New definition of complexity factor in $f(R,T,R_{\mu \nu }{T}^{\mu \nu })$ gravity

Abstract

This paper is devoted to present new definition of complexity factor for static cylindrically symmetric matter configurations in $f(R,T,R_{\mu \nu }{T}^{\mu \nu })$ gravity. For this purpose, we have considered irrotational static cylindrical spacetime coupled with a locally anisotropic relativistic fluid. After formulating gravitational field and conservation equations, we have performed orthogonal splitting of the Riemann curvature tensor. Unlike GR (for spherical case) the one of the structure scalars $X_{TF}$, has been identified to be a complexity factor. This factor contains effective forms of the energy density, and anisotropic pressure components. Few peculiar relations among complexity factor, Tolman mass and Weyl scalar are also analyzed with the modified $f(R,T,R_{\mu \nu }{T}^{\mu \nu })$ corrections.

Introduction

Albert Einstein proposed the general theory of relativity (GR) in 1915 and provided his well-known field equations in which matter and geometry were interlinked. According to his beliefs, our cosmos does not expand or contract with the passage of time, i.e., it is static in nature. After few years, in 1929, Edwin Hubble shown the accelerating expansion of our universe by performing an experiment on galaxies and found the red-shifted light coming from them which was an indication that these galaxies are moving away from each other. There have recently been some remarkable observations which pointed out the need of deeper understanding of our cosmic dynamics because of the extremely high amount of dark energy and dark matter, which is about $95$% of our whole universe. Several relativistic astrophysicists have established interests in studying different ways to describe the dark source elements of the universe [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. In a recent era, a variety of different modified gravity theories which could be used to study the dynamical properties of expanding universe have been proposed.

The $f\left(R\right)$ theory, which is the forthright generalization of GR was attained by replacing the Ricci scalar with its generic function in an action function. In contrast to GR, Nojiri and Odintsov [13] asserted the stability of $f\left(R\right)$ theory by constructing various $f\left(R\right)$ models which are consistent with specific solar system experiments to explore mysterious facets of the universe. Bamba et al. [14] examined the $\Lambda$CDM-like universe in the light of various models and also investigated some properties of dark energy. In $f\left(R\right)$ gravity, various researchers explored the role of different components which produce irregularity in a self-gravitating system and analyzed precise solutions to modified field equations by considering some applications [15], [16], [17], [18]. In accordance with GR, Yousaf and Bhatti [19] observed that $f\left(R\right)$ gravitational models are highly acceptable to host more massive compact stars.

The $f(R,T)$ theory where matter lagrangian is an arbitrary function of matter and geometry was then proposed by Harko et al. [20] in which $T$ indicates the trace of energy–momentum tensor. They analyzed the motion of some test particles in this theory and found equations of motion for them via variational principle. Baffou et al. [21] found some cosmological solutions which are in accordance with the observed data and the stability of a particular model through some methods was also checked. Haghani et al. [22] proposed the more general form of $f(R,T)$ gravity which they called as $f(R,T,Q)$ theory, in which the matter lagrangian contains the strong dependence of geometry and fluid. They also determined field equations in this theory by considering Lagrange multiplier method. Odintsov and Sáez-Gómez [23] observed the role of strong non-minimal connection of spacetime with geometry and discovered that some extra terms of $f(R,T,Q)$ gravity permit our cosmic expansion.

In $f(R,T,Q)$ gravity, some particular scalar and vector fields have been considered by Ayuso et al. [24] in which they studied the conditions for stability. Using some peculiar solutions, Baffou et al. [25] investigated the stability of $f(R,T,Q)$ theory and concluded that the extra curvature terms could be helpful to understand the early phases of our cosmic evolution. They also discussed the stability of some particular models in $f(R,T,Q)$ theory by obtaining their solution through numerical techniques. Yousaf et al. [26], [27], [28], [29] studied the gravitational collapse in self-gravitating structures and calculated their equations of motion in $f(R,T,Q)$ theory as well as some relations with Weyl tensor. Bhatti et al. [30], [31] studied the gravitational collapse in $f(R,T,Q)$ theory and calculated few constraints under which the systems would enter into the collapsing phase.

A mixture of multiple components that can cause to generate complications in any balanced self-gravitating structure in known as complexity. Here, we have to describe the role of $f(R,T,Q)$ theory on the existing outcomes of zero complexity measures in GR. Different concepts of complexity can be seen in different fields of science. Among those several definitions, López-Ruiz et al. [32], [33], [34] offered this concept via entropy and information. Entropy is the measurement of any system’s disorderness while knowledge about a system could be known as information. This concept has also been then introduced through a term which is known as disequilibrium by López-Ruiz et al. [32].

In physics, one can illustrate the concept of complexity by considering simplest systems (which have no complexity by definition), i.e., the isolated ideal gas and perfect crystal. These both models are extreme (but opposite) in nature from each other. The former model is completely disordered in its nature as it is made-up of molecules which are moving randomly, and hence it provides maximal information due to the equal participation of all molecules in this system. In a later model, the constituents which make the system are arranged in an ordered form and thus it gives minimal data set as the study of its small portion is enough to know about its nature. Hence, it could be seen that there is maximum disequilibrium in former case while zero in later case. In astrophysics, the structural properties of self-gravitating structures can also be analyzed by using the concept of complexity factor. Usually, the components which make the system more complex are pressure, heat flux and energy density etc. Without considering the pressure component in the stress energy tensor, only the energy density is not sufficient to study complexity of the system.

The nature of various physical features has usually been tested using cylindrical structures at different scales. The gravitational collapse, its radiation, spinning celestial objects and rotating fluids are especially in astrophysics, which inspire to take cylindrical symmetry into account. The Birkhoff’s principle states that there is a vacuum outside a spherical symmetric object and thus spherical fluid collapse induces no gravitational waves. That is why one switches to another basic symmetry, namely cylindrical geometry. For cylindrically symmetrical star, Einstein and Rosen [35] found the solutions for gravitational waves by assuming weak gravitational fields and also claimed that in the Minkowski space, such problem reduce to regular cylindrical waves. Regarding cylindrical symmetric propagation, several astrophysical aspects have been explored. In the collapse of cylindrical star, Herrera and Santos [36] investigated the conditions for smooth matching of inner and outer manifold and justified the radial pressure on the boundary not to be zero. But later, Herrera et al. [37] found some mistakes in calculations and claimed that the correction of those mistakes will give zero radial pressure on the boundary in a result. There has been some interesting results about the evolution and stability of relativistic systems [38], [39], [40], [41], [42], [43], [44], [45].

Herrera et al. [46] found the matching conditions for static self-gravitating cylindrical object with the Levi-Civita vacuum geometry as well as equations of motion and observed their regularity. They also shown that an incompressible fluid can be represented by a set of conformally flat solutions. Sharif and Butt [47] described the complexity factor for the static cylindrical system in GR. The expansion free condition has been investigated for anisotropic cylindrical fluid distribution by Yousaf and Bhatti [48], which produces vacuum cavity inside the fluid distribution. Recently, Herrera [49], [50] and Yousaf et al. [51], [52] described the importance of different congruences of observers for describing the very different physical phenomena of the same geometry of fluids.

The paper is listed as below. We propose some new physical parameters and field equations for $f(R,T,Q)$ gravitational theory in the next section. After this, we find four scalar functions from the Riemann tensor and claim one of them as complexity factor in Section 3. In Section 4, we design the condition for disappearing complexity factor and also provide some exact solutions of field equations in the background of $f(R,T,Q)$ theory. Finally, all these results are concluded with the effects of modified corrections in Section 5.