and $\bar{\tau}_{1}$. Placing the value of $E$
from Eq. (23) into Eq. (21), we get

$\displaystyle m(t,r)=\frac{r^{3}\kappa
T^{0}_{0}}{6f_{R}}-\frac{1}{6}\int^{r}_{0}\frac{r^{3}}{f_{R}}\left[\kappa(T^{0}_{0})^{\prime}+\frac{3\bar{\tau}_{0}}{r}\right]dr.$
(24)

It is clear from the above expression that the interior geometric mass of the
solid sphere depends upon the energy density along with the curvature
emendations resulting from the Palatini’s gravitational technique. Due to this
reason, our result is different from the mass function appearing in Eq. (47)
in [48]. The components of stress-energy tensor assumes the values as
$T^{0}_{0}=\tilde{\rho}$ and $T^{2}_{2}-T^{1}_{1}=\Pi$ for the following three
particular situations:

Static Regime

(i) In this situation, all the time derivatives and the velocity $\omega$
vanish.

Quasi-static Regime

(ii) In this situtaion, we assume
$\omega^{2}\approx\dot{\nu}\dot{\lambda}\approx\ddot{\nu}\approx\dot{\omega}\approx\ddot{\lambda}\approx\dot{\lambda}^{2}\approx
0$.

Right after the withdrawal from Equilibrium State

(iii) In such a situation, we suppose that $\dot{\omega}\neq 0$ but
$\dot{\lambda}\approx\omega\approx\dot{\nu}\approx 0$.
Bearing in mind these particular cases, we work out Eqs. (23) and (24) again
as follows

$\displaystyle E$
$\displaystyle=-\frac{1}{2r^{3}}\int^{r}_{0}\frac{r^{3}}{f_{R}}\left(\kappa\tilde{\rho}^{\prime}+\frac{3\bar{\tau}_{0}}{r}\right)dr+\frac{\kappa\Pi}{2f_{R}}+\frac{1}{2f_{R}}(\bar{\tau}_{0}+\bar{\tau}_{1}),$
$\displaystyle m(t,r)$
$\displaystyle=\frac{r^{3}\kappa\tilde{\rho}}{6f_{R}}-\frac{1}{6}\int^{r}_{0}\frac{r^{3}}{f_{R}}\left(\kappa\tilde{\rho}^{\prime}+\frac{3\bar{\tau}_{0}}{r}\right)dr.$

It is noteworthy that $E$ is linked with the physical fluid characteristics
(anisotropy and density homogeneity in this case) whereas the mass function is
connected with homogeneous energy density along with the change produced by
inhomogeneity in the value for energy density.

###  2.3 The Active Gravitational Spherical Mass

The total energy inside the fluid content can be well-defined using an
expression that incorporates fluid variables via stress-energy tensor
components. We call it the active gravitational mass of the astronomical
object under consideration. Mathematically, we write it as

$\displaystyle m_{T}$
$\displaystyle=\frac{1}{2}\int^{r_{\Sigma}}_{0}e^{(\nu+\lambda)/2}r^{2}\kappa(\left.T^{0}_{0}\right.^{eff}-\left.T^{1}_{1}\right.^{eff}-2\left.T^{2}_{2}\right.^{eff})dr+\frac{1}{2}\int^{r_{\Sigma}}_{0}e^{(\nu+\lambda)/2}r^{2}f_{R}(R_{0})$
$\displaystyle\times\frac{\partial}{\partial t}\left[\frac{\partial
L}{\frac{\partial}{\partial
t}[\partial(g^{\alpha\sigma}\sqrt{-g})]}\right]g^{\alpha\sigma}dr.$

Here, $L$ symbolize the Lagrangian density. Entirely inside the boundary
$\Sigma$, the mass of solid sphere having radius $r$ can be defined as

$\displaystyle m_{T}$
$\displaystyle=\frac{1}{2}\int^{r}_{0}e^{(\nu+\lambda)/2}r^{2}\kappa(\left.T^{0}_{0}\right.^{eff}-\left.T^{1}_{1}\right.^{eff}-2\left.T^{2}_{2}\right.^{eff})dr+\frac{1}{2}\int^{r}_{0}e^{(\nu+\lambda)/2}r^{2}f_{R}(R_{0})$
$\displaystyle\times\frac{\partial}{\partial t}\left[\frac{\partial
L}{\frac{\partial}{\partial
t}[\partial(g^{\alpha\sigma}\sqrt{-g})]}\right]g^{\alpha\sigma}dr.$

Putting into use the Palatini gravitational equational together with the
definition of interior spherical mass, we infer

$m_{T}=e^{\nu+\lambda/2}f_{R}\left[m(t,r)-\frac{r^{3}}{2f_{R}}\left(\kappa
T^{1}_{1}-\bar{\tau}_{1}\right)\right]-\frac{1}{2}\int^{r}_{0}e^{\lambda-\nu/2}\dot{\lambda}\dot{f_{R}}dr.$
(25)

Putting the value of component $T^{1}_{1}$ from Eq. (5) and $m$ from Eq. (18),
we attain

$m_{T}=\frac{r^{2}\nu^{\prime}e^{\frac{\nu-\lambda}{2}}f_{R}}{2}-\int
r^{2}\dot{f_{R}}\dot{\lambda}e^{\frac{\nu-\lambda}{2}}dr.$ (26)

For a test particle initially at rest, the static field induces the following
value for the acceleration

$a=\sqrt{\frac{m_{T}e^{-\nu/2}}{r^{2}}-\frac{e^{\frac{-\nu-\lambda}{2}}f_{R}^{\prime
2}}{4f_{R}^{2}}}.$

Taking the radial derivative of Eq. (26) and using it together with Eqs. (21)
and (25), we can write

$\displaystyle-3m_{T}+rm_{T}^{\prime}=\left(\frac{\nu^{\prime}f_{R}^{\prime}}{f_{R}}-\frac{\dot{\lambda}\dot{f_{R}}}{f_{R}}\right)\times\frac{r^{3}f_{R}e^{\frac{\nu-\lambda}{2}}}{2}+\left(\frac{\ddot{\lambda}}{2}+\frac{\dot{\lambda}^{2}}{4}-\frac{\dot{\lambda}\dot{\nu}}{4}\right)\times
r^{3}f_{R}e^{\frac{\lambda-\nu}{2}}+\frac{r^{3}\kappa
e^{\frac{\nu+\lambda}{2}}}{2}\times$
$\displaystyle\left(T^{0}_{0}-T^{1}_{1}-2T^{2}_{2}-\frac{\bar{\tau}_{0}}{\kappa}+\frac{\bar{\tau}_{1}}{\kappa}+\frac{2\bar{\tau}_{2}}{\kappa}\right)-r^{3}e^{\frac{\nu+\lambda}{2}}\left[\kappa\left(\frac{T^{0}_{0}}{2f_{R}}-\frac{T^{0}_{0}}{2f_{R}}-\frac{T^{0}_{0}}{2f_{R}}+\frac{\bar{\tau}_{1}+\bar{\tau}_{2}-\bar{\tau}_{0}}{\kappa}\right)\right.$
$\displaystyle\left.+\frac{E}{f_{R}}\right]+\frac{3}{2}\int^{r}_{0}r^{2}e^{\frac{\lambda-\nu}{2}}\dot{\lambda}\dot{f_{R}}dr.$

Employing integration w.r.t $r$, we get

$\displaystyle m_{T}$
$\displaystyle=(m_{T})_{\Sigma}\left(\frac{r}{r_{\Sigma}}\right)^{3}-r^{3}\int_{r}^{r_{\Sigma}}\frac{e^{\frac{\nu-\lambda}{2}}}{2r}\left(\nu^{\prime}f_{R}^{\prime}-\dot{\lambda}\dot{f_{R}}\right)dr-r^{3}\int_{r}^{r_{\Sigma}}r^{2}f_{R}e^{\frac{\lambda-\nu}{2}}\left(\frac{\ddot{\lambda}}{2}+\frac{\dot{\lambda}^{2}}{4}-\frac{\dot{\lambda}\dot{\nu}}{4}\right)$
$\displaystyle
dr-r^{3}\int^{r_{\Sigma}}_{r}\frac{e^{\frac{\nu+\lambda}{2}}}{r}\left(\frac{\kappa}{2}(T^{1}_{1}-T^{2}_{2}-\frac{\bar{\tau}_{2}-\bar{\tau}_{1}}{\kappa})-\frac{E}{f_{R}}\right)dr-r^{3}\int_{r}^{r_{\Sigma}}\frac{3}{2r^{4}}\int^{r}_{0}r^{2}e^{\frac{\lambda-\nu}{2}}\dot{\lambda}\dot{f_{R}}drdr.$
(27)

Previously in literature, as given in Eq. (57) in [48], the active
gravitational mass in the frame of Einstein’s GR was found to be dependent
only on the fluid variables along with the derivatives of metric coefficients.
But in our theory ($f(R)$ by Palatini’s approach), it comes out to be
dependent also on the dark source terms which means that the dark source also
affects the strength of an astronomical body’s gravitational flux. Eq. (23)
when utilized here, provides the following outcome

$\displaystyle m_{T}$
$\displaystyle=(m_{T})_{\Sigma}\left(\frac{r}{r_{\Sigma}}\right)^{3}-r^{3}\int_{r}^{r_{\Sigma}}\frac{e^{\frac{\nu-\lambda}{2}}}{2r}\left(\nu^{\prime}f_{R}^{\prime}-\dot{\lambda}\dot{f_{R}}\right)dr-r^{3}\int_{r}^{r_{\Sigma}}\frac{f_{R}e^{\frac{\lambda-\nu}{2}}}{r}\left(\frac{\ddot{\lambda}}{2}+\frac{\dot{\lambda}^{2}}{4}-\frac{\dot{\lambda}\dot{\nu}}{4}\right)dr$
$\displaystyle-r^{3}\int^{r_{\Sigma}}_{r}\frac{e^{\frac{\nu+\lambda}{2}}}{r}\left(\kappa
T^{1}_{1}-\kappa T^{2}_{2}+\bar{\tau}_{2}-\bar{\tau}_{1}+\frac{\kappa
f_{R}}{2r^{3}}\int_{0}^{r}\frac{r^{3}}{f_{R}}\left((T^{0}_{0})^{\prime}+\frac{3\bar{\tau}_{0}}{\kappa
r}\right)dr\right)dr.$

###  2.4 Analysis of Structure Formation and Evolutionary Phases of Spherical
Star

This section focuses on the manipulation of ceratin differential equations,
each one explicating the physical phenomena that the spherical object
confronts. In this regard, the first conservation equation is inferred as

$\displaystyle\tilde{\rho}^{*}+(\tilde{\rho}+\tilde{P}_{r})\left(\theta-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}-\frac{2f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right)-\frac{2}{3}\left(\theta+\frac{\sigma}{2}-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}\right.$
$\displaystyle\left.-\frac{3f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{2\omega^{2}f_{R}}\right)\Pi+\tilde{q}^{\dagger}+2\tilde{q}\left[\sqrt{a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})^{2}}+\varepsilon_{1}}\right.$
$\displaystyle\left.+\frac{s^{1}}{r}\right]+\varphi_{1}^{(D)}=0,$

with the value

$\displaystyle\varphi_{1}^{(D)}$
$\displaystyle=\mathcal{T}_{0,1}+\mathcal{T}^{0}_{0,0}+\mathcal{T}^{0}_{0}\left[\frac{\dot{\lambda}}{2}+\frac{\dot{f_{R}}}{2f_{R}}\right]+\mathcal{T}^{1}_{0}\left[\frac{\lambda^{\prime}}{2}+\frac{2}{r}+\frac{3f_{R}^{\prime}}{2f_{R}}\right]-\mathcal{T}^{0}_{1}\left[\frac{\nu^{\prime}e^{\nu-\lambda/2}}{2}+\frac{f_{R}^{\prime}e^{\nu-\lambda/2}}{2f_{R}}\right]$
$\displaystyle+T^{0}_{0}\left[\frac{\dot{f_{R}}}{2f_{R}}\right]+T^{1}_{0}\left[\frac{3f_{R}^{\prime}}{2f_{R}}\right]-\mathcal{T}^{1}_{1}\left[\frac{\dot{\lambda}}{2}+\frac{\dot{f_{R}}}{2f_{R}}\right]-T^{1}_{1}\left[\frac{\dot{f_{R}}}{2f_{R}}\right]-T^{0}_{1}\left[\frac{f_{R}^{\prime}e^{\nu-\lambda/2}}{2f_{R}}\right].$

The second conservation equation is inferred as

$\displaystyle\tilde{P}_{r}^{\dagger}+(\tilde{\rho}+\tilde{P}_{r})\sqrt{a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})}+\varepsilon_{1}}+\frac{2s^{1}}{r}\Pi=\frac{\tilde{q}}{3}\left[\sigma-4\theta+\frac{6\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}\sqrt{1-\omega^{2}}}\right.$
$\displaystyle\left.+\frac{8\dot{f_{R}}e^{-\nu/2}}{f_{R}\sqrt{1-\omega^{2}}}+\frac{3\dot{f_{R}}e^{-\nu/2}\sqrt{1-\omega^{2}}}{f_{R}\omega^{2}}\right]+\tilde{q}^{\star}-\varphi_{2}^{(D)}.$

where

$\displaystyle\varphi_{2}^{(D)}$
$\displaystyle=\mathcal{T}^{1}_{1,1}+\mathcal{T}^{0}_{0,0}+\mathcal{T}^{1}_{1}\left(\frac{\nu^{\prime}}{2}+\frac{f_{R}^{\prime}}{2f_{R}}\right)-\mathcal{T}^{0}_{0}\left(\frac{\nu^{\prime}}{2}+\frac{f_{R}^{\prime}}{2f_{R}}\right)+\tilde{T}^{1}_{1}\left(\frac{2}{r}+\frac{f_{R}^{\prime}}{f_{R}}\right)-\mathcal{T}^{1}_{0}\left(\frac{\dot{\lambda}e^{\lambda-\nu}}{2}\right.$
$\displaystyle\left.+\frac{\dot{f_{R}}e^{\lambda-\nu}}{2f_{R}}\right)-\frac{2\mathcal{T}^{2}_{2}}{r}+\mathcal{T}^{0}_{1}\left(\frac{\dot{\nu}}{2}+\frac{\dot{f_{R}}}{2f_{R}}\right).$

Both of the above equations when compared with Eqs. (59) and (60) in [48]
exhibit that the presence of dark source produces significant effects on the
stress-energy conservation of astronomical systems. The Raychaudhuri equation
that explicates the flow kinematics induces the following outcome

$\displaystyle\left[\theta-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}-\frac{2f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right]^{\star}+\frac{1}{3}\left[\theta-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}-\frac{2f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right]^{2}$
$\displaystyle+\frac{1}{6}\left[\sigma+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{f_{R}\omega^{2}}-\frac{2f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right]^{2}-\left[\sqrt{a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})}+\varepsilon_{1}}\right]^{\dagger}-$
$\displaystyle\left[a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})}+\varepsilon_{1}\right]-\frac{2s^{1}}{r}\sqrt{a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})}+\varepsilon_{1}}=-\frac{\kappa}{2}(\tilde{\rho}+3\tilde{P}_{r})+\kappa\Pi,$

which being dissimilar to Eq. (61) in [48] for GR, demonstrates that the dark
source also plays its part in characterizing the fluid flow in the spacetime
background. The Ricci identities yield

$\displaystyle\left[\frac{\sigma}{2}+\theta+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{2\omega^{2}f_{R}}-\frac{3\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}\right]^{\dagger}=-\frac{3s^{1}}{2r}$
$\displaystyle\times\left(\sigma+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{\omega^{2}f_{R}}-\frac{2\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right)+\frac{3}{2}\kappa\tilde{q},$

which, unlike Eq. (62) in [48], signify that the fluid dynamics described by
the shear and expansion scalar and the heat dissipation factor is also
influenced by the inclusion of dark source in the gravitational theory. From
Eqs. (3), (16) together with Ricci identities produce

$\displaystyle-E-\frac{\kappa\Pi}{2}=\left[\left(a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})}+\varepsilon_{1}\right)^{1/2}\right]^{\dagger}+\left[a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})}+\varepsilon_{1}\right]$
$\displaystyle+\frac{1}{2}\left[\sigma+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{\omega^{2}f_{R}}-\frac{2\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right]^{\star}+\frac{1}{3}\left[\left(\theta-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}-\frac{2f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right)\right.$
$\displaystyle\times\left.\left(\sigma+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{\omega^{2}f_{R}}-\frac{2\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right)\right]-\frac{s^{1}}{r}\sqrt{a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})}+\varepsilon_{1}}$
$\displaystyle+\frac{1}{12}\left[\sigma+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{\omega^{2}f_{R}}-\frac{2\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right].$

From the comparison of this outcome with Eq. (63) in [48], it is worth-
observing that the incorporation of dark source quantities (emerging because
of Palatini’s formalism) influences the tidal force and the anisotropy factor
for the astronomical system. The Bianchi identities in combination with Weyl
tensor produce following two differential equations

$\displaystyle\left(\frac{\kappa\tilde{P}_{r}}{2f_{R}}+\frac{3m}{r^{3}}\right)\left[\theta-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}-\frac{3f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}+\frac{\sigma}{2}+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{\omega^{2}f_{R}}\right]$
$\displaystyle+\left[E-\frac{\kappa\Pi}{2}+\frac{\kappa\tilde{\rho}}{2}\right]^{\star}=-\frac{3s^{1}}{2r}\kappa\tilde{q},$

and

$\displaystyle\left[E-\frac{\kappa\Pi}{2}+\frac{\kappa\tilde{\rho}}{2}\right]^{\dagger}=\frac{3s^{1}}{r}\left(\frac{\kappa\Pi}{2}-E\right)+\frac{\kappa\tilde{q}}{2}\left[\frac{\sigma}{2}+\theta+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{2\omega^{2}f_{R}}\right.$
$\displaystyle\left.-\frac{3\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}\right],$

each of which shows a physical meaning distinct from that provided by Eqs.
(64,65) in [48]. This particular distinction is because we are taking the
emendations in Ricci scalar curvature which, in turn, has peculiar effects on
the fluid flow. Equation (19) can be reformed as

$\frac{3m}{r^{3}}=\frac{E}{f_{R}}-\frac{\kappa\Pi}{2f_{R}}+\frac{\kappa\tilde{\rho}}{2f_{R}}-\frac{\bar{\tau}_{0}+\bar{\tau}_{1}}{2f_{R}}.$

##  3 Breaking-up the Riemann Tensor via Orthogonal Decomposition Method

Following the approach used by Bel [47], we introduce three explicit tensors
$X_{\varrho\beta},Y_{\varrho\beta}$ and $Z_{\varrho\beta}$ as

$\displaystyle
Y_{\varrho\beta}=R_{\varrho\gamma\beta\delta}u^{\gamma}u^{\delta},\quad
Z_{\varrho\beta}=^{*}R_{\varrho\gamma\beta\delta}u^{\gamma}u^{\delta}=\frac{1}{2}\eta_{\varrho\gamma\epsilon\rho}R^{\epsilon\rho}_{\beta\delta}u^{\gamma}u^{\delta},\quad
X_{\varrho\beta}=^{*}R^{*}_{\varrho\gamma\beta\delta}u^{\gamma}u^{\delta}=\frac{1}{2}\eta_{\varrho\gamma}^{\epsilon\rho}R^{*}_{\epsilon\rho\beta\delta}u^{\gamma}u^{\delta},$

with the value

$R^{*}_{\varrho\beta\gamma\delta}=\frac{1}{2}\eta_{\epsilon\omega\gamma\delta}R^{\epsilon\omega}_{\varrho\beta}.$

Putting into use the Palatini $f(R)$ field equations given in Eq. (3), we
infer

$\displaystyle
R^{\sigma\alpha}_{\pi\beta}=C^{\sigma\alpha}_{\pi\beta}+2\kappa\left.T^{(eff)}\right.^{[\sigma}_{[\pi}\delta^{\alpha]}_{\beta]}+\kappa\left.T^{(eff)}\right.\left(\frac{1}{3}\delta^{\sigma}_{[\pi}\delta^{\alpha}_{\beta]}-\delta^{[\sigma}_{[\pi}\delta^{\alpha]}_{\beta]}\right).$

The Riemann tensor is broken up into three distinct parts given as under

$R^{\sigma\alpha}_{\pi\beta}=R^{\sigma\alpha}_{(I)\pi\beta}+R^{\sigma\alpha}_{(II)\pi\beta}+R^{\sigma\alpha}_{(III)\pi\beta},$

where each part holds the following value

$\displaystyle R^{\sigma\alpha}_{(I)\pi\beta}=$ $\displaystyle
2\kappa\tilde{\rho}u^{[\sigma}u_{[\pi}\delta^{\alpha]}_{\beta]}-2\kappa\hat{P}h^{[\sigma}_{[\pi}\delta^{\alpha]}_{\beta]}+\left(\kappa\tilde{\rho}-3\kappa\check{P}\right.$
$\displaystyle\left.+2f-2Rf_{R}-3\Box f_{R}+\frac{3(\nabla
f_{R})^{2}}{f_{R}}-\frac{3\nabla^{\gamma}f_{R}\nabla_{\gamma}f_{R}}{f_{R}}\right)\left(\frac{1}{3}\delta^{\sigma}_{[\pi}\delta^{\alpha}_{\beta]}-\delta^{[\sigma}_{[\pi}\delta^{\alpha]}_{\beta]}\right)$
$\displaystyle+2\left[\nabla^{[\sigma}\nabla_{[\pi}f_{R}\delta^{\alpha]}_{\beta]}-\Box
f_{R}\delta^{[\sigma}_{[\pi}\delta^{\alpha]}_{\beta]}+(f-Rf_{R})\delta^{[\sigma}_{[\pi}\delta^{\alpha]}_{\beta]}+\frac{3}{4f_{R}}(\delta^{[\sigma}_{[\pi}\delta^{\alpha]}_{\beta]})(\nabla
f_{R})^{2}\right.$
$\displaystyle\left.-\frac{3}{2f_{R}}(\nabla^{[\sigma}f_{R}\nabla_{[\pi}f_{R}\delta^{\alpha]}_{\beta]})\right],$
$\displaystyle R^{\sigma\alpha}_{(II)\pi\beta}=$ $\displaystyle
2\kappa\Pi\left[s^{[\sigma}s_{[\pi}\delta^{\alpha]}_{\beta]}+\frac{1}{3}h^{[\sigma}_{[\pi}\delta^{\alpha]}_{\beta]}\right]+2\kappa\left(\tilde{q}s^{[\sigma}u_{[\pi}\delta^{\alpha]}_{\beta]}+\tilde{q}u^{[\sigma}s_{[\pi}\delta^{\alpha]}_{\beta]}\right),$
$\displaystyle R^{\sigma\alpha}_{(III)\pi\beta}=$ $\displaystyle
4u^{[\sigma}u_{[\pi}E^{\alpha]}_{\beta]}-\epsilon^{\sigma\alpha}_{\mu}\epsilon_{\pi\beta\nu}E^{\mu\nu}.$

Such a breakdown of the Riemann tensor was first proposed by Herrera in [48]
for GR. Eqs. (73) of this article coincide with our results if we consider the
restrictions that $f_{R}\rightarrow 1,f(R)=R$ and all the derivatives of
$f_{R}$ equal to zero. Expressions for tensorial quantities described above
take the following values

$\displaystyle X_{\varrho\beta}$ $\displaystyle=\frac{\kappa
h_{\varrho\beta}}{3}\tilde{\rho}-E_{\varrho\beta}+\frac{\kappa}{2}\Pi_{\varrho\beta}+\chi_{\varrho\beta},$
(28) $\displaystyle Y_{\varrho\beta}$ $\displaystyle=\frac{\kappa
h_{\varrho\beta}}{6}\tilde{\rho}+\frac{\kappa
h_{\varrho\beta}}{2}\hat{P}+\frac{\kappa}{2}\Pi_{\varrho\beta}+E_{\varrho\beta}+\psi_{\varrho\beta},$
(29) $\displaystyle Z_{\varrho\beta}$
$\displaystyle=\frac{\kappa\tilde{q}}{2}(s^{\rho}\epsilon_{\varrho\rho\beta})+\tilde{\xi}_{\varrho\beta},$
(30)

with the values of curvature emendation terms as follows

$\displaystyle\chi_{\varrho\beta}$
$\displaystyle=\frac{g_{a\varrho}\epsilon^{\epsilon\rho
a}}{8}\left[(\nabla^{\pi}\nabla_{\epsilon}f_{R}-\frac{3}{2f_{R}}\nabla^{\pi}f_{R}\nabla_{\epsilon}f_{R})\epsilon_{\beta\pi\rho}+(\frac{3}{2f_{R}}\nabla^{\pi}f_{R}\nabla_{\rho}f_{R}-\nabla^{\pi}\nabla_{\rho}f_{R})\epsilon_{\beta\pi\epsilon}\right.$
$\displaystyle\left.+(\frac{3}{2f_{R}}\nabla^{\theta}f_{R}\nabla_{\epsilon}f_{R}-\nabla^{\theta}\nabla_{\epsilon}f_{R})\epsilon_{\theta\beta\rho}+(\nabla^{\theta}\nabla_{\rho}f_{R}-\frac{3}{2f_{R}}\nabla^{\theta}f_{R}\nabla_{\rho}f_{R})\epsilon_{\theta\beta\epsilon}\right]+\frac{f-Rf_{R}}{6}h_{\varrho\beta}$
$\displaystyle+\frac{(\nabla
f_{R})^{2}}{2f_{R}}h_{\varrho\beta}-\frac{\nabla^{\delta}f_{R}\nabla_{\delta}f_{R}}{2f_{R}}h_{\varrho\beta},$
$\displaystyle\psi_{\varrho\beta}$
$\displaystyle=\frac{1}{2}\left(\nabla_{\varrho}\nabla_{\beta}f_{R}-u_{\beta}u^{\delta}\nabla_{\varrho}\nabla_{\delta}f_{R}-u_{\varrho}u_{\theta}\nabla^{\theta}\nabla_{\beta}f_{R}-u_{\theta}u^{\delta}g_{\varrho\beta}\nabla^{\theta}\nabla_{\delta}f_{R}\right)+\frac{3}{2f_{R}}[-\nabla_{\varrho}f_{R}\nabla_{\beta}f_{R}+$
$\displaystyle(\nabla_{\varrho}f_{R}\nabla_{\delta}f_{R})u_{\beta}u^{\delta}+(\nabla^{\theta}f_{R}\nabla_{\beta}f_{R})u_{\varrho}u_{\theta}-(\nabla^{\theta}f_{R}\nabla_{\delta}f_{R})u_{\theta}u^{\delta}g_{\varrho\beta}]-\left(\frac{f-Rf_{R}}{6}\right)h_{\varrho\beta}-$
$\displaystyle\frac{(\nabla
f_{R})^{2}}{4f_{R}}h_{\varrho\beta}+\frac{\nabla^{\beta}f_{R}\nabla_{\beta}f_{R}}{2}h_{\varrho\beta},$
$\displaystyle\tilde{\xi}_{\varrho\beta}$
$\displaystyle=-\frac{1}{2}[(\nabla^{\gamma}\nabla_{\delta}f_{R})u^{\delta}\epsilon_{\varrho\gamma\beta}]+\frac{3}{4f_{R}}(\nabla^{\gamma}\nabla_{\delta}f_{R})u^{\delta}\epsilon_{\varrho\gamma\beta}.$

###  3.1 Super-Poynting Vector and Bel Superenergy

The values for Bel superenergy and Super-Poynting vector turn out to be as
under

$\displaystyle\bar{W}=\frac{5\kappa^{2}\tilde{\rho}^{2}}{24}+\frac{\kappa^{2}\Pi^{2}}{6}+\frac{2E^{2}}{3}+\frac{\kappa^{2}\tilde{\rho}\hat{P}}{4}+\frac{3\kappa^{2}\hat{P}}{8}+\frac{\kappa^{2}\tilde{q}^{2}}{2}+\frac{M_{1}+M_{2}}{2}+M_{3},$
$\displaystyle\bar{P}_{\beta}=\frac{\kappa^{2}\tilde{q}}{2}\left(\tilde{\rho}+P_{r}\right)s_{\beta}-\varsigma_{\beta},$

where

$\displaystyle\varsigma_{\beta}$
$\displaystyle=\frac{\kappa\tilde{q}}{2}(s_{\alpha}h^{\delta}_{\beta}-s_{\beta}\delta^{\delta}_{\alpha}+u_{\alpha}u^{\delta}s_{\beta})\varphi^{\alpha}_{\delta}+\xi^{\gamma\delta}\epsilon_{\beta\gamma\alpha}\left[E^{\alpha}_{\delta}+\frac{\kappa
h^{\alpha}_{\delta}}{6}\tilde{\rho}+\frac{\kappa
h^{\alpha}_{\delta}}{2}\hat{P}+\frac{\kappa}{2}\Pi^{\alpha}_{\delta}+\varphi^{\alpha}_{\delta}\right]$
$\displaystyle+\xi^{\gamma\delta}\epsilon_{\beta\gamma\alpha}\varphi^{\alpha}_{\delta}+\xi^{\gamma\delta}\epsilon_{\beta\gamma\alpha}\left[-E^{\alpha}_{\delta}+\frac{\kappa
h^{\alpha}_{\delta}}{3}\tilde{\rho}+\frac{\kappa}{2}\Pi^{\alpha}_{\delta}+\vartheta^{\alpha}_{\delta}\right]+\frac{\kappa\tilde{q}}{2}(s_{\alpha}h^{\delta}_{\beta}-s_{\beta}\delta^{\delta}_{\alpha}+u_{\alpha}u^{\delta}s_{\beta})\vartheta^{\alpha}_{\delta}$
$\displaystyle+\xi^{\gamma\delta}\epsilon_{\beta\gamma\alpha}\vartheta^{\alpha}_{\delta}.$

The curvature emendations are as follows

$\displaystyle M_{1}$
$\displaystyle=-E_{\mu\nu}\vartheta^{\mu\nu}+\frac{\kappa\tilde{\rho}h_{\mu\nu}\vartheta^{\mu\nu}}{3}+\frac{\kappa\Pi_{\mu\nu}\vartheta^{\mu\nu}}{2}-E^{\mu\nu}\vartheta_{\mu\nu}+\frac{\kappa\tilde{\rho}h^{\mu\nu}\vartheta_{\mu\nu}}{3}+\frac{\kappa\Pi^{\mu\nu}\vartheta_{\mu\nu}}{2}+\vartheta_{\mu\nu}\vartheta^{\mu\nu},$
$\displaystyle M_{2}$
$\displaystyle=\frac{\kappa\tilde{\rho}h_{\mu\nu}\tilde{\vartheta}^{\mu\nu}}{6}+\frac{\kappa\hat{P}h_{\mu\nu}\tilde{\vartheta}^{\mu\nu}}{2}+E_{\mu\nu}\tilde{\vartheta}^{\mu\nu}+\frac{\kappa\Pi_{\mu\nu}\tilde{\vartheta}^{\mu\nu}}{2},$
$\displaystyle M_{3}$
$\displaystyle=\frac{\kappa\tilde{q}s^{\rho}\epsilon_{\mu\rho\nu}\xi^{\mu\nu}}{2}+\frac{\kappa\tilde{q}}{2}s^{\rho}\epsilon^{\mu\nu}_{\rho}\xi_{\mu\nu}+\xi_{\mu\nu}\xi^{\mu\nu}.$

The Bel Robinson tensor depicted as
$W=E^{\mu\eta}E_{\mu\eta}=\frac{2E^{2}}{3}$ when subtracted form the Bel
superenergy, the following outcome is established

$\bar{W}-W=\frac{5\kappa^{2}\tilde{\rho}^{2}}{24}+\frac{\kappa^{2}\tilde{\rho}\hat{P}}{4}+\frac{3\kappa^{2}\hat{P}}{8}+\frac{\kappa^{2}\Pi^{2}}{6}+\frac{\kappa^{2}\tilde{q}^{2}}{2}+\frac{M_{1}+M_{2}}{2}+M_{3}.$

###  3.2 Evaluation of Modified Scalar Functions

In this Section, five structure scalars are ascertained from the trace and
trace-free parts of tensorial quantities $X_{\varrho\beta},Y_{\varrho\beta}$
and $Z_{\varrho\beta}$. This analysis is carried out by following the work of
Herrera [48].
Another technique of defining the tensor $X_{\varrho\beta}$ is

$\displaystyle
X_{\varrho\beta}=\frac{h_{\varrho\beta}}{3}TrX+X_{<\varrho\beta>},\quad\textmd{where}\quad
X_{<\varrho\beta>}=h^{\gamma}_{\varrho}h^{\nu}_{\beta}\left(X_{\varrho\beta}-\frac{h_{\gamma\nu}}{3}TrX\right).$

This leads us to the following outcome

$X_{<\varrho\beta>}=X_{TF}\left(s_{\varrho}s_{\beta}+\frac{h_{\varrho\beta}}{3}\right).$

Now computing the trace component i.e. $TrX=X^{\varrho}_{\varrho}=X_{T}$ and
the trace-free component $X_{TF}$, we infer

$\displaystyle X_{T}$ $\displaystyle=\kappa\tilde{\rho}+\eta_{1},$ (31)
$\displaystyle X_{TF}$
$\displaystyle=\frac{\kappa\Pi}{2}-E+\vartheta_{1}^{(D)},$ (32)

where

$\displaystyle\eta_{1}$
$\displaystyle=\frac{1}{4}\left[\left(\frac{3}{2f_{R}}\nabla^{\nu}f_{R}\nabla_{\epsilon}f_{R}-\nabla^{\nu}\nabla_{\epsilon}f_{R}\right)h^{\epsilon}_{\nu}+\left(\frac{3}{2f_{R}}\nabla^{\nu}f_{R}\nabla_{\rho}f_{R}-\nabla^{\nu}\nabla_{\rho}f_{R}\right)h^{\rho}_{\nu}\right.$
$\displaystyle\left.+\left(\frac{3}{2f_{R}}\nabla^{\alpha}f_{R}\nabla_{\epsilon}f_{R}-\nabla^{\alpha}\nabla_{\epsilon}f_{R}\right)h^{\epsilon}_{\alpha}+\left(\frac{3}{2f_{R}}\nabla^{\alpha}f_{R}\nabla_{\rho}f_{R}-\nabla^{\alpha}\nabla_{\rho}f_{R}\right)h^{\rho}_{\alpha}\right]+\left(\frac{f-Rf_{R}}{2}\right)$
$\displaystyle+\frac{3(\nabla
f_{R})^{2}}{2f_{R}}-\frac{3\nabla^{\gamma}f_{R}\nabla_{\gamma}f_{R}}{2f_{R}}.$

Eq. (31 and (32) correspond to Eqs. (89, 91) in [48] with a significant
distinction that the influence of the dark source is incorporated in the trace
and trace free parts of $X_{\varrho\beta}$.
In the same manner, the trace and trace-free components of $Y_{\varrho\beta}$
are acquired as

$\displaystyle Y_{T}$
$\displaystyle=\frac{\kappa}{2}\left[\tilde{\rho}+3\tilde{P}_{r}-2\Pi\right]+\eta_{2},$
(33) $\displaystyle Y_{TF}$
$\displaystyle=\frac{\kappa\Pi}{2}+E+\vartheta_{2}^{(D)},$ (34)

where

$\displaystyle\eta_{2}$ $\displaystyle=\frac{\Box f_{R}}{2}-\frac{3\Box
f_{R}}{2f_{R}}+\left(\frac{3}{2f_{R}}\nabla^{\nu}f_{R}\nabla_{\delta}f_{R}-\frac{\nabla^{\nu}\nabla_{\delta}f_{R}}{2}\right)u_{\nu}u^{\delta}+\left(\frac{3}{2f_{R}}\nabla^{\pi}\nabla_{\nu}f_{R}-\frac{\nabla^{\pi}\nabla_{\nu}f_{R}}{2}\right)$
$\displaystyle
u^{\nu}u_{\pi}+2\left(\nabla^{\pi}\nabla_{\delta}f_{R}-\frac{3\nabla^{\pi}f_{R}\nabla_{\delta}f_{R}}{f_{R}}\right)u_{\pi}u^{\delta}-\left(\frac{f-Rf_{R}}{2}\right)-\frac{3(\nabla
f_{R})^{2}}{4f_{R}}+\frac{3\nabla^{\gamma}f_{R}\nabla_{\gamma}f_{R}}{2f_{R}}.$

Again, on comparison of Eq. (33) and (34 )with Eqs. (92, 94) of [48], it is
worthy to note that the trace and trace-free components of $Y_{\varrho\beta}$
are also influenced by the existence of dark source.
From the tensorial quantity $Z_{\varrho\beta}$ connected with the heat
dissipative flux, we infer the last scalar $Z$ as

$Z^{2}=Z_{\varrho\beta}Z^{\varrho\beta}=\frac{\kappa^{2}\tilde{q}^{2}}{2}+\tilde{\xi},$
(35)

with
$\vartheta_{1}^{(D)}=\frac{\chi_{\varrho\beta}}{s_{\varrho}s_{\beta}+\frac{h_{\varrho\beta}}{3}}$,
$\vartheta_{2}^{(D)}=\frac{\psi_{\varrho\beta}}{s_{\varrho}s_{\beta}+\frac{h_{\varrho\beta}}{3}}$
and $\tilde{\xi}$ being equal to

$\tilde{\xi}=\frac{\kappa\tilde{q}}{2}s^{\nu}\epsilon_{\mu\nu\alpha}\xi^{\mu\alpha}+\frac{\kappa\tilde{q}}{2}s^{\nu}\epsilon^{\mu\alpha}_{\nu}\xi_{\mu\alpha}+\xi^{\mu\alpha}\xi_{\mu\alpha}.$

###  3.3 Structure Formation and Evolution Equations in terms of Structure
Scalars

The stellar equations stated in Section 2.4 are restated in terms of five
modified structure scalars as under

$\displaystyle\frac{\kappa\tilde{\rho}^{*}}{2}+\frac{1}{3}\left[X_{T}+Y_{T}+X_{TF}+Y_{TF}-\eta_{1}-\eta_{2}-\vartheta_{1}^{(D)}-\vartheta_{2}^{(D)}\right]\left(\theta-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}\right.$
$\displaystyle\left.-\frac{2f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right)-\frac{1}{3}\left(\theta+\frac{\sigma}{2}-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}-\frac{3f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{2\omega^{2}f_{R}}\right)$
$\displaystyle\times\left(X_{TF}+Y_{TF}-\vartheta_{1}^{(D)}-\vartheta_{2}^{(D)}\right)+\frac{\sqrt{2}}{2}\left[\sqrt{Z^{2}-\tilde{\xi}}\right]^{\dagger}+2\sqrt{Z^{2}-\tilde{\xi}}$
$\displaystyle\times\left[\sqrt{a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})^{2}}+\varepsilon_{1}}+\frac{s^{1}}{r}\right]+\varphi_{1}^{(D)}=0,$
(36)
$\displaystyle\frac{\kappa\tilde{P}_{r}}{2}^{\dagger}+\frac{1}{3}\left[X_{T}+Y_{T}+X_{TF}+Y_{TF}-\eta_{1}-\eta_{2}-\vartheta_{1}^{(D)}-\vartheta_{2}^{(D)}\right]\sqrt{a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})}+\varepsilon_{1}}$
$\displaystyle+\frac{s^{1}}{r}\left(X_{TF}+Y_{TF}-\vartheta_{1}^{(D)}-\vartheta_{2}^{(D)}\right)-\frac{1}{3}\sqrt{2Z^{2}-2\tilde{\xi}}\left[\frac{\sigma}{2}-2\theta+\frac{3\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}\sqrt{1-\omega^{2}}}\right.$
$\displaystyle\left.+\frac{4\dot{f_{R}}e^{-\nu/2}}{f_{R}\sqrt{1-\omega^{2}}}+\frac{3\dot{f_{R}}e^{-\nu/2}\sqrt{1-\omega^{2}}}{f_{R}\omega^{2}}\right]+\frac{\sqrt{2}}{2}\left[\sqrt{Z^{2}-\tilde{\xi}}\right]^{\star}+\varphi_{2}^{(D)}=0,$
(37)
$\displaystyle\left[\theta-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}-\frac{2f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right]^{\star}+\frac{1}{3}\left[\theta-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}-\frac{2f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right]^{2}$
$\displaystyle+\frac{1}{6}\left[\sigma+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{f_{R}\omega^{2}}-\frac{2f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right]^{2}-\left[\sqrt{a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})}+\varepsilon_{1}}\right]^{\dagger}-$
$\displaystyle\left[a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})}+\varepsilon_{1}\right]-\frac{2s^{1}}{r}\left[\left(a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})}+\varepsilon_{1}\right)^{1/2}\right]=-Y_{T}+\eta_{2},$
(38)
$\displaystyle\left[\frac{\sigma}{2}+\theta+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{2\omega^{2}f_{R}}-\frac{3\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}\right]^{\dagger}=-\frac{3s^{1}}{2r}$
$\displaystyle\times\left(\sigma+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{\omega^{2}f_{R}}-\frac{2\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right)+\frac{3}{2}\sqrt{2Z^{2}-2\tilde{\xi}},$
(39) $\displaystyle-
Y_{TF}+\vartheta_{2}^{(D)}=\left[\left(a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})}+\varepsilon_{1}\right)^{1/2}\right]^{\dagger}+\left[a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})}+\varepsilon_{1}\right]+$
$\displaystyle\frac{1}{2}\left[\sigma+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{\omega^{2}f_{R}}-\frac{2\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right]^{\star}+\frac{1}{3}\left[\left(\theta-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}-\frac{2f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right)\right.$
$\displaystyle\times\left.\left(\sigma+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{\omega^{2}f_{R}}-\frac{2\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right)\right]-\frac{s^{1}}{r}\sqrt{a^{2}-\frac{f_{R}^{\prime
2}\omega^{2}e^{-\lambda}}{4f_{R}^{2}(1-\omega^{2})}+\varepsilon_{1}}$
$\displaystyle+\frac{1}{12}\left[\sigma+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{\omega^{2}f_{R}}-\frac{2\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}\right],$ (40)
$\displaystyle\frac{1}{3f_{R}}\left(X_{T}+Y_{T}-2X_{TF}+Y_{TF}-\eta_{1}-\eta_{2}+2\vartheta_{1}^{(D)}+\vartheta_{2}^{(D)}\right)\left[\theta-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}\right.$
$\displaystyle\left.-\frac{3f_{R}^{\prime}\omega
e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}+\frac{\sigma}{2}+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{\omega^{2}f_{R}}\right]+\left[\frac{X_{T}}{2}-X_{TF}-\frac{\theta_{1}}{2}+\vartheta_{1}^{(D)}\right]^{\star}=-\frac{3\sqrt{2}s^{1}}{2r}$
$\displaystyle\times\sqrt{Z^{2}-\tilde{\xi}},$ (41)
$\displaystyle\left[\frac{X_{T}}{2}-X_{TF}-\frac{\eta_{1}}{2}+\vartheta_{1}^{(D)}\right]^{\dagger}=\frac{3s^{1}}{r}\left(X_{TF}-\vartheta_{1}^{(D)}\right)+\frac{1}{2}\sqrt{2Z^{2}-2\tilde{\xi}}\left[\frac{\sigma}{2}+\theta\right.$
$\displaystyle\left.+\frac{3\dot{f_{R}}e^{-\nu/2}(1-\omega^{2})^{1/2}}{2\omega^{2}f_{R}}-\frac{3\omega
f_{R}^{\prime}e^{-\lambda/2}}{f_{R}(1-\omega^{2})^{1/2}}-\frac{2\dot{f_{R}}e^{-\nu/2}}{f_{R}(1-\omega^{2})^{1/2}}\right].$
(42)

The last equation becomes

$\frac{3mf_{R}}{r^{3}}=\frac{X_{T}}{2}-X_{TF}-\frac{\eta_{1}}{2}-\frac{\bar{\tau}_{0}+\bar{\tau}_{1}}{2}+\vartheta_{1}^{(D)}.$
(43)

Keeping in view Eqs. (97)-(104) for GR in [48], each of the above equation
maintains a correspondence with the said equations with a significant change
that our equations relate the kinematical functions with the modified
structure scalars (emended using the Palatini’s approach). The Super-Poynting
vector and Superenergy can also be restated as

$\displaystyle\bar{W}=\frac{1}{6}\left(X_{T}^{2}+Y_{T}^{2}-N_{1}\right)+\frac{1}{3}\left(X_{TF}^{2}+Y_{TF}-N_{2}\right)+\sqrt{Z^{2}-\tilde{\xi}},$
$\displaystyle\bar{P}_{\beta}=\frac{1}{3}\sqrt{2Z^{2}-2\tilde{\xi}}\left[X_{T}+Y_{T}+X_{TF}+Y_{TF}-\eta_{1}-\eta_{2}-\vartheta_{1}^{(D)}-\vartheta_{2}^{(D)}\right]s_{\beta}-\varsigma_{\beta}.$

The values for $N_{1}$ and $N_{2}$ are as under

$\displaystyle N_{1}$
$\displaystyle=\eta^{2}_{1}+2\kappa\rho\eta_{1}+\eta^{2}_{2}-2\left(\frac{\kappa\tilde{\rho}}{2}+\frac{3\kappa\hat{P}}{2}\right)\eta_{2},$
$\displaystyle N_{2}$
$\displaystyle={\vartheta_{1}^{(D)}}^{2}+{\vartheta_{2}^{(D)}}^{2}+\vartheta^{(D)}_{2})+2(E+\frac{\kappa\Pi}{2})\vartheta^{(D)}_{2}-2\left(\frac{\kappa\Pi}{2}-E\right)\vartheta^{(D)}_{2}.$

###  3.4 Active Gravitational Mass in terms of Structure Scalars

This section deals with restating the expression for active gravitational
spherical mass in the context of the structure scalars. Eq. (27) is re-written
as

$\displaystyle m_{T}$
$\displaystyle=(m_{T})_{\Sigma}\left(\frac{r}{r_{\Sigma}}\right)^{3}-r^{3}\int_{r}^{r_{\Sigma}}\frac{\nu^{\prime}f_{R}^{\prime}e^{\frac{\nu-\lambda}{2}}}{2r}dr-r^{3}\int_{r}^{r_{\Sigma}}\frac{f_{R}\ddot{\lambda}r^{3}e^{\frac{\lambda-\nu}{2}}}{2r}dr-r^{3}\int^{r_{\Sigma}}_{r}\frac{e^{\frac{\nu+\lambda}{2}}}{r}\times$
$\displaystyle\left(Y_{TF}-\vartheta_{2}^{(D)}+\frac{\bar{\tau}_{1}-2\bar{\tau}_{2}}{\kappa}\right)dr.$

For both equilibrium and quasi-equilibrium states, we acquire

$m_{T}=\int_{0}^{r}e^{\frac{\nu+\lambda}{2}}r^{2}(Y_{T}-\eta_{2})dr.$

##  4 Locally Anisotropic Static Spherical Systems

Sticking to static spherical configurations, we carry out three distinct
systematic approaches to arrive at three distinct line elements. Putting into
use the stellar equations derived in previous sections, we infer these three
different techniques stated below.

###  4.1 First Technique

Putting into use Eqs. (18) and (43), we infer

$e^{-\lambda}-1=-\frac{2r^{2}}{3f_{R}}\left[\frac{X_{T}}{2}-X_{TF}-\frac{\eta_{1}}{2}-\frac{\bar{\tau}_{0}+\bar{\tau}_{1}}{2}+\vartheta^{(D)}_{1}\right].$
(44)

In the context of static case, Eqs. (38) and (40) deduce the following outcome

$a=\frac{r}{3s^{1}}\left(Y_{TF}+Y_{T}-\eta_{2}-\vartheta^{(D)}_{2}\right).$
(45)

In terms of metric coefficients and Eq. (15), we infer

$a=\sqrt{\frac{e^{-\lambda}\nu^{\prime
2}}{4}-\frac{e^{\frac{-\nu-\lambda}{2}}f_{R}^{\prime 2}}{4f_{R}^{2}}};\quad
s^{1}=e^{-\lambda/2}.$ (46)

Substituting Eq. (46) back into (45), we render

$\displaystyle e^{\nu}$
$\displaystyle=c_{1}exp\left[\int\frac{2r}{3}\left[1-\frac{2r^{2}}{3f_{R}}\left(\frac{X_{T}}{2}-X_{TF}-\frac{\eta_{1}}{2}+\vartheta^{(D)}_{1}-\frac{\tilde{\tau}_{0}+\tilde{\tau}_{1}}{2}\right)\right]^{-1}\times\left[\left(Y_{T}+Y_{TF}\right.\right.\right.$
$\displaystyle\left.\left.-\eta_{2}-\vartheta^{(D)}_{2}\right)^{2}+\frac{e^{\frac{-\nu-\lambda}{2}}f_{R}^{\prime
2}}{4f_{R}^{2}}\right]^{1/2}dr.$

Junction conditions can be utilized to attain the value of the integration
constant $c_{1}$. The line element is yielded as

$\displaystyle ds^{2}$
$\displaystyle=c_{1}exp\left[\int\frac{2r}{3}\left[1-\frac{2r^{2}}{3f_{R}}\left(\frac{X_{T}}{2}-X_{TF}-\frac{\eta_{1}}{2}+\vartheta^{(D)}_{1}-\frac{\tau_{0}+\tau_{1}}{2}\right)\right]^{-1}\times\left[\left(Y_{T}+Y_{TF}\right.\right.\right.$
$\displaystyle\left.\left.\left.-\eta_{2}-\vartheta^{(D)}_{2}\right)^{2}+\frac{f_{R}^{\prime
2}e^{\frac{-\nu-\lambda}{2}}}{4f_{R}^{2}}\right]^{1/2}dr\right]dt^{2}-\left[1-\frac{2r^{2}}{3f_{R}}\left(\frac{X_{T}}{2}-X_{TF}-\frac{\eta_{1}}{2}-\right.\right.$
$\displaystyle\left.\left.\frac{\bar{\tau}_{0}+\bar{\tau}_{1}}{2}+\vartheta^{(D)}_{1}\right)\right]^{-1}dr^{2}-r^{2}d\theta^{2}-r^{2}sin^{2}\theta
d\phi^{2}.$

Comparing the above outcome with the line element in Eq. (115) in [48] it is
revealed that for static structures in the frame of $f(R)$ gravity by
Palatini’s approach, the metric also has direct dependence on the effects
generated by the existence of the dark source.

###  4.2 Second Technique

Another line element can be procured by using the following strategy.
Equations (20), (31) and (43) when combined together, produce the following
outcome

$m(r)=r^{3}\int\left(\frac{X_{TF}}{rf_{R}}+\breve{\tau}\right)dr+c_{2},$

employing the integration of which infers

$m(r)=\frac{r^{3}}{3f_{R}}\left(\frac{m^{\prime}f_{R}}{r^{2}}+\frac{\bar{\tau}_{0}}{2}\right)-\frac{r^{3}}{3f_{R}}\left(X_{TF}-\vartheta_{1}^{(D)}\right).$

From the definition of interior spherical mass,

$e^{-\lambda}=1-2r^{2}\left(\int\left[\frac{X_{TF}}{rf_{R}}+\breve{\tau}\right]dr+c_{2}\right),$

where $\breve{\tau}=-\frac{\tau_{0}}{2r}-\frac{\vartheta_{1}^{(D)}}{r}$.
Again, the junction condition can be used to determine the value for
integration constant $c_{2}$. Putting into use Eq. (5) in static form, we
render

$\frac{\kappa
P_{r}}{2}=\frac{f_{R}}{2}\left[\frac{e^{-\lambda}-1}{r^{2}}+\frac{\nu^{\prime}e^{-\lambda}}{r}\right]-\frac{\bar{\tau}_{2}}{2}.$

Using Eqs. (18), (33) and (34) in combination with (45) and (46), we attain

$\left(\frac{\kappa
P_{r}}{2}+\frac{mf_{R}}{r^{3}}\right)=-\frac{\bar{\tau}_{2}}{2}+\frac{e^{-\lambda}\nu^{\prime}f_{R}}{2r}=Y_{h},$
(47)

where

$Y_{h}=\frac{f_{R}}{3}\sqrt{\left(Y_{T}+Y_{TF}-\theta-\vartheta_{2}^{(D)}\right)^{2}+\frac{f_{R}^{\prime
2}e^{\frac{-\nu-3\lambda}{2}}}{4r^{2}f_{R}^{2}}}.$

Integrating Eq. (47), we render

$\displaystyle
e^{\nu}=c_{3}exp\left[\int\frac{2rY_{h}}{f_{R}}\left(1-2r^{2}\left[\int\left(\frac{X_{TF}}{rf_{R}}+\breve{\tau}\right)dr\right]\right)^{-1}dr\right].$

The second static line element reads

$\displaystyle ds^{2}$
$\displaystyle=\left(c_{3}exp\left[\int\frac{2rY_{h}}{f_{R}}\left(1-2r^{2}\left[\int\left(\frac{X_{TF}}{rf_{R}}+\tilde{\tau}\right)dr+c_{2}\right]\right)^{-1}dr\right]\right)dt^{2}$
$\displaystyle-\left[1-2r^{2}\left(\int\left(\frac{X_{TF}}{rf_{R}}+\tilde{\tau}\right)dr+c_{2}\right)\right]^{-1}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}).$

Comparing this result with Eq. (123) in [48], it comes out that the modified
structure scalars along with the dark source factors can be used to explicate
a static spherical object when working with theories other than GR.

###  4.3 Third Technique

From the reorganization of Weyl scalar $E$, we deduce

$E=-\frac{e^{-\lambda}}{2}\left[\left(\frac{\nu^{\prime}}{2}\right)^{2}+\frac{\nu^{\prime\prime}}{2}+\frac{\lambda^{\prime}}{2r}+\frac{1-e^{\lambda}}{r^{2}}+\frac{\nu^{\prime}}{2}\left(-\frac{\lambda^{\prime}}{2}-\frac{1}{r}\right)\right].$

Introduction of new variables in the context of metric coefficients provided
in the line element (2) infers

$\frac{\nu^{\prime}}{2}=\frac{u^{\prime}}{u};\quad\quad\quad y=e^{-\lambda}.$

Under these new variables, a differential equation is ascertained as

$y^{\prime}+2\left(\frac{u^{\prime\prime}-\frac{u^{\prime}}{r}+\frac{u}{r^{2}}}{u^{\prime}-\frac{u}{r}}\right)y=\frac{2(1-2r^{2}E)u}{\left(u^{\prime}-\frac{u}{r}\right)r^{2}}.$

Employing integration w.r.t. $r$ yields

$y=e^{-\int\tilde{k}(r)dr}\left(\int
e^{-\int\tilde{k}(r)dr}\tilde{f}(r)dr+c_{4}\right).$

The functions $\tilde{k}(r)$ and $\tilde{f}(r)$ are given as

$\tilde{k}(r)=2\frac{d}{dr}\left[ln\left(u^{\prime}-\frac{u}{r}\right)\right],\quad\quad\tilde{f}(r)=\frac{2(1-2r^{2}E)u}{\left(u^{\prime}-\frac{u}{r}\right)r^{2}}.$

Switching to old variables, we obtain a first order differential equation in
$\nu$ as

$-\frac{1}{r}+\frac{\nu^{\prime}}{2}=\frac{e^{\lambda/2}}{r}\sqrt{(1-2Er^{2})+c_{4}r^{2}e^{-\nu}+r^{2}e^{-\nu}\int\frac{e^{\nu}}{r^{2}}(2r^{2}E)^{\prime}dr}.$
(48)

Introducing another new variable $z$ in terms of $\nu$ as

$e^{\nu}=\frac{e^{2\int zdr}}{r^{2}},$ (49)

which deduces

$z(r)=\frac{1}{r}+\frac{\nu^{\prime}}{2}.$ (50)

Using Eqs. (49) and (50) into (48), we render

$z(r)-\frac{2}{r}=\frac{e^{\lambda/2}}{r}\sqrt{(1-2Er^{2})+c_{4}r^{4}e^{-\int
2z(r)dr}+r^{4}e^{-\int 2z(r)dr}\int\frac{e^{-\int
2z(r)dr}}{r^{4}}(2r^{2}E)^{\prime}dr}.$ (51)

Eqs. (3) yield

$\displaystyle\kappa\Pi$
$\displaystyle=f_{R}\left[e^{-\lambda}\left(-\frac{\nu^{\prime\prime}}{2}-\left(\frac{\nu^{\prime}}{2}\right)^{2}+\frac{\nu^{\prime}}{2r}+\frac{1}{r^{2}}\right)+\frac{\lambda^{\prime}e^{-\lambda}}{2}\left(\frac{\nu^{\prime}}{2}+\frac{1}{r}\right)-\frac{1}{r^{2}}\right]+(\bar{\tau}_{2}-\bar{\tau}_{1}).$

Placing the values for $y$ and $z$ above, we deduce

$\displaystyle
y^{\prime}+y\left[\frac{2z^{\prime}}{z}+2z-\frac{6}{r}+\frac{4}{r^{2}z}\right]=-\frac{2}{z}\left[\frac{\kappa\Pi}{f_{R}}+\frac{1}{r^{2}}-\frac{(\bar{\tau}_{2}-\bar{\tau}_{1})}{f_{R}}\right].$
(52)

For the metric coefficient $\lambda$, we attain

$\displaystyle
e^{\lambda(r)}=\frac{z^{2}e^{\int\left(2z+\frac{4}{zr^{2}}\right)dr}}{r^{6}\left[-2\int\frac{z}{r^{8}}\left(\frac{\kappa\Pi
r^{2}}{f_{R}}+1-\frac{(\bar{\tau}_{2}-\bar{\tau}_{1})r^{2}}{f_{R}}\right)e^{\int\left(2z+\frac{4}{zr^{2}}\right)dr}dr+c_{5}\right]}.$
(53)

The scalar variables $X_{TF}$ and $Y_{TF}$ are utilized and Eqs. (51) and (53)
are restated as

$\displaystyle z(r)$
$\displaystyle=\frac{2}{r}+\frac{e^{\lambda/2}}{r}\left[1-r^{2}\left(Y_{TF}-X_{TF}-\vartheta^{(D)}_{1}-\vartheta^{(D)}_{2}\right)+r^{4}e^{-\int
2zdr}\left(c_{4}+\int\frac{e^{\int 2zdr}}{r^{4}}\times\right.\right.$
$\displaystyle\left.\left.\left(r^{2}(Y_{TF}-X_{TF}-\vartheta^{(D)}_{1}-\vartheta^{(D)}_{2})\right)^{\prime}dr\right)\right]^{1/2},$
(54) $\displaystyle e^{\lambda(r)}$
$\displaystyle=\frac{z^{2}e^{\int\left(2z+\frac{4}{zr^{2}}\right)dr}}{r^{6}\left[-2\int\frac{z}{r^{8}}\left(\frac{r^{2}}{f_{R}}(Y_{TF}+X_{TF}+\vartheta^{(D)}_{1}+\vartheta^{(D)}_{2})+1-\frac{(\bar{\tau}_{2}-\bar{\tau}_{1})r^{2}}{f_{R}}\right)e^{\int\left(2z+\frac{4}{zr^{2}}\right)dr}dr+c_{5}\right]}.$
(55)

The value for the variable $z$ can be ascertained by using Eq. (55) into (54)
from which the metric coefficient $\nu$ is easily determinable. Also,
utilizing the value for $z$ in Eq. (55), we can determine the metric
coefficient $\lambda$. The last outcome develops correspondence with Eq. (136)
of [48] with a difference that in our case, the metric coefficient
$e^{\lambda(r)}$ can be fully described by the modified structure scalars and
the dark source factors.

##  5 Outline of the Results

From the recently conducted astrophysical research on Cosmic Microwave
Background, Supernovae type Ia and numerous large scale configurations, it has
been revealed that our Universe is enduring an accelerated expansion phase and
its explication demands the inclusion of Dark Energy in gravity theories. In
this regard, mostly $f(R)$ theories are exploited because they open up a new
way to contemplate mysterious forces in nature by putting into use a generic
function of curvature invariant, i.e., Ricci scalar in the Lagrangian.
Variation of such a kind of Lagrangian along with matter action renders second
order gravitational equations that are free from any singularity in addition
with few supplementary terms arising due to $f(R)$ emendations. Palatini
$f(R)$ theory is supreme when compared to other approaches because is meets
all the criteria of Solar system tests and supplies accurate Newtonian limit.

Summing up the findings of the article, we can say that we have inquired the
contribution of structure scalars in the assessment of composition and time-
evolution of a spherically symmetric star. To achieve this objective, we
devise the devise the modified gravitational field equations by putting into
use the Palatini’s technique of variating the Einstein-Hilbert action. We pay
heed to a non-comoving reference frame because it has a substantial role in
explicating the mysterious secrets of evolution of our Universe. The
relativistic fluid content under consideration is assumed to stress anisotropy
attribute with emission of heat radiations. We carry out the evaluation of
various fluid kinematical variables and devised numerous equations that assist
in the inspection of structure composition and its evolution as the time
passes.

By orthogonally decomposing the Riemann tensor, we developed three tensorial
quantities which in turn generate five modified scalar functions
$X_{T},~{}Y_{T},~{}X_{TF},~{}Y_{TF}$ and $Z$. The scalar $X_{T}$ deduced from
te trace component of $X_{\varrho\beta}$ is associated with the description of
energy density with some supplementary terms emerging due to Palatini $f(R)$
emendations. Its trace-free component $X_{TF}$ administers the energ density
inhomogeneity as noticed from Eq. (42) (When $Z=0$, we acquire a differential
equation in $\tilde{\rho}$ which by making use of the regular center condition
implies that the change in energy density is zero if and only if $X_{TF}$ is
zero). From the trace and trace-free components of $Y_{\varrho\beta}$ i.e.,
$Y_{T}$ and $Y_{TF}$ it is inferred that $Y_{TF}$ explicates the impact of
anisotropic stresses as well as inhomogeneous energy density on the active
gravitational mass of the sphere. $Y_{T}$ appears in the expression for
$m_{T}$ so it can be described as the active gravitational mass density. The
scalar $Z$ is connected with the heat dissipative flux of the fluid source.
All these five scalars facilitate in demonstrating the singularity formation
in the astronomical objects like Blackholes. Taking account of spherical
static symmetry, we developed three distinct line elements stemming from the
general one (provided in Eq. (2)) to elucidate static spherical stars by
employing structure scalar functions.

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