Elsevier

Annals of Physics

Volume 451, April 2023, 169240
Annals of Physics

Study of decoupled gravastars in energy–momentum squared gravity

https://doi.org/10.1016/j.aop.2023.169240Get rights and content

Highlights

  • We generate an exact anisotropic gravastar model by gravitational decoupling method.

  • Developed stellar model satisfies essential features of a physically acceptable model.

  • All the energy bounds are satisfied except strong energy condition inside the ultra-compact stellar structure.

Abstract

In this paper, we generate an exact anisotropic gravastar model using gravitational decoupling technique through minimal geometric deformation in the framework of f(,T2) gravity. This novel model explains an ultra-compact stellar configuration whose internal region is smoothly matched to the exterior region. The developed stellar model satisfies some of the essential characteristics of a physically acceptable model such as a positive monotonically decreasing profile of energy density from the center to the boundary and monotonically decreasing behavior of the pressure. The anisotropic factor and Schwarzschild spacetime follows physically acceptable behavior. We find that all the energy bounds are satisfied except strong energy condition inside the ultra-compact stellar structure for the coupling constant of this theory, which is compatible with the regularity condition.

Introduction

The current accelerated expansion of the cosmos has been the most significant development in recent years. This expansion is thought to be the result of a mysterious force having repulsive nature and is described as dark energy (DE). Several researchers have put forward their efforts to reveal its unknown features. The first proposal explaining the hidden characteristics of DE is the cosmological constant, but it suffers issues like fine-tuning and coincidence. Different modified versions have been presented to overcome these problems that are assumed to be fascinating approaches in revealing mysteries of the universe. The first modification is f() theory of gravity which has a significant literature [1] to understand its physical properties. This modified theory was further generalized by incorporating the idea of curvature–matter coupling. Such coupling theories are non-conserved with an additional force that alters the path of the particle. The minimal coupling is introduced as f(,T) theory [2] whereas the non-minimally coupled theory is f(,T,σγTσγ) gravity [3].

Different cosmic studies have been presented to understand the beginning of the universe. One of the extensively accepted proposals termed as big-bang theory is a remarkable framework describing evolutionary processes. According to this idea, all the matter in the cosmos expanded from a single point, referred to as a singularity. This theory portrays the beginning of the universe but this proposal suffers flatness problem, horizon problem and monopole problem. The research community is always curious and dedicates its effort to find answers to cosmological issues. The fascinating notion of bounce theory (based on repeated cosmic expanding and contracting behavior) serves as a backbone for introducing a new theory. This newly introduced concept resolves big bang issues by resolving singularity and provides a better explanation of accelerated expanding cosmic behavior.

Katrici and Kavuk [4] constructed an extension of f() theory that incorporates the concept of bounce theory. They developed a particular coupling between matter and gravity through self-contraction of the energy–momentum tensor (EMT) referred to as energy–momentum squared gravity (EMSG) or known as f(,T2) theory, where T2=TσγTσγ. This proposal with a minimum scale factor as well as finite maximum energy density is considered to be a suitable framework to overcome big-bang problem. The cosmological constant resolves the big-bang singularity by providing the repulsive force in the background of this theory. This theory follows the true sequence of cosmological eras and effectively describes cosmic behavior. The field equations involve squared and product components of matter variables, which are useful in studying different cosmological scenarios.

Roshan and Shojai [5] computed the exact solution by solving EMSG field equations with homogeneous isotropic spacetime and demonstrated the possibility of a bounce in the early universe. Board and Barrow [6] calculated the range of exact solutions for an isotropic expanding universe in reference to early and late-time evolution. Nari and Roshan [7] determined a connection between mass and the radius of neutron stars. They found that a smaller or larger value of mass is governed by the central pressure of these stars and the value of model parameter of this gravity. Bahamonde et al. [8] used different coupling models to study the expansion of the universe and concluded that these models help in understanding the present accelerated cosmic expansion. We have studied various attributes of charged as well as uncharged gravastar solutions [9]. Recently, Sharif and his collaborators studied decoupled solutions [10] as well as the impact of charge on complexity of static sphere in the same theory [11].

Different cosmic phenomena, such as the origin and evolution of celestial bodies have captivated the interest of various researchers. Among all the cosmic objects, stars are the core components of galaxies, which are organized systematically in a cosmic web. When a star runs out of fuel, its outward pressure vanishes, leading to gravitational collapse and hence compact objects are formed. A black hole is such a stellar remnant which is a totally collapsed entity with a singularity hidden behind an event horizon. Mazur and Motolla [12] developed a compact model (gravastar) as an alternative to black hole to avoid singularity and event horizon. Motolla [13] discussed in detail dark energy and condensate stars. The resulting gravitational condensate star configuration resolves all black hole paradoxes, and provides a testable alternative to black holes as the final state of complete gravitational collapse. Mazur and Motolla [14] studied pressure and negative pressure interior of a non-singular black hole. In contrast to black holes, the significant feature of this hypothetical object is its singularity-free nature. Three regions constitute the complete gravastar structure, internal and external regions separated by a thin shell. The DE in the inner domain causes a repulsive force which contributes as a main barrier to overcome singularity formation. The intermediate shell surrounding the inner boundary exerts an inward force and thus hydrostatic equilibrium is maintained while the Schwarzschild metric characterizes the exterior region. Moreover, each region is described by a specific equation of state (EoS).

Visser and Wiltshire [15] examined the stability of gravastars towards radial perturbations and concluded that a viable EoS results in the stability of gravastar in GR. This work was extended by examining appropriate constraints for the stability of gravastar solutions [16]. Cattoen et al. [17] studied the usual gravastar structures and concluded that the cosmic configuration has anisotropy in the absence of intermediate shell. Bilic et al. [18] formulated gravastar solutions by replacing the inner Born–Infeld phantom metric with the de Sitter geometry which represent compact objects at the galactic center. Horvat and Ilijić [19] discussed the stability of gravastars by employing the speed of sound criteria on thin-shell to determine compactness bounds. Ovalle [20] discussed anisotropic gravastars by means of decoupling approach.

Many researchers have also investigated the formation as well as fundamental physical characteristics of gravastars in modified theories of gravity. Das et al. [21] discussed isotropic gravastar structure in the realm of f(,T) gravity. They obtained a linear profile of physical features in relation to shell thickness. Shamir and Ahmad [22] constructed spherically symmetric gravastar models whose characteristics obey an increasing trend with respect to the thickness. Different physical features of charged/uncharged gravastar model were studied in f(T) theory (f(T) is an arbitrary function of the torsion scalar) [23]. Abbas and Majeed [24] discussed isotropic gravastar structure in the background of Rastall gravity. Yousaf et al. [25] examined the gravastar model in modified theory and found that length, energy and entropy present an increasing trend with respect to thickness. In the background of f(,G) gravity (G defines the Gauss–Bonnet invariant), Bhatti et al. [26] investigated different physical characteristics related to intrinsic shell thickness and found accepted behavior. Ray et al. [27] provided a very good review on the entire aspects of gravastar as envisioned by Mazur and Mottola starting from the black hole physics to its present state and enlightening the future works. The gravastar models in a number of modified gravity models starting from f(R,T) to Rastall–Rainbow gravity have been reviewed. Bhatti and his collaborators [28] discussed charged/uncharged gravastar model in f(G) gravity to analyze different features. Bhar and Rej [29] studied the role of electromagnetic field in the stability of gravastar structure.

Cosmological solutions are very important to comprehend the structural properties as well as the mechanism of celestial structures. However, it is usually difficult to obtain exact solutions due to the presence of highly non-linear terms in the field equations. In this regard, the gravitational decoupling technique by minimal geometric deformation (MGD) has recently been developed to determine the astrophysical and cosmic solutions. Ovalle [30] firstly introduced this concept from the viewpoint of braneworld. This approach connects a new gravitational source to the EMT of the seed matter distribution through a non-dimensional parameter. In this scheme, the deformation is employed only on the radial metric function while leaving the temporal metric component unaltered. Consequently, two sets of nonlinear field equations are formed, one for the new source and the other for the seed source. The decoupling method has extensively been used to transform isotropic spherical solutions into anisotropic ones.

The isotropic, anisotropic, and charged fluid configurations are the key topics for researchers in the context of compact celestial bodies. Ovalle et al. [31] extended isotropic solutions by assuming Tolman IV spacetime in the inner region. Ovalle and his collaborators [32] investigated ultra-compact stars through decoupling technique without altering any of the fundamental property. Gabbanelli et al. [33] computed acceptable anisotropic solutions by considering the Durgapal–Fuloria stellar configuration in the interior region. Graterol [34] implemented this technique to determine anisotropic solutions by considering Buchdahl isotropic solution. The extensions of isotropic solutions were obtained by using Krori–Barua (KB) spacetime [35]. Morales and Tello-Ortiz [36] studied charged Heintzmann anisotropic solution and graphically examined matter variables for various stars. Maurya et al. [37] explored the influence of anisotropy by taking Korkina–Orlyanskii isotropic matter distribution.

Many viable and stable solutions have been studied by using the decoupling technique in modified theories. Sharif and Majid [38] obtained anisotropic solutions by using different well-known isotropic solutions in Brans–Dicke theory. Maurya et al. [39] explored anisotropic solutions by assuming Korkina–Orlyanskii spacetime in the inner domain in f(,T) theory. Zubair and Azmat [40] extended Tolman VII solution in the same theory. Sharif and Naseer [41] evaluated charged/uncharged anisotropic extensions of KB metric in non-minimal curvature–matter coupled theory. Recently, Azmat et al. [42] employed this technique to investigate physical features of anisotropic gravastar in f(,T) theory. The departure from the isotropic scenario leads to an interesting situation, giving rise to an intriguing phenomenon inside the stellar interior, i.e., the system experiences a repulsive force that counteracts the gravitational gradient, which allows the construction of more compact and massive objects. This inspired us to develop anisotropic gravastars in the background of f(,T2) providing interesting results.

The aim of this article is to study the anisotropic gravastar in the background of f(,T2) theory. We use gravitational decoupling through MGD scheme to include the impact of anisotropy in the fluid configuration of the gravastar. The paper is organized as follows. Section 2 demonstrates the complete array of the field equations corresponding to both sources (seed and source). The solutions are decoupled in Section 3 which are smoothly joined at the stellar surface. Section 4 describes MGD gravastar configuration which meets some of the viable characteristics of a stellar structure. We summarize our results in the last section.

Section snippets

Basic formalism

The action of this theory involving matter Lagrangian density (Łm) and the Lagrangian density (Łτ) of an additional source (τ) is characterized by S=d4x12k2f(,T2)+Łm+Łτg.The coupling constant is k2=8π, determinant of the metric tensor is denoted by g and symbolizes the decoupling parameter. The action (1) provides the following field equations σγf+gσγfσγf12gσγf=8πTσγ+τσγfT2σγ,here f=f and fT2=fT2. The self contraction of del operator defines d’Alembert operator (=σσ),

Minimal geometric decoupling scheme

In this section, the solution of the field equations are discussed through MGD approach. To encode the influence of extra source in isotropic matter distribution, the metric potentials (ξ and ϖ) are modified in the following manner ξφ=ξ+j(r),ϖeψ=ϖ+ς,where j(r) and ς(r) present temporal and radial deformations, respectively. Employing MGD scheme (alters only radial function through geometric deformation while keeping the temporal metric function unchanged, i.e., j(r)=0), we have two sets of

MGD gravastars in EMSG

Following Mazur and Mottola gravastar model [12], we use the EoS, Wϱ=P with W describing the EoS parameter. The inner gravastar domain follows DE EoS ϱ=P (for W=1). The negative pressure in this region provides repulsive force. Employing DE EoS along with Eq. (14) yields ϱ=ϱ0(constant) which gives ϖ=12ρ0r23(4π+ζϱ0)+C1r,In order to avoid singularity, we take the arbitrary constant C1 to be zero at core. Thus we have ϖ=12ϱ0r23(4π+ζϱ0).The temporal and radial metric functions are connected as

Concluding remarks

In this paper, we have constructed anisotropic version of the gravastar in EMSG by employing gravitational decoupling technique. The system of field equations is separated into two different arrays: one belongs to the standard version of f(,T2) equations, whereas the other involves an additional source. A gravastar solution describing the ultra-compact structure with isotropic matter distribution is used to evaluate the first system. The second set involves four unknowns, i.e., three extra

CRediT authorship contribution statement

M. Sharif: Proposed the problem and finalized the manuscript. Saba Naz: Did the calculations and prepared the draft.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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