Solutions Of Einstein Field Equations Research Articles

Study of compact objects: a new analytical stellar model

In this paper, we obtain analytical solutions of Einstein field equations for a spherically symmetric anisotropic matter distributions. For this purpose physically meaningful metric potential corresponds to grr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g_{rr}$$\\end{document} and a particular choice of the anisotropy has been utilized to obtain the solutions in closed form. This class of solution has been used to characterized observed pulsars from different aspects. Smooth matching of interior spacetime metric with the exterior Schwarzschild metric and in addition with the condition of vanishing radial pressure across the boundary leads us to determine the model parameters. 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}$$4U1820-30$$\\end{document} with its current estimated data for mass and radius (Mass =1.58M⊙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$=1.58 M_\\odot $$\\end{document} and radius =9.1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$=9.1$$\\end{document} km) has been allowed for testing the physical acceptability of our developed model. We have graphically analyzed the gross physical features of the observed pulsar. The stability of the model is also discussed under the conditions of causality, adiabatic index and generalized Tolman–Oppenheimer–Volkov (TOV) equation under the forces acting on the system. Few more pulsars with their have been considered, to show that this model is compatible with observational data, and all the requirements of a realistic star are highlighted. Mass-radius (M–R) relationship have been generated for our model. The impact of anisotropy on the gross physical features of stars have been explored with the graphical presentation.

Statefinder Diagnostic of Interacting Dark Fluid in Bianchi Type-IX Universe

In this paper, we deal with spatially homogeneous and anisotropic Bianchi types-IX and Universe filled with Holographic dark energy and cold dark matter. Assuming the condition that the shear scalar is proportional to expansion scalar, we have obtained solutions of Einstein field equations. The Statefinder diagnostic pair i.e. {r‚s} is adopted to distinguish our dark energy models from other dark energy models. Some important physical features of the models have been discussed.

On curvature related geometric properties of Hayward black hole spacetime

This paper is devoted to the study of curvature properties of Hayward black hole (briefly, HBH) spacetime, which is a solution of Einstein field equations (briefly, EFE) having non-vanishing cosmological constant. We have proved that the HBH spacetime is an Einstein manifold of level 2, 2-quasi Einstein, generalized quasi-Einstein and Roter type manifold. Also, it is shown that the nature of the HBH spacetime is pseudosymmetric and it obeys several types of pseudosymmetries, such as, pseudosymmetry due to concircular, conformal and conharmonic curvature (i.e., F⋅F=LQ(g,F) for F=W,C,K with a smooth scalar function L), and it also possesses the relation R⋅R−LQ(g,C)=Q(S,R). It is engrossing to mention that the nature of energy momentum tensor of the HBH spacetime is pseudosymmetric. On the basis of curvature related properties, we have made a comparison among Reissner–Nordström spacetime, interior black hole spacetime and HBH spacetime. It is noteworthy to mention that the gravitational force of the point-like global monopole spacetime is much stronger than that of HBH spacetime. Also, it is shown that the HBH spacetime admits an almost η-Ricci soliton as well as an almost η-Ricci-Yamabe soliton. Finally, an elegant comparative study is delineated between the HBH spacetime and the point-like global monopole spacetime with respect to different kinds of symmetry, such as, motion, curvature collineation, curvature inheritance etc.

Maximum mass of charged strange quark star in presence of strange quark mass (m s )

In this manuscript, we present an approach to calculate maximum mass of strange quark star having net charge inside. For this purpose we took the modified MIT bag model equation of state in presence of non-zero strange quark mass (m s ). The general solution of Einstein field equations in presence of charge is obtained by considering a specific form of the g rr component of the line element according to Vaidya & Tikekar. Such metric ansatz describes a homogeneous fluid distribution which has a departure from the spherical geometry determined by the two parameters: spheroidal (λ) and curvature (R). In this approach, we find that maximum mass as well as radius both decreases with the increase of strange quark mass (m s ). Also maximum mass increases with charge and obtained from our model is as high as 4.383 M ⊙ for maximum allowed value of charge with m s = 0. The stability is also studied in this model and note that our model is stable for the constraint value of parameters.

A regular interior solution of Einstein field equations

Starting from the solution of the Einstein field equations in a static and spherically symmetric space–time that contains an isotropic fluid, we construct a model to represent the interior of compact objects with compactness rate [Formula: see text]. The solution is obtained by imposing the isotropy condition for the radial and tangential pressures; this generates an ordinary differential equation of second order for the temporal gtt and radial grr metric potentials, which can be solved for a specific function of gtt. The graphic analysis of the solution shows that it is physically acceptable, which is to say that the density, pressure, and speed of sound are positive, regular and monotonically decreasing functions; also, the solution is stable due to meeting the criteria of the adiabatic index. When taking the data of mass [Formula: see text] and radius [Formula: see text] that corresponds to the estimations of the star PSR J0030+045, we obtain values of central density ρc = 7.5125 × 1017 kg/m3 for the maximum compactness u = 0.19628 and of ρc = 2.8411 × 1017 kg/m3 for the minimum compactness u = 0.13460, which are consistent with those expected for this type of star.

Impacts of complexity factor on the transition of fluid configurations from isotropic to anisotropic environment and vice versa

This article presents a study on the process of isotropization and properties of stellar models with different complexity factors, which was introduced by Herrera (Phys Rev D 97:044010, 2018). We consider gravitational decoupling via MGD (Minimal Geometric Deformation) approach for spherically symmetric systems with static background and explore the scenarios where the isotropization of an anisotropic solution of Einstein field equations is possible. Moreover, we work on generating the new analogues of Durgapal–Fuloria model under different conditions associated with complexity factor. The physical existence and stability of the new stellar models have been discussed in details with the help of different potent tools.

Proposed method of combining continuum mechanics with Einstein Field Equations

The paper proposes an amendment to the relativistic continuum mechanics which introduces the relationship between density tensors and the curvature of spacetime. The resulting formulation of a symmetric stress–energy tensor for a system with an electromagnetic field leads to the solution of Einstein Field Equations indicating a relationship between the electromagnetic field tensor and the metric tensor. In this EFE solution, the cosmological constant is related to the invariant of the electromagnetic field tensor, and additional pulls appear, dependent on the vacuum energy contained in the system. In flat Minkowski spacetime, the vanishing four-divergence of the proposed stress–energy tensor expresses relativistic Cauchy’s momentum equation, leading to the emergence of force densities which can be developed and parameterized to obtain known interactions. Transformation equations were also obtained between spacetime with fields and forces, and a curved spacetime reproducing the motion resulting from the fields under consideration, which allows for the extension of the solution with new fields.

Creation field cosmological models for dust distribution and time-dependent cosmological term (Λ) in Bianchi type II space-time

In this paper, we present solution of Einstein field equations which admit dust distribution and negative energy massless scalar field as a source with cosmological term ([Formula: see text]) in the frame work of Bianchi type II space-time. The models satisfy conservation equation and creation field increases with time satisfying the result as investigated by Hoyle and Narlikar [A new theory of gravitation, Proc. Roy. Soc. A 282 (1964) 191]. The deceleration parameter ([Formula: see text]) for the model (22) shows a transition from decelerating to accelerating regime with creation. A comparison with other cosmological models shows a transition from decelerating to accelerating regime with and without creation is mentioned. The present model is free from singularity, particle horizon and provides a natural explanation for inflationary scenario and isotropization. These features make the creation field cosmological models theoretically superior to the Big Bang models. Creation field and Einstein field equation are derived using principle of least action and the Lagrangian formulation of variable cosmological term is derived as mentioned by Moffat [Lagrangian formulation of a solution to the cosmological constant problem (1996), arXiv:astro-ph/9608202v1]. The current creation rate with other physical consequences is also discussed. For illustrative purposes, evolutionary behaviors of some cosmological parameters are shown graphically. In the figures, the cosmic time [Formula: see text] is given in gravitational units assuming the velocity of light [Formula: see text]. This [Formula: see text] and [Formula: see text]. Therefore, by multiplying cosmic time [Formula: see text] by (light years)−1, we obtain the coordinate time in years.

Static and stationary dark fluid universes: a gravitoelectromagnetic perspective

The usual characterization of exact solutions of Einstein field equations, including cosmological solutions, is based on the symmetry properties of their corresponding metrics which is obviously mathematically involved. Here we present a physical characterization of the static and stationary perfect fluid solutions of the Einstein field equations by employing the 1+3 formulation of spacetime decomposition which introduces the so-called quasi-Maxwell form of the Einstein field equations in the broader context of gravitoelectromagnetism. These solutions have a single or 2-component perfect fluid sources, and are characterized according to their gravitoelectric and gravitomagnetic fields which are the gravitational analogs of the electromagnetic fields. It is shown that the absence or presence of either or both of these fields could restrict the equations of state of the contributing perfect fluid sources. As the representative of each family of solutions, we consider those spaces that include the cosmological term as a dark fluid source with the equation of state p=-rho = constant.

Equivalence between two charged black holes in dynamics of orbits outside the event horizons

Using the Fermi–Dirac distribution function, Balart and Vagenas gave a charged spherically symmetric regular black hole, which is a solution of Einstein field equations coupled to a nonlinear electrodynamics. In fact, the regular black hole is a Reissner–Nordström (RN) black hole when a metric function is given a Taylor expansion to first order approximations. It does not have a curvature singularity at the origin, but the RN black hole does. Both black hole metrics have horizons and similar asymptotic behaviors, and satisfy the weak energy conditions everywhere. They are almost the same in photon effective potentials, photon circular orbits and photon spheres outside the event horizons. Due to the approximately same photon spheres, the two black holes have no explicit differences in the black hole shadows and constraints of the black hole charges based on the First Image of Sagittarius A $$^{*}$$ . There are relatively minor differences between effective potentials, stable circular orbits and innermost stable circular orbits of charged particles outside the event horizons of the two black holes immersed in external magnetic fields. Although the two magnetized black holes allow different construction methods of explicit symplectic integrators, they exhibit approximately consistent results in the regular and chaotic dynamics of charged particles outside the event horizons. Chaos gets strong as the magnetic field parameter or the magnitude of negative Coulomb parameter increases, but becomes weak when the black hole charge or the positive Coulomb parameter increases. A variation of dynamical properties is not sensitive dependence on an appropriate increase of the black hole charge. The basic equivalence between the two black hole gravitational systems in the dynamics of orbits outside the event horizons is due to the two metric functions having an extremely small difference. This implies that the RN black hole is reasonably replaced by the regular black hole without curvature singularity in many situations.

Asymptotic structure with a positive cosmological constant

This is the second of two papers (Fernández-Álvarez F and Senovilla J M M 2021 Class. Quantum Grav. 39 165011) that study the asymptotic structure of space–times with a non-negative cosmological constant Λ. This paper deals with the case Λ > 0. Our approach is founded on the ‘tidal energies’ built with the Weyl curvature and, specifically, we use the asymptotic super-Poynting vector computed from the rescaled Bel–Robinson tensor at infinity to provide a covariant, gauge-invariant, criterion for the existence, or absence, of gravitational radiation at infinity. The fundamental idea we put forward is that the physical asymptotic properties are encoded in , where the first element of the triplet is a three-dimensional manifold, the second is a representative of a conformal class of Riemannian metrics on , and the third element is a traceless symmetric tensor field on . The full set of physically relevant properties of the space–time cannot be characterised at infinity without taking D ab into consideration, and our radiation criterion takes this fully into account. We similarly propose a no-incoming radiation criterion based also on the triplet and on radiant supermomenta deduced from the rescaled Bel–Robinson tensor too. We search for news tensors encoding the two degrees of freedom of gravitational radiation and argue that any news-like object must be associated to, and depends on, two-dimensional cross-sections of . We identify one component of news for every such cross-section and present a general strategy to find the second component, which depends on the particular physical situation. We put in connection the radiation condition and the news-like tensors with the directional structure of the gravitational field at infinity and the criterion of no-incoming radiation. We also introduce the concept of equipped by endowing the conformal boundary with a selected congruence of curves which may be determined by the algebraic structure of the asymptotic Weyl tensor. We also define a group of asymptotic symmetries preserving the new structures. We consider the limit Λ → 0, and apply all our results to selected exact solutions of Einstein field equations in order to illustrate their validity.

Yukawa black holes from interacting vacuum

In this paper we obtain an exact solution of Einstein field equations assuming an interaction between a vacuum component and the Maxwell field. The key feature of such interaction refers to a simple stress exchange so that the electromagnetic field naturally incorporates the Yukawa potential. It is shown that the resulting spacetime thus obtained can either be a naked singularity or a black hole with an inner Cauchy horizon R − and an exterior event horizon R +. For this latter configuration we examine the group velocity of test photons in the region R > R +. Beyond a lower bound for the frequency we show that superluminal velocities arise in a neighbourhood of the event horizon and that the coupling parameter of the interaction is actually connected to a nonvanishing rest mass for the photon.

Constant Gaussian curvature, FLRW conditions and a universe without matter

The problem of FLRW universes can be seen in the excellent works of A. Friedmann, G. Lemaître, H. P. Robertson, A. G. Walker. The key of obtaining FLRW universes with or without cosmological constant is related to the special form of the contravariant energy–momentum tensor which describes a perfect fluid with components [Formula: see text] because such a “matter” leads to FLRW conditions [Formula: see text] which allow to obtain a texture compatible with Hubble law. The results obtained in the works presented as references show very complete analysis about how the constant curvature 2D metric [Formula: see text] can create possible universes and their evolution. Can we expect to have some other FLRW universes if we use a 2D spatial part as [Formula: see text]? The answer is yes and we prove the special role of the constant ratio [Formula: see text] in the creation of FLRW universes. The solutions obtained lead to FLRW universes which come from surfaces having constant Gaussian curvature related by the previous [Formula: see text] from the rule [Formula: see text]. Therefore, in the case of the 2D metrics [Formula: see text], only the ones having constant Gaussian curvature create FLRW universes. In fact, we find all possible FLWR-type solutions of Einstein field equations coming from constant Gaussian curvature geometries. We complete our study about possible FLRW universes with an example of a spacetime that cannot be “filled” with matter using the above FLRW conditions. This becomes a new example of universe without matter satisfying the Einstein’s field equation with cosmological constant.

Bi-Quadratic Varying Deceleration Parameter to Study the Cosmological Model

This paper investigates a cosmological model employing a four-degree function of cosmic time t in the f(R, T) gravity theory using an extended form of deceleration parameter. Under the constraints of flat, diagonal, and Bianchi type-1 metrics, we develop the exact solution of Einstein field equations (EFE). We research early time deceleration and accomplish the definite solution of EFE using an extended form of deceleration parameter, as well as the always super-exponential expansion behaviour of bi-quadratically variable deceleration parameter using a graph. We explore the model’s efficacy by examining the relationship between the deceleration parameter and red shift, which offers a valid result concerning the FRW metric’s consistent homogeneous structure in light of recent cosmological observations.

Curvature properties of Morris-Thorne wormhole metric

The Morris-Thorne wormhole is a spherically symmetric solution of Einstein field equations with cosmological constant. The present article aims to investigate the geometric properties in terms of curvatures admitted by this spacetime. It is found that such a spacetime possesses several kinds of symmetries, such as, Ricci generalized pseudosymmetry, Ricci generalized projectively pseudosymmetry, pseudosymmetry due to Weyl conformal curvature, semisymmetry due to conharmonic curvature etc. Also, it is an Einstein manifold of level 2 as well as special quasi-Einstein manifold. The Tachibana tensor due to energy momentum tensor of the wormhole satisfies some conditions of pseudosymmetric type and also the energy momentum tensor is Weyl compatible and Riemann compatible. Finally, this spacetime is compared with the Gödel spacetime with respect to their admitting geometric structures.

Zipoy-Voorhees Gravitational Object as a Source of High-Energy Relativistic Particles

The Zipoy-Voorhees solution is known as the γ-metric and/or q-metric being static and axisymmetric vacuum solution of Einstein field equations which becomes strong curvature naked singularity. The metric is characterized by two parameters, namely, the mass M and the dimensionless deformation parameter γ. It is shown that the velocity of test particle orbiting around the central γ-object can reach the speed of light, consequently, the total energy of the particle will be very high for a specific value the deformation parameter of the spacetime. It is also shown that causality problem arises in the interior region of the physical singularity for the specific value of the deformation parameter when test particles can move with superluminal velocity being greater than the speed of light that might be an additional tool for explaining the existence of tachyons for γ>1/2 which are invisible for an observer.

Invariant solutions of Einstein field equations in pure radiation fields

The exact static accelerating solutions of Einstein’s equations in the non-comoving pure radiation fields, with an indefinite non-degenerate stationary metric in cylindrical coordinates, are obtained. The considered space-time is of Petrov type I and is not conformally flat. Moreover, the classical symmetry method is used for symmetry reduction and finding the solutions in terms of hyperbolic functions. The conservation laws are found by using the multiplier approach.

New cylindrically symmetric solution of Einstein field equations their conservation laws and the particles dynamics

The study explores the particle dynamics, asymptotic behavior, and flatness in Riemann curvature corresponding to the solutions obtained from the Ricci tensor of cylindrical spacetimes. Since these solutions are non-Minkowskian and are proposed to have curvature in almost all the cases, that is why some cylindrically symmetric static vacuum solutions of Einstein Field Equations have been worked out. We explore the Noether symmetries along with their conservation laws of the corresponding static-spacetime and calculate the equation of motions by taking into account Noether symmetries. We study the dynamics of particle in the form of effective potential, effective force.

An analytical anisotropic compact stellar model of embedding class I

A class of solutions of Einstein field equations satisfying Karmarkar embedding condition is presented which could describe static, spherical fluid configurations, and could serve as models for compact stars. The fluid under consideration has unequal principal stresses i.e. fluid is locally anisotropic. A certain physically motivated geometry of metric potential has been chosen and codependency of the metric potentials outlines the formation of the model. The exterior spacetime is assumed as described by the exterior Schwarzschild solution. The smooth matching of the interior to the exterior Schwarzschild spacetime metric across the boundary and the condition that radial pressure is zero across the boundary lead us to determine the model parameters. Physical requirements and stability analysis of the model demanded for a physically realistic star are satisfied. The developed model has been investigated graphically by exploring data from some of the known compact objects. The mass-radius (M-R) relationship that shows the maximum mass admissible for observed pulsars for a given surface density has also been investigated. Moreover, the physical profile of the moment of inertia (I) thus obtained from the solutions is confirmed by the Bejger–Haensel concept.

Invariant solutions and conservation laws of Einstein field equations in non-comoving radiation fields

The exact non-static accelerating solutions of Einstein field equations in non-comoving pure radiation fields, corresponding to an indefinite non-degenerate metric in cartesian coordinates, are obtained. The gravitational field is of Petrov type D and the considered spacetime is not conformally flat. Lie symmetry method is used for symmetry reduction of nonlinear partial differential equations and for finding the exact solutions, comprising both arbitrary functions and known analytic functions. Conservation laws are obtained by using multiplier approach. The graphical representations of solutions are shown by considering different possibilities of the arbitrary functions.