Complexity Factor of Static Axial Complex Structures in f(R, T) Gravity

Abstract

This article investigates the physical features of static axial sources that produce complexity within the matter configuration within the perspective of $f(R, T)$ theory, where R is the curvature invariant and T identifies the trace of matter energy tensor. In this case, the contracted Bianchi identities of effective as well as normal matter are used to develop the conservation equations. We split the curvature tensor to compute structure scalars, involving the physical aspects of the source in the influence of modified factors. We explore the evolving source and compute the complexity of the system. Three complexity factors are determined by using structure scalars; after that, the corresponding propagation equations are explored to investigate the intense gravitational consequences. Finally, the outcomes of irregular anisotropic spheroids are presented using the criterion of vanishing complexity. The $f(R, T)$ corrections are shown to be an additional source of complexity for the axial anisotropic configuration.

1. Introduction

The recent observation regarding the universe’s rapid expansion in the present epoch could be found in information from a variety of sources, such as type-Ia Supernovae, huge scale structures, and the cosmic-microwave-background, etc. [1,2,3]. This universe’s behavior is considered to be caused by dark energy, which exerts strong repulsive pressure. Several astronomical studies, such as the sloan-digital sky survey, the two degree field galactic red-shift survey, and the huge synoptic-survey telescope, have shown how much galaxies and stars affect the evolution of the universe. It is therefore desirable to analyze these constituents to better comprehend the universe and its formation. The analysis of such gravitating systems illustrates the physics of dark energy and dark matter. Due to the notion of dark energy and the experimental evidence of an expanding cosmos, theoretical cosmology has been in a dilemma. Thus, modified gravity theories (MGTs) are an effective approach to point out the concerns of dark energy and the inflationary epochs accompanying late-time acceleration.

In order to execute the cosmic-acceleration by various approaches, we shall utilize $f\left(R\right)$ gravity, which is a simple extension of the Einstein–Hilbert (EH) action. Fay et al. [4] discussed the dynamics of the various cosmic configurations in the setting of $f\left(R\right)$ techniques. They adopted the numerical simulations as well as the phase-space analysis for the systematic evaluation. Amendola et al. [5] studied the feasibility of the $f\left(R\right)$ cosmic configurations and argued that the power-law models in the $f\left(R\right)$ scenario are not associated with late-time acceleration and hence are not physically acceptable. Trembly and Faraoni [6] studied the problem of initial value by utilizing the dynamical association between Palatini/metric $f\left(R\right)$ formalisms and Brans-Dicke approach. They found that the Cauchy problem under matter configuration is well-posed in the metric approach in contrast to the Palatini approach. Bamba et al. [7] discussed the non-equilibrium/equilibrium thermodynamic descriptions of the Palatini $f\left(R\right)$ apparent horizon. They showed that the entropy of the horizon is easier to comprehend in an equilibrium scenario rather than in a non-equilibrium one. Bamba et al. [8] reviewed distinct dark-energy cosmologies for various matter contents to analyze their physical aspects under different physical frameworks by cosmography. Yousaf and his collaborators [9,10,11,12,13] explored the effects of Palatini $f\left(R\right)$ factors on the formulation of irregular constituents for relativistic spheres.

The matter–geometry coupling notion has attracted a great deal of consideration as a mode to investigate the mystery of cosmic evolution and its concerns. Bertolami et al. [14] extended $f\left(R\right)$ gravity by proposing an explicit coupling between scalar curvature R and Lagrangian matter-density ${\mathcal{L}}^m$, thereby providing a geometry–matter coupling in a non-minimal context at action level. Harko [15] proposed $f(R,{\mathcal{L}}^m)$ gravity as the extension of EH action, where the Lagrangian was regarded to be a generic function of matter-density and a curvature invariant. It is considered that every aspect of matter is encrypted in ${\mathcal{L}}^{\left(m\right)}$. Harko and Lobo [16] presented the continuation of this notion to an arbitrary geometry–matter coupling. Afterwards, Harko et al. [17] proposed another extension of $f\left(R\right)$ theory, denoted $f(R,T)$ theory. The trace of the matter energy tensor is represented by T. Its variations are caused by the quantum consequences of a non-ideal exotic fluid configuration. They introduced a Lagrangian formulation of matter–geometry-coupled configurations bearing the influence of additional force owing to the non-geodesics motion of test particles and corresponding sources, and they obtained the field equations for various cosmic models within the metric approach. The investigation of the universe’s gravitational aspects through $f(R,T)$ theory has drawn the interest of many researchers. Houndjo [18] examined the cosmic reconstruction of the model $f(R, T)=f_1\left(R\right)+f_2\left(T\right)$ in order to describe the accelerated or matter-dominated epochs. Alvarenga et al. [19] displayed the feasibility settings for some $f(R,T)$ paradigms with the help of energy conditions. Sun and Huang [20] explored the cosmic evolution without dark energy by concentrating on particular configuration of $f(R,T)$ gravity. They presented certain plots for red-shift analysis that were compatible with the astrophysical observational data. Mishra et al. [21] constructed the anisotropic cosmic configurations in $f(R,T)$ gravity. For the late-time cosmological acceleration, they analyzed the impact of cosmic anisotropy as well as the coupling invariant on the cosmological dynamics. They found that the dynamics of the cosmos are significantly influenced by cosmic anisotropy. Shabani and Farhoudi [22] discussed the consequences of solar and cosmic configurations within the influence of $f(R,T)$ modifications. Zaregonbadi et al. [23] investigated the effects of dark matter on a galactic scale. They proposed the static spheres in the locality of Einstein’s gravity solutions and discussed the minimal matter–curvature-coupled configuration. They determined the geometric constituents in the scenario of a galactic halo. Sahoo et al. [24] explored the energy conditions for anisotropic sources by considering distinct $f(R,T)$ cosmic models. Various cosmic effects of $f(R,T)$ gravity have been extensively presented in the literature; see, for instance, [25,26,27,28].

The search for physically significant and exact analytic findings to gravitational field equations, demonstrating the configuration other than spherical sources, is a key endeavor in Einstein’s theory and MGTs. Generally, it is proposed that the notion of axially symmetric configurations is based on the assertion that cosmic dense structures deviate from spherical symmetry in incidental aspects. This strengthens the need for having non-spherical models of interior regions to analyze the physical aspects of highly compact structures. Axially symmetric configurations have a great impact on the study of the cosmic scenario, where the effects of anisotropic distributions of energy and matter can never be ignored. The various systematic analyses of axially symmetric sources have been presented extensively in the scenario of Einstein’s theory [29,30,31,32]. However, addressing it in MGTs is a key problem. In the context of bimetric gravitation theory, Jain et al. [33] studied the effects of wet dark fluid on the gravitating axial cosmic system. They showed that the wet dark fluid does not appear to contribute to this notion. Aygün et al. [34] investigated the localization of the stress-energy tensor within Marder’s axially symmetric metric. Rao and his collaborators [35] proposed axial cosmic models with the description of perfect matter from the perspective of Einstein’s theory and $f(R,T)$ theory. In last few years, there has also been a discussion of a numerous relativistic consequences in distinct symmetric configurations (spherical and axial, cylindrical, etc.) within the context of MGTs [36,37,38,39,40]. Herrera et al. [41] evaluated scalar variables (structure scalars) for static axially symmetric and anisotropic fluids. They obtained significant findings about the evolving structures and identified the factors causing energy-density inhomogeneity. Moreover, they constructed some exact analytic solutions for the isotropic as well as anisotropic spheroids. It was shown that the class of anisotropic findings could be matched to the Weyl exterior spacetime. Herrera et al. [42] investigated the collapsing axial symmetric model in Einstein’s theory. They derived the heat-conduction equation to discuss the thermodynamic consequences of the source. In this case, they found that the scalar functions had a significant effect on the evolving system.

Sahoo et al. [43] studied the impact of $f(R,T)$ corrections on the axial system by considering the influence of perfect matter constituents. They analyzed the physical and geometric aspects of some specific cosmological models. They found certain analytical consequences of the field equations by considering the mean Hubble-parameter variation. Tariq et al. [44] analyzed the effects of Palatini $f\left(R\right)$ corrections on the stability of isotropic axial and spherical configurations. They found that in non-spherical collapsing models, both the physical parameters and modified terms make a significant contribution to the evolution pattern. The evolution of the cosmic structures is significantly influenced by physical factors, including density inhomogeneity, and the anisotropy of the fluids [45,46,47]. In modern relativistic astrophysics, the scalar variables are important for describing the physical aspects of galaxy configurations that have their own gravity (for more description, see [48,49,50,51]). The idea of junction conditions has captured great concern to glue (match) cosmic configurations together. For this purpose, both the system’s internal and external configurations must be carefully examined. The vacuum solution explains the unique outward gravitational structure of galactic bodies as deriving from certain interior findings for relativistic structures.

Gad [52] evaluated the stress-energy densities for the Weyl metrics through Einstein stress-energy complexes. He discovered that stress-density is provided for the Weyl metrics scenarios. The same energy density is provided, with the exception of the case $R\to \infty$. Herrera et al. [53] provided the matching conditions for generic distribution of anisotropic and dissipative spheres. They integrated the zero-expansion configuration and demonstrated that the emergence of a void is assured without an expansion evolution case. The matching conditions have been the subject of extensive research in both Einstein’s theory [54,55,56] and in MGTs [57,58,59,60,61,62]. Goswami et al. [63] studied the collapsing phenomena of massive star-like configurations in the direction of $f\left(R\right)$ gravity. They exhibited the additional matching criteria results due to modified corrections that enforced significant limitations on celestial systems and thermodynamical aspects. They showed that an analytic finding for an irregular collapsing sphere within the Starobinski-model satisfies all matching and energy requirements. In the event of a cosmic scalar-field, Maharaj et al. [64] deployed $f\left(R\right)$ technique in order to figure out some analytical consequences for the Friedmann universe. Rosa [65] constructed the $f(R, T)$ version of the matching conditions to match the two configurations across a hyper-surface separation for the perfect fluids.

Any self-gravitating model’s homogenous distribution may experience issues due to a variety of physical reasons, which is referred to as complexity. The study of complexity has drawn significant attention across a number of research fields. Herrera [66] devised an innovative concept for analyzing the density-inhomogeneities as a continuation of the examination of the regular distribution of non-dynamic stars with the aid of a factor. He represented this factor through structure-scalars and dubbed it the complexity factor (CF). Then, Herrera and his co-workers [67] extended this notion to a completely dynamic scenario. In this case, they considered both the requirements of the CF of the matter structure and the minimally complex evolution pattern of the dynamical spheres. They also constructed specific stellar consequences to demonstrate the system’s stability through the vanishing of the CF.

In this paper, we continue the concept of complexity factors (CFs) initiated in [68] to the non-dynamic axially symmetric case from the setting of $f(R, T)$ theory. This study is assigned to understanding the effects of $f(R, T)$ constituents in the theoretical modeling of non-dynamic axially symmetric structures. To do so, we will first provide the CFs and then these factors in the form of corresponding scalar variables. Following is an outline of the paper:

In the following section, we provide the $f(R, T)$ formalism for gravitational equations and the description of matter configuration for axial symmetric systems in a static case. We determine the conservation and $f(R, T)$ equations of the proposed system. After computing the $f(R, T)$ version of structure scalars, we explain the zero CFs scenario to simplify our complex system. We develop analytical solutions for isotropic and anisotropic spheroids that satisfy the vanishing CFs criteria in the direction of $f(R, T)$ theory in Section 3. The Section 4 contains the discussion of our main results. Finally, we present the Appendix A containing the $f(R, T)$ terms that reveal the impact of high curvature realms.

2. $\mathit{f}(\mathit{R}, \mathit{T})$ Formalism and Matter Variables

This section reviews the field equations of $f(R, T)$ gravity. The $f(R, T)$ formalism has a significant impact on understanding both the physical aspects and analytical findings of stellar structures based on high curvature epochs. This theory results from the extension of Einstein–Hilbert action supplied by the usual matter Lagrangian $S_m$ as [17]

$$\begin{array}{c}\hfill I_{f(R, T)}=\frac{1}{16\pi }\int f(R, T)\sqrt{-g}{d}^{4}x+\int \sqrt{-g}{S}_m{d}^{4}x,\end{array}$$

(1)

here g is the determinant of the metric tensor. We are using normalized units G, c both equal to one and the gravitational coupling constant $\kappa$ is chosen to be $8\pi$ in our scenario. The tensor for the description of the matter and energy is provided by

$$\begin{array}{cc}\hfill & T_{\omega \beta }^{\left(m\right)}=-\frac{2}{\sqrt{-g}}\frac{\delta \left(\sqrt{-g}{S}_m\right)}{\delta g^{\omega \beta }}.\hfill \end{array}$$

(2)

Taking variation of Equation (1) with $g^{\omega \beta }$, we obtain the gravitational equations

$$\begin{array}{c}\hfill R_{\omega \beta }{f}_R-\frac{1}{2}{g}_{\omega \beta }f+(g_{\omega \beta }\square -{\nabla }_{\omega }{\nabla }_{\beta })f_R=\kappa T_{\omega \beta }^{\left(m\right)}-f_T({\Theta }_{\omega \beta }+T_{\omega \beta }),\end{array}$$

(3)

where operators □, $f_R(R, T)$, and $f_T(R, T)$ stand for ${\nabla }_{\omega }{\nabla }^{\omega }$, $\frac{df(R, T)}{dR}$ and $\frac{df(R, T)}{dT}$, respectively, while ${\nabla }^{\omega }$ represents covariant derivative. The quantity ${\Theta }_{\omega \beta }$ is to be described

$$\begin{array}{c}\hfill {\Theta }_{\omega \beta }=-2T_{\omega \beta }-2g^{\alpha \gamma }\frac{{\partial }^2{S}_m}{\partial g^{\omega \beta }\partial g^{\alpha \gamma }}+g_{\omega \beta }{S}_m.\end{array}$$

For our problem, we take the matter-energy tensor as the originator of locally anisotropic fluid composition. We now provide the detailed explanation of the source. We will use the Bondi-technique [69] to describe the physical meaning of the stress-energy tensor’s constituents. This technique includes matter profile in a thoroughly local-Minkowskian-frame $(\tau , x, y, z)$ where the first variations of the metric are no longer present, or, likewise, assume a tetrad-field associated with such a locally-Minkowski-frame. For the system under discussion, the most generic stress-energy tensor in a locally specified frame is provided by

$$\begin{array}{c}\hfill {\widehat{T}}_{\omega \beta }=\left(\begin{array}{cccc}\widehat{\mu }& 0& 0& 0\\ 0& {\widehat{P}}_{xx}& {\widehat{P}}_{xy}& 0\\ 0& {\widehat{P}}_{yx}& {\widehat{P}}_{yy}& 0\\ 0& 0& 0& {\widehat{P}}_{zz}\end{array}\right)\end{array}$$

(4)

here ${\widehat{P}}_{xx},{\widehat{P}}_{xy},{\widehat{P}}_{zz},{\widehat{P}}_{yy}$ and $\widehat{\mu }$ identify the various pressure components and the energy density, respectively, as computed by an observer co-moving with the fluid in a locally specified Minkowskian frame. It is noticed that ${\widehat{P}}_{xy}={\widehat{P}}_{yx}$, generally, ${\widehat{P}}_{xx}\ne {\widehat{P}}_{yy}\ne {\widehat{P}}_{zz}$. Introducing

$$\begin{array}{cc}\hfill & {\widehat{V}}_{\omega }=(-1,0,0,0);{\widehat{K}}_{\omega }=(0,1,0,0);{\widehat{L}}_{\omega }=(0,0,1,0),\hfill \end{array}$$

(5)

so for, we have

$$\begin{array}{cc}\hfill {\widehat{T}}_{\omega \beta }& =(\widehat{\mu }+{\widehat{P}}_{zz}){\widehat{V}}_{\beta }{\widehat{V}}_{\omega }+{\widehat{P}}_{zz}{\eta }_{\omega \beta }-({\widehat{P}}_{zz}-{\widehat{P}}_{xx}){\widehat{K}}_{\beta }{\widehat{K}}_{\omega }-({\widehat{P}}_{zz}-{\widehat{P}}_{yy}){\widehat{L}}_{\beta }{\widehat{L}}_{\omega }+2{\widehat{P}}_{xy}{\widehat{K}}_{(\omega }{\widehat{L}}_{\beta )},\hfill \end{array}$$

(6)

where ${\eta }_{\omega \beta }$ specifies the Minkowskian line element. Switching back to our coordinate system, we then get the content of matter tensor related to the physical quantities as described in a locally specified Minkowskian frame.

$$\begin{array}{cc}\hfill T_{\omega \beta }& =(\mu +P_{zz})V_{\beta }{V}_{\omega }+P_{zz}{g}_{\omega \beta }-(P_{zz}-P_{xx})K_{\beta }{K}_{\omega }-(P_{zz}-P_{yy})L_{\beta }{L}_{\omega }+2P_{xy}{K}_{(\omega }{L}_{\beta )},\hfill \end{array}$$

(7)

where

$$\begin{array}{cc}\hfill & V_{\omega }=-A{\delta }_{\omega }^0;K_{\omega }=B{\delta }_{\omega }^1;L_{\omega }=Br{\delta }_{\omega }^2;S_{\omega }=C{\delta }_{\omega }^3,\hfill \end{array}$$

(8)

We consider observers who are at rest concerning the matter content. As an alternative, we represent the matter-energy tensor in the following way

$$\begin{array}{cc}\hfill T_{\omega \beta }& =(\mu +P)V_{\beta }{V}_{\omega }+P{g}_{\omega \beta }+{\Pi }_{\omega \beta },\hfill \end{array}$$

(9)

where P is the isotropic pressure while ${\Pi }_{\omega \beta }, \mu$, and $V_{\omega }$ are the anisotropic tensor, energy density, and the fluid’s 4-velocity, accordingly, provided by

$$\begin{array}{c}\hfill {\Pi }_{\omega \beta }=(P_{xx}-P_{zz})\left(K_{\omega }{K}_{\beta }-\frac{h_{\omega \beta }}{3}\right)+(P_{yy}-P_{zz})\left(L_{\omega }{L}_{\beta }-\frac{h_{\omega \beta }}{3}\right)+2P_{xy}{K}_{(\omega }{L}_{\beta )},\end{array}$$

(10)

where $h_{\omega \beta }=g_{\omega \beta }+V_{\omega }{V}_{\beta }$ and Equation (10) may be rewritten as

$$\begin{array}{c}\hfill {\Pi }_{\omega \beta }=\frac{1}{3}(2{\Pi }_I+{\Pi }_{II})\left(K_{\omega }{K}_{\beta }-\frac{h_{\omega \beta }}{3}\right)+\frac{1}{3}({\Pi }_I+2{\Pi }_{II})\left(L_{\omega }{L}_{\beta }-\frac{h_{\omega \beta }}{3}\right)+2{\Pi }_{KL}{K}_{(\omega }{L}_{\beta )},\end{array}$$

(11)

where

$$\begin{array}{cc}\hfill & {\Pi }_{KL}=T_{\omega \beta }{K}^{\omega }{L}^{\beta },{\Pi }_{II}=(2L^{\omega }{L}^{\beta }-K^{\omega }{K}^{\beta }-S^{\omega }{S}^{\beta })T_{\omega \beta },{\Pi }_I=(2K^{\omega }{K}^{\beta }-L^{\omega }{L}^{\beta }-S^{\omega }{S}^{\beta })T_{\omega \beta }.\hfill \end{array}$$

The content $P_{yy},P_{zz},P_{xx},P_{xy}$ and the anisotropic components ${\Pi }_{KL},{\Pi }_{II},{\Pi }_I$ of the tensor ${\Pi }_{\omega \beta }$ have a relationship expressed as

$$\begin{array}{cc}\hfill & {\Pi }_3=P_{yy}-P_{zz}=\frac{1}{3}({\Pi }_I+2{\Pi }_{II});{\Pi }_2=P_{xx}-P_{zz}=\frac{1}{3}(2{\Pi }_I+{\Pi }_{II});{\Pi }_{KL}=P_{xy},\hfill \end{array}$$

(12)

or inversely

$$\begin{array}{cc}\hfill & P_{yy}=\frac{1}{3}(2{\Pi }_3-{\Pi }_2)+P;P_{zz}=-\frac{1}{3}({\Pi }_3+{\Pi }_2)+P;P_{xx}=\frac{1}{3}(2{\Pi }_2-{\Pi }_3)+P,\hfill \end{array}$$

(13)

together with

$$\begin{array}{c}\hfill 3P=P_{xx}+P_{yy}+P_{zz}.\end{array}$$

(14)

Through an Einstein’s tensor, we write Equation (3) as

$$\begin{array}{cc}\hfill & G_{\omega \beta }=\kappa {T_{\omega \beta }}^{\mathrm{eff}},\hfill \end{array}$$

(15)

where $G_{\omega \beta }=R_{\omega \beta }-\frac{g_{\omega \beta }}{2}R$ and

$$\begin{array}{cc}\hfill {T_{\omega \beta }}^{\mathrm{eff}}& =\frac{1}{f_R}\left[\mu g_{\omega \beta }{f}_T+\left(\frac{f}{2}-\frac{R}{2}{f}_R\right)g_{\omega \beta }+(1+f_T)T_{\omega \beta }+{\nabla }_{\omega }{\nabla }_{\beta }{f}_R-g_{\omega \beta }\square f_R\right]\hfill \end{array}$$

is the effective matter tensor comprising the additional curvature components owing to $f(R,T)$ gravity. Our system under analysis is configured as a static axially symmetric relativistic body whose spacetime is provided by [68]

$$\begin{array}{c}\hfill d{\mathfrak{s}}^2-=-A^2(r,\theta )d{t}^2+B^2(r,\theta )(r^{2}d{\theta }^2+d{r}^2)+C^2(r,\theta )d{\varphi }^2,\end{array}$$

(16)

The $\varphi$ and t coordinates are disclosed to the killing vectors revealed by spacetime, resulting in geometric variables that are only functions of $\theta$ and r. When the appearance of a body remains unchanged when rotating across an axis, it is referred to as being axially symmetric. In mathematical study, symmetry frequently results in a design that is true even after being resized, translated, rotated, and reflected. Consequently, symmetry has evolved to stand for invariance, or the absence of alteration under a specific modification, in physics description. Since it has been established beyond doubt that practically all laws of cosmology operate in symmetrical frameworks, this concept has indeed been among the most valuable directions of astrophysics. Our findings would apply to axial-symmetric stellar systems with anisotropic fluids for astrophysical implementations.

2.1. Modified Einstein’s and Conservation Equations

We have considered axially symmetric space-time and assume that it is coupled with anisotropic fluid configuration as defined in Equation (9). The field equations are calculated using Equations (9), (15) and (16) as

$$\begin{array}{cc}\hfill G_{00}& =\frac{\kappa }{f_R}\left[A^2\left\{\mu +\frac{1}{2}(R{f}_R-f)+{\chi }_1\right\}\right],\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill G_{11}& =\frac{\kappa }{f_R}\left[B^2\left\{(1+f_T)P_{xx}+\mu f_T+\frac{1}{2}(R{f}_R-f)+{\chi }_2\right\}\right],\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill G_{22}& =\frac{\kappa }{f_R}\left[r^2{B}^2\left\{(1+f_T)P_{yy}+\mu f_T-\frac{1}{2}(R{f}_R-f)+{\chi }_3\right\}\right],\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill G_{33}& =\frac{\kappa }{f_R}\left[D^2\left\{(1+f_T)P_{zz}+\mu f_T-\frac{1}{2}(R{f}_R-f)+{\chi }_4\right\}\right],\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill G_{12}& =\frac{\kappa }{f_R}\left[(1+f_T)B^{2}r{P}_{xy}-f_{R,\theta }^{\prime }+\frac{B_{\theta }}{B}{f}_R^{\prime }+\frac{{\left(Br\right)}^{\prime }}{Br}{f}_{R,\theta }\right].\hfill \end{array}$$

(21)

In the above equations, the subscript $\theta$ and prime signify the variations with $\theta$ and r coordinates, successively. The additional curvature content arising because the gravitational consequences of $f(R, T)$ gravity are represented by ${\chi }_{i^{\prime }s}$ which are given in Appendix A. The contracted Bianchi identities demonstrating the conservation of matter tensor of the source are computed as ${T_{\omega \beta ;\beta }}^{\mathrm{eff}}\ne 0$. Its corresponding components are evaluated as given below

$$\begin{array}{cc}\hfill & {\dot{\mu }}^{\mathrm{eff}}=0,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & \frac{1}{f_R}\left[(1+f_T)\right\{{\left(P-\frac{1}{3}({\Pi }_3-2{\Pi }_2)\right)}^{\prime }-\frac{B^{\prime }}{B}({\Pi }_3-{\Pi }_2)+\frac{C^{\prime }}{C}{\Pi }_2+\frac{A^{\prime }}{A}\left(\mu +P-\frac{1}{3}({\Pi }_3-2{\Pi }_2)\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\frac{1}{r}\left({\Pi }_{KL}\left\{2\frac{B_{\theta }}{B}+\frac{A_{\theta }}{A}+\frac{C_{\theta }}{C}\right\}+{\Pi }_{KL,\theta }-{\Pi }_3+{\Pi }_2\right)\left\}\right]=\frac{1}{f_R}{\chi }_6,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & \frac{1}{f_R}\left[(1+f_T)\right\{{\left(P-\frac{1}{3}({\Pi }_2-2{\Pi }_3)\right)}^{\prime }+3\frac{B_{\theta }}{B}(\frac{{\Pi }_3}{3}-\frac{{\Pi }_2}{3})+\frac{C^{\prime }}{C}{\Pi }_3+\frac{A_{\theta }}{3A}\left(\frac{\mu }{3}+\frac{P}{3}-({\Pi }_2-2{\Pi }_3)\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +r\left(2{\Pi }_{KL}\left\{\frac{B^{\prime }}{B}+\frac{A^{\prime }}{2A}+\frac{C^{\prime }}{2C}\right\}+{\Pi }_{KL}^{\prime }\right)+2{\Pi }_{KL}\left\}\right]=\frac{1}{f_R}{\chi }_7.\hfill \end{array}$$

(24)

Equation (22) is the trivial effect of the staticity of the source where the time derivative is represented by the dot. However, Equations (23) and (24) are the requirements for hydro-static equilibrium of the relevant configuration. The higher curvature terms emerging due to the consequences of minimal-coupled gravity are expressed in form of ${\chi }_6$ and ${\chi }_7$. Their expressions are provided in Appendix A.

2.2. $f(R, T)$ Structure Scalars and Related Differential Equations

Now, we would want to compute the scalar variables for an axial source within the approach of $f(R, T)$ theory. For their computation, firstly, we require the electric division of the conformal curvature tensor, whose content may be derived simply from its defined expression. The impact of tidal forces on an anisotropic self-gravitating configuration is determined by the conformal tensor [70]. To analyze the connection between the density inhomogeneity and the conformal tensor, we compute the conformal tensor using the decomposition of the Riemann tensor as

$$\begin{array}{c}\hfill C_{\omega \beta \gamma }^{\alpha }=R_{\omega \beta \gamma }^{\alpha }-\frac{1}{2}{R}_{\beta }^{\alpha }{g}_{\omega \gamma }+\frac{1}{2}{R}_{\omega \beta }{\delta }_{\gamma }^{\alpha }-\frac{1}{2}{R}_{\omega \gamma }{\delta }_{\beta }^{\alpha }+\frac{1}{2}{R}_{\gamma }^{\alpha }{g}_{\omega \beta }+\frac{1}{6}R({\delta }_{\beta }^{\alpha }{g}_{\omega \gamma }-g_{\omega \beta }{\delta }_{\gamma }^{\alpha }),\end{array}$$

(25)

where $R,C_{\omega \beta \gamma }^{\alpha },R_{\omega \beta }$, and $R_{\omega \beta \gamma }^{\alpha }$ describe the Ricci-scalar, conformal tensor, the Ricci tensor, and the Riemann tensor, accordingly. Moreover, the Weyl curvature tensor is decomposed into magnetic and electric components. For the symmetry under consideration, the magnetic component disappears identically. This fact suggests that the closest flow-lines scatter independently with one another. Consequently, propagation is locally dependent on matter content, whereas the electric component is defined as

$$\begin{array}{cc}\hfill & E_{\omega \beta }=C_{\omega \nu \beta \gamma }{V}^{\nu }{V}^{\gamma },\hfill \end{array}$$

(26)

whose non-vanishing components are provided in the Appendix A. Inserting the conformal tensor in Equation (26), it reads for our static axial source as

$$\begin{array}{cc}\hfill & E_{\omega \beta }={\epsilon }_1(K_{\omega }{L}_{\beta }+L_{\beta }{K}_{\omega })+{\epsilon }_2\left(K_{\omega }{K}_{\beta }-\frac{h_{\omega \beta }}{3}\right)+{\epsilon }_3\left(L_{\omega }{L}_{\beta }-\frac{h_{\omega \beta }}{3}\right).\hfill \end{array}$$

(27)

The Weyl scalars are symbolized by ${\epsilon }_1,{\epsilon }_2$, and ${\epsilon }_3$, and their expressions are given in Appendix A. In our case, these could be rewritten in the perspective of Equations (17)–(21) as

$$\begin{array}{cc}\hfill & {\epsilon }_1=\frac{E_{12}}{r{B}^2}=\frac{\kappa }{2f_R}\left[(1+f_T){\Pi }_{KL}+2{\chi }_5\right]-\frac{2}{B^{2}r}\left[\frac{A^{\prime }}{2A}\frac{B_{\theta }}{B}+\frac{A_{\theta }}{A}\left(\frac{1}{2r}+\frac{B^{\prime }}{2B}\right)-\frac{A_{\theta }^{\prime }}{2A}\right],\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & {\epsilon }_2=-2\left[\frac{E_{33}}{C^2}+\frac{E_{22}}{2r^2{B}^2}\right]=\frac{\kappa }{2f_R}\left[(1+f_T)(\mu +3P+{\Pi }_2)+2\mu f_T+{\chi }_1+{\chi }_2+2{\chi }_3\right]+\frac{2}{r^2{B}^2}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \times \frac{A_{\theta }}{A}\left(\frac{B_{\theta }}{2B}-\frac{C_{\theta }}{C}\right)-\frac{2}{B^2}\frac{A^{\prime }}{3A}\left(\frac{6}{r}+\frac{3C^{\prime }}{C}+\frac{3B^{\prime }}{2B}\right)-\frac{1}{r^2{B}^2}\left(\frac{A_{\theta \theta }}{A}\right),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & {\epsilon }_3=-2\left[\frac{E_{33}}{C^2}-\frac{E_{22}}{2r^2{B}^2}\right]=\frac{\kappa }{2f_R}\left[(1+f_T){\Pi }_3+{\chi }_3-{\chi }_4\right]+\frac{A_{\theta }}{2A{r}^2{B}^2}\left\{\frac{-2B_{\theta }}{B}-\frac{2C_{\theta }}{C}\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\frac{3A^{\prime }}{2A{B}^2}\left(\frac{2}{3r}-2\frac{C^{\prime }}{3C}+\frac{23B^{\prime }}{B}\right)+\left(\frac{A_{\theta \theta }}{A}\right)\frac{1}{r^2{B}^2}.\hfill \end{array}$$

(30)

Now, we want to present the propagation equations related to the conformal tensor of the system via Bianchi identities. The Bianchi identities serve as a demonstration of the link between the conformal curvature tensor, the matter profile, and the minimal-coupled gravity corrections. Therefore, the corresponding equations for our matter configuration are found to be

$$\begin{array}{cc}\hfill & \frac{1}{3r}{(2{\epsilon }_3-{\epsilon }_2)}_{,\theta }+{\epsilon }_1^{\prime }+2{\epsilon }_1\left(\frac{1}{r}+\frac{C^{\prime }}{2C}+\frac{B^{\prime }}{B}\right)-\frac{{\epsilon }_2}{r}\frac{B_{\theta }}{B}+2\frac{{\epsilon }_3}{r}\left(\frac{B_{\theta }}{2B}+\frac{C_{\theta }}{2C}\right)=\frac{\kappa }{6r{f}_R}[(3P+2\mu )\hfill \end{array}$$

$$\begin{array}{cc}\hfill \times & (1+f_T)+3\mu f_T-\frac{1}{2}(R{f}_R-f){]}_{,\theta }+\frac{\kappa }{2r{f}_R}\{\left(\mu +P-\frac{1}{3}({\Pi }_2-2{\Pi }_3)\right)(1+f_T)+\mu f_T\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\chi }_1+{\chi }_3\}\frac{A_{\theta }}{A}+\frac{\kappa }{2f_R}\left[{\Pi }_{KL}(1+f_T)+{\chi }_3\right]\frac{A^{\prime }}{A},\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \frac{{\epsilon }_{1,\theta }}{r}-\frac{2}{3}{(\frac{{\epsilon }_3}{2}-{\epsilon }_2)}^{\prime }+2\frac{{\epsilon }_1}{r}\left(\frac{B_{\theta }}{B}+\frac{C_{\theta }}{2C}\right)+\frac{{\epsilon }_2}{3}\left(\frac{3}{r}+\frac{3C^{\prime }}{C}+\frac{3B^{\prime }}{B}\right)-\frac{{\epsilon }_3}{2}\left(\frac{2}{r}+2\frac{B^{\prime }}{B}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\frac{\kappa }{6f_R}{\left[(2\mu +3P)(1+f_T)+3\mu f_T-\frac{1}{2}(R{f}_R-f)+{\chi }_6\right]}^{\prime }+\frac{4\pi }{f_R}\{\mu +P+\mu f_T+{\chi }_1+{\chi }_2\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -\frac{1}{3}({\Pi }_3-2{\Pi }_2)(1+f_T)\}\frac{A^{\prime }}{A}+\frac{\kappa }{2r{f}_R}\left\{{\Pi }_{KL}(1+f_T)+{\chi }_5\right\}\frac{A_{\theta }}{A},\hfill \end{array}$$

(32)

revealing the effects of the tidal forces, the spatial derivatives of the geometric and matter profile together with the consequences of the strong gravitational zone interactions. It is worth mentioning that all of the consequences are converted to the previous ones presented in Einstein’s theory [68] under usual bounds i.e., $f(R, T)⟶R$. Structure scalars are helpful to obtain a better understanding of the fundamental constituents of the fluid configuration of the self-gravitating source. We apply Herrera’s approach [48,68,71] for the computation of the electric portion of the Riemann tensor to determine the structure scalars that have a great influence on the formulation of the fluid structures. Further, these scalars help us to compute the CFs of the system. Substituting the $f(R, T)$ equations in Equation (25), the intrinsic curvature tensor could be manipulated in the form of the effective stress-energy tensor and the conformal tensor as

$$\begin{array}{cc}\hfill & R_{\beta \gamma }^{\omega \alpha }=C_{\beta \gamma }^{\omega \alpha }+2\kappa T_{[\beta }^{{}^{\mathrm{eff}}[\omega }{\delta }_{\gamma ]}^{\alpha ]}+\kappa T^{\mathrm{eff}}\left(\frac{1}{3}{\delta }_{[\beta }^{\omega }{\delta }_{\gamma ]}^{\alpha }-{\delta }_{[\beta }^{[\omega }{\delta }_{\gamma ]}^{\alpha ]}\right).\hfill \end{array}$$

(33)

In this manner, the Riemann tensor is associated with the conformal tensor and the matter content coupled with the consequences of $f(R, T)$ theory. The conformal tensor provides the illustration about the geometrical configuration of the system, which is influenced by the effects of tidal forces. In Einstein’s theory, Herrera et al. [48] proposed the Riemann tensor’s decomposition orthogonally to formulate the five scalars for dissipative and anisotropic spheres. Furthermore, they formulated some analytical findings with the help of these scalar quantities. These scalars are the outcomes of trace-free and trace components of the relevant tensorial terms. This approach has been shown to be quite beneficial for analyzing the basic constituents of the self-gravitating matter configurations. The aspects of the sources are directly impacted by these scalars. Herrera et al. [72] analyzed the evolution of viscous and dissipative stellar structures to analyze the instability within the shear-free evolution. They found that the departure nature of the shear-free evolution is governed by the structure scalar as well as the expansion parameter. In their case, the structure scalar $Y_{TF}$ revealed the effect of the physical parameters such as dissipation, density inhomogeneity, and the local anisotropy of the matter configuration. Bhatti et al. [73] discussed the dynamical consequences of the evolving spheres with imperfect matter content within high curvature realms. They developed propagation equations for the expansion factors under shearing effects. They found a significant influence of $f(R, T)$ corrections as well as dynamical variables on the evolutionary stages of stellar structures. To continue our study, we consider the two tensors as [48]

$$\begin{array}{c}{X}_{\omega \beta }{=}^{*}{R}_{\omega \alpha \beta \lambda }^{*}{V}^{\alpha }{V}^{\lambda }=\frac{1}{2}{\eta }_{\omega \alpha }^{ϵ\rho }{R}_{ϵ\rho \beta \lambda }^{*}{V}^{\lambda }{V}^{\alpha },\hfill \end{array}$$

(34)

$$\begin{array}{c}{Y}_{\omega \beta }=R_{\omega \alpha \beta \lambda }{V}^{\alpha }{V}^{\lambda },\hfill \end{array}$$

(35)

here, ${}^{*}{R}_{\alpha \lambda \omega \beta }=\frac{1}{2}{\eta }_{\delta \gamma \omega \beta }{R}_{\alpha \lambda }^{\delta \gamma }$ and $R_{\omega \alpha \beta \lambda }^{*}=\frac{1}{2}{\eta }_{\omega \alpha \gamma \delta }{R}_{\beta \lambda }^{\gamma \delta }$ are the left and right duals of the intrinsic curvature tensor, respectively. Applying Equation (33) in Equation (35) and after some lengthy computations, the subsequent form of $Y_{\omega \beta }$ is obtained.

$$\begin{array}{c}\hfill Y_{\omega \beta }=E_{\omega \beta }+\frac{\kappa }{f_R}\left[(1+f_T)\frac{{\Pi }_{\omega \beta }}{2}+(\mu +3P)\frac{h_{\omega \beta }}{6}+(R{f}_R-f)\frac{h_{\omega \beta }}{6}\right]+{\vartheta }_{\omega \beta }.\end{array}$$

(36)

The quantity ${\vartheta }_{\omega \beta }$ illustrates the results of additional curvature constituents that are provided in the Appendix A. The aforementioned tensor’s trace and trace free elements can be separated as follows:

$$\begin{array}{cc}\hfill & Y_{\omega \beta }=\frac{1}{3}{Y}_T{h}_{\omega \beta }+Y_{TF1}(K_{\omega }{L}_{\beta }+L_{\beta }{K}_{\omega })+Y_{TF2}(K_{\omega }{K}_{\beta }-\frac{h_{\omega \beta }}{3})+Y_{TF3}(L_{\omega }{L}_{\beta }-\frac{h_{\omega \beta }}{3}).\hfill \end{array}$$

(37)

Consequently, we obtain

$$\begin{array}{c}{Y}_T=(\mu +3P)\frac{\kappa }{2f_R}-\frac{1}{2f_R}(R{f}_R-f)+{\vartheta }^{*},\hfill \end{array}$$

(38)

$$\begin{array}{c}{Y}_{TF1}={\epsilon }_1-\kappa (1+f_T)\frac{{\Pi }_{KL}}{2f_R},\hfill \end{array}$$

(39)

$$\begin{array}{c}{Y}_{TF2}={\epsilon }_2-\kappa (1+f_T)\frac{{\Pi }_2}{2f_R},\hfill \end{array}$$

(40)

$$\begin{array}{c}{Y}_{TF3}={\epsilon }_3-\frac{\kappa }{2f_R}(1+f_T){\Pi }_3,\hfill \end{array}$$

(41)

where ${\vartheta }^{*}$ is the trace of the quantity ${\vartheta }_{\omega \beta }$. The trace and trace-free components are indicated here by the subscripts T and $TFi$’s ($i=1,2,3$), respectively. The scalar $Y_T$ is directly linked to the influence of the physical parameters such as energy density, pressure anisotropy, and heat dissipation. In our scenario, this factor explains the results of the anisotropic pressure, energy-density irregularities, and $f(R, T)$ modifications as described in Equation (38). Further, $Y_{TFi}$’s are associated with the influence of the tidal forces and anisotropy of the source under $f(R, T)$ gravity. Moreover, these dynamical variables would also be useful for measuring the system’s complexity. Analogously, $Y_{\omega \beta }$, $X_{\omega \beta }$ could be decomposed in trace-free and trace components, as presented below.

$$\begin{array}{cc}\hfill & X_{\omega \beta }=\frac{1}{3}{X}_T{h}_{\omega \beta }+X_{TF1}(K_{\omega }{L}_{\beta }+L_{\beta }{K}_{\omega })+X_{TF2}(K_{\omega }{K}_{\beta }-\frac{h_{\omega \beta }}{3})+X_{TF3}(L_{\omega }{L}_{\beta }-\frac{h_{\omega \beta }}{3}).\hfill \end{array}$$

(42)

By using the $f(R, T)$ equations, we obtain the subsequent expressions as

$$\begin{array}{cc}\hfill & X_{\omega \beta }=-E_{\omega \beta }-\frac{\kappa }{f_R}\left[(1+f_T)\frac{{\Pi }_{\omega \beta }}{2}-\frac{\mu }{3}{h}_{\omega \beta }+(R{f}_R-f)\frac{h_{\omega \beta }}{6}\right]+{\psi }_{\omega \beta },\hfill \end{array}$$

(43)

with the four scalar variables

$$\begin{array}{cc}\hfill & X_T=\frac{\kappa }{f_R}\mu +\frac{\kappa }{2f_R}(R{f}_R-f)+{\psi }^{*},\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill & X_{TF1}=-{\epsilon }_1-\kappa \left(1+f_T\right)\frac{{\Pi }_{KL}}{2f_R},\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & X_{TF2}=-{\epsilon }_2-\kappa \left(1+f_T\right)\frac{{\Pi }_2}{2f_R},\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill & X_{TF3}=-{\epsilon }_3-\kappa \left(1+f_T\right)\frac{{\Pi }_3}{2f_R}.\hfill \end{array}$$

(47)

The tensorial quantity ${\psi }_{\omega \beta }$ shows the strong gravitational influence at high curvature realms, whereas ${\psi }^{*}$ is the trace of the respective quantity. It is important to note that $X_T$ is associated with the consequences of irregularities of the energy density and the extra curvature terms emerging due to geometry–matter coupling as presented in Equation (44). On the other hand, dynamical variables related to the spatial vectors, i.e., $X_{TFi}$’s depict the effects of the conformal tensor and anisotropic fluid configuration under higher-order gravitational aspects. The presence of complexity can be observed in the sources whose structure formation depends upon numerous physical aspects. These aspects may reveal the consequences of the anisotropy, heat dissipation, viscosity, and inhomogeneity of the matter configuration. In collapsing phenomena, the density inhomogeneity is an important factor. To measure the complexity of the systems, first, we would want to obtain the particular relations of the physical and geometric factors that are helpful to identify the uneven distribution of the physical variables. In our study, the evolution of the conformal tensor could be manipulated through dynamical variables. These variables describe the impact of anisotropic pressure and tidal forces under the influence of higher-order gravitational influence in the subsequent results. Thereby, using Equations (38)–(41) and (44)–(47), Equations (31) and (32) may also be manipulated in the form of an extended version of the dynamical variables as

$$\begin{array}{cc}\hfill & \frac{2\kappa }{6f_R}\left[{\mu }^{\prime }+\frac{1}{2}{(R{f}_R-f)}^{\prime }+{\chi }_1^{\prime }\right]=\frac{1}{r}[Y_{TF1,\theta }+{\left\{\frac{\kappa }{f_R}(1+f_T){\Pi }_{KL}\right\}}_{,\theta }+(Y_{TF1}+\frac{\kappa }{f_R}(1+\hfill \end{array}$$

$$\begin{array}{cc}\hfill & f_T){\Pi }_{KL}\left){(ln{B}^{2}C)}_{,\theta }\right]+[\frac{2}{3}\left(Y_{TF2}^{\prime }+{\left\{\frac{\kappa }{f_R}(1+f_T){\Pi }_2\right\}}^{\prime }\right)+\left(Y_{TF2}+\frac{\kappa }{f_R}(1+f_T){\Pi }_2\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \times {(lnBrC)}^{\prime }]-\left[\frac{1}{3}\left(Y_{TF3}^{\prime }+{\left\{\frac{\kappa }{f_R}(1+f_T){\Pi }_3\right\}}^{\prime }\right)+\left(Y_{TF3}+\left\{\frac{\kappa }{f_R}(1+f_T){\Pi }_3\right\}\right){(lnBr)}^{\prime }\right],\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill & \frac{2\kappa }{6f_R}\left[{\mu }_{,\theta }+\frac{1}{2}{(R{f}_R-f)}_{,\theta }+{\chi }_{1,\theta }\right]=-\frac{1}{r}[\frac{1}{3}\left(Y_{TF2,\theta }+{\left\{\frac{\kappa }{f_R}(1+f_T){\Pi }_2\right\}}_{,\theta }\right)+(Y_{TF2}+\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \frac{\kappa }{f_R}(1+f_T){\Pi }_2\left){(lnB)}_{,\theta }\right]+\frac{1}{r}[\frac{2}{3}\left(Y_{TF3,\theta }+{\left\{\frac{\kappa }{f_R}(1+f_T){\Pi }_3\right\}}_{,\theta }\right)+(Y_{TF3}+\frac{\kappa }{f_R}(1+\hfill \end{array}$$

$$\begin{array}{cc}\hfill & f_T){\Pi }_3\left){(lnBC)}_{,\theta }\right]+[\left(Y_{TF1}^{\prime }+{\left\{\frac{\kappa }{f_R}(1+f_T){\Pi }_{KL}\right\}}^{\prime }\right)+\left(Y_{TF1}+\left\{\frac{\kappa }{f_R}(1+f_T){\Pi }_{KL}\right\}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \times {(ln{B}^{2}D{r}^2)}^{\prime }].\hfill \end{array}$$

(49)

These results describe the inhomogeneity of the matter configuration that is induced by the tidal forces, pressure anisotropy, and the $f(R, T)$ terms. It is important to mention that the aforesaid consequences are governed by the variables $Y_{TF1}$, $Y_{TF2}$, and $Y_{TF3}$. In this way, the evolution of the conformal tensor might be tuned by the influence of the fluid’s content and higher-order curvature terms. One may describe the propagation of Equations (48) and (49) through the variables $X_{TFi}$’s as well.

3. Fluid Configurations with Vanishing CFs Conditions

One of the main concerns in physical cosmology is the formation and propagation of the stellar objects that are driven towards their cores continuously. Several astronomical studies of such self-gravitating bodies provide insights into the formation of the structures and their physical aspects. The mechanism of stellar configurations is indeed not straightforward and is affected by several physical factors, including temperature, pressure, and energy density. It is challenging to identify the effects of a minor perturbation within the configuration forms on the interconnected controlling factors. As a result, defining CFs that evaluate the significance of each factor and linking them using a mathematical representation becomes important. These factors offer a standard for evaluating the system’s stability, in addition to serving as a benchmark for comparing the complexity of distinct structures as provided below:

• In the case of self-gravitating spheres [66], CF is a scalar quantity that aims to quantify the degree of complexity of the matter composition. This scalar might be recognized as one of the scalar variables that emerge through the breakdown of the electric portion of the Riemann tensor. However, the axially symmetric scenario is somehow complicated compared to the spherical one. Consequently, the structure scalars in this case are greater in number. Nonetheless, the generic criteria for defining the variables that assess the fluid’s complexity would remain the same. It is convenient to choose the most feasible and simplest distribution of the fluid. Therefore, we choose the incompressible (with constant energy-density) fluid’s composition. It has been established that the vanishing of the $X_{TF1}$, $X_{TF3}$, and $X_{TF2}$ criterion is both essential and sufficient for the spatial variations of the density to vanish. This implies

  $$\begin{array}{cc}\hfill X_{TF2}& =X_{TF1}=X_{TF3}=0\iff {\mu }_{,\theta }^{\mathrm{eff}}={\mu }^{{}^{\prime }\mathrm{eff}}=0\hfill \end{array}$$

  (50)
  Nevertheless, $X_{TF1}=X_{TF3}=X_{TF2}=0$ corresponds to the even distribution of the energy density. In this scenario,

  $$\begin{array}{cc}\hfill & Y_{TF1}=-\frac{\kappa }{f_R}(1+f_T){\Pi }_{KL},Y_{TF2}=-\frac{\kappa }{f_R}(1+f_T){\Pi }_2,Y_{TF3}=-\frac{\kappa }{f_R}(1+f_T){\Pi }_3.\hfill \end{array}$$

  (51)
  In our case, CFs are caused by the uneven distribution of the pressure and the energy density, as well as by the matter–geometry association.
• In our propagation equations, some unknown factors are involved. We confine our system to being in a less complex condition to deal with this kind of situation. Therefore, from Equations (17)–(21), (31), and (32), we are in a position to formulate the zero CFs constraints presented in a subsequent way

  $$\begin{array}{cc}\hfill Y_{TF1}=& \frac{2}{B^{2}r}\left[-\frac{A^{\prime }}{2A}\frac{B_{\theta }}{B}-\frac{A_{\theta }}{A}\left(\frac{1}{2r}+\frac{B^{\prime }}{2B}\right)+\frac{A_{\theta }^{\prime }}{2A}\right]+\frac{\kappa }{2f_R}{\chi }_5=0,\hfill \end{array}$$

  (52)

  $$\begin{array}{cc}\hfill Y_{TF2}=& \frac{A^{″}}{A{B}^2}+\frac{2}{r^2{B}^2}\frac{A_{\theta }}{A}\left\{\frac{B_{\theta }}{2B}-\frac{C_{\theta }}{2C}\right\}-\frac{2}{B^2}\frac{A^{\prime }}{A}\left\{\frac{B^{\prime }}{2B}+\frac{C^{\prime }}{2C}\right\}+\frac{\kappa }{2f_R}\left[{\chi }_1+{\chi }_2+2{\chi }_3\right]=0,\hfill \end{array}$$

  (53)

  $$\begin{array}{cc}\hfill Y_{TF3}=& -\frac{2}{r^2{B}^2}\frac{A_{\theta }}{A}\left\{\frac{B_{\theta }}{2B}+\frac{C_{\theta }}{2C}\right\}+\frac{2}{r^2{B}^2}\left(\frac{A_{\theta \theta }}{2A}\right)+\frac{A^{\prime }}{2A{B}^2}\left\{\frac{2}{r}-2\frac{C^{\prime }}{C}+2\frac{B^{\prime }}{B}\right\}\hfill \end{array}$$

  $$\begin{array}{cc}& +\frac{\kappa }{2f_R}\left[{\chi }_3-{\chi }_4\right]=0.\hfill \end{array}$$

  (54)
  In the above equations, the terms ${\chi }_i$’s comprise the ingredients of higher-order gravity, illustrating the strong gravitational influence. These terms are listed in the Appendix A. Instead of analysing the static spheres [66], we have determined the three CFs in the axially symmetric scenario on the bounds of $f(R, T)$ gravity that are provided in Equations (52)–(54). The absence of the CFs has a substantial impact on the system’s complexity. Our evolving system is made more stable under these conditions. It is worth highlighting that the CFs given in Equations (52)–(54) depict the impact of geometrical variables as well as $f(R, T)$ ingredients at large galactic-scales. The above presented consequences can be obtained in the context of Einstein’s theory [68] within the usual bounds.

Next, we will derive a few analytical solutions that, by imposing structural scalars, accommodate the zero CFs criteria. It is worth noting that our goal is to show how these kinds of configurations might be possible even when the effects of significant curvature realms are assumed. The zero CFs condition is entertained by the isotropic fluids with even energy distribution; however, this is not the only possible scenario. This is also feasible for a matter configuration with anisotropic pressure and energy-density inhomogeneity, where these factors counteract each other’s effects upon the CFs. Here, we determine the circumstances in which the source carries an even energy distribution in view of various structures.

3.1. Isotropic Spheroid with Constant Energy-Density

In this subsection, we discuss certain analytic findings for isotropic fluids together with the constant energy-density within the axially symmetric static geometry. Subsequently, the fluid under analysis is incompressible and reveals the isotropic aspects of pressure. Herrera et al. [68] proposed the matter spheroid with distinct characteristics. They derived a few analytical solutions, accommodating the zero CFs constraint. They concluded that one of those findings, within the spherical bounds, transforms into the well-established Schwarzschild interior solution that models the spheroid with isotropic and constant energy-density. They also derived another class of analytical solutions, matching to the Weyl exteriors that model the spheroid with the uneven distribution of the energy density and pressure. We retrieve them simply by applying the same technique to $f(R, T)$ theory. The conformally flat formulations are effective in order to study the consequential impact of the physical factors in the evolving self-gravitating sources within the high curvature epochs. The following requirements define the conformally flat and isotropic configurations:

• From Equations (44)–(47) and (50), assuming ${\mu }^{\mathrm{eff}}$ and ${\mu }_0^{\mathrm{eff}}$ to be constant also ${\epsilon }_1={\epsilon }_3={\epsilon }_2=0$, $P_{xy}=0$ and $P=P_{yy}=P_{xx}=P_{zz}$. In the domain of astrophysics, the idea of matching contexts has captured a great attention. With the assist of such conditions, two distinct geometrical systems could be smoothly glued. The stellar system’s internal and external structures must be carefully examined in light of these constraints. Surely, in our study, more generalized equations of surface are also attainable. However, that would present a complex situation. We will suppose, for the sake of simplicity, that the surface boundary $\Delta$ for our axial geometry is specified as follows

  $$\begin{array}{c}\hfill r_1=r=constant.\end{array}$$

  (55)
  The Darmois-conditions are the only matchings scenario for the axially symmetric configurations in the metric-based approach [68]. To do this, we require that all $\theta$ and r derivatives of geometrical functions be continuous through $\Delta$. By use of the Equations (18), (19) and (55), we achieve the following matchings where the notation $\stackrel{\Delta }{=}$ specifies that the computations are made over the surface $\Delta$.

  $$\begin{array}{c}\hfill P^{eff}\stackrel{\Delta }{=}0.\end{array}$$

  (56)
  The term $P^{eff}$ includes the usual matter and the $f(R, T)$ factors due to the matter–curvature coupling. For any vacuum external spacetime, Equation (56) needs to be accommodated. In our coordinates configuration, Equations (23) and (24) are integrated to construct

  $$\begin{array}{c}\hfill P+{\mu }_0=\frac{\varsigma \left(\theta \right)}{A}-{\int }_0^r\frac{{\chi }_6}{f_R}dr,\end{array}$$

  (57)

  $$\begin{array}{c}\hfill P+{\mu }_0=\frac{\zeta \left(r\right)}{A}-{\int }_0^{\theta }\frac{{\chi }_7}{f_R}d\theta ,\end{array}$$

  (58)

  where $\zeta \left(r\right)$ and $\varsigma \left(\theta \right)$ both are the arbitrary integration-functions. As the curvature on the surface-boundary is assumed to be invariant, so, in Equation (56) the impact of $P^{\mathrm{eff}}$ over the external region of $\Delta$ vanishes. However, if this does not go away then it imposes the diverge scenario of $P^{{}^{\prime }\mathrm{eff}}$ across $\Delta$ which is dissatisfactory. As then the equilibrium of hydrostatic scenario might not be entertained. Putting the aforesaid matchings in Equations (57) and (58), the outcomes next in order are

  $$\begin{array}{c}\hfill \varsigma =const:A(r_1,\theta )=\frac{\alpha }{{\mu }_0}=const.\end{array}$$

  (59)
  Consequently, the conformally flat configurations made up of the homogeneous distribution of the pressure and energy-density can be expressed as follows:

  $$\begin{array}{cc}\hfill & d{s}^2=\frac{1}{{(\lambda r_1^2+\delta +urcos\theta )}^2}\left[-{(\alpha r^2+\sigma +arcos\theta )}^{2}d{t}^2+d{r}^2+(d{\theta }^2+{sin}^2\theta d{\varphi }^2)r^2\right].\hfill \end{array}$$

  (60)
  The matter profile could then be determined with ease. Its energy density under the influence of $f(R, T)$ is expressed as

  $$\begin{array}{cc}\hfill \frac{\kappa }{f_R}\left[\mu +\frac{1}{2}(f-R{f}_R)\right]=& -\frac{\kappa }{f_R}\left[\frac{1}{B^2}\left\{\frac{A^{\prime }}{A}{f}_R^{\prime }+\frac{f_{R,\theta }}{r^2}\left(\frac{A_{\theta }}{A}\right)+{\chi }_0\right\}\right]+4\left[3\lambda \delta -\frac{3}{4}{u}^2\right].\hfill \end{array}$$

  (61)

The quantity ${\chi }_0$ is enlisted in Appendix A; it reveals the outcomes of higher-order gravitational corrections. Next, we utilize Equations (57) and (59) with a view to achieve pressure as

$$\begin{array}{cc}\hfill & \frac{\kappa }{f_R}\left[(1+f_T)P+\frac{1}{2}(R{f}_R-f)-\mu f_T\right]=\frac{\kappa }{f_R}\left[\frac{1}{B^2}\left\{f_R^{″}-\left(\frac{B}{B}{f}_R^{\prime }\frac{B_{\theta }}{B{r}^2}{f}_{R,\theta }\right)-{\chi }_0\right\}\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -4(3\lambda \delta -\frac{3}{4}{u}^2)\left[1-\frac{\alpha r_1^2+\sigma \lambda r^2+\delta +ur(cos\theta )}{\lambda r_1^2+\alpha \delta r^2+\sigma +ur(cos\theta )}\right].\hfill \end{array}$$

(62)

$\delta ,\sigma$, and u are constants in Equation (62). To accomplish (56), the expressions for a and $\varsigma$ are formulated as

$$\begin{array}{cc}\hfill & \varsigma =\frac{\alpha r_1^2+\sigma }{\lambda r_1^2+\delta }{\mu }_0;a=\frac{\alpha r_1^2+\sigma }{\lambda r_1^2+\delta }u.\hfill \end{array}$$

(63)

It is important to emphasize this finding since it could not be glued to any Weyl exteriors. However, the scenario would be changed under spherical bounds, despite having a surface of disappearing effective pressure as presented in Equation (56). One can accomplish the aforesaid finding in Einstein’s theory [68] under usual bounds, which is consistent with the case demonstrating that perfect and static matter (with isotropic aspects of pressure) configurations in nature are spherical [74].

3.2. Anisotropic Spheroid with Inhomogeneous Energy-Density

In this subsection, we would like to construct a model describing the spheroid whose pressure and energy-density are not distributed homogeneously. Although the zero CFs constraints are not usually satisfied by this type of matter spheroids [41], the solution can have an inhomogeneous-energy distribution that eliminates the impact of the system’s anisotropy and tidal forces. The metric parameters, for such a solution, might be written as [68]

$$\begin{array}{cc}\hfill A(r,\theta )& =a_{1}r\frac{sin\theta }{u_1{r}^2+u_2};B(r,\theta )=\frac{1}{u_1{r}^2+u_2},\hfill \end{array}$$

(64)

$$\begin{array}{cc}\hfill C(r,\theta )& =\frac{u_1{r}^2-u_2}{u_1{r}^2+u_2}F\left(\frac{rcos\theta }{u_1{r}^2-u_2}\right),\hfill \end{array}$$

(65)

satisfying Equations (52)–(54). In these equations, $u_1,u_2$ and $a_1$ are treated as constants. The matter profile is then determined by the metric above and implementing Equations (17)–(21), we obtain

$$\begin{array}{cc}\hfill & \frac{\kappa }{f_R}\left[\mu -\frac{1}{2}(R{f}_R-f)+{\chi }_1\right]=12u_1{u}_2-\frac{{(u_1{r}^2+u_2)}^2}{{(u_1{r}^2-u_2)}^2}\left(1+\frac{4u_1{u}_2{r}^2{cos}^2\theta }{{(u_1{r}^2-u_2)}^2}\right)\frac{F_{zz}}{F},\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill & \frac{\kappa }{f_R}\left[(1+f_T){\Pi }_2+{\chi }_2-{\chi }_4\right]=\frac{F_{zz}}{4F}\frac{{(u_1{r}^2+u_2)}^2}{{(u_1{r}^2-u_2)}^2}{sin}^2\theta ,\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill & \frac{\kappa }{f_R}\left[(1+f_T){\Pi }_3+{\chi }_3-{\chi }_4\right]=\frac{F_{zz}}{4F}\frac{{(u_1{r}^2+u_2)}^2}{{(u_1{r}^2-u_2)}^2}{cos}^2\theta ,\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill & \frac{\kappa }{f_R}\left[(1+f_T){\Pi }_{KL}+{\chi }_5\right]=-\frac{F_{zz}}{2F}\frac{r{(u_1{r}^2+u_2)}^3}{{(u_1{r}^2-u_2)}^2}sin2\theta ,\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill & \frac{\kappa }{f_R}\left[(1+f_T)P-\frac{1}{2}(R{f}_R-f)+\mu f_T+{\chi }_2\right]=\frac{{(u_1{r}^2+u_2)}^2}{{(u_1{r}^2-u_2)}^2}\left[\frac{4u_1{u}_2{r}^2{cos}^2\theta }{3{(u_1{r}^2-u_2)}^2}+1\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \times \frac{F_{zz}}{F}-12u_1{u}_2,\hfill \end{array}$$

(70)

together with

$$\begin{array}{cc}\hfill z& =\left(\frac{rcos\theta }{u_1{r}^2-u_2}\right).\hfill \end{array}$$

(71)

The quantities ${\chi }_i$’s in Equations (66)–(70) are mentioned in the Appendix A, which shows the results of higher-curvature corrections. These factors represent the system’s inhomogeneity. The systematic nature of this solution is physically acceptable and admits the zero CFs conditions. For a range of parameters’s values, the spacetime could be matched to a Weyl solution across hyper-surface $\Delta$.

The analysis of the solution for the isotropic spheroid cannot correspond to any Weyl exterior, except in the case of spherical symmetric configuration despite having a surface with vanishing pressure (as presented in [67] in detail). However, to put it another way, the current solution (60) can correspond to any spherically symmetric solution except for the Weyl exterior because of the nature of the Weyl exterior solution. Recovering the spherically symmetric situation may provide relevant information (the interior Schwarzschild solution); in this instance, Equation (60) holds for $a=u=0$ [41]. Since the surface boundary’s curvature is presumed to be constant, the effect of the $P^{\mathrm{eff}}$ over the external region of $\Delta$ disappears, as shown in Equation (56). However, if this does not vanish then it provides the diverge context of $P^{{}^{\prime }\mathrm{eff}}$ across $\Delta$, which is unacceptable. Thus, the equilibrium of the hydrostatic case might not be addressed. It is well understood that smooth gluing is only achievable when the inner and outer regions have the same-coordinates system. So, Equation (56) must be satisfied for any vacuum exterior metric. Moreover, if the presence of the charge is considered within the inner region then the suggested solution would be changed and its gluing may be achieved with charged Vaidya, the Reissner–Nordström solution, and charged anti de-Sitter/de-Sitter solutions. Unlike the spherically symmetric case, the axially symmetric case is so complicated that the matching of the axial geometry cannot be possible in a formal way. However, the physical nature of the matter variables is acceptable, and the spacetime may be matched to a Weyl solution. For the record, there has been a lot of debate on this matter in the literature. Herrera et al. [41] studied certain fundamental concepts on axially symmetric static sources and identified some exact analytical solutions. They discussed that the solution related to anisotropic matter contents can match to the Weyl exterior. Further, Sharif and Bhatti [75] identified the effects of the electromagnetic field on this concept and concluded that only the anisotropic solution has relevance with the Weyl metrics. They also demonstrated that the charged isotropic solution does not correspond to the Weyl metrics. Herrera and his colleagues [68] considered the CFs for the axially symmetric static source in Einstein’s theory and found some analytical solutions corresponding to the anisotropic as well as isotropic spheroids. They identified that the influence of pressure over the hyper-surface vanishes for any vacuum exterior metric. They also discussed that the anisotropic solution in the axial case may be matched to a Weyl solution. Later on, this concept was generalized in modified gravity [76]. Observational findings suggest that departures from spherical symmetry in compact self-gravitating bodies are more likely to be incidental features than the fundamentals of these configurations. In account of these facts, the sources of distinct Weyl metrics have already been discussed by various researchers [30,77,78,79], particularly in relation to the deviation of spherical-symmetry induced by various physical aspects, namely, magnetic fields [80,81]. By taking the motivation from [68], our aim is to generalize the definition of the complexity for the axial static scenario within the context of the matter-field coupling. The analytical solution corresponding to the isotropic and anisotropic spheroids are formulated, which satisfy the vanishing CFs conditions. It is identified that the isotropic solution (60) corresponds to spherical symmetry for $a=u=0$ [41], while the anisotropic solution may correspond to a Weyl solution [68].

4. Discussion

The current observations regarding the universe’s rapid expansion in the present era could be found from a variety of sources. It is believed that dark energy, which exerts considerable negative pressure, accounts for this aspect of the cosmos. To better explain the cosmos and its formation, it is worthwhile exploring these components. As a result, MGTs are the most meaningful tools for addressing dark energy and the inflationary eras that entail late-time acceleration; one such effective way is $f(R, T)$ theory. The cosmic acceleration in the $f(R, T)$ scenario may be produced by the geometric as well as matter contribution to the entire cosmic energy-density. In this study, we have figured out the significance of higher-order gravitational aspects on axially symmetric complex anisotropic structures. Axially symmetric structures have a meaningful impact on the study of the cosmic scenario because it is impossible to disregard the consequences of the uneven distribution of matter and energy-density. To do so, we have assumed anisotropic matter composition. The computed results are summarized as

• The modified conservation and field equations are figured out for systematic study. Equation (22) arises due to the trivial effects of the static source. However, the requirements for hydro-static equilibrium are reported in Equations (23) and (24). It is noted that both the system’s uneven energy composition and pressure aspects are impacted by the $f(R, T)$ factors as shown in Equations (23) and (24).
• The impact of tidal-force on the propagating axial source is regulated by the conformal tensor. Therefore, to explore the link between the conformal tensor and the inhomogeneity of the structure, we have formed the results in terms of the conformal tensor via the splitting of the Riemann tensor. The magnetic portion identically disappears due to spherical-symmetry, as shown in Equation (16). This fact suggests that the closest flow-lines scatter independently of one another. Subsequently, evolution is locally dependent on matter configuration. Equations (31) and (32) show that the system’s tidal forces become strong due to the contribution of $f(R, T)$ factors.
• An intuitive depiction of the source is complex, indicating the demand to examine the relevant CFs. Complexity refers to a set of physical parameters that might interfere with the uniform distributions of any self-gravitating structure. The quantity CF aims to specify the degree of complexity of the matter composition. The axially symmetric case is complex compared to the spherical one [66]. Therefore, in the axial case three CFs are computed, unlike in the spherical case. In this scenario, the electric part ($Y_{\omega \beta }$) of the Riemann tensor is relevant for devising CFs. These CFs are pointed out by $Y_{TFi}$’s ($i=1,2,3)$ that are associated with the influence of the tidal forces and anisotropy of the source under $f(R, T)$ gravity, whereas $Y_T$ manipulated the aspects of pressure anisotropy, energy-density irregularities, and higher-order theory. Moreover, the inhomogeneity of the source is controlled by the factors $Y_{TFi}$’s, as shown in Equations (48) and (49) that endorsed the relevance of these factors.
• The CFs offer a standard for evaluating the system’s stability, in addition to serving as a benchmark for comparing the complexity of distinct structures. Therefore, we have considered the vanishing CFs constraints (52)–(54) to configure the lack of complexity of our source. In our study, CFs are caused by the uneven distribution of the pressure and energy-density, as well as by the curvature–matter coupling terms. The complex structures are significantly affected by the zero CF constraints that have increased the stability of our evolving object.
• A few analytical results are derived by imposing the scalar functions that admit the zero CFs criteria. The solution corresponds to isotropic fluids where the energy is distributed homogeneously in the absence of CFs are explored. On the other side, we have also formulated the outcomes for anisotropic spheroids with non-homogeneous energy density (described in Equations (64) and (65)), whose systematic nature is physically acceptable enough that Equations (52) and (53) are admitted). Such spacetime could be glued to a Weyl exterior. The higher-curvature factors that appeared in the related matter profile (66)–(70) are another cause of the system’s inhomogeneity.
• Under the usual bounds, i.e., $f(R, T)=R$, all such findings could be reduced to Einstein’s theory [68].