Role of $f(R,T,R_{\mu\nu}T^{\mu\nu})$ Model on the Stability of Cylindrical Stellar Model

TL;DR

This paper examines the stability and instability conditions of cylindrical stellar models within a modified gravity framework, highlighting the influence of dark source terms and the adiabatic index on stellar stability.

Contribution

It introduces a perturbation analysis of non-static anisotropic cylindrical stars in $f(R,T,R_{ ext{μν}}T^{ ext{μν}})$ gravity, emphasizing the role of dark sources and adiabatic index in stability.

Findings

Dark source terms significantly affect stability conditions.

Adiabatic index determines stability regimes in modified gravity.

Instability constraints are derived for Newtonian and post-Newtonian limits.

Abstract

The aim of this paper is to investigate the stable/unstable regimes of the non-static anisotropic filamentary stellar models in the framework of $f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity. We construct the field equations and conservation laws in the perspective of this gravity. The perturbation scheme is applied to analyze the behavior of a particular $f(R,T,R_{\mu\nu}T^{\mu\nu})$ cosmological model on the evolution of cylindrical system. The role of adiabatic index is also checked in the formulations of instability regions. We have explored the instability constraints at Newtonian and post-Newtonian limits. Our results reinforce the significance of adiabatic index and dark source terms in the stability analysis of celestial objects in modified gravity.

Equations

$$I_{f(R,T,Q)}=\frac{1}{2}\int d^{4}x\sqrt{-g}[f(R,T,R_{\lambda \sigma }{T}^{\lambda \sigma })+{\text{L}}_m],$$

$$T_{\lambda \sigma }^{(m)}=-\frac{2}{\sqrt{-g}}\frac{\delta (\sqrt{-g}{\text{L}}_m)}{\delta g^{\lambda \sigma }}.$$

$$-G_{\lambda \sigma }(f_Q{\mathcal{L}}_m-f_R)-g_{\lambda \sigma }\left\{\frac{f}{2}-\square f_R-\frac{R}{2}{f}_R-\frac{1}{2}{\nabla }_{\pi }{\nabla }_{\rho }(f_Q{T}^{\pi \rho })-{\mathcal{L}}_m{f}_T\right\}$$

$$+2f_Q{R}_{\pi (\lambda }{T}_{\sigma )}^{\pi }+\frac{1}{2}\square (f_Q{T}_{\lambda \sigma })-{\nabla }_{\pi }{\nabla }_{(\lambda }[T_{\sigma )}^{\pi }{f}_Q]-2\left(f_T{g}^{\pi \rho }+f_Q{R}^{\pi \rho }\right)\frac{{\partial }^2{\mathcal{L}}_m}{\partial g^{\lambda \sigma }\partial g^{\pi \rho }}$$

$$-T_{\lambda \sigma }^{(m)}(f_T+\frac{R}{2}{f}_Q+1)-{\nabla }_{\lambda }{\nabla }_{\sigma }{f}_R=0,$$

$$R_{\lambda \sigma }-\frac{R}{2}{g}_{\lambda \sigma }=G_{\lambda \sigma }=\stackrel{\text{eff}}{T_{\lambda \sigma }},$$

$${T_{\lambda \sigma }}^{\text{eff}}$$

$$\times {\nabla }_{\pi }{\nabla }_{\rho }(f_Q{T}^{\pi \rho })\}{g}_{\lambda \sigma }-\frac{1}{2}\square (f_Q{T}_{\lambda \sigma })-(g_{\lambda \sigma }\square -{\nabla }_{\lambda }{\nabla }_{\sigma })f_R-2f_Q{R}_{\pi (\lambda }{T}_{\sigma )}^{\pi }$$

$$+{\nabla }_{\pi }{\nabla }_{(\lambda }[T_{\sigma )}^{\pi }{f}_Q]+2\left(f_Q{R}^{\pi \rho }+f_T{g}^{\pi \rho }\right)\frac{{\partial }^2{\text{L}}_m}{\partial g^{\lambda \sigma }\partial g^{\pi \rho }}].$$

$$d{s}_{-}^2=-A^2(t,r)(d{t}^2-d{r}^2)+B^2(t,r)d{z}^2+C^2(t,r)d{\varphi }^2.$$

$$-\infty \le t\le \infty ,0\le r,-\infty <z<\infty ,0\le \varphi \le 2\pi .$$

$$d{s}_{+}^2=-e^{2(\gamma -\upsilon )}(d{\nu }^2-d{\rho }^2)+e^{-2\upsilon }{\rho }^{2}d{\varphi }^2+e^{2\upsilon }d{z}^2,$$

$$\rho ({\upsilon }_{\nu }^2+{\upsilon }_{\rho }^2)=\frac{\tilde{f}-\tilde{R}{\tilde{f}}_R}{2{\tilde{f}}_R}{e}^{2(\gamma -\upsilon )},$$

$$2\rho {\upsilon }_{\nu }{\upsilon }_{\rho }={\gamma }_{\nu },$$

$${\upsilon }_{\nu \nu }-\frac{{\upsilon }_{\rho }}{\rho }-{\upsilon }_{\rho \rho }=\frac{e^{2(\gamma -\upsilon )}}{4\rho }\left(\frac{\tilde{f}-\tilde{R}{\tilde{f}}_R}{{\tilde{f}}_R}\right)\left\{\rho e^{-4\upsilon }+\frac{e^{2\gamma }}{\rho }\right\},$$

$$T_{\lambda \sigma }=(P_r+\mu )V_{\lambda }{V}_{\sigma }+P_r{g}_{\lambda \sigma }-K_{\lambda }{K}_{\sigma }(P_r-P_{\varphi })-S_{\lambda }{S}_{\sigma }(P_r-P_z),$$

$$V^{\lambda }{V}_{\lambda }=-1,K^{\lambda }{K}_{\lambda }=1=S^{\lambda }{S}_{\lambda },V^{\lambda }{K}_{\lambda }=V^{\lambda }{S}_{\lambda }=K^{\lambda }{S}_{\lambda }=0.$$

$$V_{\lambda }=-A{\delta }_{\lambda }^0,K_{\lambda }=C{\delta }_{\lambda }^3,L_{\lambda }=A{\delta }_{\lambda }^1\text{and}{S}_{\lambda }=B{\delta }_{\lambda }^2,$$

$$\Theta =\frac{1}{A}\left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right),$$

$${\sigma }_{\lambda \sigma }=V_{(\lambda ;\sigma )}+a_{(\lambda }{V}_{\sigma )}-\frac{1}{3}\Theta h_{\lambda \sigma },$$

$${\sigma }_{\lambda \sigma }={\sigma }_s\left(S_{\lambda }{S}_{\sigma }-\frac{h_{\lambda \sigma }}{3}\right)+{\sigma }_k\left(K_{\lambda }{K}_{\sigma }-\frac{h_{\lambda \sigma }}{3}\right),$$

$${\sigma }_s=-\frac{1}{A}\left(\frac{\dot{A}}{A}-\frac{\dot{B}}{B}\right),{\sigma }_k=-\frac{1}{A}\left(\frac{\dot{A}}{A}-\frac{\dot{C}}{C}\right).$$

$$\frac{1}{A^2}\left[\frac{\dot{C}\dot{B}}{BC}-\frac{C^{\prime \prime }}{C}-\frac{B^{\prime \prime }}{B}-\frac{B^{\prime }{C}^{\prime }}{BC}+{\alpha }_1\right]=\stackrel{\text{eff}}{\mu },$$

$$\left(\frac{B^{\prime }}{B}+\frac{C^{\prime }}{C}\right)\frac{\dot{A}}{A}-\frac{\dot{C^{\prime }}}{C}-\frac{\dot{B^{\prime }}}{B}+\left(\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right)\frac{A^{\prime }}{A}=0,$$

$$\frac{B^{\prime }{C}^{\prime }}{BC}-\frac{\ddot{B}}{B}-\frac{\dot{B}\dot{C}}{BC}-\frac{\ddot{C}}{C}+{\alpha }_1=\stackrel{\text{eff}}{P_r},$$

$${\left(\frac{B}{A}\right)}^2\left[{\beta }_1+\frac{C^{\prime \prime }}{C}-\frac{\ddot{C}}{C}\right]=\stackrel{\text{eff}}{P_z},{\left(\frac{C}{A}\right)}^2\left[{\beta }_1+\frac{B^{\prime \prime }}{B}-\frac{\ddot{B}}{B}\right]=\stackrel{\text{eff}}{P_{\varphi }},$$

$${\alpha }_1$$

$$\stackrel{\text{eff}}{\mu }$$

$$+{\mu }^{\prime }{\chi }_3+\frac{P_r^{\prime \prime }}{2A^2}{f}_Q+P_r{\chi }_4+P_r^{\prime }\left\{\frac{f_Q^{\prime }}{A^2}-\frac{5A^{\prime }}{2A^3}{f}_Q\right\}-\frac{f_Q}{2A^{2}B}(\dot{P_z}\dot{B}+P_z^{\prime }{B}^{\prime })$$

$$-\dot{P_r}\frac{\dot{A}}{A^3}{f}_Q+P_z{\chi }_5+P_{\varphi }{\chi }_6-\frac{f_Q}{2A^2}\left(\dot{P_{\varphi }}\frac{\dot{C}}{C}-P_{\varphi }^{\prime }\frac{C^{\prime }}{C}\right)-\frac{\dot{f_R}}{A^2}\left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right)$$

$$-\frac{f_R^{\prime }}{A^2}\left(\frac{A^{\prime }}{A}-\frac{B^{\prime }}{B}-\frac{C^{\prime }}{C}\right)+\frac{f_R^{\prime \prime }}{A^2}],$$

$$\stackrel{\text{eff}}{P_r}$$

$$-f_R^{\prime }{\psi }_2+P_r{\chi }_7-\frac{\ddot{\mu }}{2A^2}{f}_Q+\mu {\chi }_8+\frac{{\mu }^{\prime }{A}^{\prime }}{2A^3}{f}_Q-\frac{\ddot{P_r}}{2A^2}{f}_Q+P_z{\chi }_9+P_{\varphi }{\chi }_{10}$$

$$+\frac{f_Q}{2A^2}\left\{\dot{P_z}\frac{\dot{B}}{B}-P_z^{\prime }\frac{B^{\prime }}{B}+\dot{P_{\varphi }}\frac{\dot{C}}{C}-P_{\varphi }^{\prime }\frac{C^{\prime }}{C}\right\}+\dot{P_r}{\chi }_{11}+P_r^{\prime }{\chi }_{12}],$$

$$\stackrel{\text{eff}}{P_z}$$

$$+P_r{\chi }_{15}+\frac{\dot{A}\dot{P_r}}{2A^3}{f}_Q+P_r^{\prime }\left(\frac{5A^{\prime }}{2A^3}{f}_Q-\frac{f_Q^{\prime }}{A^2}\right)-\frac{P_r^{\prime \prime }}{2A^2}{f}_Q+P_z{\chi }_{16}+P_z^{\prime }{\chi }_{17}-\dot{P_z}$$

$$\times {\chi }_{18}-\frac{\ddot{P_z}}{2A^2}{f}_Q+\frac{P_z^{\prime \prime }}{2A^2}{f}_Q+P_{\varphi }{\chi }_{19}+\frac{f_Q}{2A^2}\left(\dot{P_{\varphi }}\frac{\dot{C}}{C}-P_{\varphi }^{\prime }\frac{C^{\prime }}{C}\right)+{\psi }_3],$$

$$\stackrel{\text{eff}}{P_{\varphi }}$$

$$+P_r{\chi }_{15}+\frac{\dot{A}\dot{P_r}}{2A^3}{f}_Q+P_r^{\prime }\left(\frac{5A^{\prime }}{2A^3}{f}_Q-\frac{f_Q^{\prime }}{A^2}\right)-\frac{P_r^{\prime \prime }}{2A^2}{f}_Q+P_z{\chi }_{20}+P_{\varphi }^{\prime }{\chi }_{23}+\dot{P_{\varphi }}$$

$$\times {\chi }_{22}-\frac{\ddot{P_{\varphi }}}{2A^2}{f}_Q+\frac{P_{\varphi }^{\prime \prime }}{2A^2}{f}_Q+P_{\varphi }{\chi }_{21}+\frac{f_Q}{2A^2}\left(\dot{P_z}\frac{\dot{B}}{B}-P_z^{\prime }\frac{B^{\prime }}{B}\right)+{\psi }_4],$$

$$R$$

Full text

Keywords: $f(R,T,Q)$ gravity; Non-dissipative fluid; Adiabatic index.

PACS: 04.20.-q; 04.40.-b; 04.40.Dg; 04.40.Nr.

1 Introduction

The accelerated expansion of the cosmos is strongly manifested after the discovery of unexpected reduction in the detected energy fluxes coming from Supernovae type Ia [1, 2]. Other observational data like cosmic microwave background radiations, large scale structures and galaxy red shift surveys [3, 4] also provide evidence in this favor. These observational data propose an enigmatic form of force, dubbed as dark energy (DE) which takes part in the expansion phenomenon and dominates overall energy density of the cosmos. Despite some very solid claims about the existence of DE, its unknown nature is the substantial puzzle in cosmology. The idea of modified gravitational theories is obtained by extending the standard Einstein-Hilbert (EH) action, which has gained much fame in order to demonstrate the secrets of cosmic accelerating expansion.

There exists various theories of modified gravity such as $f(R)$ gravity with $R$ as the Ricci scalar, $f(\mathcal{T})$ gravity in which $\mathcal{T}$ is a torsion scalar, $f(R,T)$ gravity with $T$ as the trace of energy-momentum tensor, $f(G)$ gravity in which $G$ represents Gauss-Bonnet invariant and $f(R,T,Q)$ gravity (where $Q=R_{\lambda\sigma}T^{\lambda\sigma}$) etc. Nojiri and Odintsov [5] reviewed various versions of modified gravity models that could explain DE dominance in this accelerating cosmos. Cognola et al. [6] introduced some viable formulations of $f(R)$ DE models and classified them into four main-streams. Nojiri and Odintsov [7] studied some important aspects of $f(R)$ gravity in order to make them well-consistent with observational data. Bamba et al. [8] discussed the role of DE, through modified cosmic models, on the expansion of our accelerating cosmos. Durrer and Maartens [9] investigated that some $f(R)$ models could lead to new schemes to test out the credibility of general relativity itself on cosmological scales. Bhatti et al. [10] discussed dynamical instability of non-static cylindrical cosmic configuration by using $f(\mathcal{T})$ gravity and found that additional curvature conditions generate the stability of expanding stellar frame.

Harko et al. [11] used $f(R,T)$ theory of gravity and presented the corresponding equations of motion for the massive particles through variational principle in $f(R,T)$ theory. The generalization of $f(R,T)$ gravity is $f(R,T,Q)$ gravity, where $Q=R_{\lambda\sigma}T^{\lambda\sigma}$ shows the non-minimal coupling between matter and geometry [12]. Haghani et al. [13] obtained the field equations by using Lagrange multiplier in $f(R,T,Q)$ theory of gravity. Odintsov and Sáez-Gómez [14] studied $f(R,T,Q)$ gravity with non-minimal association between matter and gravitational fields and concluded that respective modified gravity contains additional points which would recast the possible cosmological evolution. Elizalde and Vacaru [15] evaluated some exact off-diagonal cosmological models in $f(R,T,Q)$ gravity. Baffou et al. [16] used perturbation technique and performed stability analysis with the help of de-Sitter and power law models through numerical simulations in $f(R,T,Q)$ gravity.

Gravitational collapse is the fundamental and highly dissipative phenomena for structure formation in our universe. Chandrasekhar [17] studied the dynamical instability of oscillating spherically symmetric model by using perfect fluid and found instability limits in terms of adiabatic index. Herrera et al. [18] analyzed the dynamical instability of dense stars with zero expansion scalar in spherically symmetric configuration and found instability limits that are independent of adiabatic index. Cembranos et al. [19] studied gravitational collapse in $f(R)$ gravity and found this phenomenon as a key tool to constrain modified gravity models that depict late time cosmological acceleration. Yousaf et al. [20] investigated the irregularity constituents for spherical self-gravitating stars in the presence of imperfect matter distribution within $f(R,T)$ gravity and found that the complexity of matter increases with the increase of anisotropic stresses. Yousaf [21] explored collapsing spherical models supporting vacuum core in a $\Lambda$-dominated era within the stellar interior.

The subject of exploring the cosmic filamentary celestial objects has been a source of great attention, motivated by many relativistic astrophysicists [22, 23, 24, 25, 26, 27]. On large cosmic scale, it has been analyzed that matter is usually configured to make large filaments. These stellar structures have been found to be very clear characteristics of the interstellar medium. They may give rise to galaxies upon contraction. Motivated from several simulations and observational outcomes, the stability analysis of cosmic filaments with more realistic assumptions has received great interest. Binney and Tremaine [28] have linearized the Vlasov equation about the steady phase of the relativistic interior and solved the resulting eigenvalue equation in order to discuss the dynamical stability of collision-less celestial structures supported by the Vlasov-Poisson formulations. Chavanis [29] has extended their results in the context of non-linear dynamical stability and explored the problem of stability of barotropic as well as collision less stellar systems via the maximization of a Casimir functional (or H-function) with fixed values of energy and mass. Quillen and Comparetta [30] assumed a constant linear mass density and approximately evaluated a dispersion relation in the background of the tidal galaxy tail.

Myers [31] has discussed the evolution of some observed characteristics of cores and filaments and concluded that during the contraction of host filaments, the core grows in mass and radius and this phenomenon stops if the surrounding filament gas will no longer exists for further accretion. Breysse et al. [32] carried out analytical approach with the detailed perturbation background and investigated the stability of polytropic fluid filaments. They found that the instabilities of the cosmic fluid filaments could be enhanced by introducing tangential fluid motion of the system. Sharif and Manzoor [33] studied the dynamical instability of axially symmetric stellar structure with reflection degrees of freedom coupled with locally anisotropic fluid configurations in self-interacting Brans-Dicke gravity and obtained stability conditions through adiabatic index at both N and pN approximations. Birnboim et al. [34] performed stability analysis in planar, filamentary and spherical infall geometries for the existence of virialized gas in one, two and three-dimensional (3D) gravitational collapse and concluded that cosmic filaments are likely to host halos under some constraints.

Recently, we have investigated the anisotropic spherical collapse in the background of $f(R,T,Q)$ gravity and discussed the stability of compact stars by taking into account the particular viable model with perturbation technique. We also examined that adiabatic index $\Gamma_{1}$ has significant role in the dynamical instability of these massive stars [35]. The motivation of this paper is to explain the mathematical as well as physical features of self gravitating cylindrical celestial objects within the framework of $f(R,T,Q)$ theory of gravity. Particularly, some properties of viable modified gravity model are discussed to create the expansion and DE consequences in cosmos. This paper is organized as follows: We provide the basic formalism of $f(R,T,Q)$ gravity in section 2. Section 3 deals with the dynamics of cylindrical self-gravitating collapsing model in which formation of field equation and conservation laws by linear perturbation technique and instability constraints at Newtonian (N) and post-Newtonian (pN) limits are investigated. Finally, we conclude our main results in the last section.

2 The Formalism of $f(R,T,Q)$ Gravity

The formalism of $f(R,T,Q)$ gravity is based on the contribution of non-minimal coupling of geometry and matter. Where the $R$ in EH action is replaced with an arbitrary function of $R,~{}T$ and $R_{\gamma\delta}T^{\gamma\delta}$. In [13] modified EH action is demonstrated in the following way

[TABLE]

where, $\textit{L}_{m}$ expresses the relative Lagrangian density of matter distribution then the respective energy momentum tensor is expressed as

[TABLE]

On varying the modified action Eq.(1), with metric tensor $g_{\lambda\sigma}$, the following field equations are obtained

[TABLE]

where $\nabla_{\pi}$ and $G_{\lambda\sigma}$ indicates covariant derivative and Einstein tensor, respectively, with $\Box=g^{\lambda\sigma}\nabla_{\lambda}\nabla_{\sigma}$ as a d’Alembert’s operator. From Eq.(3), one can obtain the expression of trace as in [35]. In the framework of [11] the matter Lagrangian has no specific distinction for perfect fluid, and the corresponding second variation was neglected in their calculations. Equation (3) can be rewritten in GR perspective as follows

[TABLE]

where the effective energy momentum tensor ${{T}_{\lambda\sigma}}^{\textrm{eff}}$ has the following form

[TABLE]

On taking $Q=0$ in above equation, then $f(R,T,Q)$ gravity would reduce to $f(R,T)$ theory. However, in case of vacuum it leads to $f(R)$ gravity theory and consequently we will obtain GR results whenever $f(R)=R$.

3 Anisotropic Matter Distribution and Cylindrical Field Equations

We consider the 3-dimensional (3D) timelike hypersurface, $\Delta$, that would demarcate the 4D manifold $\mathcal{W}$ into couple of regions, i.e., exterior $\mathcal{W}^{+}$ and interior $\mathcal{W}^{-}$. The interior region of relativistic stellar system is given by the following cylindrically symmetric spacetime

[TABLE]

For the representation of cylindrical symmetry, the following ranges are imposed on the coordinates

[TABLE]

We number the respective coordinates $x^{0}=t$, $x^{1}=r$, $x^{2}=z$ and $x^{3}=\phi$. We assumed $C=0$ at $r=0$ that represents a non-singular axis. The spacetime for $\mathcal{W}^{+}$ is [36]

[TABLE]

where $\gamma$ and $\upsilon$ are the functions of $\nu$ and $\rho$, while the coordinates are numbered as $x^{\beta}=(\nu,~{}\rho,~{}\phi,~{}z)$. The corresponding vacuum field equations provide

[TABLE]

where subscripts $\rho$ and $\nu$ show partial differentiations with respect to $\rho$ and $\nu$, respectively and tilde indicates that the corresponding values are evaluated with constant $R,~{}T$ and $Q$ conditions. It has been proved by Senovilla [37] that modified extra curvature terms on the boundary surface should be constant. Due to this reason, we have evaluated above equations with constant $R,~{}T$ and $Q$. These equations suggest the existence of gravitational field. We assume anisotropic and non-dissipative collapsing matter in cylindrical geometry, whose energy momentum tensor is

[TABLE]

where $\mu$ is the energy density which is the eigenvalue of $T_{\lambda\sigma}$ for eigenvector $V_{\lambda}$, while $P_{\phi},~{}P_{z},~{}P_{r}$ are the principal stresses. The spacetime (5) is the canonical form for cylindrical symmetry, defined as usual by the 2D group that defines the cylindrical symmetry. The unitary vectors $V_{\lambda},~{}L_{\lambda},~{}S_{\lambda},~{}K_{\lambda}$ are configuring to make a canonical orthonormal tetrad in which a hypersurface orthogonal 4-velocity vector is $V_{\lambda}$. Further, the two vectors $S_{\lambda}$ and $K_{\lambda}$ are tangent to the orbits of the 2D group that preserves cylindrical geometry and $L_{\lambda}$ is orthogonal to 4-velocity $V_{\lambda}$ and to these orbits. It is worthy to stress that we are considering an Eckart frame where fluid elements are at the state of rest. The four vectors obey the following relations

[TABLE]

We choose the fluid to be comoving in a given coordinate system, therefore, we have

[TABLE]

The four acceleration vector is $a_{\lambda}=V_{\lambda;\sigma}V^{\sigma}$, with $a=\frac{A^{\prime}}{A^{2}}$ as a scalar associated with the four-acceleration. The expansion scalar, ($\Theta=V^{\lambda}_{~{}~{};\lambda}$), for our cylindrical spacetime leads to

[TABLE]

where over dot represents the time derivative. The shear tensor $\sigma_{\lambda\sigma}$ is

[TABLE]

where $h_{\lambda\sigma}$ is a projection tensor with $h_{\lambda\sigma}=g_{\lambda\sigma}+V_{\lambda}V_{\sigma}$. The shear tensor can also be expressed as follows

[TABLE]

where

[TABLE]

The non zero modified gravitational field equations for our cylindrical line element associated with matter distribution (10) take the form

[TABLE]

where

[TABLE]

where prime stands for $\frac{\partial}{\partial r}$ operator and the quantities $\chi_{i}$’s contain combinations of metric variables and their derivatives are mentioned in Appendix. The value of $R$ for respective spacetime is given as

[TABLE]

3.1 Viability Of $f(R,T,Q)$ Model and Junction Conditions

In this subsection, we shall deal with the hydrodynamics of cylindrical stellar collapse by using dynamical equations. The expression of covariant derivative of effective energy momentum tensor is

[TABLE]

which would provide two equations of motion in $f(R,T,Q)$ theory. Making use of $G^{\lambda\sigma}_{~{};\sigma}=0$ and Eqs.(15)-(18) along with $\lambda=0,~{}1$, the above equation gives

[TABLE]

where superscript $``\textrm{eff}"$ indicates the presence of $f(R,T,Q)$ terms in the matter variables and the expressions of $Z_{1}$ and $Z_{2}$ are mentioned in Appendix as Eqs.(A1) and (A2). The quantities $Z_{1}$ and $Z_{2}$ are coming due to the non-conserved divergence of energy momentum tensor. The dynamical equations could help to explain hydrodynamics of locally anisotropic cylindrical relativistic massive bodies. It is worthy to mention that the theoretically designed stellar models are of worth importance if they are stable against instabilities and fluctuations. Now, we will explain the dynamic instability of anisotropic and non dissipative relativistic cylindrical geometry by using particular $f(R,T,Q)$ model [38].

[TABLE]

where $\alpha$ and $\beta$ are constants. The model, $\alpha R^{n}+\beta Q^{m}$ is the generalization of above mentioned $f(R,T,Q)$ model in which $m$ and $n$ are constants. In order to deal this theory free from Ostrogradski instabilities, one should take $n\neq 1$. However, this model will generate stable theory for $m=1$, by giving EH term including canonical scalar field having non-minimal variation coupling of Einstein tensor. The model with $n=2$ and $m=1$ along with constant $\beta$ could help to understand the dynamics and evolution of inflationary cosmos. For the particular value of constant $\alpha$, i.e., $\alpha=\frac{1}{6M^{2}}$ [39] with $M=2.7\times 10^{-12}$ GeV, this model behaves as a substitute of DM. In case of $\alpha=0$, there is geometry-matter association on behalf of coupling between stress-energy tensor and the Ricci scalar. Yousaf et al. [35] studied this model with $n=2,~{}m=1$ and discussed the stability of compact stars in anisotropic spherical configuration by taking $\beta>0$ along with $\alpha=\frac{1}{6M^{2}}$.

For the smooth matching of Eqs.(5) and (6) over $\Delta$, we shall use junction conditions proposed by Darmois [40] as well as Senovilla [37] for $f(R,T,Q)$ theory. Since we have assumed a timelike hypersurface, therefore we impose $r=$constant in Eq.(5) and $\rho(\nu)$ in the exterior metric (10). In this framework, the first fundamental form provides

[TABLE]

with $1-\left(\frac{d\rho}{d\nu}\right)^{2}>0$. Here, the notation overset $\Delta$ indicates that the corresponding equations and quantities are evaluated on the hypersurface, $\Delta$. The second fundamental form yields

[TABLE]

By making use of Eqs.(28)-(31), field equations and after some manipulations, we obtain

[TABLE]

From Eq.(4), one can write the following form

[TABLE]

which can be transformed as

[TABLE]

where $\Omega_{\lambda\sigma}$ indicates tensor associated with bulk matter. In a Gaussian normal coordinates system, we have

[TABLE]

in Which the boundary surface us at $y=0$. In this context, the Ricci scalar takes the form

[TABLE]

where $K_{ab}$ is the extrinsic curvature at the hypersurface, tilde shows the constant choice of the Ricci scalar evaluated through induced spacetime, while $K^{*}_{ab}$ and $K$ are the trace-less and trace components of the extrinsic curvature respectively. The value of the extrinsic curvature can be expressed through $\gamma_{ab}$ as $K_{ab}=-1/2\times\partial_{y}\gamma_{ab}$. The Einstein tensor provides

[TABLE]

Now, we split Eq.(33) into two tensorial quantities as

[TABLE]

where

[TABLE]

The components of Eq.(36) are obtained as follows

[TABLE]

while Eq.(37) provides the following relations

[TABLE]

Now, we compute the $ya$ and $yy$ components of Eq.(35), which after some simplifications give rise to

[TABLE]

Upon integration across the hypersurface, Eq.(38) yields

[TABLE]

The integration of Eq.(34) gives $R|_{-}^{+}=0$, while the trace and traceless components of Eq.(39) gives rise to

[TABLE]

along with

[TABLE]

provide the matching conditions for $f(R,T,Q)$ theory of gravity in which $f_{,RR}\neq 0$ and $f_{,QQ}\neq 0$ should be satisfied. The details of this approach in $f(R)$ gravity has been mentioned in [37, 41, 42]. Equation (32) arises due to Darmois junction conditions that indicates that effective radial pressure on $\Delta$ is zero. The obeying of Eqs.(40) and (41) over $\Sigma$ is required for the continuity of $R$ and $Q$ invariants even for matter thin shells.

3.2 Perturbation Scheme

In order to discuss the stability of cylindrical celestial objects, we shall explore the perturbed form of field as well as dynamical equations in this section. As perturbation deals with small variations in a physical system resulted by gravitational effects of other stellar objects. Therefore, in recent few decades, researchers are very keen to analyze the stability of the cosmic stellar filaments against oscillatory motion induced by perturbations. Here, we use the linear perturbation scheme with very small perturbation parameter $\epsilon$ so that one can neglect its second and higher powers. Initially, the celestial system is considered to be in hydrostatic equilibrium, but with the passage of time passes, it is subjected to the oscillatory motion. All the metric functions and fluid parameters can be perturbed as [18]

[TABLE]

By using above perturbation technique along with junction conditions (32), (40) and (41), Eq.(17) can be executed in terms of second order partial differential equation as

[TABLE]

where

[TABLE]

The most general solution of the above equation is given by

[TABLE]

where $\mathfrak{c}_{1}$ and $\mathfrak{c}_{2}$ are arbitrary constants. Equation (44) indicates two solutions that are independent to each other. Here, we wish to explore unstable regimes of collapsing stellar anisotropic system. Due to this reason, we consider that our stellar filament is in static equilibrium at large past time, i.e., $\omega(-\infty)=0$, then with the passage of time it enters into the present state and goes forward in the phase of gravitational collapse by decreasing its areal radius. Such model could be achieved only by taking $\mathfrak{c}_{1}=-1$ along with $\mathfrak{c}_{2}=0$. This would describes the monotonically decreasing configuration of the solution as the time proceed forward.

The most general solution of Eq.(43) includes oscillating and non-oscillating functions that correspond to stable and unstable configurations of stellar anisotropic filament, respectively. The choice $\mathfrak{c}_{1}=+1$ is exactly equivalent to the case if one absorbs the sign in $a,~{}b,~{}c$ and $d$. Our aim is to explore instability regimes of collapsing stellar interiors, therefore we have to restrict our perturbations $a,~{}b,~{}c$ and $d$ on the boundary surface as a positive definite in order to make $\chi^{2}>0$. (This assumption has been taken by number of relativistic astrophysicists [18, 43, 44, 45, 46, 33, 47, 48] to discuss unstable limits of collapsing stellar populations). The required solution associated with Eq.(43) can be achieved by taking $\mathfrak{c}_{1}=-1$ and $\mathfrak{c}_{2}=0$ as

[TABLE]

The perturbed configuration of $f(R,T,Q)$ model is

[TABLE]

where

[TABLE]

By using above perturbation scheme, the static forms of $f(R,T,Q)$ field equations are

[TABLE]

where superscript $(S)$ indicates static form of Einstein tensors. Their expressions are given in Appendix as Eqs.(A3)-(A5). However, the perturbed configuration of these equations are

[TABLE]

where over bar shows perturbed form of Einstein tensors and are written in Appendix as Eqs.(A6)-(A9). In case of hydro-static equilibrium, the second dynamical equation has the following form

[TABLE]

while, their non-static forms are

[TABLE]

where

[TABLE]

Under non-static environment, the scalar variables associated with expansion and shear tensors are found as follows

[TABLE]

3.3 Stability Analysis

Here, we want to discuss about the stability of cylindrical anisotropic compact objects in terms of the stiffness parameter $\Gamma_{1}$. The Harrison-Wheeler equation of state [49] has a great impact in this context which forms a relationship between pressure components and energy density given as

[TABLE]

Then, Eq.(55) can be rewritten as follows

[TABLE]

The integration of this equation gives

[TABLE]

Using the above value of $\bar{\mu}^{\textrm{eff}}$ in Eq.(57), we obtain

[TABLE]

Substituting the values from Eqs.(58) and (59) in Eq.(54), the corresponding modified collapse equation turns out to be

[TABLE]

In a given equation, the terms including adiabatic index $\Gamma_{1}$ would generate pressure and counter gravitational effects while the remaining terms work as the generator of the gravity force. The effects, produced by principal stresses and $f(R,T,Q)$ gravity terms intervened by fluid have greatest relevance in the analysis of gravity forces.

3.3.1 N Approximations

Here, we compute the instability for cylindrical interior system at N limit with the theory of gravity induced by $\alpha R^{2}+\beta Q$ model. In N regime, we shall consider a flat background metric, that provide weak field approximations. Therefore, we take

[TABLE]

Since, we are dealing with the compact configurations of cosmic stellar filament, therefore, we assume that the energy density of the matter content is much greater than the pressure components. Due to this reason, we shall consider the following constraint in our calculation with N limit

[TABLE]

It was demonstrated by Chandrasekhar [17] and Herrera et al. [18] that all the terms coming in the stability conditions should be positive definite. Therefore, to attain the instability regions of cylindrical stellar system, we are considering each term in the respective collapse equation to be positive. The collapse equation (60) takes the form

[TABLE]

where

[TABLE]

includes anisotropic effects for onset of instability regimes in cylindrical compact objects.

Now, we recall the work of Chandrasekhar [17], who checked the collapsing behavior of a perfect spherical star with the help of numerical value of $\Gamma_{1}$. He found three possibilities about the N limits of the star. These are

The effects of the star weight will be stronger than pressure, once the system satisfies $\Gamma_{1}<4/3$ condition. This would eventually lead the body to enter into collapse state. 2. 2.

The initial compression would lead the system towards hydrostatic equilibrium, if $\Gamma_{1}=4/3$. 3. 3.

Further, the limit $\Gamma_{1}>4/3$ indicates that the influence of pressure on the stellar dynamics is much greater than the star weight, thereby increasing the resulting outward force. Then, the system will move towards equilibrium and is said to be dynamical stable.

Keeping in mind the same analysis for $f(R,T,Q)$ theory of gravity, the evolving cylindrical anisotropic stellar object will be in phase of hydrostatic equilibrium whenever it satisfies

[TABLE]

If the effects of $|\overset{~{}~{}\textrm{eff}}{\mu_{o}}(a^{\prime}/a)+\Pi+\bar{Z_{2}}|$ and $|\overset{~{}~{}\textrm{eff}}{\mu_{o}}\left(a+b+\frac{c}{C_{o}}\right)+\bar{Z_{1}}|$ are equal, then

[TABLE]

will give us the condition of hydrostatic equilibrium for the cylindrically symmetric anisotropic interiors. However, if the role of $|\overset{~{}~{}\textrm{eff}}{\mu_{o}}(a^{\prime}/a)+\Pi+\bar{Z_{2}}|$ is lesser than $|\overset{~{}~{}\textrm{eff}}{\mu_{o}}\left(a+b+\frac{c}{C_{o}}\right)+\bar{Z_{1}}|$, then the relation

[TABLE]

shows that the given system is in unstable region and the range of adiabatic index would belongs to $(0,1)$. If the modified gravity forces generated by $|\overset{~{}~{}\textrm{eff}}{\mu_{o}}(a^{\prime}/a)+\Pi+\bar{Z_{2}}|$ are higher than that of $|\overset{~{}~{}\textrm{eff}}{\mu_{o}}\left(a+b+\frac{c}{C_{o}}\right)+\bar{Z_{1}}|$, then this will make the system enter into the stable window. This means that the forces of anti-gravity and principal stresses produce the stability constraint at N region as

[TABLE]

This state is said to be the dynamical stable.

3.3.2 pN Approximations

In order to attain the pN instability constraints, we consider $A_{o}(r)=1-\phi,~{}B_{o}(r)=1+\phi$ with effects upto $O(\phi)$, where $\phi(r)=\frac{m_{0}}{r}$. In this context, the collapse equation (60) provides the following value of $\Gamma_{1}$

[TABLE]

where

[TABLE]

The anisotropic cosmic filament will enter into the window of stable configurations, once the modified gravity forces generated by $F_{pN}$ are greater than that of $E_{pN}$. In that case, the stability of the relativistic system is ensured by the following pN limit

[TABLE]

However, if during evolution, the system attains the state at which $F_{pN}=E_{pN}$, then the system will cease in the regime of equilibrium. At that time, the cylindrical system will no longer be in the evolutionary phases. One can deal with such situation by considering Eq.(65). The constraint for instability can be entertained by the anisotropic cylindrical compact system, if the impact of $F_{pN}$ is less than $E_{pN}$. This would give

[TABLE]

This pN instability limit depends upon the contribution of principal stresses and counter gravity terms related with $\Gamma_{1}$ and $f(R,T,Q)$ gravity. This also indicates the significance of hydrostatic equilibrium factors in the study of dynamical unstable regimes of our system.

4 Concluding Remarks

In the framework of modified gravity, the stability problem of massive objects has appeared as a main concern in relativistic astrophysics. In this paper, we have analyzed the instability ranges of self-gravitating cylindrical collapsing model in $f(R,T,Q)$ gravity structure. We have investigated the field equations for cylindrical symmetric spacetime within anisotropic and non-dissipative matter distribution. In this aspect, the dynamical equations are developed by using the contraction of Bianchi identities. The perturbed profile of the field, dynamical equations and kinematical quantities are evaluated by imposing the perturbation scheme on material and geometric variables.

Initially, we have supposed that our cylindrical system is in hydrostatic equilibrium position. However, as time passes, it undergoes into the oscillating phase. Therefore, the resulting equations are applied to construct the collapse equation, which is further analyzed at N and pN limits. In this background, adiabatic index assisted by equation of state has been used to quantify the stiffness of matter composition. Also, we have considered a feasible model of $f(R,T,Q)$ theory and examined its impact in the dynamical evolution of locally anisotropic celestial system. It is noticed that additional curvature terms are appearing because of the modification in the gravity model, which are the major cause of obstacles in evolving celestial objects. Consequently, forming the evolving cosmic filament system more stable due to their non-attractive behavior.

It is noted that, for the stability of isotropic spherical relativistic bodies, the particular numerical value of stiffness parameter, i.e., $\frac{4}{3}$, was calculated by Chandrasekhar [17]. Since then, many astrophysicists have tried to examine the instability regimes for various celestial geometries. We have observed the critical role of adiabatic index in the description of un-stable/stable regimes. We also examined that $\Gamma_{1}$ depends upon the static configuration of geometry and matter as well as on the additional terms which appear due to matter curvature coupling. It is noted that the system will remain unstable whenever it holds up its range as specified in expressions (64) and (67) for N and pN limits, respectively. When the system unable to follow the above-mentioned ranges, it will enter into the stable or equilibrium phase. It should be remarkably noted that in the absence of non-minimal coupling of matter and geometry, these results mark down to $f(R,T)$ outcomes. However, in case of vacuum, one can get result of $f(R)$ gravity.

Acknowledgments

The authors would like to thank the anonymous reviewer for valuable and constructive comments and suggestions to improve the quality of the paper.

Appendix A

The quantities $\chi_{i}$’s and $\psi$’s appearing in Eqs.(19)-(22) are

[TABLE]

The expressions $S_{i}$’s appearing in Eq.(65) are

[TABLE]

The quantities $Z_{1}$ and $Z_{2}$ coming in Eqs.(25) and (26) are

[TABLE]

The static configurations of Einstein tensors appearing in Eqs.(47)-(49) are

[TABLE]

The non-static perturbed configurations of Einstein tensors appearing in Eqs.(50)-(53) are

[TABLE]