The dynamical instability of relativistic polytropic spheres, embedded in a spacetime with a repulsive cosmological constant, is studied in the framework of general relativity. We apply the methods used in our preceding paper to study the trapping polytropic spheres with $Λ= 0$, namely, the critical point method and the infinitesimal and adiabatic radial perturbations method developed by Chandrasekhar. We compute numerically the critical adiabatic index, as a function of the parameter $σ= p_{\mathrm{c}}/(ρ_{\mathrm{c}} c^2)$, for several values of the cosmological parameter $λ$ giving the ratio of the vacuum energy density to the central energy density of the polytrope. We also determine the critical values for the parameter $σ_{\mathrm{cr}}$, for the onset of instability, by using both approaches. We found that for large values of the parameter $λ$, the differences between the values of $σ_{\mathrm{cr}}$ calculated by the critical point method differ from those obtained via the radial perturbations method. Our results, given by both applied methods, indicate that large values of the cosmological parameter $λ$ have relevant effects on the dynamical stability of the polytropic configurations.
We compute the equilibrium, the fundamental eigenfrequency of oscillations modes, and quadrupolar tidal deformability of anisotropic polytropic spheres. These studies are respectively performed through the numerical solution of the Tolman-Oppenheimer-Volkoff equation, Chandrasekhar radial oscillation equations, and nonlinear first-order Riccati equation for tidal deformability, all modified from their original version to include the anisotropic effects. For the polytropic exponent $γ=2$ and the anisotropic model of Cattoen, Faber, and Visser, we show that the anisotropy could be reflected in t…
We study the static stellar equilibrium configurations ofuncharged and charged spheres composed by a relativistic polytropic fluid, and compare with those of spheres composed by a non-relativistic polytropic fluid, the later case already being studied in a previous work [J. D. Arbañil, P. S. Lemos, V. T. Zanchin, Phys. Rev. D \textbf{88}, 084023 (2013)]. In the relativistic fluid case, a relativistic polytropic equation of state, $p=ωδ^γ$, is assumedd. Here, $δ=ρ-p/(γ-1)$, with $δ$ and $ρ$ being the rest mass density and the energy density, respectively, and $ω$ and $γ$ are respectively the po…
In this work we evaluate the physical acceptability of relativistic anisotropic spheres modeled by two polytropic equations of state -- with the same newtonian limit -- commonly used to describe compact objects in General Relativity. We integrate numerically the corresponding Lane-Emden equation in order to get density, mass and pressure profiles. An ansatz is used for the anisotropic pressure allowing us to have material configurations slightly deviated from isotropic condition. Numerical models are classified in a parameter space according to the number of physical acceptability conditions t…
We examine static spherically symmetric polytropic spheres in Palatini f(R) gravity and show that no regular solutions to the field equations exist for physically relevant cases such as a monatomic isentropic gas or a degenerate electron gas, thus casting doubt on the validity of Palatini f(R) gravity as an alternative to General Relativity.
Within $R^2$ gravity, we study the linear stability of strongly gravitating spherically symmetric configurations supported by a polytropic fluid. All calculations are carried out in the Jordan frame. It is demonstrated that, as in general relativity, the transition from stable to unstable systems occurs at the maximum of the curve mass-central density of the fluid.
We use standard polynomial expansion technique to show the existence of a relation between polytropic model and the description of gas spheres at finite temperature. A numerical analysis is made concerning the obtained perturbative parameters. It is shown that there is a correspondence between polytropic and gas sphere thermal models for fermions and bosons. For instance, the polytropic index $n$ displays evident correlation with temperature and chemical potential.
We investigate spherically symmetric, static matter configurations with polytropic equation of state for a class of f(R) models in Palatini formalism and show that the surface singularities recently reported in the literature are not physical in the case of Planck scale modified lagrangians. In such cases, they are just an artifact of the idealized equation of state used. In fact, we show that for the models f(R)=R\pmλR^2, with λon the order of the Planck length squared, the presence of a single electron in the Universe would be enough to cure all stellar singularities of this type. From our a…
We explore a class of compact charged spheres made of a charged perfect fluid with a polytropic equation of state. The charge density is chosen to be proportional to the energy density. The study is performed by solving the Tolman-Oppenheimer-Volkoff (TOV) equation which describes the hydrostatic equilibrium. We show the dependence of the structure of the spheres for several characteristic values of the polytropic exponent and for different values of the charge densities. We also study other physical properties of the charged spheres, such as the total mass, total charge, radius and sound spee…
We complete the stability study of general relativistic spherically symmetric polytropic perfect fluid spheres, concentrating attention to the newly discovered polytropes containing region of trapped null geodesics. We compare the methods of treating the dynamical stability based on the equation governing infinitesimal radial pulsations of the polytropes and the related Sturm-Liouville eigenvalue equation for the eigenmodes governing the pulsations, to the methods of stability analysis based on the energetic considerations. Both methods are applied to determine the stability of the polytropes …
Analogue systems are a powerful instrument to investigate and understand in a controlled setting many general-relativistic effects. Here, we focus on superradiant-triggered instabilities and quasi-normal modes. We consider a compressible hydrodynamic vortex characterized by a polytropic equation of state, the {\it polytropic hydrodynamic vortex}, a purely circulating system with an ergoregion but no event horizon. We compute the quasinormal modes of this system numerically with different methods, finding excellent agreement between them. When the fluid velocity is larger than the speed of soun…
We demonstrate that in the framework of standard general relativity polytropic spheres with properly fixed polytropic index $n$ and relativistic parameter $σ$, giving ratio of the central pressure $p_\mathrm{c}$ to the central energy density $ρ_\mathrm{c}$, can contain region of trapped null geodesics. Such trapping polytropes can exist for $n > 2.138$ and they are generally much more extended and massive than the observed neutron stars. We show that in the $n$--$σ$ parameter space the region of allowed trapping increase with polytropic index for interval of physical interest $2.138 < n < 4$. …
We study behaviour of gravitational waves in the recently introduced general relativistic polytropic spheres containing a region of trapped null geodesics extended around radius of the stable null circular geodesic that can exist for the polytropic index $N>2.138$ and the relativistic parameter, giving ratio of the central pressure $p_\mathrm{c}$ to the central energy density $ρ_\mathrm{c}$, higher than $σ= 0.677$. In the trapping zones of such polytropes, the effective potential of the axial gravitational wave perturbations resembles those related to the ultracompact uniform density objects, …
In this paper we investigate gravitationally bound, spherically symmetric equilibrium configurations consisting of ordinary (polytropic) matter nonminimally coupled to an external chameleon scalar field. We show that this system has static, regular, asymptotically flat general relativistic solutions. The properties of these spherical configurations, such as total mass, distribution of matter, and size, depend strongly on the surrounding scalar field. The mass is found in terms of the parameter $σ$, which is proportional to the central mass density of the ordinary matter. We perform a stability…
Non-vacuum static spherically-symmetric solutions in Palatini f(R) gravity are examined. It is shown that for generic choices of f(R), there are commonly-used equations of state for which no satisfactory physical solution of the field equations can be found within this framework, apart from in the special case of General Relativity, casting doubt on whether Palatini f(R) gravity can be considered as giving viable alternatives to General Relativity.
Dark matter halos modeled by polytropic spheres influenced by the relict cosmological constant and trapping polytropes forming supermassive black holes
We determine the onset point of secular instability for the nonaxisymmetric bar mode in rigidly rotating equilibrium configurations in the Post-Newtonian approximation, in order to apply it to neutron stars. The treatment is based on a precedent Newtonian analytic energy variational method which we have extended to the Post-Newtonian case. This method, based upon Landau's theory of second-order phase transitions, provides the critical value of the ellipsoid polar eccentricity e at the onset of viscosity-driven instability and it is valid for any equation of state. The extension of this method …
Ultra-compact objects describe horizonless solutions of the Einstein field equations which, like black-hole spacetimes, possess null circular geodesics (closed light rings). We study {\it analytically} the physical properties of spherically symmetric ultra-compact isotropic fluid spheres with a polytropic equation of state. It is shown that these spatially regular horizonless spacetimes are generally characterized by two light rings $\{r^{\text{inner}}_γ,r^{\text{outer}}_γ\}$ with the property ${\cal C}(r^{\text{inner}}_γ)\leq{\cal C}(r^{\text{outer}}_γ)$, where ${\cal C}\equiv m(r)/r$ is the …
The main point of this paper is a suggestion about the proper treatment of the photon gas in a theory of stellar structure and other plasmas. This problem arises in the study of polytropic gas spheres, where we have already introduced some innovations. The main idea, already advanced in the contextof neutral, homogeneous, polytropic stellar models, is to base the theory firmly on a variational principle. Another essential novelty is to let mass distribution extend to infinity, the boundary between bulk and atmosphere being defined by an abrupt change in the polytropic index, triggered by the d…
The effects of the cosmological constant on the static equilibrium configurations and stability against small radial perturbations of relativistic polytropic spheres are investigated. This study numerically solves the hydrostatic equilibrium equation and the radial stability equation, both of which are modified from their standard form to introduce the cosmological constant. For the fluid, we consider a pressure $p$ and an energy density $ρ$, which are connected through the equation of state $p=κδ^Γ$ with $δ=ρ-p/(Γ-1)$, where $κ$, $Γ$ and $δ$ represent the polytropic constant, adiabatic index …
We study the dynamical instability against bar-mode deformation of differentially rotating stars. We performed numerical simulation and linear perturbation analysis adopting polytropic equations of state with the polytropic index $n=1$. It is found that rotating stars of a high degree of differential rotation are dynamically unstable even for the ratio of the kinetic energy to the gravitational potential energy of $O(0.01)$. Gravitational waves from the final nonaxisymmetric quasistationary states are calculated in the quadrupole formula. For rotating stars of mass $1.4M_{\odot}$ and radius se…
The Sturm-Liouville eigenvalue equation for eigenmodes of the radial oscillations is determined for spherically symmetric perfect fluid configurations in spacetimes with a nonzero cosmological constant and applied in the cases of configurations with uniform distribution of energy density and polytropic spheres. It is shown that a repulsive cosmological constant rises the critical adiabatic index and decreases the critical radius under which the dynamical instability occurs.
We present results on the effect of the stiffness of the equation of state on the dynamical bar-mode instability in rapidly rotating polytropic models of neutron stars in full General Relativity. We determine the change in the threshold for the emergence of the instability for a range of the adiabatic $Γ$ index from 2.0 to 3.0, including two values chosen to mimic more realistic equations of state at high densities.
As an extension of our previous work, we investigate the dynamical instability against nonaxisymmetric bar-mode deformations of differentially rotating stars in Newtonian gravity varying the equations of state and velocity profiles. We performed the numerical simulation and the followup linear stability analysis adopting polytropic equations of state with the polytropic indices n=1, 3/2, and 5/2 and with two types of angular velocity profiles (the so-called j-constant-like and Kepler-like laws). It is confirmed that rotating stars of a high degree of differential rotation are dynamically unsta…
In this paper, we study the dynamical instability of gaseous sphere under radial oscillations approaching the Reissner-Nordström limit. For this purpose, we derive linearized perturbed equation of motion following the Eulerian and Lagrangian approaches. We formulate perturbed pressure in terms of adiabatic index by employing the conservation of baryon numbers. A variational principle is established to evaluate characteristic frequencies of oscillations which lead to the criteria for dynamical stability. The dynamical instability of homogeneous sphere as well as relativistic polytropes with dif…
We present results about the effect of the use of a stiffer equation of state, namely the ideal-fluid $Γ=2.75$ ones, on the dynamical bar-mode instability in rapidly rotating polytropic models of neutron stars in full General Relativity. We determine the change on the critical value of the instability parameter $β$ for the emergence of the instability when the adiabatic index $Γ$ is changed from 2 to 2.75 in order to mimic the behavior of a realistic equation of state. In particular, we show that the threshold for the onset of the bar-mode instability is reduced by this change in the stiffness…
The orbital separation of compact binary stars will shrink with time due to the emission of gravitational radiation. This inspiralling phase of a binary system's evolution generally will be very long compared to the system's orbital period, but the final coalescence may be dynamical and driven to a large degree by hydrodynamic effects, particularly if there is a critical separation at which the system becomes dynamically unstable toward merger. Indeed, if weakly relativistic systems (such as white dwarf-white dwarf binaries) encounter a point of dynamical instability at some critically close s…
With the question, ``Can relativistic charged spheres form extremal black holes?" in mind, we investigate the properties of such spheres from a classical point of view. The investigation is carried out numerically by integrating the Oppenheimer-Volkov equation for relativistic charged fluid spheres and finding interior Reissner-Nordström solutions for these objects. We consider both constant density and adiabatic equations of state, as well as several possible charge distributions, and examine stability by both a normal mode and an energy analysis. In all cases, the stability limit for these s…
A main question in astrophysics and cosmology has been the severe stability of the astrophysical objects, whether a particular equilibrium configuration is stable. In this article, we study the relativistic self-gravitating, hydrostatic spheres with a polytropic equation of state , considering structures with the polytropic indices and illustrates the results for the relativistic parameters . We determined the critical relativistic parameter at which the mass of the polytrope has a maximum value and represents the first mode of radial instability. For n=1 (0.5)-2.5, stable relativistic polytro…
The objective of this paper is to discuss the dynamical instability in the context of Newtonian and post Newtonian regimes. For this purpose, we consider non-viscous heat conducting charged isotropic fluid as a collapsing matter with cylindrical symmetry. Darmois junction conditions are formulated. The perturbation scheme is applied to investigate the influence of dissipation and electromagnetic field on the dynamical instability. We conclude that the adiabatic index $Γ$ has smaller value for such a fluid in cylindrically symmetric than isotropic sphere.
The equations describing the adiabatic, small radial oscillations of general relativistic stars are generalized to include the effects of a cosmological constant. The generalized eigenvalue equation for the normal modes is used to study the changes in the stability of the homogeneous sphere induced by the presence of the cosmological constant. The variation of the critical adiabatic index as a function of the central pressure is studied numerically for different trial functions. The presence of a large cosmological constant significantly increases the value of the critical adiabatic index. The…
We investigate the viscosity driven instability in rotating relativistic stars by means of an iterative approach. We focus on polytropic rotating equilibrium stars and impose an m=2 perturbationin the lapse. We vary both the stiffness of the equation of state and the compactness of the star to study those effects on the value of the threshold. For a uniformly rotating star, the criterion T/W, where T is the rotational kinetic energy and W is the gravitational binding energy, mainly depends on the compactness of the star and takes values around 0.13 ~ 0.16, which differ slightly from that of Ne…
In this note, I derive the Chandrasekhar instability of a fluid sphere in ($N$+1)-dimensional Schwarzschild-Tangherlini spacetime and take the homogeneous (uniform energy density) solution for illustration. Qualitatively, the effect of positive (negative) cosmological constant tends to destabilize (stabilize) the sphere. In the absence of cosmological constant, the privileged position of (3+1)-dimensional spacetime is manifest in its own right. As it is the marginal dimensionality in which a monatomic ideal fluid sphere is stable but not too stable to trigger the onset of gravitational collaps…
We explore the effects of a positive cosmological constant on astrophysical and cosmological configurations described by a polytropic equation of state. We derive the conditions for equilibrium and stability of such configurations and consider some astrophysical examples where our analysis may be relevant. We show that in the presence of the cosmological constant the isothermal sphere is not a viable astrophysical model since the density in this model does not go asymptotically to zero. The cosmological constant implies that, for polytropic index smaller than five, the central density has to e…
We show that for every asymptotically AdS solution compactified on a torus there is a corresponding Ricci-flat solution obtained by replacing the torus by a sphere, performing a Weyl rescaling of the metric and appropriately analytically continuing the dimension of the torus/sphere (as in generalized dimensional reduction). In particular, it maps Minkowski spacetime to AdS on a torus, the holographic stress energy tensor of AdS to the stress energy tensor due to a brane localized in the interior of spacetime and AdS black branes to (asymptotically flat) Schwarzschild black branes. Applying it …
The transonic phenomenon of black hole accretion and the existence of the photon sphere are the characteristics of strong gravitational fields near a black hole horizon. In this work, we study spherical accretion flow onto a general parametrized spherically symmetric black hole spacetimes. For this purpose, we analyze the accretion process of various perfect fluids, such as the isothermal fluid of ultra-stiff, ultra-relativistic, and sub-relativistic types and polytropic fluid, respectively. The influences of extra parameters beyond the Schwarzschild black hole in the general parameterized sph…
