Spheres In General Relativity Research Articles

Static Charged Fluid Sphere in General Relativity

Static Charged Fluid Sphere in General Relativity

A study of charged stellar structure in modified f (R,ϕ,χ) gravity

The aim of this paper is to investigate and analyze the behavior of charged compact stars in the modified [Formula: see text] theory of gravity. When solving the field equations, we assume that Krori– Barua space-time describes the geometry of the inner space [K. Krori and J. Barua, A singularity-free solution for a charged fluid sphere in general relativity, J. Phys. A Math. Gen. 8 (1975) 508]. We further analyzed various physical characteristics for our model, including energy density, pressure components, anisotropy parameter, mass function, compactness element, and redshift through their graphical representations. Additionally, we have checked the equilibrium position by taking Tolman–Oppenheimer–Volkoff equation and stability by using Abreu’s criterion [H. Abreu, H. Hernandez, and L. Nunez, Sound speeds, cracking and the stability of self-gravitating anisotropic compact objects, Class. Quantum Grav. 24 (2007) 4631] and relativistic adiabatic index. For our study, we choose Hercules XI, Centaurus XIII, EXO 1785-248, LMC XIV stars for analysis.

Geometry of static w=-1/5 perfect fluid spheres in general relativity

We discuss the physical features of two recent classes of analytical solutions of the Einstein equations sourced by an exotic perfect fluid with equation of state P=-rho /5. These geometries depend on up to four parameters and are static and spherically symmetric. They describe compact spaces with naked central singularities.

Retraction Note to: Luminosity profiles and the contraction of radiating spheres in general relativity

Retraction Note to: Luminosity profiles and the contraction of radiating spheres in general relativity

Isothermal spheres in general relativity and Hořava-type gravity

We construct a toy model for isothermal spheres in Horava gravity, which includes Einstein's gravity if a parameter is appropriately chosen. The equations for the isothermal spheres are derived from the partition function of the gravitating particle system. We confirm that the Newtonian limit of the system coincides with the model of the well-known isothermal sphere. The stability of the isothermal sphere is found to be sensitive to the energy density at the center of the sphere.

A simple geometry to model fluid spheres in general relativity

A simple geometry which is static and spherically symmetrical is proposed, useful to model a fluid sphere in general relativity. This allows us to describe the interior of stellar objects both neutral and charged. It is shown that in both cases the solution is stable, which means that the behavior of the density and radial pressure is monotonic decreasing functions. Both models depend on two parameters, in the neutral case, these are the compactness rate $$u=GM/c^2R$$ and w related to the monotonically increasing or decreasing behavior of the speed of sound, and in the charged case, they are the compactness rate and the rate between the charge Q and the radius R, $$q=Q/R$$ . A comparison is done between the case of a charged perfect fluid and the case of an anisotropic fluid; from there we can show that, as a result of the presence of the charge and its repulsion effect that it generates at a local and universal level, the density for the anisotropic case is greater than the density in the charged case. For both models, we obtain physical values of objects with a compactness rate $$u=0.35578$$ for the particular cases of masses and radios for $$(M_\odot ,R=4150.1\,\mathrm{m})$$ , $$(1.5M_\odot ,R=6225\,\mathrm{m})$$ , $$(2M_\odot ,R=8300 m)$$ . In the neutral case, the greater central density $$\rho _c=5.0840\times 10^{19} \,\mathrm{kg/m}^3$$ matches the object of $$1.5M_\odot $$ and for the charged case, the maximum central density is of $$\rho _c=8.8439\times 10^{18} \,\mathrm{kg/m}^3$$ , associated with the star with $$1M_\odot $$ . Additionally, by using the proposed model, we obtain the values of density and pressure for the star PSR J1614–223, the maximum value of the central density $$\rho _c=1.0573\times 10^{18} \,\mathrm{kg/m}^3$$ .

Convection and cracking stability of spheres in general relativity

In the present paper we consider convection and cracking instabilities as well as their interplay. We develop a simple criterion to identify equations of state unstable to convection, and explore the influence of buoyancy on cracking (or overturning) for isotropic and anisotropic relativistic spheres. We show that a density profile rho (r), monotonous, decreasing and concave , i.e. rho ' < 0 and rho '' < 0, will be stable against convection, if the radial sound velocity monotonically decreases outward. We also studied the cracking instability scenarios and found that isotropic models can be unstable, when the reaction of the pressure gradient is neglected, i.e. delta mathcal {R}_p = 0; but if it is considered, the instabilities may vanish and this result is valid, for both isotropic and anisotropic matter distributions.

All static charged dust spheres in general relativity

Exact solutions for distributions of static charged dust in general relativity are sought in a systematic way in curvature coordinates. An algorithm is proposed to identify all dust spheres subject to carrying out a single integration after specifying a gravitational potential. It is argued that charged dust distributions are described by metrics that are inherently singular. The algorithm is illustrated for a simple case that generalises several known cases treated historically. Further new dust models are generated by specifying the other gravitational potential and well known metric Ansätze, namely, the generalised Schwarzschild and Finch–Skea prescriptions are treated.

A new solution of embedding class I representing anisotropic fluid sphere in general relativity

In this paper, we are willing to develop a model of an anisotropic star by choosing a new [Formula: see text] metric potential. All the physical parameters like the matter density, radial and transverse pressure are regular inside the anisotropic star, with the speed of sound less than the speed of light. So the new solution obtained by us gives satisfactory description of realistic astrophysical compact stars. The model of this paper is compatible with observational data of compact objects like RX J1856-37, Her X-1, Vela X-12 and Cen X-3. A particular model of Her X-1 (Mass 0.98 [Formula: see text] and radius[Formula: see text]=[Formula: see text]6.7 km.) is studied in detail and found that it satisfies all the condition needed for physically acceptable model. Our model is described analytically as well as with the help of graphical representation.

Relativistic model of anisotropic charged fluid sphere in general relativity

In this present paper, we present a class of static, spherically symmetric charged anisotropic fluid models of super dense stars in isotropic coordinates by considering a particular type of metric potential, a specific choice of electric field intensity $E$ and pressure anisotropy factor $\Delta$ which involve parameters $K$ (charge) and $\alpha$ (anisotropy) respectively. The solutions so obtained are utilized to construct the models for super-dense stars like neutron stars and strange quark stars. Our solutions are well behaved within the following ranges of different constant parameters. In the absence of pressure anisotropy and charge present model reduces to the isotropic model Pant et al. (Astrophys. Space Sci. 330:353–359, 2010). Our solution is well behaved in all respects for all values of $X$ lying in the range $0< X \leq 0.18$ , $\alpha$ lying in the range $0 \leq \alpha \leq6.6$ , $K$ lying in the range $0< K \leq 6.6$ and Schwarzschild compactness parameter “ $u$ ” lying in the range $0< u \leq 0.38$ . Since our solution is well behaved for a wide ranges of the parameters, we can model many different types of ultra-cold compact stars like quark stars and neutron stars. We have shown that corresponding to $X=0.088$ , $\alpha=0.6$ and $K=4.3$ for which $u=0.2054$ and by assuming surface density $\rho_{b} = 4.6888 \times 10^{14}~\mbox{g/cm}^{3}$ the mass and radius are found to be $1.51~M_{\varTheta}$ and 10.90 km respectively. Assuming surface density $\rho_{b} = 2 \times 10^{14}~\mbox{g/cm}^{3}$ the mass and radius for a neutron star candidate are found to be $2.313~M_{\varTheta}$ and 16.690 km respectively. Hence we obtain masses and radii that fall in the range of what is generally expected for quark stars and neutron stars.

Cracking of Anisotropic Spheres in General Relativity Revisited

We have studied the stability in some anisotropic and isotropic matter configurations, by using the concept of cracking in General Relativity, which was conceived as an alternative and different approach to detect instabilities. It is based in the analysis of the radial forces that appear on the system after a perturbation capable of taking it out of its equilibrium state. We have studied the effects of local density perturbations on systems with “barotropic” equations of state. The considerations of the perturbations on the gradient of pressure lead to very different results than the previous works, and have found that not only anisotropic models may present cracking (or overturning), but it can also occur in isotropic matter configurations.

Three new exact solutions for charged fluid spheres in general relativity

In the present article three new exact solutions of Einstein’s field equations for charged fluid spheres are derived as per Ivanov’s classification (ν,q). Where ‘ν’ stands for ‘logg 44’ and q represents total charge inside the sphere of radius r. The solutions are obtained by considering ν=nlogB 1/n (1+br 2) with b 0 and assuming suitable charged function ‘q’. The solutions so obtained are utilized to construct the super dense star models with surface density ρ a =2×1014 g cm−3 and possessing the maximum mass M and the corresponding radius a as: (i) n=−1, M=3.9678M Θ , a=22.9698 km, (ii) n=−2, M=4.9489M Θ , a=18.332 km, (iii) n=−3, M=0.9643M Θ , a=3.1698 km. The red shift, adiabatic index, variation of velocity of light throughout the stars have been studied carefully and kept the every entity within the physical range.

Addendum to ‘Coherent Lagrangian vortices: the black holes of turbulence’

AbstractIn Haller &amp; Beron-Vera (J. Fluid Mech., vol. 731, 2013, R4) we developed a variational principle for the detection of coherent Lagrangian vortex boundaries. The solutions of this variational principle turn out to be closed null geodesics of the Lorentzian metric induced by a generalized Green–Lagrange strain tensor family. This metric interpretation implies a mathematical analogy between coherent Lagrangian vortex boundaries and photon spheres in general relativity. Here, we give an improved discussion of this analogy.

On the Solution of Spherically Symmetric Static Problem for a Fluid Sphere in General Relativity

The paper is devoted to the spherically symmetric static problem of General Theory of Relativity (GTR) originally solved by K. Schwarzschild in 1916 for a particular form of the line element. This classical solution specifies the metric tensor for the external and internal semi-Riemannian spaces for a perfect fluid sphere with constant density and includes the so called gravitational radius rg which is associated with the singular behavior of the solution. The Schwarzschild solution for the external space becomes singular if the sphere radius reaches rg which is referred to as the radius of the Black Hole event horizon. The solution for the internal space gives infinitely high fluid pressure at the center of sphere with radius equal to 9/8 rg. In contrast to the classical solution, the solution presented in the paper is based on the general form of line element for spherically symmetric Riemannian space in which the circumferential component of the metric tensor ρ 2 (r) is an arbitrary function of the radial coordinate. As shown, the solution of the static problem exists for a whole class of functions ρ(r). The particular form of this function is determined in the paper under the assumption according to which the gravitation, changing the Euclidean space to the Riemannian space inside the sphere in accordance with GTR equations, does not affect the sphere mass. The solution obtained for the proposed particular form of the line element cannot be singular neither on the sphere surface nor at the sphere center. Direct comparison with the Schwarzschild solution for external and internal spaces is presented.

A class of well-behaved generalized charged analogues of Vaidya-Tikekar type fluid sphere in general relativity

In this paper first ever we have developed a class of well behaved charged fluid spheres expressed by a space time with its hypersurfaces $t = \operatorname {const}$ . as spheroid for the case 0<K<1 with surface density 2×1014 gm/cm3. The same utilized to construct a superdense star and seen that star satisfies all well behaved condition for 0<K≤0.038. The maximum mass occupied and the corresponding radius are found to be 4.830982M Θ and 20.7612 km respectively. The redshift at the center and on the surface is given z 0=0.425367 and z a =0.240901.

Some static relativistic compact charged fluid spheres in general relativity

In this work a new family of relativistic models of electrically charged compact star has been obtained by solving Einstein–Maxwell field equations with preferred form of one of the metric potentials and a suitable form of electric charge distribution function. The resulting equation of state (EOS) has been calculated. The relativistic stellar structure for matter distribution obtained in this work may reasonably models an electrically charged compact star whose energy density associated with the electric fields is on the same order of magnitude as the energy density of fluid matter itself (e.g. electrically charged bare strange stars). Based on the analytic model developed in the present work, the values of the relevant physical quantities have been calculated by assuming the estimated masses and radii of some well known strange star candidates like X-ray pulsar Her X-1, millisecond X-ray pulsar SAX J 1808.4-3658, and 4U 1820-30.

Incompressible Einstein–Maxwell fluids with specified electric fields

The Einstein–Maxwell equations describing static charged spheres with uniform density and variable electric field intensity are studied. The special case of constant electric field is also studied. The evolution of the model is governed by a hypergeometric differential equation which has a general solution in terms of special functions. Several classes of exact solutions are identified which may be considered as charged generalizations of the incompressible Schwarzschild interior model. An analysis of the physical features is undertaken for the uniform case. It is demonstrated that uniform density spheres with constant electric field intensity are not realizable with isotropic pressures. This highlights the necessity of studying the criteria for physical admissability of gravitating spheres in general relativity which are solutions to the Einstein–Maxwell equations.

Thermodynamical instabilities of perfect fluid spheres in General Relativity

For a static, perfect fluid sphere with a general equation of state, we obtain the relativistic equation of hydrostatic equilibrium, namely the Tolman–Oppenheimer–Volkov equation, as the thermodynamical equilibrium in the microcanonical, as well as the canonical, ensemble. We find that the stability condition determined by the second variation of entropy coincides with the dynamical stability condition derived by the variations to first order in the dynamical Einstein’s equations. Thus, we show the equivalence of microcanonical thermodynamical stability with linear dynamical stability for a static, spherically symmetric field in General Relativity. We calculate the Newtonian limit and find the interesting property that the microcanonical ensemble in General Relativity transforms to the canonical ensemble for non-relativistic dust particles. Finally, for specific kinds of systems, we study the effect of the cosmological constant on the microcanonical thermodynamical stability of fluid spheres.

ISOTROPIC CASES OF STATIC CHARGED FLUID SPHERES IN GENERAL RELATIVITY

In this paper we study the isotropic cases of static charged fluid spheres in general relativity. For this purpose we consider two different specializations and under these we solve the Einstein–Maxwell field equations in isotropic coordinates. The analytical solutions thus obtained are matched to the exterior Reissner–Nordström solutions which concern the values for the metric coefficients eν and eμ. We derive the pressure, density and pressure-to-density ratio at the center of the charged fluid sphere and boundary R of the star. Our conclusion is that static charged fluid spheres provide a good connection to compact stars.

Variety of Well behaved parametric classes of relativistic charged fluid spheres in general relativity

The paper presents a variety of classes of interior solutions of Einstein–Maxwell field equations of general relativity for a static, spherically symmetric distribution of the charged fluid with well behaved nature. These classes of solutions describe perfect fluid balls with positively finite central pressure, positively finite central density; their ratio is less than one and causality condition is obeyed at the center. The outmarch of pressure, density, pressure–density ratio and the adiabatic speed of sound is monotonically decreasing for these solutions. Keeping in view of well behaved nature of these solutions, two new classes of solutions are being studied extensively. Moreover, these classes of solutions give us wide range of constant K for which the solutions are well behaved hence, suitable for modeling of super dense star. For solution (I1) the mass of a star is maximized with all degree of suitability and by assuming the surface density ρb=2×1014 g/cm3 corresponding to K=1.19 and X=0.20, the maximum mass of the star comes out to be 2.5MΘ with linear dimension 25.29 Km and central redshift 0.2802. It has been observed that with the increase of charge parameter K, the mass of the star also increases. For n=4,5,6,7, the charged solutions are well behaved with their neutral counterparts however, for n=1,2,3, the charged solution are well behaved but their neutral counterparts are not well behaved.