# Complexity analysis of Cylindrically Symmetric Self-gravitating Dynamical
System in $f(R,T)$ Theory of Gravity

M. Zubair<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore-Pakistan
Hina Azmat<EMAIL_ADDRESS>Department of Mathematics, COMSATS
University Islamabad, Lahore Campus, Lahore-Pakistan

###### Abstract

In this article, we have studied a cylindrically symmetric self-gravitating
dynamical object via complexity factor which is obtained through orthogonal
splitting of Reimann tensor in $f(R,T)$ theory of gravity. Our study is based
on the definition of complexity for dynamical sources, proposed by Herrera 12b
. We actually want to analyze the behavior of complexity factor for
cylindrically symmetric dynamical source in modified theory. For this, we
define the scalar functions through orthogonal splitting of Reimann tensor in
$f(R,T)$ gravity and work out structure scalars for cylindrical geometry. We
evaluated the complexity of the structure and also analyzed the complexity of
the evolutionary patterns of the system under consideration. In order to
present simplest mode of evolution, we explored homologous condition and
homogeneous expansion condition in $f(R,T)$ gravity and discussed dynamics and
kinematics in the background of a generic viable non-minimally coupled
$f(R,T)=\alpha_{1}R^{m}T^{n}+\alpha_{2}T(1+\alpha_{3}T^{p}R^{q})$ model. In
order to make a comprehensive analysis, we considered three different cases
(representing both minimal and non-minimal coupling) of the model under
consideration and found that complexity of a system is increased in the
presence of higher order curvature terms, even in the simplest modes of
evolution. However, higher order trace terms affects the complexity of the
system but they are not crucial for simplest modes of evolution in the case of
minimal coupling. The stability of vanishing of complexity factor has also
been discussed.

Keywords: Relativistic systems; $f(R,T)$ gravity; complexity factors; modes of
evolution, dynamical and kinematical considerations.

##  I Introduction

The characterization of cosmological structures on the basis of their degree
of complexity is an interesting physical phenomenon as it is directly related
to the complications appearing in a stellar system due to its physical and
structural characteristics. The measure of degree of complexity can be helpful
in the study of the formation and evolution of stellar objects.All those
factors that triggers the complications in a system are also involved in the
different evolutionary phases of a system. However, a precise and concise
definition of complexity in the field of astrophysics is still an ambition and
aspiration of the community of astrophysicists. In past, many sincere efforts
have been made in order to define an appropriate criterion for the measure of
degree of the complexity in different branches of science 1 -10 . Among the
many definitions that have been proposed so far, most of them are related to
the concepts of information and disequilibrium. These are based on the natural
thought that complexity is related to a basic property describing the
structures existing within the system. In physics, perfect crystals and ideal
gas represent two different but simplest models, so they can be considered as
systems with zero complexity. Nevertheless, study of both of these systems
ensure that definition of complexity cannot be confined to the concepts of
entropy and information, rather it includes some other factors which was going
to be ignored.

A definition of complexity for self-gravitating systems is introduced in 11
which is based on the work developed by Lopez-Ruiz and his collaborators 6 .
In this definition, probability distribution which appear in the definition of
entropy and information is replaced by energy density of the fluids, which has
been justified by the argument that this physical quantity is related to the
probability of finding some particles at given specified location inside the
star, or it proved difficult to suggest a better alternative from the
available physical quantities. Nevertheless, this definition has some
important drawbacks, as it only encompasses the role of energy density whereas
other physical parameters like pressure isotropy or anisotropy which are
expected to play an important role in structure formation of a system are
completely ignored. In order to avoid these drawbacks, an entirely new
definition of complexity 12  has been proposed for spherically symmetric and
static self-gravitating source. It is devised on the basic assumption that
less complex systems correspond to the fluid configurations with homogeneous
energy density and isotropic pressure. Such distributions are assigned with
zero degree of complexity factor which appears in orthogonal splitting of
Riemann tensor. Herrera and his collaborators extended this concept of
complexity from static to non-static scenario, where they consider not only
the complexity factor of structure of fluid configuration but also discuss the
conditions of minimum complexity of evolutionary patterns 12a . For an axially
symmetric static source, they explored three different complexity factors and
found that all the three factors vanish for simplest fluid configuration 12b .
The significance of this definition motivated the researchers to explore it
for different scenarios. Herrera et al. extended this fascinating concept to
the vacuum solutions with the help of the Bondi metric which covers a huge
number of spacetimes like Minkowski spacetime, the static Weyl metrics,
gravitationally radiating metrics, non radiative and non static metrics LH ,
while Casadio et al. studied isotropization and change of complexity employing
gravitational decoupling approach for a static and spherically symmetric
system CAS . In 13 , the effects of electric charge has been incorporated and
complexity factor has been analyzed. They have found that electromagnetic
field has considerable impacts on the complexity of a cosmological structure.
The same authors discussed the complexity of cylindrically symmetric static
source in 14 .

Cosmological objects are usually studied with the consideration of spherical
symmetry because observational evidences show that deformations in spherical
symmetry are very rare, however non-spherical symmetries may also exit and
provide significant information about celestial objects. Different
cosmological issues have also been discussed by assuming non-spherical
situations. Cylindrically symmetric sources grasped the attention of the
community of the relativists when Levi-Civita found its vacuum solution.
Cylindrically symmetric thin-shell wormholes, cylindrical polytropes with
generalized polytropic equation of state, charged non-adiabatic and perfect
fluid with cylindrically symmetric background were studied in 17 -21 .
Cylindrically symmetric self-gravitating objects with anisotropic background
evolving under different conditions 22 -24  have been explored in order to
understand different phases of evolution. Herrera and his collaborators 25
developed structure scalars for cylindrically symmetric and studied
dissipative fluid distribution with the help of these scalars.

General Relativity (GR) is still one of the most comprehensive theory in order
to understand the dynamics of the universe in accordance with its matter
components. However, some cosmological issues like unification of gravitation
and quantum mechanics covering the singularity problem and late time
accelerated expansion of the universe necessitated the improvements in
theoretical framework of GR through modified theories. A large number of class
of modified theories has been introduced to overcome the cosmological issues.
Due to their significance, different cosmological phenomena like anisotropy,
luminosity and stability analysis of celestial objects have been discussed and
presented in literature with the help of these theories 15 ; 16 ; 22 ; 23 ; 24
. Reverberi 27  studied the contracting dust cloud with the $f(R)$ model and
found that increase in energy density bring about the curvature singularity.
Cembranos and his co-authors 28  explored the inflation candidate in the
context of a spherically symmetric self-gravitating collapsing dust cloud in
$f(R)$ gravity. Gravitational collapse and stability constraints for spherical
and axial symmetry has been studied via $f(R,T)$ gravity in 37 -39 .

Compact stars have also been examined and explored in details with the help of
modified gravities. Strong gravity regime in viable models of $f(R)$ gravity
has been studied 40  and the claim that stars with relativistically deep
potentials cannot exist in $f(R)$ gravity has been disproved 41 . Capozziello
et al. 42  considered modified Lane-Emden equation that comes out from $f(R)$
gravity and discussed the hydrostatic equilibrium of stellar structures. Some
interior models of compact stars like $4U1820-30,HerX-1,SAXJ1808-3658$ has
been studied using Krori and Barua analytical solution to the static spacetime
with fluid source in modified $f(R)$ gravity 43 . Polytropic stars in Palatini
$f(R)$-theories has also been investigated and it was shown how findings rely
on regularity of the function $f(R)$ 44 . The possible formation of compact
astrophysical objects and their physical features has been discussed in the
framework of $f(R,T)$ and $f(G,T)$ gravity 45 -49 . Abbas and Nazar 15 ; 16
discussed the effects of $f(R)$ gravity on complexity factor for spherically
symmetric static source with anisotropic background. In s1 ; s2 , complexity
factor has been explored for static anisotropic sources in the context of
Brans-Dick theory, while in s3 ; s4  it has been examined for a dynamical
sphere and a class of compact stars in the framework of $f(R,T)$ gravity. The
physical properties of compact objects were analyzed and their numerical
outputs were found for different values of coupling parameter. Keeping in view
the significance of modified theories, we intended to explore the definition
of complexity factor for a cylindrically symmetric dynamical source with
anisotropic fluid distribution in $f(R,T)$ gravity.

This article has been organized as follows: Next section presents basic
notations, variables and set of field equations for cylindrically symmetric
self-gravitating dynamical source in $f(R,T)$ gravity. Section III and IV
entail the construction of structure scalars for self-gravitating cylinder in
$f(R,T)$ gravity and the identification of complexity factor, respectively.
Section V covers the discussion regarding simplest modes of evolution, while
the section VI comprises the $f(R,T)$ model which is followed by the
discussion about kinematics and dynamics of the system for three different
cases of $f(R,T)$ model. Section VII explores the stability of vanishing
complexity factor condition. Last section concludes our results.

##  II Basic notations, variables and relativistic system of Equations in
$f(R,T)$ theory of gravity

We have considered an anisotropic stellar configuration with cylindrical
geometry which experiences the dissipation in the form of heat flux and its
interior region is given by the following expression

$ds^{2}_{-}=-F^{2}(t,r)dt^{2}+G^{2}(t,r)dr^{2}+H^{2}(t,r)\left(d\theta^{2}+dz^{2}\right),$
(1)

Here, $F$ and $G$ are dimensionless, while $H$ has the same dimension as $r$
has.

The $f(R,T)$ modification of Einstein-Hilbert action is given by 1*

$\int dx^{4}\sqrt{-g}\left[\frac{f(R,T)}{16\pi G}+\mathcal{L}_{(m)}\right].$
(2)

Here, action due to matter is described by $\mathcal{L}_{(m)}$, whose
different choices can be made, each choice represents a particular form of
fluid. Varying the modified action given in Eq.$(\ref{2})$ with respect to
metric $g_{\alpha\beta}$, we have the following set of field equations

$\displaystyle
R_{\alpha\beta}f_{R}(R,T)-\frac{1}{2}g_{\alpha\beta}f(R,T)+(g_{\alpha\beta}\Box-\nabla_{\alpha}\nabla_{\beta})f_{R}(R,T)=8\pi
GT_{\alpha\beta}^{(m)}-f_{T}(R,T)T_{\alpha\beta}^{(m)}-f_{T}(R,T)\Theta_{\alpha\beta},$
(3)

where $f_{R}(R,T)=\frac{\partial f(R,T)}{\partial R}$,
$f_{T}(R,T)=\frac{\partial f(R,T)}{\partial T}$, while $\nabla_{\alpha}$ is
the covariant derivative and $\Box$ is four-dimensional Levi-Civita covariant
derivative. The term $\Theta_{\alpha\beta}$ has the following mathematical
representation

$\displaystyle\Theta_{\alpha\beta}=\frac{g^{\rho\nu}\delta T_{\mu\nu}}{\delta
g^{\alpha\beta}}=-2T_{\alpha\beta}+g_{\alpha\beta}\mathcal{L}_{(m)}-2g^{\rho\nu}\frac{\partial^{2}\mathcal{L}_{(m)}}{\partial
g^{\alpha\beta}\partial g^{\rho\nu}}.$ (4)

With the choice of $\mathcal{L}_{(m)}=\mu$ (energy density) and $8\pi G=1$,
$\Theta_{\alpha\beta}$ takes the form as

$\displaystyle\Theta_{\alpha\beta}=-2T_{\alpha\beta}+\mu g_{\alpha\beta}.$ (5)

Using Eq.$(\ref{5})$, the modified field equations given in (3) become

$\displaystyle G_{\alpha\beta}$ $\displaystyle=$ $\displaystyle
T_{\alpha\beta}^{eff},$ (6)

where

$\displaystyle T^{eff}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{f_{R}}\left[(f_{T}+1)T^{(m)}_{\alpha\beta}-\mu
g_{\alpha\beta}f_{T}+\frac{f-Rf_{R}}{2}g_{\alpha\beta}+(\nabla_{\alpha}\nabla_{\beta}-g_{\alpha\beta}\Box)f_{R}\right],$
(7)

where $T^{(m)}_{\alpha\beta}$ represents the energy momentum tensor for the
usual matter.

We have taken into account a fluid distribution which is locally anisotropic
and suffering dissipation in the form of heat flux. Its energy momentum tensor
has the following mathematical expression

$\displaystyle
T^{(m)}_{\alpha\beta}=(\mu+P_{\perp})V_{\alpha}V_{\beta}-P_{\perp}g_{\alpha\beta}+(P_{r}-P_{\perp})\chi_{\alpha}\chi_{\beta}+q_{\alpha}V_{\beta}+V_{\alpha}q_{\beta},$
(8)

where $P_{r}$ and $P_{\perp}$ are two principal stresses. In most general
cylindrical case, one has three principal stresses which leads to an
anisotropic tensor depending on two independent scalar functions 25 . However,
in our case it will lead to an anisotropic tensor depending on one scalar
function. It will ultimately lead to a single scalar dependent structure
scalar, which can further be analyzed for complexity factor. We are interested
in structure scalar depending on single scalar instead of two, thus our
assumption for fluid distribution is based on two unequal principal stresses.
Here $V_{\beta}$, $\chi_{\beta}$ and $q_{\alpha}$ denote four-velocity, unit
four-vector along radial direction and heat flux respectively. Under co-moving
relative motion, these quantities are defined as

$V_{\alpha}=F^{-1}\delta^{0}_{\alpha},\quad\chi_{\alpha}=G^{-1}\delta^{1}_{\alpha},\quad
q^{\alpha}=qG^{-1}\delta_{1}^{\alpha},$ (9)

and satisfy the following relations

$\displaystyle
V^{\alpha}V_{\alpha}=-1,\quad\chi^{\alpha}\chi_{\alpha}=1,\quad\chi^{\alpha}V_{\alpha}=0,\quad
V^{\alpha}q_{\alpha}=0.$

Eq.$(\ref{9})$ can also be expressed as

$\displaystyle T_{\alpha\beta}^{(m)}$ $\displaystyle=$ $\displaystyle\mu
V_{\alpha}V_{\beta}+Ph_{\alpha\beta}+\Pi_{\alpha\beta}+q(V_{\alpha}\chi_{\beta}+\chi_{\alpha}V_{\beta}),$
(10)

where

$\displaystyle P$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(P_{r}+2P_{\perp}\right),\quad
h_{\alpha\beta}=g_{\alpha\beta}+V_{\alpha}V_{\beta},\quad\Pi_{\alpha\beta}=\Pi\left(\chi_{\alpha}\chi_{\beta}-\frac{1}{3}h_{\alpha\beta}\right),\quad\Pi=P_{r}-P_{\perp}.$
(11)

while the components of shear tensor $\sigma_{\alpha\beta}$ are defined as

$\displaystyle\sigma_{\alpha\beta}=V_{(\alpha;\beta)}-a_{(\alpha}V_{\beta)}-\frac{1}{3}\Theta\left(g_{\alpha\beta}-V_{\alpha}V_{\beta}\right),$
(12)

Here $a_{\alpha}$ is four-acceleration and $\Theta$ is expansion scalar which
defines the rate of infinitesimal change of matter distribution. These two
quantities are defined by the following mathematical formulae

$a_{\alpha}=V_{(\alpha;\beta)}V^{\beta},\quad\Theta=V^{\alpha}_{;\alpha}.$
(13)

Four acceleration $a_{\alpha}$, expansion scalar $\Theta$ and non- zero
components of shear tensor $\sigma_{\alpha\beta}$ can easily be calculated for
given fluid distribution of cylindrically symmetric gravitational source and
these are given below

$\displaystyle
a_{1}=\frac{F^{\prime}}{F},\quad\Theta=\frac{1}{F}\left(\frac{\dot{G}}{G}+\frac{2\dot{H}}{H}\right),$
(14)
$\displaystyle\sigma_{11}=\frac{2}{\sqrt{3}}G^{2}\sigma,\quad\quad\sigma_{22}=\sigma_{33}=-\frac{1}{\sqrt{3}}H^{2}\sigma$
(15)

where

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}F}\left(\frac{\dot{G}}{G}-\frac{\dot{H}}{H}\right),$
(16)

and its scalar value takes the form as

$\displaystyle\sigma^{\alpha\beta}\sigma_{\alpha\beta}$ $\displaystyle=$
$\displaystyle 2\sigma^{2}.$ (17)

The set of $f(R,T)$ field equations for the given cylindrically symmetric
interior metric is given by

$\displaystyle G_{00}$ $\displaystyle=$
$\displaystyle\frac{A^{2}}{f_{R}}\left[\mu+\Psi+\psi_{00}\right],$ (18)
$\displaystyle G_{01}$ $\displaystyle=$
$\displaystyle\frac{FG}{f_{R}}\left((1+f_{T})(-q)+\frac{\psi_{01}}{FG}\right),$
(19) $\displaystyle G_{11}$ $\displaystyle=$
$\displaystyle\frac{G^{2}}{f_{R}}\left[(1+f_{T})\left(P_{r}\right)+\mu
f_{T}-\Psi+\psi_{11}\right],$ (20) $\displaystyle G_{22}$ $\displaystyle=$
$\displaystyle\frac{H^{2}}{f_{R}}\left[(1+f_{T})\left(P_{\perp}\right)+\mu
f_{T}-\Psi+\psi_{22}\right],$ (21)

where

$\displaystyle\Psi$ $\displaystyle=$
$\displaystyle\frac{f-Rf_{R}}{2},\quad\psi_{00}=\frac{f^{\prime\prime}_{R}}{G^{2}}-\frac{\dot{f_{R}}}{F^{2}}\left(\frac{\dot{G}}{G}+2\frac{\dot{H}}{H}\right)+\frac{f_{R}^{\prime}}{G^{2}}\left(2\frac{H^{\prime}}{H}-\frac{G^{\prime}}{G}\right),$
(22) $\displaystyle\psi_{01}$ $\displaystyle=$
$\displaystyle\dot{f^{\prime}_{R}}-\frac{F^{\prime}}{F}\dot{f_{R}}-\frac{\dot{G}}{G}f^{\prime}_{R},$
(23) $\displaystyle\psi_{11}$ $\displaystyle=$
$\displaystyle\frac{\ddot{f_{R}}}{F^{2}}-\frac{\dot{f_{R}}}{F^{2}}\left(\frac{\dot{F}}{F}-2\frac{\dot{H}}{H}\right)-\frac{f^{\prime}_{R}}{B^{2}}\left(\frac{F^{\prime}}{F}+2\frac{H^{\prime}}{H}\right),$
(24) $\displaystyle\psi_{22}$ $\displaystyle=$
$\displaystyle\frac{\ddot{f_{R}}}{F^{2}}-\frac{f^{\prime\prime}_{R}}{G^{2}}-\frac{\dot{f_{R}}}{F^{2}}\left(\frac{\dot{F}}{F}-\frac{\dot{G}}{G}-\frac{\dot{H}}{H}\right)-\frac{f^{\prime}_{R}}{G^{2}}\left(\frac{F^{\prime}}{F}-\frac{G^{\prime}}{G}+\frac{H^{\prime}}{H}\right).$
(25)

Thorne s5  proposed the idea that total amount of energy in a cylindrical
celestial object can defined through the gravitational C-energy, which takes
the following form for the case under consideration

$\displaystyle m(t,r)$ $\displaystyle=$
$\displaystyle\left\\{\left(\frac{\dot{H}}{F}\right)^{2}-\left(\frac{H^{\prime}}{G}\right)^{2}\right\\}\frac{H}{2}+\frac{l}{8}.$
(26)

Before moving towards the further computations, it is worthwhile to define
some notations. The $D_{T}$ and $D_{H}$ are operators which represent proper
time and radial derivatives, respectively, and are defined as

$\displaystyle D_{T}=\frac{1}{F}\frac{\partial}{\partial t},\quad
D_{H}=\frac{1}{H^{\prime}}\frac{\partial}{\partial r},$ (27)

whereas relativistic velocity of interior of collapsing fluid is given by

$\displaystyle U=D_{T}H=\frac{\dot{H}}{F}<0,$ (28)

From Eq.$(\ref{24})$, we can obtain

$\displaystyle\tilde{E}=\frac{H^{\prime}}{G}=\sqrt{\frac{l}{4H}+U^{2}-\frac{2}{H}m(t,r)}.$
(29)

Using above equation together with Eq.$(\ref{17})$, we can develop the
expression given below

$\displaystyle\tilde{E}\left(\sqrt{3}\frac{\sigma}{H}-\frac{1}{3}D_{H}(\Theta-\sqrt{3}\sigma)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2f_{R}}\left(-q(1+f_{T})+\frac{\psi_{01}}{FG}\right).$
(30)

Eq.$(\ref{24})$ together with $(\ref{16})-(\ref{19})$ and $(\ref{25})$
provides

$\displaystyle D_{T}m$ $\displaystyle=$
$\displaystyle\frac{H^{2}}{2f_{R}}\left\\{-(1+f_{T})\tilde{E}q-(1+f_{T})UP_{r}+f_{T}\mu
U+\frac{\tilde{E}}{FG}\pi_{01}-U\left(\Psi+\psi_{11}\right)\right\\},$ (31)

whereas radial derivative of mass provides

$\displaystyle D_{H}m$ $\displaystyle=$
$\displaystyle\frac{H^{2}}{2f_{R}}\left\\{\mu+\frac{U}{\tilde{E}}(1+f_{T})q+\pi+\pi_{00}-\frac{U}{\tilde{E}}\frac{\psi_{01}}{FG}\right\\},$
(32)

which further leads towards the following expression

$\displaystyle m$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{r}\frac{H^{2}}{f_{R}}\left\\{\mu+\frac{U}{\tilde{E}}(1+f_{T})q+\psi+\psi_{00}-\frac{U}{\tilde{E}}\frac{\psi_{01}}{FG}\right\\}H^{\prime}dr,$
(33)

It can also be written as

$\displaystyle\frac{3m}{H^{3}}$ $\displaystyle=$
$\displaystyle\frac{3}{2H^{3}}\int_{0}^{r}\frac{H^{2}}{f_{R}}\left\\{\mu+\frac{U}{\tilde{E}}(1+f_{T})q+\psi+\psi_{00}-\frac{U}{\tilde{E}}\frac{\psi_{01}}{FG}\right\\}H^{\prime}dr.$
(34)

##  III Weyl tensor and Structure scalars

In order to define the structures scalars, we first need to find the weyl
tensor which has two parts, i.e, electric and magnetic parts. The electric
part is given below

$\displaystyle E_{\alpha\beta}$ $\displaystyle=$ $\displaystyle
C_{\alpha\mu\beta\nu}V^{\mu}V^{\nu},$ (35)

The non-trivial components of electric component of weyl tensor are

$\displaystyle E_{11}=\frac{2}{3}G^{2}\eta,\quad
E_{22}=-\frac{1}{3}H^{2}\eta=E_{33},$ (36)

where

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{1}{2F^{2}}\left\\{\frac{\ddot{H}}{H}-\frac{\ddot{G}}{G}-\left(\frac{\dot{H}}{H}-\frac{\dot{G}}{G}\right)\left(\frac{\dot{F}}{F}+\frac{\dot{H}}{H}\right)\right\\}$
(37)
$\displaystyle+\frac{1}{2B^{2}}\left\\{\frac{F^{\prime\prime}}{F}-\frac{H^{\prime\prime}}{H}+\left(\frac{G^{\prime}}{G}+\frac{H^{\prime}}{H}\right)\left(\frac{H^{\prime}}{H}-\frac{F^{\prime}}{F}\right)\right\\}-\frac{1}{2H^{2}}.$

With the help of $(\ref{24})$ and $(\ref{32})$, we can find the expression for
the above scalar value

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{1}{2f_{R}}\left[\mu-(1+f_{T})\Pi+\Psi+\psi_{00}-\psi_{11}+\psi_{22}\right]$
(38)
$\displaystyle-\frac{3}{2H^{3}}\int_{0}^{r}\frac{H^{2}}{f_{R}}\left\\{\mu+\frac{U}{E}(1+f_{T})q-\Psi+\psi_{00}-\frac{U}{E}\frac{\psi_{01}}{FG}\right\\}H^{\prime}dr,$

The electric component $E_{\alpha\beta}$, in view of unit four-velocity and
four-vectors can be given by

$\displaystyle E_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\eta(\chi_{\alpha}\chi_{\beta}-\frac{1}{3}h_{\alpha\beta})$ (39)

Following Bel Bel  and Herrera et al. H1 -HRv , we develop formalism for
structure scalars in $f(R,T)$ gravity and introduce a couple of tensors, named
$Y_{\alpha\beta}$ and $X_{\alpha\beta}$. For this, we orthogonally decompose
the Riemann curvature tensor and find that

$\displaystyle X_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{3f_{R}}\left[\mu+\Psi+\psi_{00}\right]h_{\alpha\beta}-\frac{1}{2f_{R}}\left[(1+f_{T})\Pi+\psi_{11}-\psi_{22}\right]\left(\chi_{\alpha}\chi_{\beta}-\frac{1}{3}h_{\alpha\beta}\right)-E_{\alpha\beta},$
(40) $\displaystyle Y_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{6f_{R}}\left[\mu+3f_{T}\mu+(1+f_{T})(3P_{r}-2\Pi)+\Psi+\psi_{00}+\psi_{11}+2\psi_{22}\right]h_{\alpha\beta}-\frac{1}{2f_{R}}\left[(1+f_{T})\Pi\right.$
(41)
$\displaystyle+\left.\psi_{11}-\psi_{22}\right]\left(\chi_{\alpha}\chi_{\beta}-\frac{1}{3}h_{\alpha\beta}\right)+E_{\alpha\beta}.$

For detailed discussion of these quantities, one can see HRv . These tensors
can be written in the combination of structure scalars ($X_{T}$, $X_{TF}$,
$Y_{T}$ and $Y_{TF}$).

$\displaystyle X_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{3}X_{T}h_{\alpha\beta}+X_{TF}\left(\chi_{\alpha}\chi_{\beta}-\frac{1}{3}h_{\alpha\beta}\right),$
(42) $\displaystyle Y_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{3}Y_{T}h_{\alpha\beta}+Y_{TF}\left(\chi_{\alpha}\chi_{\beta}-\frac{1}{3}h_{\alpha\beta}\right).$
(43)

By making use of Eqs.$(\ref{16})$, $(\ref{18})$, $(\ref{19})$, $(\ref{24})$
and $(\ref{ex})$, we have the following expression

$\displaystyle\frac{3}{H^{3}}\left(m-\frac{l}{8}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2f_{R}}\left(\mu+\psi-(1+f_{T})\Pi+\psi_{00}-\pi_{11}+\psi_{22}\right)-\eta,$
(44)

which makes it possible to produce the following expression with the help of
Eqs.(34) and (44)

$\displaystyle Y_{TF}$ $\displaystyle=$
$\displaystyle\frac{1}{2f_{R}}\left\\{\mu-2(1+f_{T})\Pi+\pi+\pi_{00}-2\psi_{11}+2\pi_{22}\right\\}-\frac{3}{2H^{3}}\int_{0}^{r}\frac{H^{2}}{f_{R}}\left\\{\mu+\psi+\psi_{00}\right.$
(45)
$\displaystyle+\left.\frac{U}{E}q(1+f_{T})-\frac{U}{E}\frac{\psi_{01}}{FG}\right\\}H^{\prime}dr+\frac{3l}{8H^{3}},$

whereas $X_{TF}$ takes the form as

$\displaystyle X_{TF}$ $\displaystyle=$
$\displaystyle-\frac{1}{2f_{R}}\left(\mu+\psi+\psi_{00}\right)+\frac{3}{2H^{3}}\int_{0}^{r}\frac{H^{2}}{f_{R}}\left\\{\mu+\psi+\psi_{00}+\frac{U}{E}q(1+f_{T})-\frac{U}{E}\frac{\psi_{01}}{FG}\right\\}H^{\prime}dr+\frac{3l}{8H^{3}}.$
(46)

Thus, we are now able to construct a differential equation which shows a
relationship between energy density inhomogeneity and weyl tensor.

$\displaystyle\left(X_{TF}+\mu+\frac{1}{2f_{R}}(\mu+\psi+\psi_{00})\right)^{\prime}$
$\displaystyle=$
$\displaystyle-3\frac{H^{\prime}}{H}X_{TF}+\frac{(\Theta-\sigma)}{2f_{R}}\left(q(1+f_{T})G+\frac{\psi_{01}}{G}\right),$
(47)

If we choose $X_{TF}=0$ in the absence of dark source and dissipation, then we
have

$\displaystyle(\mu+\psi+\psi_{00})^{\prime}$ $\displaystyle=$ $\displaystyle
0,$ (48)

however, in general dissipative case it assumes the form

$\displaystyle(\mu+\psi+\psi_{00})^{\prime}$ $\displaystyle=$
$\displaystyle\frac{(\Theta-\sigma)}{2f_{R}}\left(q(1+f_{T})G+\frac{\psi_{01}}{G}\right).$
(49)

It shows that $X_{TF}$ controls the energy density homogeneity along with dark
source terms.

##  IV The Complexity Factor

The definition of quantity measuring the complexity of a dynamical system is
more generalized than for the static one as it faces two additional factors.
In static case, only fluid parameters are involved, while in non-static case,
complexity of structure of system and of patterns of evolution also contribute
to the situation. For static case, definition is based on the assumptions that
homogenous energy density and isotropic pressure corresponds to the simplest
system. However, for the later case, simplest possible patterns are also
considered in order to measure the degree of complexity of evolutionary
patterns.

Recently, we have analyzed definition of complexity for anisotropic fluid non-
static sphere in $f(R,T)$ gravity. We chose $Y_{TF}$ as complexity factor as
it covers all the components that contributes to the complexity of a system.
In the case under consideration, we again found $Y_{TF}$ as most suitable
scalar in order to analyze the components that trigger complications in a
system. It also incorporates the effects of dark source terms. We can see in
Eq.(45) that it also contains the term comprising length of cylinder. Thus, it
also measures the geometric variations in a system.

##  V The Homologous Evolution And The Homogeneous Expansion Condition

After making the choice of $Y_{TF}$ as complexity factor, our next task is to
analyze the complexity of evolutionary patterns of the system. Such analysis
involves two possibilities: the homologous condition and homogeneous
expansion. Homogeneous expansion corresponds to the zero value of prime
derivative of expansion scalar which measures infinitesimal changes in fluid
distribution, whereas homologous evolution corresponds to the similarity of
the patterns.

###  V.1 The Homologous Evolution

We can see that Eq.$(\ref{28})$ can be written as

$\displaystyle D_{H}\left(\frac{U}{H}\right)$ $\displaystyle=$
$\displaystyle\frac{1+f_{T}}{f_{R}}\frac{q}{\tilde{E}}-\frac{1}{FGf_{R}\tilde{E}}\pi_{01}+\sqrt{3}\frac{\sigma}{H}.$
(50)

whose integration leads to the equation

$\displaystyle\frac{U}{H}$ $\displaystyle=$
$\displaystyle\int_{0}^{r}\left(\frac{1+f_{T}}{f_{R}}\frac{q}{\tilde{E}}-\frac{1}{FGf_{R}\tilde{E}}\pi_{01}+\sqrt{3}\frac{\sigma}{H}\right)H^{\prime}dr+h(t),$
(51)

where $h(t)$ is function of integration.

$\displaystyle U$ $\displaystyle=$ $\displaystyle
H\int_{0}^{r}\left(\frac{1+f_{T}}{f_{R}}\frac{q}{\tilde{E}}-\frac{1}{FGf_{R}\tilde{E}}\psi_{01}+\sqrt{3}\frac{\sigma}{H}\right)H^{\prime}dr+Hh(t),$
(52)

which yields

$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{U_{\Sigma}}{H_{\Sigma}}H-H\int_{0}^{r}\left(\frac{1+f_{T}}{f_{R}}\frac{q}{\tilde{E}}-\frac{1}{FGf_{R}\tilde{E}}\psi_{01}+\sqrt{3}\frac{\sigma}{H}\right)H^{\prime}dr.$
(53)

If the integral in Eq.$(\ref{101})$ and Eq.$(\ref{102})$ vanishes, then $U\sim
H$ which is characteristic of homologous evolution; it would be possible if
$\sigma=0$, $q=0$ and $\psi_{01}=0$ or the terms cancel each other.
For homologous evolution, $U=h(t)H$ and $h(t)=\frac{U_{\sigma}}{H_{\sigma}}$
where $U=D_{T}H$. It makes us to follow that $H$ is separable and can be
written as

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{1}(t)H_{2}(r)$ (54)

The term with negative sign in Eq.$(\ref{102})$ shows that dissipation, shear
and dark source entities are responsible for the deviation of evolution from
being homologous. Thus, we have

$\displaystyle\frac{1+f_{T}}{f_{R}}\frac{qG}{H^{\prime}}-\frac{1}{f_{R}FH^{\prime}}\psi_{01}+\sqrt{3}\frac{\sigma}{H}$
$\displaystyle=$ $\displaystyle 0$ (55)

It represents homologous condition in general. For non-dissipative case, it
takes the form as

$\displaystyle\sqrt{3}\frac{\sigma}{H}$ $\displaystyle=$
$\displaystyle\frac{1}{f_{R}FH^{\prime}}\psi_{01}$ (56)

It is obvious that homologous evolution does not correspond to the shear free
condition in general, rather it depends on the choice of $f(R,T)$ model.

###  V.2 The homogeneous Expansion

Homogeneous expansion also represents simple pattern of evolution. Under
homogeneous expansion, Eq.$(\ref{28})$ assumes the following form

$\displaystyle
f_{R}\left(\sqrt{3}\frac{\sigma}{H}+\frac{1}{3}D_{H}\sigma\right)-\frac{1}{FH^{\prime}}\psi_{01}$
$\displaystyle=$ $\displaystyle-\frac{qG}{H^{\prime}}(1+f_{T}).$ (57)

If we analyze the Eqs.$(\ref{105})$ and $(\ref{106i})$, then it can be clearly
observe that imposition of these two conditions is followed by
$D_{H}(\sigma)=0$. Here, its implication is based on the regularity conditions
in the neighborhood of the center, that shear free condition implies zero
dissipation.

##  VI The $f(R,T)$ Model

We can see that the result depends on the choice of $f(R,T)$ model. So we need
to choose a viable $f(R,T)$ model in order to represent our results in a
meaningful way. The $f(R,T)$ model we have selected for discussion is
developed by Sharif and Zubair zub  and has the following mathematical form

$\displaystyle f(R,T)$ $\displaystyle=$
$\displaystyle\alpha_{1}R^{m}T^{n}+\alpha_{2}T(1+\alpha_{3}T^{p}R^{q}),$ (58)

where $\alpha_{i}^{\prime}s$ are positive real numbers , whereas $m,n,p,q$
assumes some value greater than or equal to $1$ . We will analyze our results
considering different cases of above mentioned model and we will proceed our
further discussion under following three cases:

  1. 1.

$f(R,T)=R+\alpha_{2}T$, for $\alpha_{1}=1,m=1,n=0,\alpha_{3}=0$

  2. 2.

$f(R,T)=\alpha_{1}R+\alpha_{2}T+\alpha_{4}T^{2}$, for
$m=1,n=0,\alpha_{4}=\alpha_{1}\alpha_{3},p=1,q=0$

  3. 3.

$f(R,T)=\alpha_{1}R+\alpha_{2}T(1+\alpha_{3}TR^{2})$, for $m=1,n=0,p=1,q=2$

##  VII Kinematics and Dynamics of Stellar systems

###  VII.1 Case I: $f(R,T)=R+\alpha_{2}T$

This form of model involves direct minimal curvature matter coupling, which
has been used widely to explore a number of cosmological phenomena because of
its theoretical and cosmological consistency 1* . Many theoretical
cosmological models of the universe have been proposed to analyze the behavior
of mysterious components and their physical and cosmological consequences are
explored s6 ; s7 . Galactic structures and their existence have also been
discussed, and results are found in agreement with previously established
solutions and assumptions s8 . For this model, homologous condition and
homogeneous expansion condition given in Eqs.$(\ref{105})$ and $(\ref{106i})$
take the form as

$\displaystyle(1+\alpha_{2})qG$ $\displaystyle=$
$\displaystyle\sqrt{3}\frac{\sigma H^{\prime}}{H},$ (59)
$\displaystyle\left(\sqrt{3}\frac{\sigma}{H}+\frac{1}{3}D_{H}\sigma\right)$
$\displaystyle=$ $\displaystyle-\frac{qG}{H^{\prime}}(1+\alpha_{2}).$ (60)

Here, Eq.$(\ref{1081})$ clearly depicts that fluid cannot be dissipative under
shear free condition and homogeneous expansion. However, if Eqs.$(\ref{1071})$
and $(\ref{1081})$ hold at the same time, then we have

$\displaystyle(\Theta-\sqrt{3}\sigma)^{\prime}$ $\displaystyle=$
$\displaystyle 0.$ (61)

By inserting the values of expansion scalar and shear scalar given in
Eqs.$(\ref{13})$ and $(\ref{sh})$, respectively, we have

$\displaystyle(\Theta-\sqrt{3}\sigma)^{\prime}$ $\displaystyle=$
$\displaystyle\left(\frac{3}{F}\frac{\dot{H}}{H}\right)^{\prime}=0.$ (62)

Eq.$(\ref{103})$ together with above equation follows that $F^{\prime}=0$
which ensures the geodesic condition of the fluid. As $F$ possesses an
arbitrary constant value, so we may choose $F=1$. With this choice, we get the
following expression from Eq.(62)

$\displaystyle\Theta-\sqrt{3}\sigma$ $\displaystyle=$ $\displaystyle
3\frac{\dot{H}}{H}.$ (63)

We analyze this expression closer to the center and obtain the condition
$(\Theta-\sqrt{3}\sigma)^{\prime}=0$. The successive derivatives of
Eq.$(\ref{1101})$ with respect to $r$, also support our argument and
strengthens the point that fluid is homologous.
Again we analyze the Eq.$(\ref{28})$, if we assume $\sigma=0$, (also fluid is
non-dissipative), then we have

$\displaystyle\Theta^{\prime}$ $\displaystyle=$ $\displaystyle 0.$ (64)

It can clearly be observe that homologous patterns of the evolution imply
homogeneity of the expansion scalar.
If we assume $\Theta^{\prime}=0$, then Eq.$(\ref{28})$ takes the following
form

$\displaystyle\left(\frac{\sqrt{3}\sigma}{H}-\frac{1}{\sqrt{3}}D_{H}(\sigma)\right)$
$\displaystyle=$ $\displaystyle 0,$ (65)

which further takes the form as

$\displaystyle\frac{\sigma^{\prime}}{\sigma}$ $\displaystyle=$
$\displaystyle\frac{3R^{\prime}}{R},$ (66)

implying

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle\frac{f_{1}(t)}{H^{3}},$
(67)

where $f_{1}(t)$ is an arbitrary function of integration. As $r$ assumes zero
value at the center, so $H$ will also be zero. Thus, we must have $f_{1}(t)=0$
in order to avoid unboundedness of the expression. This situation lead to the
vanishing of shear scalar. On the other hand, if we choose the zero value for
shear scalar, then Eq.$(\ref{28})$ ensures the homogeneous expansion. Here, it
is obvious that homogeneous expansion and homologous condition imply each
other in non dissipative case. Further, we have analyzed the situation for
homogeneous expansion in the presence of dissipation and obtained

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{2H^{3}}\int^{r}_{0}H^{3}qG(1+\alpha_{2})dr.$
(68)

This expression makes it clear that homogeneous expansion and homologous
condition are not compatible in the presence of dissipation.

Now, we consider some dynamical situations for the system under consideration.
As our previous discussion shows that fluid is geodesic under homologous
condition in both dissipative and non-dissipative cases. Thus, if we apply
homologous condition on the Eq.(A.4), we obtain

$\displaystyle D_{T}U$ $\displaystyle=$
$\displaystyle-\frac{m}{H^{2}}-\frac{H}{2}\left\\{\alpha_{2}\mu-(1+\alpha_{2})P_{r}+\frac{\alpha_{2}T}{2}\right\\}.$
(69)

This equation can be re-written in the form of $Y_{TF}$ as

$\displaystyle\frac{3D_{T}U}{H}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left\\{(1-3\alpha_{2})\mu-\alpha_{2}T-2(1+\alpha_{2})\Pi+3(1+\alpha_{2})P_{r}\right\\}+Y_{TF}-\frac{3l}{8H^{3}}.$
(70)

Now, the manipulation of Eqs.$(\ref{16})$, $(\ref{18})$ and $(\ref{19})$
provides

$\displaystyle-\frac{2\ddot{H}}{H}-\frac{\ddot{G}}{G}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{(1-3\alpha_{2})\mu-\alpha_{2}T-2(1+\alpha_{2})\Pi+3(1+\alpha_{2})P_{r}\right\\}+Y_{TF}-\frac{3l}{8H^{3}},$
(71)

while the definition of velocity ‘U’ of collapsing star provides

$\displaystyle\frac{3D_{T}U}{H}$ $\displaystyle=$
$\displaystyle\frac{3\ddot{H}}{H}.$ (72)

Insertion of above two equations into Eq.$(\ref{1151a})$ leads to

$\displaystyle Y_{TF}$ $\displaystyle=$
$\displaystyle\frac{\ddot{H}}{H}-\frac{\ddot{G}}{G}-\frac{3l}{8H^{3}}.$ (73)

If $Y_{TF}=0$, then Eq.$(\ref{1181})$ becomes

$\displaystyle\frac{3l}{8H^{3}}$ $\displaystyle=$
$\displaystyle\frac{\ddot{H}}{H}-\frac{\ddot{G}}{G}.$ (74)

Since we are working on the assumption that fluid is homologous, so we can
write the Eq.$(\ref{1151a})$ as

$\displaystyle 3\left(\dot{h}(t)+h(t)\frac{\dot{H}}{H}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left\\{(1-3\alpha_{2})\mu-\alpha_{2}T-2(1+\alpha_{2})\Pi+3(1+\alpha_{2})P_{r}\right\\}+Y_{TF}-\frac{3l}{8H^{3}},$
(75)

Now, we see the both cases when the fluid is non-dissipative or dissipative.

####  VII.1.1 The Dissipative and Non-dissipative Scenarios

Here, we assume another condition that fluid is non-dissipative. Under this
assumption and for the choice of $f(R,T)$ model in case I, Eq.(56) implies
shear free condition for fluid configuration. With this implication,
Eq.$(\ref{sh})$ leads to

$\displaystyle\frac{\ddot{H}}{H}-\frac{\ddot{G}}{G}$ $\displaystyle=$
$\displaystyle 0\quad\Rightarrow\quad Y_{TF}=\frac{3l}{8H^{3}}.$ (76)

Here, we can see that complexity factor $Y_{TF}=0$ is proportional to the
fraction of $l$ and $H^{3}$. We inserted this relation in Eq.$(\ref{Y})$ and
found $\Pi=0$, which implies that $\mu^{\prime}=0$. Thus,
$Y_{TF}=\frac{3l}{8H^{3}}$ represents simplest modes of evolution in the case
of cylinder. Further, Eqs.$(\ref{118a})$ and $(\ref{1251})$ also strengthens
our argument.

Now, we analyze the situation in the presence of dissipation. In this case,
Eqs.$(\ref{1081})$ and $(\ref{sh})$ produce the following expression

$\displaystyle\dot{\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\left\\{\left(\frac{\dot{H}}{H}\right)^{2}-\left(\frac{\dot{G}}{G}\right)^{2}+\frac{3l}{8H^{3}}-Y_{TF}\right\\}.$
(77)

The time derivative of Eq.$(\ref{1081})$ and above equation provide the
following mathematical expression

$\displaystyle Y_{TF}\frac{H^{\prime}}{H}$ $\displaystyle=$
$\displaystyle\frac{1}{2}Bq(1+\alpha_{2})\left(\frac{\dot{q}}{q}+\frac{2\dot{B}}{B}+\frac{\dot{H}}{H}\right)+\frac{3l}{8H^{3}}.$
(78)

If we assign zero value to complexity factor, then we get

$\displaystyle
Bq(1+\alpha_{2})\left(\frac{\dot{q}}{q}+\frac{2\dot{B}}{B}+\frac{\dot{H}}{H}\right)+\frac{3l}{4H^{3}}$
$\displaystyle=$ $\displaystyle 0.$ (79)

This differential equation can further be solved by using some suitable
numerical or analytical methods of integration. It actually holds to represent
simplest dissipative regime.

###  VII.2  Case-II: $f(R,T)=\alpha_{1}R+\alpha_{2}T+\alpha_{4}T^{2}$

This form of our selected model comprises linear and quadratic terms in the
trace of the energy-momentum tensor( EMT). The squared terms of EMT was first
introduced in s9 . This particular form of $f(R,T)$ model has been used to
explore non-exotic matter wormholes s10 . This type of choice usually contrast
with higher order gravity and the results provide description of the universe
that enters from a decelerated phase of expansion to an accelerated one and in
agreement with observational data. For this choice of $f(R,T)$ model,
conditions obtained against simplest modes of evolution given in
Eqs.$(\ref{105})$ and $(\ref{106i})$ take the form as

$\displaystyle\frac{1+\alpha_{2}+2\alpha_{4}T}{\alpha_{1}}qG$ $\displaystyle=$
$\displaystyle\sqrt{3}\frac{\sigma H^{\prime}}{H},$ (80)
$\displaystyle\left(\sqrt{3}\frac{\sigma}{H}+\frac{1}{3}D_{H}\sigma\right)$
$\displaystyle=$
$\displaystyle-\frac{qG}{\alpha_{1}H^{\prime}}(1+\alpha_{2}+2\alpha_{4}T)$
(81)

Here, if we assume $\sigma=0$, Eq.(81) ensures the vanishing of dissipative
variable. Thus, homogeneous expansion and shear-free condition again ceases
the fluid to be dissipative. All the situations that exits in Eqs.(67-81) are
also valid in this case. However, in the presence of dissipation, shear scalar
assume the form as

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{2H^{3}}\int^{r}_{0}\frac{H^{3}}{\alpha_{1}}qG(1+\alpha_{2}+\alpha_{4}T)dr.$
(82)

In this case, dissipative variable again affects the homogeneous expansion and
homologous condition and these are not found compatible.

Now, we consider some dynamical situations for the system under consideration.
As our previous discussion shows that fluid is geodesic under homologous
condition in both dissipative and non-dissipative cases. Thus, if we apply
homologous condition on the Eq.(A.4), we obtain

$\displaystyle D_{T}U$ $\displaystyle=$
$\displaystyle-\frac{m}{H^{2}}-\frac{H}{2\alpha_{1}}\left\\{(\alpha_{2}+2\alpha_{4}T)\mu-(1+\alpha_{2}+2\alpha_{4}T)P_{r}+\frac{(\alpha_{2}+2\alpha_{4}T)T}{2}\right\\}.$
(83)

This equation can be re-written in the form of $Y_{TF}$ as

$\displaystyle\frac{3D_{T}U}{H}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\alpha_{1}}\left\\{(1-3\alpha_{2}-6\alpha_{4}T)\mu-\alpha_{2}T-\alpha_{4}T^{2}-2(1+\alpha_{2}+\alpha_{4}T)\Pi+3(1+\alpha_{2}+\alpha_{4}T)P_{r}\right\\}$
(84) $\displaystyle+Y_{TF}-\frac{3l}{8H^{3}}.$

Now, the manipulation of Eqs.$(\ref{16})$, $(\ref{18})$ and $(\ref{19})$
provides

$\displaystyle-\frac{2\ddot{H}}{H}-\frac{\ddot{G}}{G}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{(1-3\alpha_{2}-6\alpha_{4}T)\mu-\alpha_{2}T-2\alpha_{4}T^{2}-2(1+\alpha_{2}+2\alpha_{4}T)\Pi+3(1+\alpha_{2}+2\alpha_{4}T)P_{r}\right\\}$
(85) $\displaystyle+Y_{TF}-\frac{3l}{8H^{3}},$

while the definition of velocity ‘U’ of collapsing star provides

$\displaystyle\frac{3D_{T}U}{H}$ $\displaystyle=$
$\displaystyle\frac{3\ddot{H}}{H},$ (86)

Insertion of above two equations into Eq.$(\ref{115**})$ leads to

$\displaystyle Y_{TF}$ $\displaystyle=$
$\displaystyle\frac{\ddot{H}}{H}-\frac{\ddot{G}}{G}-\frac{3l}{8H^{3}}.$ (87)

If $Y_{TF}=0$, then Eq.$(\ref{118*})$ becomes

$\displaystyle\frac{3l}{8H^{3}}$ $\displaystyle=$
$\displaystyle\frac{\ddot{H}}{H}-\frac{\ddot{G}}{G}.$ (88)

Since we are working on the assumption that fluid is homologous, so we can
write the Eq.$(\ref{115**})$ as

$\displaystyle 3\left(\dot{h}(t)+h(t)\frac{\dot{H}}{H}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left\\{(1-3\alpha_{2}-6\alpha_{4}T)\mu-\alpha_{2}T-\alpha_{4}T^{2}-2(1+\alpha_{2}+\alpha_{4}T)\Pi+3(1+\alpha_{2}+\alpha_{4}T)P_{r}\right\\}+Y_{TF}$
(89) $\displaystyle-\frac{3l}{8H^{3}},$

####  VII.2.1 The Dissipative and Non-dissipative Scenarios

In non-dissipative case, we again observe the same scenario as it is discussed
in case I. Thus, we need to discuss only dissipative case. Here, we analyze
the situation in the presence of dissipation. In this case, Eqs. $(\ref{sh})$
and $(\ref{108a})$ produce the following expression

$\displaystyle\dot{\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\left\\{\left(\frac{\dot{H}}{H}\right)^{2}-\left(\frac{\dot{G}}{G}\right)^{2}+\frac{3l}{8H^{3}}-Y_{TF}\right\\}.$
(90)

The time derivative of Eq.$(\ref{108a})$ and above equation provide the
following mathematical expression

$\displaystyle Y_{TF}\frac{H^{\prime}}{H}$ $\displaystyle=$
$\displaystyle\frac{1}{2\alpha_{1}}Bq(1+\alpha_{2}+2\alpha_{4}T)\left(\frac{\dot{q}}{q}+\frac{2\dot{B}}{B}+\frac{\dot{H}}{H}\right)+\frac{\alpha_{4}}{\alpha_{2}}\dot{T}+\frac{3l}{8H^{3}}.$
(91)

If we assign zero value to complexity factor, then we get

$\displaystyle\frac{Bq}{\alpha_{1}}(1+\alpha_{2}+2\alpha_{4}T)\left(\frac{\dot{q}}{q}+\frac{2\dot{B}}{B}+\frac{\dot{H}}{H}\right)+2\frac{\alpha_{4}}{\alpha_{2}}\dot{T}+\frac{3l}{4H^{3}}$
$\displaystyle=$ $\displaystyle 0.$ (92)

This differential equation can further be solved by using some suitable
numerical or analytical methods of integration. It actually holds to represent
simplest dissipative regime.

###  VII.3  Case III: $f(R,T)=\alpha_{1}R+\alpha_{2}T(1+\alpha_{3}TR^{2})$

This type of models offer the non-minimal coupling of curvature and matter
components. The similar type of choice has been recently used to measure the
impact of collision matter on the late-time dynamics of $f(R,T)$ gravity s11 .
In this case, homologous and homogeneous conditions will take the form as

$\displaystyle\frac{1+\gamma_{2}}{\gamma_{1}}qG+\sqrt{3}\sigma\frac{H^{\prime}}{H}$
$\displaystyle=$
$\displaystyle\frac{1}{\gamma_{1}}\left(\gamma_{5}-\frac{\dot{B}}{B}\gamma_{4}\right),$
(93)
$\displaystyle\gamma_{1}\left(\sqrt{3}\frac{\sigma}{H}+\frac{1}{\sqrt{3}}D_{H}\sigma\right)$
$\displaystyle=$
$\displaystyle\frac{1}{FH^{\prime}}\left(\gamma_{5}-\frac{A^{\prime}}{A}\gamma_{3}-\frac{\dot{B}}{B}\gamma_{4}\right),$
(94)

however, Eq.(56) takes the form as

$\displaystyle\sqrt{3}\frac{\sigma}{H}$ $\displaystyle=$
$\displaystyle\frac{1}{\gamma_{1}FH^{\prime}}\left(\gamma_{5}-\frac{A^{\prime}}{A}\gamma_{3}-\frac{\dot{B}}{B}\gamma_{4}\right),$
(95)

where

$\displaystyle\gamma_{1}$ $\displaystyle=$
$\displaystyle\alpha_{1}+2\alpha_{4}T^{2}R,$ $\displaystyle\gamma_{2}$
$\displaystyle=$ $\displaystyle\alpha_{2}+2\alpha_{4}TR^{2},$
$\displaystyle\gamma_{3}$ $\displaystyle=$
$\displaystyle\alpha_{4}(T^{2}\dot{R}+2T\dot{T}R),$ $\displaystyle\gamma_{4}$
$\displaystyle=$ $\displaystyle\alpha_{4}(T^{2}R^{\prime}+2TT^{\prime}R),$
$\displaystyle\gamma_{5}$ $\displaystyle=$ $\displaystyle
2\alpha_{4}(T^{2}\dot{R^{\prime}}+2TT^{\prime}\dot{R}+2T^{\prime}\dot{T}R+2T\dot{T^{\prime}}R+2T\dot{T}R^{\prime}).$

Here, Eq.(95) clearly shows that homologous evolution does not imply shear
free condition in non-dissipative case. Nevertheless, validity of the
Eqs.$(\ref{lab1})$ and $(\ref{lab2})$ at the same time again makes us to
believe that fluid is homologous. However, if we choose $\Theta^{\prime}=0$
and consider the Eq.$(\ref{28})$, then we have

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2H^{3}}\int\frac{H^{3}}{F\gamma_{1}}\left(\gamma_{5}-\frac{F^{\prime}}{F}\gamma_{3}-\frac{\dot{B}}{B}\gamma_{4}\right)dr.$
(96)

Under the same condition (i.e., $q=0$), if we assume $\sigma=0$, then
Eq.$(\ref{28})$ does not imply the homogeneous expansion, rather it takes the
form as

$\displaystyle\Theta^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{F\gamma_{1}}\left(\gamma_{5}-\frac{F^{\prime}}{F}\gamma_{3}-\frac{\dot{G}}{G}\gamma_{4}\right).$
(97)

It is obvious from Eqs.$(\ref{lab3})$ and $(\ref{lab4})$ that homologous and
homogeneous expansion conditions do not imply each other in non-dissipative
case. In the presence of dissipation, $\Theta^{\prime}=0$ produce the
following result

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2H^{3}}\int\frac{H^{3}}{F\gamma_{1}}\left\\{\left(\gamma_{5}-\frac{F^{\prime}}{F}\gamma_{3}-\frac{\dot{B}}{B}\gamma_{4}\right)-FGq(1+\gamma_{2})\right\\}dr.$
(98)

Again, we take into account dynamical considerations and apply homologous
condition and obtain from Eq.$(\ref{B4})$

$\displaystyle D_{T}U$ $\displaystyle=$
$\displaystyle-\frac{m}{H^{2}}-\frac{H}{2\gamma}\left\\{\gamma_{2}\mu+(1+\gamma_{2})P_{r}+\phi+\phi_{11}\right\\},$
(99)

which further takes the form as

$\displaystyle\frac{3D_{T}U}{H}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\gamma_{1}}\left\\{(1+3\gamma_{2})\mu-2\phi-2(1+\gamma_{2})\Pi+\phi_{11}+2\phi_{22}+3(1+\gamma_{2})P_{r}\right\\}+Y_{TF}-\frac{3l}{8H^{3}},$
(100)

where $\phi$ and $\phi_{ii}$ occur due to $f(R,T)$ extra degrees of freedom
involved in the evolution. The above expression can further be re-written as

$\displaystyle 3\left(\dot{h}(t)+h(t)\frac{\dot{H}}{H}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2\gamma_{1}}\left\\{(1+3\gamma_{2})\mu-2\phi-2(1+\gamma_{2})\Pi+\phi_{11}+2\phi_{22}+3(1+\gamma_{2})P_{r}\right\\}+Y_{TF}-\frac{3l}{8H^{3}}.$
(101)

Further, from field equations we have extracted the following result

$\displaystyle\frac{1}{2\gamma_{1}}\left\\{(1+3\gamma_{2})\mu-2\phi-2(1+\gamma_{2})\Pi+\phi_{11}+2\phi_{22}+3(1+\gamma_{2})P_{r}\right\\}+Y_{TF}-\frac{3l}{8H^{3}}$
$\displaystyle=$ $\displaystyle-\frac{2\ddot{H}}{H}-\frac{\ddot{G}}{G},$ (102)

Using the definition of velocity ‘$U$’ of collapsing star together with above
equation, we again obtain the same result as given in Eq.(73) and vanishing of
complexity factor $Y_{TF}$ yields the same expression as given in Eq.(74).

####  VII.3.1 The Dissipative and Non-dissipative Scenarios

In this case, homologous condition does not imply shear free condition for
non-dissipative scenario. Thus, under homologous condition, we have
$Y_{TF}=\frac{\ddot{H}}{H}-\frac{\ddot{G}}{G}-\frac{3l}{8H^{3}}$, which shows
that complexity of the system is increased in the presence of higher order
curvature terms. Even in simplest modes of evolution, system have enough
complexity index and fluid configuration does not corresponds to the isotropic
pressure and homogenous energy density.

Using Eqs.$(\ref{sh})$ and $(\ref{lab2})$, we find the relation for
dissipative case as

$\displaystyle Y_{TF}\frac{H^{\prime}}{H}$ $\displaystyle=$
$\displaystyle\frac{1}{2\alpha_{1}}Gq\frac{1+\gamma_{2}}{\gamma_{1}}\left(\frac{\dot{q}}{q}+\frac{2\dot{G}}{G}+\frac{\dot{H}}{H}\right)+\left(\frac{\gamma_{2}}{\gamma_{1}}\right)_{,0}qG-\frac{1}{\gamma_{1}}\left(\gamma_{5}-\frac{F^{\prime}}{F}\gamma_{3}-\frac{\dot{G}}{G}\gamma_{4}\right)+\frac{3l}{8H^{3}}.$
(103)

We can see dark source terms play important role and vanishing of complexity
factor yields the following expression

$\displaystyle\frac{1}{2\alpha_{1}}Gq\frac{1+\gamma_{2}}{\gamma_{1}}\left(\frac{\dot{q}}{q}+\frac{2\dot{G}}{G}+\frac{\dot{H}}{H}\right)+\left(\frac{\gamma_{2}}{\gamma_{1}}\right)_{,0}qG-\frac{1}{\gamma_{1}}\left(\gamma_{5}-\frac{F^{\prime}}{F}\gamma_{3}-\frac{\dot{G}}{G}\gamma_{4}\right)+\frac{3l}{8H^{3}}$
$\displaystyle=$ $\displaystyle 0.$ (104)

##  VIII Stability of The Vanishing Complexity Factor Condition

In this section, our task is to find and analyze the conditions which are
responsible for an initial state of vanishing complexity factor under
homologous condition. For this, we need to develop the the evolution equation
for structure scalar $Y_{TF}$ with the help of Eqs.$(\ref{xx})$, $(\ref{yy})$,
and (A.2) which takes the form as

$\displaystyle\dot{Y}_{TF}+(1+\alpha_{2})\dot{\Pi}+\frac{3\dot{H}}{H}Y_{TF}+2(1+\alpha_{2})\Pi\frac{\dot{H}}{H}+\frac{(1+\alpha_{2})}{2}(\mu+P_{r})\sigma-\frac{1+\alpha_{2}}{2}(\mu+P_{r})\frac{\dot{G}}{G}-(1-\alpha_{2}^{2})(\mu+P_{\perp})\frac{\dot{H}}{H}$
$\displaystyle+\alpha_{2}\frac{\dot{G}}{G}(\mu+P_{\perp})-\frac{3}{2}\frac{H^{\prime}}{H}(1+\alpha_{2})q\frac{1-\alpha_{2}^{2}}{2G^{2}}(Gq)^{\prime}+\frac{q(1-\alpha^{2})}{G^{2}}\left(\frac{2H^{\prime}}{H}-\frac{G^{\prime}}{G}\right)+\alpha_{2}q^{\prime}+\Lambda=0,$
(105)

where $\Lambda$ contains dark source entities. Now, we analyze this equation
for dissipative and non-dissipative cases turn by turn. First we consider the
non-dissipative case at some initial moment where $Y_{TF}=q=\sigma=\Pi=0$,
then previous equation takes the form as

$\displaystyle\dot{Y}_{TF}+(1+\alpha_{2})\dot{\Pi}-\frac{1-\alpha_{2}^{2}}{2}(\mu+P_{\perp})\frac{\dot{H}}{H}+\Lambda$
$\displaystyle=$ $\displaystyle 0.$ (106)

In the most general case, when the system is dissipative, we have at the
initial moment

$\displaystyle\dot{Y}_{TF}+(1+\alpha_{2})\dot{\Pi}+2(1+\alpha_{2})\Pi\frac{\dot{H}}{H}+\frac{1+\alpha_{2}}{2}(\mu+P_{r})\sigma-\frac{1+\alpha_{2}}{2}(\mu+P_{r})\frac{\dot{G}}{G}-(1-\alpha_{2}^{2})(\mu+P_{\perp})\frac{\dot{H}}{H}$
$\displaystyle+\alpha_{2}\frac{\dot{G}}{G}(\mu+P_{\perp})-\frac{3}{2}\frac{H^{\prime}}{H}(1+\alpha_{2})q\frac{1-\alpha_{2}^{2}}{2G^{2}}(Gq)^{\prime}+\frac{q(1-\alpha^{2})}{G^{2}}\left(\frac{2H^{\prime}}{H}-\frac{G^{\prime}}{G}\right)+\alpha_{2}q^{\prime}+\Lambda=0.$
(107)

It is obvious from above equation that pressure anisotropy, energy density
inhomogeneity, dissipative variable and dark source entities all are crucial
for complexity of a cylindrical system. Last term $\Lambda$ on the right hand
side of the above equation incorporates the effects of dark source terms which
can be measured for any particular model.

##  IX Conclusion

The study of complexity on astrophysical scales is an interesting concept
which helps the researchers to explore and identify the factors that are
responsible for the emergence of complexity in a system. In the pursuance of a
complete and comprehensive definition of complexity, Herrera 12b  proposed a
new definition for an anisotropic fluid sphere. Following Herrera’s work, we
have explored the behavior of complexity factor for a cylindrically symmetric
dynamical object in $f(R,T)$ gravity. For this, we have explored field
equations for cylindrical symmetry and developed mass function using C-energy
expression. We have constructed structure scalars for cylindrical geometry in
$f(R,T)$ gravity and identified $Y_{TF}$ as complexity factor among these
scalars which incorporates the effects of inhomogeneous energy density,
anisotropic pressure and dissipative variable together with dark source terms.

The definition of complexity for a dynamical system involves not only the
complexity of the structure of the system, but also considers degree of
complexity of the pattern of evolution of the system. Thus, we have studied
the complexity of patterns of evolution by assuming two simplest modes of
evolution,i.e., homologous evolution and homogeneous expansion. We have worked
out the conditions representing homogeneous expansion and homologous
evolution, where we have found that shear free condition implies zero
dissipation if both of these conditions hold at the same time. In order to
present detailed study in $f(R,T)$ gravity, we have selected a generic non-
minimally coupled $f(R,T)$ model and discussed our findings for three
different cases of model under consideration.

  * •

In the first case, our $f(R,T)$ model is linear combination of first order
terms of curvature ‘$R$’ and trace of energy momentum tensor ‘$T$’. Here, we
have found that homogeneous expansion and shear free condition cease the fluid
to be dissipative. We have also found that shear free condition and homologous
condition imply each other for this particular choice. In non-dissipative
scenario, simplest modes of evolution are found to be compatible with each
other. However, in the presence of dissipative variable, they are not found
compatible. We have discussed dissipative and non-dissipative scenarios and
found complexity factor is proportional to the fraction of length of cylinder
and curvature term of cubic order which represents minimum value of complexity
factor when total amount of energy in a cylindrical system is represented
through gravitational C-energy.

  * •

In the second case, our model is of the form $f(R,T)=f_{1}(R)+f_{2}(T)$ ,
where $f_{1}(R)$ is linear function, while $f_{2}(T)$ is quadratic one. All
the results that are obtained in the first case are same for the second choice
of $f(R,T)$ model. However in dissipative case, complexity of the system is
increased due to the presence of quadratic $T$ terms.

  * •

In the third case, our assumptions yields the form which consists on two
parts. The first part coincides with case I, however second part contains
product of quadratic terms of ‘$R$’ and ‘$T$’. In this case, shear free
condition and homologous condition are not found to be compatible for both
dissipative and non-dissipative cases. It is observed that complexity of a
system is increased in the presence of dark source entities, even in the
simplest modes of evolution it does not attain the value that represents
minimum complexity of the system.

In the end, we analyzed the stability of vanishing complexity condition and
observed different outcomes in both dissipative and non-dissipative scenarios.
In the absence of dissipation, significance of pressure isotropy and dark
source entities is obvious from Eq.(106), however in general scenario,
Eq.(107) shows that multiple factors are involved.

We have compared our results with previous literature 12a ; s3  and found that
complexity factor for cylindrical system contains some extra terms because of
the geometrical difference of the self gravitating systems, which ceases it to
attain zero value in simplest modes of evolution. The complexity system for
static self-gravitating systems has been explored extensively, but it needs
more study for dynamical systems. We intend to extend our work for dynamical
self-gravitating systems in the presence of charge.

## Appendix A

The divergence of the energy-momentum tensor is nonzero in $f(R,T)$ gravity
and is found as

$\displaystyle\nabla{{}^{\alpha}}T_{\alpha\beta}=\frac{f_{T}}{1-f_{T}}\left[(\Theta_{\alpha\beta}-T_{\alpha\beta})\nabla{{}^{\alpha}}lnf_{T}-\frac{1}{2}g_{\alpha\beta}\nabla{{}^{\alpha}}T+\nabla^{{}^{\alpha}}\Theta_{\alpha\beta}\right].$
(A.1)

Its divergence yields the following two equations:

$\displaystyle\dot{\mu}\left(\frac{1-f_{T}-f_{R}f_{T}}{f_{R}(1-f_{T})}\right)-\mu\frac{\dot{f_{R}}}{f_{R}^{2}}+\frac{\dot{G}}{G}\frac{1}{f_{R}}(1+f_{T})(\mu+P_{r})+\frac{\dot{2H}}{H}\frac{1}{f_{R}}(1+f_{T})(\mu+P_{\perp})+\left(\frac{\dot{T}}{2}\right)\frac{f_{T}}{1-f_{T}}$
$\displaystyle-q^{\prime}\frac{F}{G}\frac{(1+f_{T})}{f_{R}}-\frac{q}{G^{2}}\left\\{\frac{FG(1+f_{T})}{f_{R}}\right\\}_{,1}-\frac{F}{G}\left(\frac{F^{\prime}}{F}-\frac{G^{\prime}}{G}+\frac{2H^{\prime}}{H}\right)(1+f_{T})q+\left\\{\frac{1}{f_{R}}\left(\Psi+\psi_{00}\right)\right\\}_{,0}$
$\displaystyle-\frac{1}{G^{2}}\left\\{\frac{\psi_{01}}{f_{R}}\right\\}_{,1}+{\frac{\psi_{01}}{G^{2}f_{R}}}\left(\frac{F^{\prime}}{F}-\frac{G^{\prime}}{G}+\frac{2H^{\prime}}{H}\right)+\frac{\psi_{00}}{f_{R}}\left(\frac{\dot{G}}{G}+\frac{\dot{2H}}{H}\right)+\frac{\dot{G}}{G}\frac{\psi_{11}}{f_{R}}+\frac{\dot{2H}}{H}\frac{\psi_{22}}{f_{R}}$
$\displaystyle-\left(\frac{2\dot{G}}{G}(\mu+P_{r})+\frac{F}{G}2q^{\prime}+4q\frac{F^{\prime}}{G}\right)\frac{f_{T}}{1-f_{T}}-\frac{F}{G}q\frac{f^{\prime}_{T}}{1-f_{T}}=0,$
(A.2) $\displaystyle
P^{\prime}_{r}\left(\frac{1-f^{2}_{T}+2f_{R}f_{T}}{f_{R}(1-f_{T})}\right)+\frac{P_{r}}{f_{R}}\left(f^{\prime}_{T}-\frac{f^{\prime}_{R}(1+f_{T})}{f_{R}}\right)+\mu^{\prime}\frac{f_{T}}{f_{R}}+\frac{\mu}{f_{R}}\left(f^{\prime}_{T}-\frac{f_{T}f^{\prime}_{R}}{f_{R}}\right)+\frac{F^{\prime}}{F}\frac{(1+f_{T})}{f_{R}}(\mu+P_{r})$
$\displaystyle+\frac{2H^{\prime}}{H}\frac{1}{f_{R}}(1+f_{T})(P_{r}-P_{\perp})+\frac{f^{\prime}_{T}}{1-f_{T}}(\mu+P_{r})-\frac{f_{T}}{1-f_{T}}\mu^{\prime}+\frac{G}{F}\frac{(1+f_{T})}{f_{R}}\dot{q}+\frac{q}{F^{2}}\left\\{\frac{FG(1+f_{T})}{f_{R}}\right\\}_{,0}$
$\displaystyle+\left\\{\frac{1}{f_{R}}\left(\Psi+\psi_{11}\right)\right\\}_{,1}-\frac{1}{F^{2}}\left\\{\frac{\psi_{01}}{f_{R}}\right\\}_{,0}+{\frac{\psi_{01}}{G^{2}f_{R}}}\left(\frac{\dot{F}}{F}-\frac{\dot{G}}{G}+\frac{\dot{2H}}{H}\right)+q\frac{G}{F}(1+f_{T})\left(\frac{\dot{F}}{F}-\frac{\dot{G}}{G}+\frac{\dot{2H}}{H}\right)$
$\displaystyle+\frac{f_{T}}{1-f_{T}}\left(\frac{T^{\prime}}{2}+\mu^{\prime}\right)+\frac{F^{\prime}}{F}\frac{1}{f_{R}}\left(\psi_{00}+\pi_{11}\right)+\frac{2H^{\prime}}{H}\frac{1}{f_{R}}\left(\pi_{11}-\psi_{22}\right)+\frac{2G}{F}q\left(1+\frac{F^{\prime}}{F}\right)\frac{f_{T}}{1-f_{T}}$
$\displaystyle+\frac{G}{F}\left(q\frac{\dot{f_{T}}}{f_{R}}+\frac{\dot{G}}{F}(\mu_{+}2P_{r})\right)\frac{f_{T}}{1-f_{T}}+\frac{f_{T}}{1-f_{T}}\frac{2}{A}(Bq)_{,0}=0.$
(A.3)

The acceleration $D_{T}U$ of a collapsing star can be obtained by manipulating
the Eqs. $(\ref{13})$, $(\ref{17})$, $(\ref{24})$ and $(\ref{27})$:

$\displaystyle D_{T}U$ $\displaystyle=$
$\displaystyle\frac{m}{H^{2}}-\frac{R}{2f_{R}}\left(1+f_{T})P_{r}+f_{T}\mu+\pi+\pi_{11}\right)+Ea.$
(A.4)

## Acknowledgments

“Authors thank the Higher Education Commission, Islamabad, Pakistan for its
financial support under the NRPU project with grant number
5329/Federal/NRPU/R&D/HEC/2016”.

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