General Relativistic Stars: Polytropic Equations of State (original) (raw)
Related papers
Annals of Physics, 2003
We investigate static spherically symmetric perfect fluid models in Newtonian gravity for barotropic equations of state that are asymptotically polytropic at low and high pressures. This is done by casting the equations into a three-dimensional regular dynamical system with bounded dependent variables. The low and high central pressure limits correspond to two two-dimensional boundary subsets, described by homology invariant equations for exact polytropes. Thus the formulation naturally places work about polytropes in a more general context. The introduced framework yields a visual aid for obtaining qualitative information about the solution space and is also suitable for numerical investigations. Moreover, it makes a host of mathematical tools from dynamical systems theory available, which allows us to prove several theorems about the relationship between the equation of state and properties concerning total masses and radii.
General Relativistic Stars: Linear Equations of State
Annals of Physics, 2000
In this paper Einstein's field equations, for static spherically symmetric perfect fluid models with a linear barotropic equation of state, are recast into a 3-dimensional regular system of ordinary differential equations on a compact state space. The system is analyzed qualitatively, using the theory of dynamical systems, and numerically. It is shown that certain special solutions play important roles as building blocks for the solution structure in general. In particular, these special solutions determine many of the features exhibited by solutions with a regular center and large central pressure. It is also shown that the present approach can be applied to more general classes of barotropic equations of state.
Relativistic stars with polytropic equation of state
The European Physical Journal Plus, 2015
We consider the Einstein-Maxwell system of equations in the context of isotropic coordinates for matter distributions with anisotropy in the presence of an electric field. We assume a polytropic equation of state for the matter configuration. New classes of exact solutions are generated for different polytropic indices. The model is well behaved and we can regain masses of several observed objects, in particular we obtain the mass of the star PSR J1903+327.
Dynamical systems approach to relativistic spherically symmetric static perfect fluid models
Classical and Quantum Gravity, 2003
We investigate relativistic spherically symmetric static perfect fluid models in the framework of the theory of dynamical systems. The field equations are recast into a regular dynamical system on a 3dimensional compact state space, thereby avoiding the non-regularity problems associated with the Tolman-Oppenheimer-Volkoff equation. The global picture of the solution space thus obtained is used to derive qualitative features and to prove theorems about mass-radius properties. The perfect fluids we discuss are described by barotropic equations of state that are asymptotically polytropic at low pressures and, for certain applications, asymptotically linear at high pressures. We employ dimensionless variables that are asymptotically homology invariant in the low pressure regime, and thus we generalize standard work on Newtonian polytropes to a relativistic setting and to a much larger class of equations of state. Our dynamical systems framework is particularly suited for numerical computations, as illustrated by several numerical examples, e.g., the ideal neutron gas and examples that involve phase transitions. We will see in Sec. 3 that, e.g., C 2 is sufficient, although this restriction can be weakened. Below we show how to handle even less restrictive situations like phase transitions. 7 However, note that ρ stands for the rest-mass density in the Newtonian case. 8 The region where m(r) < 0 can be analyzed with the same dynamical systems methods that are going to be used in the following. The treatment turns even out to be considerably simpler, cf. the case of Newtonian perfect fluids . If a solution to (3) satisfies 1 − 2Gm/(rc 2 ) > 0 initially at r = r 0 for initial data (m 0 , p 0 ), then this condition holds everywhere. This has been proved (for regular solutions), e.g., in [1]. Within the dynamical systems formulation this result can be established quite easily as we will see in Sec. 4. Solutions violating the condition 1 − 2Gm/(rc 2 ) > 0 could be treated with the dynamical systems methods presented in this paper as well. However, we refrain from a discussion of such solutions here.
Relativistic polytropic equations of state in Hořava gravity and Einstein-æther theory
The equations of state for a characteristic spacetime are studied in the context of the spherically symmetric interior exact and analytical solutions in Hořava gravity and Einstein-æther theory in which anisotropic fluids are considered. In particular, for a given anisotropic interior solution, the equations of state relating the density to the radial and tangential pressure are derived, by means of a polynomial best fit. Moreover, the well-known relativistic polytropic equations of state are used in order to obtain the profile of the thermodynamical quantities inside the stellar object as provided by the specific exact solution considered. It is then shown that these equations of state need to be modified in order to account for the profiles of density and pressures.
Phys Rev D, 2002
This is the second in a series of papers on the construction and validation of a three-dimensional code for the solution of the coupled system of the Einstein equations and of the general relativistic hydrodynamic equations, and on the application of this code to problems in general relativistic astrophysics. In particular, we report on the accuracy of our code in the long-term dynamical evolution of relativistic stars and on some new physics results obtained in the process of code testing. The following aspects of our code have been validated: the generation of initial data representing perturbed general relativistic polytropic models (both rotating and nonrotating), the long-term evolution of relativistic stellar models, and the coupling of our evolution code to analysis modules providing, for instance, the detection of apparent horizons or the extraction of gravitational waveforms. The tests involve single nonrotating stars in stable equilibrium, nonrotating stars undergoing radial and quadrupolar oscillations, nonrotating stars on the unstable branch of the equilibrium configurations migrating to the stable branch, nonrotating stars undergoing gravitational collapse to a black hole, and rapidly rotating stars in stable equilibrium and undergoing quasiradial oscillations. We have carried out evolutions in full general relativity and compared the results to those obtained either with perturbation techniques, or with lower dimensional numerical codes, or in the Cowling approximation (in which all the perturbations of the spacetime are neglected). In all cases an excellent agreement has been found. The numerical evolutions have been carried out using different types of polytropic equations of state using either the rest-mass density only, or the rest-mass density and the internal energy as independent variables. New variants of the spacetime evolution and new high resolution shock capturing treatments based on Riemann solvers and slope limiters have been implemented and the results compared with those obtained from previous methods. In particular, we have found the ``monotonized central differencing'' limiter to be particularly effective in evolving the relativistic stellar models considered. Finally, we have obtained the first eigenfrequencies of rotating stars in full general relativity and rapid rotation. A long standing problem, such frequencies have not been obtained by other methods. Overall, and to the best of our knowledge, the results presented in this paper represent the most accurate long-term three-dimensional evolutions of relativistic stars available to date.
A Spherical Relativistic Anisotropic Compact Star Model
International Journal of Astronomy and Astrophysics, 2018
We provide solutions to Einsteins field equations for a model of a spherically symmetric anisotropic fluid distribution, relevant to the description of compact stars. The central matter-energy density, radial and tangential pressures, red shift and speed of sound are positive definite and are decreasing monotonically with increasing radial distance from the center of matter distribution of astrophysical object. The causality condition is satisfied for complete fluid distribution. The central value of anisotropy is zero and is increasing monotonically with increasing radial distance from the center of the distribution. The adiabatic index is increasing with increasing radius of spherical fluid distribution. The stability conditions in relativistic compact star are also discussed in our investigation. The solution is representing the realistic objects
General solution for a relativistic star
Classical and Quantum Gravity, 1997
In a spherically symmetric spacetime we find the general solution for a relativistic star in hydrostatic equilibrium having the spheroidal geometry for the 3-space embedded in 4-Euclidean space. The parameter λ is the measure of spheroidal character and determines the physical properties of the star. It has a lower bound > 3/17; stars with smaller mass to radius ratio can occur for all allowed λ while ultracompact stars, having ratios between 1/3 and 1/2, will have λ 0.9. It turns out that a wide range of values of λ can represent objects with the fluid density of a neutron star. The model is shown to possess all the desirable physical features.
The European Physical Journal C
We present an algorithm to generalize a plethora of well-known solutions to Einstein field equations describing spherically symmetric relativistic fluid spheres by relaxing the pressure isotropy condition on the system. By suitably fixing the model parameters in our formulation, we generate closed-form solutions which may be treated as an anisotropic generalization of a large class of solutions describing isotropic fluid spheres. From the resultant solutions, a particular solution is taken up to show its physical acceptability. Making use of the current estimate of mass and radius of a known pulsar, the effects of anisotropic stress on the gross physical behaviour of a relativistic compact star is also highlighted.
Generalized relativistic anisotropic models for compact stars
arXiv: General Relativity and Quantum Cosmology, 2015
We present new anisotropic generalization of Buchdahl [1] type perfect fluid solution by using the method of earlier work [2]. In similar approach we have constructed the new pressure anisotropy factor Delta{\Delta}Delta by the help both the metric potential elambdae^{\lambda}elambda and enue^{\nu}enu. The metric potential elambdae^{\lambda}elambda same as Buchdahl [1] and enue^{\nu}enu is monotonic increasing function as suggested by Lake [3]. After that we obtain new well behaved general solution for anisotropic fluid distribution. We calculated the physical quantities like energy density, radial and tangential pressures, velocity of sound and red-shift etc. We observe that these quantities are positive and finite inside the compact star. Also note that mass and radius of our models can represent the structure of realistic astrophysical objects such as like Her X-1 and RXJ1856-37.