Prepared for submission to JCAP Study of Gravastars in Rastall Gravity Shounak Ghosh,a Sagar Dey b Amit Das,a Anirban Chanda,b and B C Paul.c;1 aDepartment of Physics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India bDepartment of Physics, University of North Bengal, West Bengal 734014, India. cIUCAA Centre for Astronomy Research and Development (ICARD) and Department of Physics, University of North Bengal, West Bengal 734014, India. E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract. Gravastars have been considered as a feasible alternative to black holes in the past couple of decades. Stable models of gravastar have been studied in many of the alternative gravity theories besides standard General Relativity (GR). The Rastall theory of gravity is a popular alternative to GR, specially in the cosmological and astrophysical context. Here, we propose a stellar model under the Rastall gravity following Mazur-Mottola’s [1,2] conjecture. The gravastar consists of three regions, viz., (I) Interior region, (II) Intermediate shell region, and (III) Exterior region. The pressure within the interior core region is assumed with a constant negative matter-energy density which provides a repulsive force over the entire thin shell region. The shell is assumed to be made up of fluid of ultrarelativistic plasma which follows the Zel’dovich’s conjecture of stiff fluid [3,4]. It is also assumed that the pressure is proportional to the matter-energy density according to Zel’dovich’s conjecture, which cancel the repulsive force exerted by the interior region. The exterior region is completely vacuum which is described by the Schwarzschild-de Sitter solution. Under all these specifications we obtain a set of exact and singularity-free solutions of the gravastar model presenting several physically valid features within the framework of Rastall gravity. The physical properties arXiv:2102.01524v1 [gr-qc] 1 Feb 2021 of the shell region namely, the energy density, proper length, total energy and entropy are explored. The stability of the gravastar model is investigated using the surface redshift against the shell thickness and maximizing the entropy of the shell within the framework of Rastall gravity. Keywords: Gravastars, modified gravity, Rastall gravity 1Corresponding author Contents 1 Introduction1 2 The Field Equations geometry in Rastall theory3 3 The gravastar models5 3.1 Interior spacetime5 3.2 Intermediate thin shell6 3.3 Exterior spacetime6 4 Boundary Condition7 5 Physical features of the model7 5.1 Pressure and Matter density8 5.2 Proper length8 5.3 Energy 9 5.4 Entropy 10 6 Junction Condition 11 7 Stability 13 7.1 Surface Redshift 13 7.2 Entropy maximization 13 8 Discussions and Conclusions 15 8.1 Possibility of detection of gravastar: 17 9 Acknowledgements 17 1 Introduction The end point of the stellar evolution of a massive object always attract the attention of the astrophysicists. It is conjectured that in the final stage of stellar evolution if the mass of the stellar remnant after forming planetary nebula or initiating supernova explosion is greater than three solar mass, then the collapsing phase of the star due its self gravitational pull leads to a highly dense object, i.e., black hole. In 1916, Schwarzschild obtained the simplest black hole solution from the Einstein field equation in vacuum which is considered as the im- portant solution of a classical black hole (static as well as uncharged). But the Schwarzschild solution has some shortcomings due to (i) presence of singularity inside the black hole, (ii) existence of event horizon which leads to major unsolved issues. Mazur and Mottola [1,2] first proposed a new idea of collapsing stellar object by considering an extended idea of Bose Einstein Condensation (BEC) in the gravitating system. In the model it was proposed that one can construct a cold, dark and compact object, they nomenclature it as a Gravitational Vacuum Condensed Star or Gravastar. The model of gravastar satisfies all the theoretical criterion for a stable end point of the stellar evolution and provides solution to the problems of classical black hole. After the successful detection of gravitation wave (GW) in 2015 [5], it – 1 – is assumed that the GWs are originated due to merging of two massive black holes. As the observational signals does not confirm the recovery of the basic problems of black holes, it demands an alternative approach to describe the shortcomings. In this context, the gravastar may play an important role to describe the final state of the stellar evolution. Though there are no sufficient observational evidences in favour of gravastars directly for their existence, it is important to study the concept of gravastar that can be claimed as a viable alternative to resolve the conceptual issues in understanding the black holes. The gravastar consists of three regions namely interior region, intermediate thin shell, and exterior region. As proposed by Mazur and Mottola [1,2] the interior region of the gravastar being in de Sitter condensate phase is assumed to be filled with vacuum energy, the exterior region is completely vacuum known as Schwarzschild vacuum, and those two regions are separated by a thin shell of ultra-relativistic matter of very high density. The barotropic equation of state (EOS) is p = !ρ, where p and ρ are the pressure and matter-energy density respectively and ! is the EOS parameter. In a gravastar the EOS parameter !, is different in different regions as follows: (I) Interior Region : 0 ≤ r < R1, ! = −1, i:e:, EOS is p = −ρ , (II) Thin Shell Region: R1 < r < R2, ! = 1, i:e:, EOS is p = ρ , (III) Exterior Region : R2 < r, ! = 0, i:e:, EOS is p = ρ = 0 . where R1 and R2 represent the radii of the interior and exterior boundary of the gravastar respectively. Therefore, thickness of the intermediate shell region is given by R2 − R1 = (say), where 1 because in a gravastar the thickness of the shell is very small compared to its size. The idea of gravastar is discussed in the literature [6–31] as an alternative to BH. Mazur and Mottola proposed a three layer structure of the gravastar and studied the thermody- namical stability employing the entropy maximization technique, whereas Visser [6] showed the dynamical stability of the system. Based on the three layer system it is now established that the theoretical acceptance of gravastar model in both lower and higher dimensions with or without charge is important in astrophysics. Usmani et al. [17] shown the possibility of existence of higher dimensional gravastar, and later on Bhar [24] and Ghosh et al. [26] studied the higher dimensional gravastar with and without conformal motion respectively. The possibility of gravastar in lower dimension is also studied [28, 29]. Chan and Silva [30] have investigated the effect of charge on the stability of the gravastar formation considering a charged shell configuration. Again, considering the Vaidya exterior spacetime, Chan et al: [31] have studied the dynamical models of prototype radiating gravastars and showed that the final output may corresponds to a number of objects, a black hole, an unstable gravastar, a stable gravastar or a “bounded excursion” gravastar depending on the mass of the shell that evolves in time, the cosmological constant, and the initial position of the dynamical shell. All of these works are mainly carried out in the framework of Einstein’s general theory of relativ- ity (GR). Though it is well known that Einstein’s GR is one of the most promising theory for revealing most of the hidden mysteries of the Nature it is realized that a modification of the GR is inevitable to wipe out shortcomings in the theoretical as well as observational aspects. The confirmation of accelerated universe based on some observational evidences as well as the existence of dark matter and dark energy led to a theoretical challenge to GR [32–42]. To accommodate the present acceleration of the universe a number of alternative theories of gravity, namely f(R) gravity, f(T) gravity, f(R; T ) gravity, f(T; T ) gravity are developed successively with a modification either in the geometrical part or in the matter-energy part – 2 – of Einstein field equations. In most of these theories different functions of the Ricci scalar with or without other scalar quantities are incorporated in the gravitational Lagrangian of the corresponding Einstein-Hilbert action. A lot of works have been performed on compact stars as well as gravastars under the framework of modified theories of gravity as available in the literature [43–50]. But, in most these theories the energy-momentum tensor (EMT) is not generally conserved unlike the case of general relativity. This interesting aspect of modified gravity motivates us to make a general study of gravastar under the Rastall theory of gravity as proposed by Rastall in 1972 [51, 52] based on the idea that the usual law of conservation ν of energy-momentum tensor, i.e., rνTµ = 0 is violated when a curved spacetime is taken into account. The violation of conservation law of EMT is the key point of this modification where the covariant derivative of EMT is directly proportional to the divergence of scalar ν curvature. Mathematically, it can be written as rνTµ /R; µ, which further takes the form ν rνTµ = R; µ, where is the constant of proportionality. Various cosmological as well as astrophysical aspects have been investigated under the framework of Rastall theory due its numerous appealing features and simpler form of field equations as available in the literatures [53–62]. Recently, in astrophysical context Rastall gravity has been extensively investigated in the context of rotating black holes [58], Kerr- Newman-AdS black hole [59], non commutative geometry [60].