# Complexity factor for static cylindrical system

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## Abstract

In this paper, we investigate the complexity factor for static cylindrical configuration with anisotropic fluid distribution. We establish field equations, Tolman–Opphenheimer–Volkoff equation, and mass function. We also evaluate structure scalars using orthogonal splitting of the Riemann tensor which leads to the complexity factor. Finally, we deduce some results about stellar objects for vanishing complexity condition.

## 1 Introduction

A system is defined as the structure in which all the constituents of the system are organized in a specific way and slight disturbance generates complications in it. The combination of different factors which produce complications in any system is called complexity. The phenomenon of complexity has been discussed in many fields of science [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. It varies from field to field and hence an appropriate definition of complexity that fulfils the requirement of all sectors of science has not been attained yet.

The definition of complexity is introduced in the pioneer work of Lopez-Ruiz et al. [7, 8, 9] via the terms, information and entropy. Information can be a specific data set or knowledge related to a certain object while entropy describes the behavior of a system which is either organized or not. In a system, there may exist many other basic factors through which complexity of the system can be defined, thus it is not sufficient to describe complexity only by the terms, information and entropy. Lopez-Ruiz et al. [7] also introduced the idea of disequilibrium to check complexity instead of these terms and showed that it is more suitable to discuss complexity of the system.

In physics, the complexity factor can be illustrated through ideal systems such as isolated ideal gas and perfect crystal. The isolated ideal gas is completely disordered because it is made of a system of random moving particles. All the particles equally participate and give maximum information about the ideal gas. On the other hand, perfect crystal is entirely arranged in a specific manner and a small amount of portion is enough to describe the behavior of perfect crystal and hence provides minimum information. These are the examples of elementary models which are extreme in order and information. In these models, the behavior of the system can be illustrated without creating any complication, so they have zero complexity. Similarly, disequilibrium would be maximum in the case of perfect crystal while for isolated ideal gas, it would be zero.

The complexity factor is also important on astrophysical scales for self-gravitating systems. These systems have various characteristics such as energy density, pressure, stability, mass-radius ratio and luminosity which have been studied frequently. The phenomenon complexity has been examined in literature [12, 13, 14, 15, 16, 17] for neutron stars and white dwarfs by using the concept of disequilibrium and information. These terms include the probability distribution which is restated in the form of energy density to investigate complexity for these objects. However, energy density itself is not sufficient to describe complexity because the main factor pressure component of the energy-momentum tensor is missing which plays an important role in the evolution of self-gravitating systems.

Recently, a quite different way is introduced by Herrera [18] to examine the complexity for astrophysical systems. He involved both energy density and pressure to define the complexity for the spherical self-gravitating system instead of information and disequilibrium. In Herrera’s approach, the complexity factor appears in the structure scalars formed by orthogonal splitting of the Riemann tensor. He also used active gravitational mass (or Tolman mass) of the fluid distribution which contains inhomogeneous energy density and anisotropic pressure. The combination of these two terms is summarized in a single scalar function (structure scalar) and is named as complexity factor. This factor vanishes if the energy density is homogenous and pressures are isotropic or the terms including inhomogeneous energy density and anisotropic pressure that cancel the effect of each other. We have extended the work of Herrera by adding the effect of electromagnetic field [19] and found that complexity of the system decreases in the presence of charge.

Generally, cylindrical systems have been used on different scales to examine the behavior of various physical aspects. Particularly in astrophysics, the phenomenon like gravitational collapse, its radiation, rotating celestial objects and rotating fluids (that help to observe the beam of light produced by stars) provide motivation to consider cylindrical symmetry. According to Birkhoff’s theorem, spherically symmetric spacetime is vacuum outside and no gravitational waves are produced from spherical fluid collapse. For this reason, one moves towards another simple symmetry, i.e., cylindrical geometry. Einstein and Rosen [20] found the solutions for gravitational waves in the case of cylindrically symmetric star. Many astrophysical issues have been discussed for cylindrically symmetric distribution. Herrera and Santos [21] analyzed the matching conditions for collapsing cylindrical fluid distributions. They concluded that the radial pressure is non-zero for this cylinder and the time dependent part of radial pressure is proportional to collapsing fluid. Herrera et al. [22] observed the matching conditions and regularity in equations for cylindrical system and showed that incompressible fluid is obtained for conformally flat solutions. Sharif and Yousaf [23] analyzed the expansion-free condition for cylindrically symmetric distribution including anisotropic pressure and concluded that some solutions satisfy the Darmois junction condition and some solutions show the existence of thin-shell on the boundary.

A wide range of relativistic phenomena have been considered for cylindrical configuration. Sharif and Azam [24] constructed thin-shell wormhole for the cylindrically symmetric black string and concluded that for specific values of different parameters static solutions exist. The same authors [25] studied the instability of cylindrical system with anisotropic fluid distribution and expansion-free condition. They found that stability of the fluid distribution depends upon anisotropic pressure and inhomogeneous energy density. Sharif and Bhatti [26] investigated the instability of charged cylindrical system with expansion-free anisotropic geometry and found that stability of the cylinder is controlled by energy density, electric charge and principal stresses of the fluid. Sharif and Sadiq [27] examined cylindrically symmetric distribution with anisotropic fluid for polytropic equation of state and deduced that among two polytropic models, only one is physically applicable.

In this paper, we consider static cylindrical configuration to discuss the complexity factor. This paper has the following format. In Sect . 2, we formulate some basic equations related to stellar objects while Sect. 3 defines the structure scalars. In Sect. 4, we define the complexity factor and also discuss some astrophysical systems with vanishing complexity. Section 5 provides a brief summary of our results.

## 2 Basic equations of stellar system

*X*,

*Y*,

*Z*are functions of

*r*and \(\alpha \) shows the arbitrary constant. For anisotropic fluid distribution, we consider the energy-momentum tensor in the following form

*r*. The corresponding exterior geometry is considered as [28]

*M*denotes the total mass in the exterior. On the hypersurface \(\Sigma \), the necessary and sufficient conditions for the smooth matching of two metrics (1) and (9) are given in [28]. We take \(Z(r)=r\) as our Schwarzschild coordinate [27], hence the field equations become

*m*

*r*, the total energy for the fluid is

## 3 Structure scalars

## 4 The complexity factor

Here, we consider two models among various stellar structures models. In the first one, we fix the energy density and radial pressure while in the second the polytropic equation of state is used.

### 4.1 The first model

*C*shows the central pressure. Using the value of \(\mu \) from Eq. (48) in (16), we obtain

*Y*from Eq. (51) in (49), we obtain

*Y*, radial pressure,

*X*and tangential pressure, respectively.

### 4.2 The second model

*K*is known as polytropic constant, \(\rho \) is known as polytropic exponent and

*n*is known as polytropic index. Further, we convert the TOV Eq. (13) into dimensionless form by introducing the following dimensionless variables

*c*represents that the term is determined at the center. On the boundary, \(r=r_{\Sigma }\) (\(\varsigma =\varsigma _{\Sigma }\)) and \(\varphi (\varsigma _{\Sigma })=0\). Using Eqs. (25) and (55)–(57) in (13), we obtain

*n*. Physically, this system describes stellar configuration including vanishing complexity condition and satisfies polytropic equation of state.

## 5 Conclusions

The study of astrophysical objects is a captivating phenomenon for researchers which motivates them to explore physical properties of these objects. The physical properties like energy density, anisotropy, stability/instability and luminosity of stellar structure have widely been studied in literature. However, for compact objects, the phenomenon of complexity is not discussed in detail. In this paper, the complexity factor is studied for static cylindrical astrophysical system. We have also investigated vanishing complexity condition using some examples. We have established the field equations and calculated the mass function through two different formalisms namely, C-energy and Tolman mass. We have studied structure scalars and found the complexity factor via these scalars. The complexity factor is given in Eq. (42) which depends upon the inhomogeneous energy density as well as anisotropic pressure implying that these terms produce complexity in the system.

We have also examined the vanishing complexity condition (47) by using \(Y_{TF}=0\) for two types of self-gravitating system. In the first case, we have discussed stellar cylindrical objects with specific energy density and radial pressure introduced by Gokhroo and Mehra [37]. Equations (48), (52) and (54) have been obtained for these objects which interpret the behavior of energy density, radial pressure and tangential pressure, respectively. Next, we have studied the cylindrical stellar configuration satisfying the polytropic equation of state and found a set of differential equations namely, TOV equation, mass equation and vanishing complexity condition in the form of dimensionless variables. With the vanishing complexity condition, this set of differential equations provide the solution for some suitable initial conditions for better understanding of the system.

Finally, we have compared the complexity factor for cylindrical system with the spherical one [18]. The comparison between both the expressions shows that there are some additional terms related to energy density in cylindrical complexity factor which are not present in spherical one.

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