Exterior Schwarzschild Solution Research Articles

Compact stellar model with vanishing complexity under Vaidya–Tikekar background geometry

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.

Complete solution of the Einstein field equations for a spherical distribution of polytropic matter

We determine the complete solution of the Einstein field equations for the case of a spherically symmetric distribution of gaseous matter, characterized by a polytropic equation of state. We show that the field equations automatically generate two sharp boundaries for the gas, an inner one and an outer one, given by radial positions $$r_{1}$$ and $$r_{2}$$ , and thus define a shell of gaseous matter outside of which the energy density is exactly zero. Hence this shell is surrounded by an outer vacuum region, and surrounds an inner vacuum region. Therefore, the solution is given in three regions, one being the well-known analytical Schwarzschild exterior solution in the outer vacuum region, one being determined analytically in the inner vacuum region, and one being determined partially analytically and partially numerically, within the matter region, between the two boundary values $$r_{1}$$ and $$r_{2}$$ of the Schwarzschild radial coordinate r. This solution is therefore somewhat similar to the one previously found for a spherically symmetric shell of liquid fluid, and is in fact exactly the same in the cases of the inner and outer vacuum regions. The main difference is that here the boundary values $$r_{1}$$ and $$r_{2}$$ are not chosen arbitrarily, but are instead determined by the dynamics of the system. As was shown in the case of the liquid shell, also in this solution there is a singularity at the origin, that just as in that case does not correspond to an infinite concentration of matter, but in fact to zero matter energy density at the center. Also as in the case of the liquid shell, the spacetime within the spherical cavity is not flat, so that there is a non-trivial gravitational field there, in contrast with Newtonian gravitation. This inner gravitational field has the effect of repelling matter and energy away from the origin, thus avoiding a concentration of matter at that point.

A study on the effect of anisotropy under Finch–Skea geometry

The popularity of the Finch-Skea ansatz to describe relativistic stellar model have encouraged us to study the analytic solutions of the Einstein field equation. We have presented a class of exact solutions to the field equations after considering the corresponding two cases: (i) positive value of anisotropic parameter, and (ii) absence of any anisotropy. Smooth matching of the interior solutions with the Schwarzschild exterior solution helped us to determine constants. The physical features of the solutions thus obtained have been studied both graphically and numerically with the specific pulsar $4U~1608-52$ (Mass = $1.57^{+0.30}_{-0.29}~M_\odot$ and radius = $9.8\pm 1.8~km$). The stability conditions for the model have also been discussed, however the model is found to be stable for zero anisotropy.

An extensive analysis of Schwarzschild exterior solution

We pursue a detailed analysis of the Schwarzschild geometry around a spherically symmetric, non-rotating, uncharged source and aim to construct the Schwarzschild metric by considering trigonometric, hyperbolic and logarithmic functions of position and time and solving the Einstein field equation. We investigate whether the Schwarzschild exterior solution is indeed independent of the choice of the nature of the function for the first two metric elements in the general expression for the Schwarzschild metric, and is solely dependent upon the centrally symmetric nature of the geometry taken into account.

Pulsar PSR B0943$$+$$10 as an isotropic Vaidya–Tikekar-type compact star

In this paper, we have constructed a model for well behaved isotropic compact star in the presence of charged perfect fluid, by considering a static and spherically symmetric metric in Schwarzschild's canonical coordinate system. To put the resulting differential equations into a closed system, we have employed the Vaidya & Tikekar (J. Astrophys. Astron. 3:325, 1982) form of the metric potential grr. The resulting energy-momentum components, i.e., energy density and pressure contain six constants; two of these are determined through the junction condition (matching the interior with the exterior Schwarzschild solution) and by the property of vanishing pressure on the boundary. The remaining constants are constrained by requirements of a real compact star. The physical acceptability of our model is tested using the data of the pulsar PSR B0943+10. Using graphical analysis and tabular information we have shown that our model obeys all the physical requirements. The stability of this model is evaluated using the Tolman-Oppenheimer-Volkoff equation, the adiabatic index and the Harrison-Zeldovich-Novikov Criterion and it has passed the evaluation.

Study of anisotropic and non-uniform Gravastars by gravitational decoupling in [formula omitted] gravity

Adopting gravitational decoupling through minimal geometric deformation (MGD) procedure, we develop an analytical version of gravastar model with non-uniform and anisotropic features, in the framework of f(R,T) gravity. This new non-uniform model describes an ultracompact stellar structure of radius RS=2M, whose interior solution smoothly joins a conformally deformed Schwarzschild exterior solution, and also it is matched to the standard Schwarzschild exterior solution under some restrictions on f(R,T) coupling constant λ. The constructed solution presents a family of stellar models satisfying some of the fundamental properties of a stable configuration, including a positive energy density everywhere with monotonically decreasing behavior from the center to surface. Besides, a non-uniform pressure is observed with monotonic behavior. All the energy conditions except the strong one are satisfied inside the ultracompact stellar configuration for all the values of f(R,T) coupling constant λ, which are compatible with regularity condition.

Wave packet treatment of neutrino flavor oscillations in various spacetimes

We study the effect of gravity on neutrino flavor oscillations when each mass eigenstate is described by a wave packet instead of a plane wave. Two different approaches for implementing the wave packet formalism in the study of neutrino flavor oscillations in curved spacetime are examined. We work with a general static and spherically symmetric spacetime before applying our results to a few spacetime metrics of interest. We first focus on general relativity by examining the effect of the exterior and interior Schwarzschild solutions, as well as the de Sitter-Schwarzschild metric, and then we examine selected metrics from modified gravity models.

Anisotropic Compact Stars in the Mimetic Gravitational Theory

Abstract In this paper, we consider the mimetic gravitational theory to derive a novel category of anisotropic star models. To end and to put the resulting differential equations into a closed system, the form of the metric potential g rr as used by Tolman is assumed, as well as a linear form of the equation of state. The resulting energy momentum components, energy density, and radial and tangential pressures contain five constants: three of these are determined through the junction condition, matching the interior with the exterior Schwarzschild solution; the fourth is constrained by the vanishing of the radial pressure on the boundary; and the fifth is constrained by a real compact star. The physical acceptability of our model is tested using the data of the pulsar 4U 1820–30. The stability of this model is evaluated using the Tolman–Oppenheimer–Volkoff equation and the adiabatic index, and it is shown to be stable. Finally, our model is challenged with other compact stars, demonstrating that it is consistent with those stars.

Strange star with Krori–Barua potential in the presence of anisotropy

In this paper, a well-behaved new model of anisotropic compact star in (3+1)-dimensional spacetime has been investigated in the background of Einstein’s general theory of relativity. The model has been developed by choosing [Formula: see text] component as Krori–Barua (KB) ansatz [Krori and Barua in J. Phys. A, Math. Gen. 8 (1975) 508]. The field equations have been solved by a proper choice of the anisotropy factor which is physically reasonable and well behaved inside the stellar interior. Interior spacetime has been matched smoothly to the exterior Schwarzschild vacuum solution and it has also been depicted graphically. Model is free from all types of singularities and is in static equilibrium under different forces acting on the system. The stability of the model has been tested with the help of various conditions available in literature. The solution is compatible with observed masses and radii of a few compact stars like Vela X-1, 4U [Formula: see text], PSR J[Formula: see text], LMC X [Formula: see text], EXO [Formula: see text].

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.

The Hilbert manifold of asymptotically flat metric extensions

In [Commun Anal Geom 13(5):845–885, 2005], Bartnik described the phase space for the Einstein equations, modelled on weighted Sobolev spaces with local regularity (g,pi )in H^2times H^1. In particular, it was established that the space of solutions to the constraints form a Hilbert submanifold of this phase space. The motivation for this work was to study the quasi-local mass functional now bearing his name. However, the phase space considered there was over a manifold without boundary. Here we demonstrate that analogous results hold in the case where the manifold has an interior compact boundary, and the metric is prescribed on the boundary. Then, still following Bartnik’s work, we demonstrate the critical points of the mass functional over this space of extensions correspond to stationary solutions with vanishing Killing vector on the boundary. Furthermore, if this solution is smooth then it is in fact a static black hole solution. In particular, in the vacuum case, critical points only occur at exterior Schwarzschild solutions; that is, critical points of the mass over this space do not exist generically. Finally, we briefly discuss a version of the result when the boundary data is related to Bartnik’s geometric boundary data. In particular, by imposing different boundary conditions on the Killing vector, we show that stationary solutions in this case correspond to critical points of an energy resembling the difference between the ADM mass and the Brown–York mass of the boundary.

Anisotropic spheres via embedding approach in R+βR2 gravity with matter coupling

The manifesto of the current article is to investigate the compact anisotropic matter profiles in the context of one of the modified gravitational theories, known as f(R,T) gravity, where R is a Ricci Scalar and T is the trace of the energy–momentum tensor. To achieve the desired goal, we capitalized on the spherical symmetric space–time and utilized the embedding class-1 solution via Karmarkar’s condition in modeling the matter profiles. To calculate the unidentified constraints, Schwarzschild exterior solution along with experimental statistics of three different stars LMC X-4, Cen X-3, and EXO 1785-248 are taken under consideration. For the evaluation of the dynamical equations, a unique model f(R,T)=R+βR2+λT has been considered, with β and λ being the real constants. Different physical aspects have been exploited with the help of modified dynamical equations. Conclusively, all the stars under observations are realistic, stable, and are free from all singularities.

On Schwarzschild’s Interior Solution and Perfect Fluid Star Model

We solve the boundary value problem for Einstein’s gravitational field equations in the presence of matter in the form of an incompressible perfect fluid of density ρ and pressure field p(r) located in a ball r≤r0. We find a 1-parameter family of time-independent and radially symmetric solutions ga,ρa,pa:−2m<a<a1 satisfying the boundary conditions g=gS and p=0 on r=r0, where gS is the exterior Schwarzschild solution (solving the gravitational field equations for a point mass M concentrated at r=0) and containing (for a=0) the interior Schwarzschild solution, i.e., the classical perfect fluid star model. We show that Schwarzschild’s requirement r0>9κM/(4c2) identifies the “physical” (i.e., such that pa(r)≥0 and pa(r) is bounded in 0≤r≤r0) solutions {pa:a∈U0} for some neighbourhood U0⊂(−2m,+∞) of a=0. For every star model {ga:a0<a<a1}, we compute the volume V(a) of the region r≤r0 in terms of abelian integrals of the first, second, and third kind in Legendre form.

Anisotropic compact stellar model of embedding class-I satisfying Karmarkar’s condition in Vaidya and Tikekar spheroidal geometry

We present a class of solutions for a spherically symmetric anisotropic matter distribution in Vaidya and Tikekar spheroidal geometry. Making use of the Vaidya and Tikekar (VT) metric ansatz (J Astrophys Astron 3:325, 1982) for one of the metric functions $$g_{rr}$$ , we obtain the unknown metric function $$g_{tt}$$ by utilizing the Karmakar’s embedding condition which makes the master equation tractable. The model parameters are fixed by utilizing the matching conditions of the interior spacetime and the exterior Schwarzschild solution at the boundary of the star where the radial pressure vanishes. The closed-form interior solution of the Einstein field equations thus obtained is shown to be capable of describing ultra-compact stellar objects where anisotropy might develop. The current estimated masses and radii of some other pulsars are utilized to show that the developed model meets all the requirements of a realistic star. The stability of the model is analyzed. The dependence of the curvature parameter K of the VT model, which characterizes a departure from homogeneous spherical distribution, is also investigated.

Existence and stability of relativistic fluid spheres supported by thin shells

We propose two models for constant density relativistic perfect-fluid spheres supported by thin shell configurations. These models are obtained from the Schwarzschild constant density star solution: the first via the collapse of the external layers of the fluid into a thin shell by performing a matching with the exterior Schwarzschild solution at a matching radius smaller than the star radius; and the second via the creation of a vacuum bubble inside the star by matching it with an interior Minkowski spacetime. Both models are shown to satisfy both the weak and the strong energy conditions (WEC and SEC) and can have a compactness arbitrarily close to that of a black-hole without developing singularities at the center, thus being exceptions to the Buchdahl limit. We compute the stability regimes of the models proposed and we show that there are combinations of the star radius $R$ and the matching radius $R_\Sigma$ for which the solutions are stable, the dominant energy condition (DEC) is satisfied, and the radius of the object is smaller than $3M$, implying that these models could be used as models for dark matter or exotic compact objects.

Gravitational field of a pit and maximal mass defects

A general relativistic solution, composed of a Zel'dovich-Letelier interior made of radial strings matched through a spherical thin shell at radius $r_0$ to an exterior Schwarzschild solution with mass $m$, is presented. It is the Zel'dovich-Letelier-Schwarzschild star. When the radius $r_0$ of the star is shrunk to its gravitational radius $2m$, the solutions have interesting properties. There are solutions with $m=0$ and $r_0=0$ that obey $\frac{2m}{r_0}=1$. The solutions have a horizon, but are not black holes, they are quasiblack holes, though atypical. The proper mass $m_p$ of the interior is nonzero and made of one string. Hence, a Minkowski exterior space hides an interior with matter in a pit. These are the pit solutions and show a maximal mass defect. There are two classes of pit solutions, one a finite string and the other a semi-infinite one. These pits are really string pits, that can be seen as Wheeler bags of gold, albeit squashed bags. There is another class, which is a compact stringy star at the $\frac{2m}{r_0}=1$ limit with $m$ nonzero. It is a typical quasiblack hole and it has maximal mass defect. Generically pit solutions with $\frac{2m}{r_0}=1$ and $m=0$ can exist with maximal mass defects. The Zel'dovich-Letelier-Schwarzschild star at the $r_0=2m$ limit is an instance of it. These three classes of static solutions yield the same spectrum of solutions that appear in critical gravitational collapse, there are solutions that yield naked null singularities, which here are the two string pit classes, there are solutions that yield black holes, which here are represented by the compact stringy stars at the quasiblack hole limit, and the solutions that disperse away in critical collapse here are the static Zel'dovich-Letelier-Schwarzschild stars. Thermodynamics of the string pit and stringy star quasiblack hole solutions is provided, and other connections are mentioned.

Linking Conditions for Models with Geometrical Basis

Usually it is demanded that the metric and its 1st derivatives have to match at the boundary of two adjacent regions which are solutions to Einstein’s field equation. We propose a new linking condition concerning gravitational models based on surfaces which could be embedded into a higher dimensional flat space. We probe this condition for the Schwarzschild interior and exterior solution.

Стационарное вакуумное решение уравнений Эйнштейна

We investigate a stationary generalization of the static metric. The static -metric is a variant of the Zipoy-Voorhees metric and simplest generalization of the Schwarzschild metric, containing a quadrupole parameter. In the present work, we introduce the stationary version of the -metric, and this stationary metric find by using the complex Ernst potential . The metric function determined by two first-order differential equations that can be integrated by quadratures once  is known. To obtain an explicit form of the new Ernst potential, we use the solution generation techniques that allows us to generate stationary solutions from a static solution. It possesses three independent parameters related to the mass, quadrupole moment and angular momentum. We investigate the geometric and physical properties of this exact stationary solution of Einstein’s vacuum equations and show that it can be used to describe the exterior gravitational field of rotating, axially symmetric, compact objects. According to the relativistic invariant Geroch definition, we analyze multipole structure using the corresponding Ernst function and we compute the lowest ten relativistic multipole moments for the static quadrupole metric. The particular choice of parameters we obtain the known solutions. i.e., the exterior Schwarzschild Solution find with the vanishing quadrupole and rotating parameter, Corresponding static -metric find with the vanishing rotating parameter and non-zero quadrupole parameter. The multipole moments of the well-known Kerr solution are given by the vanishing quadrupole parameter and non-zero rotating parameter.

Mathematical Model of Relativistic Anisotropic Compact Stellar Model Admitting Linear Equation of State

In present paper, a static, spherically symmetric, anisotropic stellar object has been discussed by assuming a linear relationship between the matter density ρ and radial pressure pr. The interior solution is continuously matched with the exterior Schwarzschild vacuum solution at the junction interface. Various physical features viz. energy conditions, mass-radius relationship, stability are analyzed for our stellar model. By assigning some particular value to the arbitrary constants we have obtained a model of compact star of radius 6.7 km. and mass 1.148 M⊙, which is very close to the observational data of the compact star Her X-1 proposed by Rawls et al. [Rawls, et al., Astrophys. J. 730 (2011) 25]. We have obtained that the model satisfies all the regularity conditions. We have found that our proposed model is stable as well as singularity-free.

A generalized family of anisotropic compact object in general relativity

We present model for anisotropic compact star under the general theory of relativity of Einstein. In the study a 4-dimensional spacetime has been considered which is embedded into the 5-dimensional flat metric so that the spherically symmetric metric has class 1. A set of solutions for the field equations are found depending on the index n involved in the physical parameters. The interior solutions have been matched smoothly at the boundary of the spherical distribution to the exterior Schwarzschild solution which necessarily provides values of the unknown constants. We have chosen the values of n as n=2 and n=10 to 20000 for which interesting and physically viable results can be found out. The investigations on the physical features of the model include several astrophysical issues, like (i) regularity behaviour of stars at the centre, (ii) well behaved condition for velocity of sound, (iii) energy conditions, (iv) stability of the system via the following three techniques — adiabatic index, Herrera cracking concept and TOV equation, (v) total mass, effective mass and compactification factor and (vi) surface redshift. Specific numerical values of the compact star candidates LMC X-4 and SMC X-1 are calculated for central and surface densities as well as central pressure to compare the model value with actual observational data.