Impact of Pressure Anisotropy on a Compact Stellar Model in General Relativity

In this article, we introduce a new compact stellar model by deriving simple, exact analytical solutions to the Einstein field equations in the presence of anisotropy. We focus on an anisotropic star, employing the quasi-local method suggested by Horvat et al. (Class Quantum Gravity 28: 025009, 2011) to describe fluid anisotropy within a spherical symmetry using quasi-local variables, whose values are determined from physics within a very small region around a spacetime point. We then ensure smooth matching between the interior spherically symmetric spacetime and the exterior Schwarzschild spacetime. To confirm the physical validity of our model, we analyze various parameters both analytically and graphically. To further support our solution’s applicability to compact stellar objects, we use observational data from the well-known pulsar 4U1608-52 in our graphical analysis.

In this paper, we develop a new relativistic compact stellar model for a spherically symmetric anisotropic matter distribution. The model has been obtained through generating a new class of solutions by invoking the Tolman ansatz for one of the metric potentials [Formula: see text] and a physically reasonable selective profile of radial pressure. We have matched our obtained interior solution to the Schwarzschild exterior spacetime over the bounding surface of the compact star. These matching conditions together with the condition of vanishing the radial pressure across the boundary of the star have been utilized to determine the model parameters. We have shown that the central pressure of the star depends on the parameter [Formula: see text]. We have estimated the range of [Formula: see text] by using the recent data of compact stars 4U 1608-52 and Vela X-1. The effect of [Formula: see text] on different physical parameters, e.g. pressure anisotropy, the subliminal velocity of sound, relativistic adiabatic index, etc., has also been discussed. The developed model of the compact star is elaborately discussed both analytically and graphically to justify that it satisfies all the criteria demanded by a realistic star. From our analysis, we have shown that the effect of anisotropy becomes small for higher values of [Formula: see text]. The mass–radius (MR) relationship which indicates the maximum mass admissible for observed pulsars for a given surface density has also been investigated in our model. Moreover, the variation of radius and mass with central density has been shown which allows us to estimate central density for a given radius (or mass) of a compact star. We have explored the tidal deformabilities by calculating the Love numbers and showed the variation of tidal Love numbers with the central pressure of a star.

A new model of anisotropic compact star is obtained in our present paper by assuming the pressure anisotropy. The proposed model is singularity free. The model is obtained by considering a physically reasonable choice for the metric potential $g_{rr}$ which depends on a dimensionless parameter `n'. The effect of $n$ is discussed numerically, analytically and through plotting. We have concentrated a wide range for n ($10\leq n \leq 1000$) for drawing the profiles of different physical parameters. The maximum allowable mass for different values of $n$ have been obtained by M-R plot. We have checked that the stability of the model is increased for larger value of $n$. For the viability of the model we have considered two compact stars PSR J1614-2230 and EXO 1785-248. We have shown that the expressions for the anisotropy factor and the metric component may serve as generating functions for uncharged stellar models in the context of the general theory of relativity.

Seven new solutions to the interior static and spherically symmetric Einstein's field equations (EFE) are found and investigated. These new solutions are a generalisation of the quadratic density fall-off profile of the Tolman VII solution. The generalisation involves the addition of anisotropic pressures and electric charge to the density profile. Of these new solutions three are found to obey all the necessary conditions of physical acceptability, including linear stability under radial perturbations, and causality of the speed of pressure waves inside the object. Additionally an equation of state can be found for all the physically viable solutions. The generalised pulsation equation for interior solutions to the EFE that include both electric charge and pressure anisotropy is derived and used to determine the stability of the solutions. However the pulsation equation found is general and can be used for all new solutions that contain these ingredients.

In this present work, we have obtained a singularity-free spherically symmetric stellar model with anisotropic pressure in the background of Einstein’s general theory of relativity. The Einstein’s field equations have been solved by exploiting Tolman ansatz [Richard C Tolman, Phys. Rev. 55:364, 1939] in (3+1)-dimensional space-time. Using observed values of mass and radius of the compact star PSR J1903+327, we have calculated the numerical values of all the constants from the boundary conditions. All the physical characteristics of the proposed model have been discussed both analytically and graphically. The new exact solution satisfies all the physical criteria for a realistic compact star. The matter variables are regular and well behaved throughout the stellar structure. Constraints on model parameters have been obtained. All the energy conditions are verified with the help of graphical representation. The stability condition of the present model has been described through different testings.

We report a generalization of our earlier formalism [Pramana, 54, 663 (1998)] to obtain exact solutions of Einstein-Maxwell’s equations for static spheres filled with a charged fluid having anisotropic pressure and of null conductivity. Defining new variables: w=(4π/3)(ρ+e)r2, u=4πξr2, vr=4πprr2, v⊥=4πp⊥r2[ρ, ξ(=−(1/2)F14F14), pr, p⊥ being respectively the energy densities of matter and electrostatic fields, radial and transverse fluid pressures whereas e denotes the eigenvalue of the conformal Weyl tensor and interpreted as the energy density of the free gravitational field], we have recast Einstein’s field equations into a form easy to integrate. Since the system is underdetermined we make the following assumptions to solve the field equations (i) u=vr=(a2/2κ)rn+2, v⊥=k1vr, w=k2vr; a2, n(>0), k1, k2 being constants with κ=((k1+2)/3+k2) and (ii) w+u=(b2/2)rn+2, u=vr, v⊥−vr=k, with b and k as constants. In both cases the field equations are integrated completely. The first solution is regular in the metric as well as physical variables for all values of n>0. Even though the second solution contains terms like k/r2 since Q(0)=0 it is argued that the pressure anisotropy, caused by the electric flux near the centre, can be made to vanish reducing it to the generalized Cooperstock-de la Cruz solution given in [14]. The interior solutions are shown to match with the exterior Reissner-Nordstrom solution over a fixed boundary.

In the present investigation, compact stellar models are dealt with in the framework of the modified gravity theory, specifically of [Formula: see text] type. We have considered that the compact objects are following a spherically symmetric static metric and obtained the Einstein field equations in the spacetime of [Formula: see text]. To make the Einstein equations solvable, we employ the methodology of conformal Killing vectors. Thereafter by using the MIT bag equation of state to the compact stars, considering that the stars are formed by strange quark, we find the solutions set. The solutions are examined via several physical testings which exhibit viability of the model.

We have modelled a neutral non-homogeneous anisotropic stellar compact object with two distinct equations of state in general relativity framework. We have considered the macroscopic features of a general relativistic gravitating compact object. The equation of state is quadratic in the core and linear in the envelope. There is smooth matching between the core, envelope and the vacuum exterior regions. We found the masses, radii and compactness of some compact objects such PSR J1614-2230, PSR J1903+0327, Vela X-1, SMC-X-1, Cen X-3; which are in agreement with previous investigations. The gravitational potentials and the matter variables are well behaved throughout the stellar structure. We present in particular the variation of the radius in the core and the envelope of the star by changing some parameters values. Physical features of the pulsars PSR J1614-2230 are presented in more details. It observed that the radial pressure in the core is higher than the radial pressure in the envelope. The investigation reveals that the model is physically relevant for the study of observed compact stars.

We make use of the condition of vanishing complexity, based on the current definition proposed by Herrera (Phys Rev D 97:044010, 2018), to find exact interior solutions to the Einstein equations for describing compact stellar objects. In the framework of general relativity, the complexity factor is an outcome of the orthogonal splitting of the Riemann tensor from which structure scalars are obtained. By using the Vaidya–Tikekar (V–T) metric ansatz (J Astrophys Astron 3:325, 1982) for the spacetime of a static spherically symmetric matter distribution, we model superdense, relativistic stars. The interior spacetime is matched to the exterior Schwarzschild solution across the boundary of the star where the radial pressure vanishes. The physical viability of the model has been tested following the current data corresponding to the pulsar 4U1820-30\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ 4U 1820-30 $$\\end{document}. The stability of the model fulfilled the given criteria, namely the Tolman–Oppenheimer–Volkoff equation, the adiabatic index and the causality conditions.

The main aim of this study is to examine the behavior of physical parameters of an anisotropic compact star model demonstrating spherical symmetry in [Formula: see text] modified gravity. To evaluate the behavior and the stability of an anisotropic compact star model, we utilize the measured mass and radius of an anisotropic compact star model. This study obtained an anisotropic compact star model by solving Einstein’s field equations. The field equations have been simplified by an appropriate selection of the metric element [Formula: see text] and the Karmarkar condition. We solve field equation to develop a differential equation that establishes a relationship between two essential components of spacetime, namely [Formula: see text] and [Formula: see text]. A physical analysis of this model reveals that the resulting stellar structure for anisotropic matter distribution is a physically plausible representation of a compact star with an energy density of order [Formula: see text]. Using the Tolman–Oppenheimer–Volkoff equation, causality condition and Harrison–Zeldovich–Novikov Condition, we investigate the hydrostatic equilibrium and stability of the compact star Cen X-3. We further determined the mass–radius relation of this compact star for different values of [Formula: see text].

In this study, we address the issue of a spherically symmetrical interior solution to the quadratic form of f(T)=T+ϵT2 gravitational theory using a physical tetrad that provides vanishing components of the off-components of the field equation, in contrast to what exists in the current literature. To be able to formulate the resulting differential equation in a closed form, we employ the Krori-Barua (KB) ansatz. Using the KB spacetime form, we derive the analytic form of the energy-density, radial, and tangential pressures and the anisotropic form. All of these quantities are affected by the dimensional parameter ϵ, which causes them to have a noted difference from those given in the frame of Einstein general relativity. The derived model of this study exhibits a non-trivial form of torsion scalar, and it also contains three constants that we drew from the matching of the boundary condition with a line element that also features a non-trivial form of torsion scalar. Having established the physical conditions that are needed for any real stellar, we check our model and show in detail that it bypasses all of these. Finally, we analyze the model's stability utilizing the Tolman-Oppenheimer-Volkoff equation and adiabatic index and show that our model satisfies these.

In this paper, we consider wormhole geometries in the context of teleparallel equivalent of general relativity (TEGR) as well as $f(T)$ gravity. The TEGR is an alternative geometrical formulation of Einstein's general relativity, where modified teleparallel gravity or $f(T)$ gravity has been invoked as an alternative approach for explaining an accelerated expansion of the universe. We present the analytical solutions under the assumption of spherical symmetry and the existence of a conformal Killing vectors to proceed a more systematic approach in searching for exact wormhole solutions. More preciously, the existence of a conformal symmetry places restrictions on the model. Considering the field equations with a diagonal tetrad and anisotropic distribution of the fluid, we study the properties of traversable wormholes in TEGR that violates the weak and the null energy conditions at the throat and its vicinity. In the second part, wormhole solutions are constructed in the framework of $f(T)$ gravity, where $T$ represents torsion scalar. As a consistency check, we also discuss the behavior of energy conditions with a viable power-law $f(T)$ model and the corresponding shape functions. In addition, a wide variety of solutions are deduced by considering a linear equation of state relating the density and pressure, for the isotropic and anisotropic pressure, independently of the shape functions, and various phantom wormhole geometries are explored.

Is Rastall's theory of gravity equivalent to Einstein's general relativity theory? This question has sparked a significant debate, prompting researchers to delve into the topic. To investigate further, we apply Rastall's theory's field equation to a spacetime characterized by spherical symmetry. This leads us to encounter a system of non-linear differential equations that is overdetermined. To address this, we make assumptions about the specific form of the metric potential's temporal component, denoted as gtt. Additionally, we impose constraints to eliminate the anisotropic condition, resulting in a vanishing effect. These steps allow us to determine the form of grr and ultimately achieve an isotropic spacetime. Furthermore, our investigation focuses on the potential of obtaining a set of parameters that align with the observed behavior of pulsars. To achieve this, we employ junction conditions to match the interior spacetime with the exterior Schwarzschild configuration, thereby constraining the model's relevant constants. Subsequently, we employ the pulsar SAXJ1748.9−2021, characterized by a measured mass of M=1.81±0.31,M⊙ and a radius of R=11.7±1.7 km, to numerically explore the physical properties of the model. Stability is assessed using the Tolman-Oppenheimer-Volkoff equation and the adiabatic index. Our findings suggest that Rastall's parameter, a key distinction of Rastall's theory from Einstein's general relativity, can play a crucial role in forming a realistic, compact object consistent with observational data. Furthermore, we verify the model's validity by comparing it with various observed masses and radii of different pulsars, ensuring a satisfactory fit between the model proposed in this study and the observed data.

We investigated Rastall gravity, for an anisotropic star with a static spherical symmetry, whereas the matter-geometry coupling as assumed in Rastall Theory (RT) is expected to play a crucial role in differentiating RT from General Relativity (GR). Indeed, all the obtained results confirm that RT is not equivalent to GR, however, it produces the same amount of anisotropy as GR for static spherically symmetric stellar models. We used the observational constraints on the mass and the radius of the pulsar \textit{Her X-1} to determine the model parameters confirming the physical viability of the model. We found that the matter-geometry coupling in RT allows slightly less size than GR for a given mass. We confirmed the model viability via other twenty pulsars' observations. Utilizing the strong energy condition we determined an upper bound on compactness $U_\text{max}\sim 0.603$, in agreement with the Buchdahl limit, whereas Rastall parameter $ε=-0.1$. For a surface density compatible with a neutron core at nuclear saturation density, the mass-radius curve allows masses up to $3.53 M_\odot$. We note that there is no equation of state is assumed, however, the model fits well with a linear behavior. We split the twenty pulsars into four groups according to the boundary densities. Three groups are compatible with neutron cores while one group fits perfectly with higher boundary density $8\times 10^{14}$ g/cm$^3$ which suggests that those pulsars may have quark-gluon cores.

We investigated Rastall gravity, for an anisotropic star with a static spherical symmetry, whereas the matter-geometry coupling as assumed in Rastall Theory (RT) is expected to play a crucial role differentiating RT from General Relativity (GR). Indeed, all the obtained results confirm that RT is not equivalent to GR, however, it produces same amount of anisotropy as GR for static spherically symmetric stellar models. We used the observational constraints on the mass and the radius of the pulsar Her X-1 to determine the model parameters confirming the physical viability of the model. We found that the matter-geometry coupling in RT allows slightly less size than GR for a given mass. We confirmed the model viability via other twenty pulsars’ observations. Utilizing the strong energy condition we determined an upper bound on compactness U_text {max}sim 0.603, in agreement with Buchdahl limit, whereas Rastall parameter epsilon =-0.1. For a surface density compatible with a neutron core at nuclear saturation density the mass-radius curve allows masses up to 3.53 M_odot . We note that there is no equation of state is assumed, however the model fits well with linear behaviour. We split the twenty pulsars into four groups according to the boundary densities. Three groups are compatible with neutron cores while one group fits perfectly with higher boundary density 8times 10^{14}hbox {g/cm}^3 which suggests that those pulsars may have quark-gluon cores.