Buchdahl Limit Research Articles

On the radial stability of ultra-compact Schwarzschild stars beyond the Buchdahl limit

In this paper we used the theory of adiabatic radial oscillations developed by Chandrasekhar to study the conditions for dynamical stability of constant energy-density stars, or Schwarzschild stars, in the unstudied ultra compact regime beyond the Buchdahl limit, that is, for configurations with radius R in the range , where is the Schwarzschild radius of the star. These recently found analytical solutions exhibit a negative pressure region in their centre and, in the limit when , the full interior region of the star becomes filled with negative pressure. Here we present a systematic analysis of the stability of these configurations against radial perturbations. We found that, contrary to the usual expectation found in many classical works, the ultra compact Schwarzschild star is stable against radial oscillations. We computed values of the critical adiabatic index for several stellar models with varying radius and found that it also approaches a finite value as .

Bootstrapping Newtonian gravity

A nonlinear equation obtained by adding gravitational self-interaction terms to the Poisson equation for Newtonian gravity is here employed in order to analyze a static spherically symmetric homogeneous compact source of given proper mass and radius and the outer vacuum. The main feature of this picture is that, although the freedom of shifting the potential by an arbitrary constant is of course lost, the solutions remain qualitatively very close to the Newtonian behavior. We also notice that the negative gravitational potential energy is smaller than the proper mass for sources with small compactness, but for sources that should form black holes according to general relativity, the gravitational potential energy becomes of the same order of magnitude of the proper mass, or even larger. Moreover, the pressure overcomes the energy density for large values of the compactness, but it remains finite for finite compactness; hence there exists no Buchdahl limit. This classical description is meant to serve as the starting point for investigating quantum features of (near) black hole configurations within the corpuscular picture of gravity in future developments.

A new generalised solution to generate anisotropic compact star models in the Karmarkar space-time manifold

In the present article a new generalised solution is obtained for anisotropic matter configuration using Karmarkar’s condition. The solution is used to model the interior structure of anisotropic relativistic objects as it satisfies all necessary physical conditions. The pressure, density and metric potentials are free from any singularities and exhibit well behaved nature inside the anisotropic fluid sphere. The TOV equation is well maintained within the stellar configuration and all energy conditions hold good. The causality condition is well satisfied for our stellar models and stability of compact star models is further verified via Herrera’s cracking method. Harrison-Zeldovich-Novikov criterion for stability is also satisfied by our model. The adiabatic index is greater than $\frac{4}{3}$ throughout the stellar interior and the compactification factor also lies within the Buchdahl limit i.e. $M/R \le 4/9$ . We investigate the models for two compact stars PSRJ0348+0432 and SAX J1808.4-3658 within the framework of the general theory of relativity. The estimated mass and radius are in close agreement with the observational data. we extensively study the solutions corresponding to compact star PSRJ0348+0432 for $ n = 0, 1, 2, 3, 4, 4.5$ and the detailed graphical analysis is provided to substantiate the viability of the compact star model. One specific feature of our solution is that for large values of n, i.e. for $ n > 5$ solution reduces to Finch and Skea type solution.

A new model for strange stars

In the present work, we attempt to find a new class of solutions for the spherically symmetric perfect fluid sphere by employing the Homotopy Perturbation Method (HPM), a new tool via which the mass polynomial function facilitates to tackle the Einstein field equations. A set of interior solutions found on the basis of the simplest MIT bag model equation of state (EOS) in the form $p=\frac{1}{3}(\rho-4B)$ where $B$ is the bag constant. The proposed interior metric for the stellar system is consistent with the exterior Schwarzschild spacetime on the boundary. In addition, we also conduct a detailed study on different tests, viz. the energy conditions, TOV equation, adiabatic index, Buchdahl limit, etc., to verify the physical validity of the proposed model. The numerical value of the used parameters is predicted for different strange star candidates, for different chosen values of the bag constant. In a nutshell, by exploiting HPM technique first time ever in the field of relativistic astrophysics, we have predicted in the present literature a singularity-free and stable stellar model which is suitable to describe ultra-dense objects, like strange (quark) stars.

On gravitational-wave echoes from neutron-star binary coalescences

A tentative detection of gravitational-wave echoes in the post-merger signal of GW170817 has been recently claimed at significance level. It has been speculated that the signal might provide evidence for near-horizon quantum structures in the remnant exotic object. We point out that if the remnant object is an ultracompact neutron star, echoes are expected for objects with radius only slightly smaller than that of an ordinary neutron star. The reported echoes at ≈72 Hz are compatible with a toy model of incompressible star with mass approximately and radius close to the Buchdahl limit, . If confirmed, low-frequency gravitational-wave echoes would be in tension with all current neutron-star models and would have dramatic implications for nuclear physics and gravity.

Packing extra mass in compact stellar structures: an interplay between Kalb-Ramond field and extra dimensions

We have derived the Buchdahl's limit for a relativistic star in presence of the Kalb-Ramond field in four as well as in higher dimensions. It turns out that the Buchdahl's limit gets severely affected by the inclusion of the Kalb-Ramond field. In particular, the Kalb-Ramond field in four spacetime dimensions enables one to pack extra mass in any compact stellar structure of a given radius. On the other hand, a completely opposite picture emerges if the Kalb-Ramond field exists in higher dimensions, where the mass content of a compact star is smaller compared to that in general relativity. Implications are discussed.

Physical plausibility of cold star models satisfying Karmarkar conditions

In the present article, we have obtained a new well behaved solution to Einstein’s field equations in the background of Karmarkar spacetime. The solution has been used for stellar modelling within the demand of current observational evidences. All the physical parameters are well behaved inside the stellar interior and our model satisfies all the required conditions to be physically realizable. The obtained compactness parameter is within the Buchdahl limit, i.e. $ 2M/R \le 8/9$ . The TOV equation is well maintained inside the fluid spheres. The stability of the models has been further confirmed by using Herrera’s cracking method. The models proposed in the present work are compatible with observational data of compact objects 4U1608-52 and PSRJ1903+327. The necessary graphs have been shown to authenticate the physical viability of our models.

Expansion of Magnetic Neutron Stars in an Energy (in)Dependent Spacetime

Abstract Regarding the strong magnetic field of neutron stars and the high-energy regime scenario that is based on the high-curvature region near the compact objects, one is motivated to study magnetic neutron stars in an energy-dependent spacetime. In this paper, we show that such a strong magnetic field and energy dependency of spacetime have considerable effects on the properties of neutron stars. We examine the variations of maximum mass and related radius, Schwarzschild radius, average density, gravitational redshift, Kretschmann scalar, and Buchdahl theorem due to the magnetic field and energy dependency of the metric. First, it will be shown that the maximum mass and radius of neutron stars are increasing functions of the magnetic field, while average density, redshift, strength of gravity, and Kretschmann scalar are decreasing functions of it. These results are due to a repulsive-like force behavior for the magnetic field. Next, the effects of gravity’s rainbow will be studied, and it will be shown that by increasing the rainbow function, the neutron stars could enjoy an expansion in their structures. Then, we obtain a new relation for the upper mass limit of a static spherical neutron star with uniform density in gravity’s rainbow (Buchdahl limit) in which such an upper limit is modified as . In addition, stability and energy conditions for the equation of state of neutron star matter are investigated, and a comparison with empirical results is done. It is notable that the numerical study in this paper is conducted by using the lowest-order constrained variational approach in the presence of a magnetic field employing AV 18 potential.

New interior solution describing relativistic fluid sphere

A new exact solution of embedding class I is presented for a relativistic anisotropic massive fluid sphere. The new exact solution satisfies Karmarkar condition, is well-behaved in all respects, and therefore is suitable for the modelling of superdense stars. Consequently, using this solution, we have studied in detail two compact stars, namely, XTE J1739-289 (strange star $$1.51M_{\odot }$$ , 10.9 km) and PSR J1614-2230 (neutron star $$1.97M_{\odot }$$ , 14 km). The solution also satisfies all energy conditions with the compactness parameter lying within the Buchdahl limit.

Buchdahl compactness limit for a pure Lovelock static fluid star

We obtain the Buchdahl compactness limit for a pure Lovelock static fluid star and verify that the limit following from the uniform density Schwarzschild's interior solution, which is universal irrespective of the gravitational theory (Einstein or Lovelock), is true in general. In terms of surface potential $\Phi(r)$, it means at the surface of the star $r=r_{0}$, $\Phi(r_{0}) < 2N(d-N-1)/(d-1)^2$ where $d$, $N$ respectively indicate spacetime dimensions and Lovelock order. For a given $N$, $\Phi(r_{0})$ is maximum for $d=2N+2$ while it is always $4/9$, Buchdahl's limit, for $d=3N+1$. It is also remarkable that for $N=1$ Einstein gravity, or for pure Lovelock in $d=3N+1$, Buchdahl's limit is equivalent to the criteria that gravitational field energy exterior to the star is less than half its gravitational mass, having no reference to the interior at all.

A charged anisotropic well-behaved Adler–Finch–Skea solution satisfying Karmarkar condition

In the present paper, we discover a new well-behaved charged anisotropic solution of Einstein–Maxwell’s field equations. We ansatz the metric potential [Formula: see text] of the form given by Maurya et al. (Eur. Phys. J. C 76(12) (2016) 693) with [Formula: see text]. In their paper, it is mentioned that for [Formula: see text], the solution is not well-behaved for neutral configuration as the speed of sound is nondecreasing radially outward. However, the solution can represent a physically possible configuration with the inclusion of some net electric charge, i.e. the solution can become a well-behaved solution with decreasing sound speed radially outward for a charged configuration. Due to the inclusion of electric charge, the solution leads to a very stiff equation-of-state (EoS) with the velocity of sound at the center [Formula: see text] [Formula: see text] and the compactness parameter [Formula: see text] is close to the Buchdahl limit 0.889. This stiff EoS support a compact star configuration of mass [Formula: see text] and radius of 10.1[Formula: see text]km.

A comparative study between EGB gravity and GTR by modeling compact stars

In this paper we utilise the Krori–Barua ansatz to model compact stars within the framework of Einstein–Gauss–Bonnet (EGB) gravity. The thrust of our investigation is to carry out a comparative analysis of the physical properties of our models in EGB and classical general relativity theory with the help of graphical representation. From our analysis we have shown that the central density and central pressure of EGB star model is higher than the GTR star model. The most notable feature is that for both GTR and the EGB star model the compactness factor crosses the Buchdahl (Phys Rev 116:1027, 1959) limit.

A new relativistic stellar model with anisotropic fluid in Karmarkar space–time

We are presenting a new class of well-behaved solutions in embedding class-I. We proceed our calculations by assuming a new type of grr metric potential and solved for the other metric gtt using Karmarkar condition. The necessary condition that any solutions of Einstein’s field equations to be class-I is to satisfy Karmarkar condition and sufficient condition is to satisfy Pandey–Sharma condition i.e. R2323≠0. The solution also satisfies strong energy condition, null energy condition, dominant energy condition and weak energy condition. The obtained compactness parameter is within Buchdahl limit i.e. 2M/R≤8/9. The solution also satisfies the causality condition and can represent stable stellar fluid system as the adiabatic index Γ>4/3 and the stability factor holds −1≤vt2−vr2≤0 good. Finally, we have tuned our solution for two compact stars PSR J1614-2230 and 4U1608-52 which are well-behaved in all respects.

A family of well-behaved Karmarkar spacetimes describing interior of relativistic stars

We are presenting a family of new exact solutions for relativistic anisotropic stellar objects by considering four dimensional spacetime embedded in five dimensional Pseudo Euclidean space known as Class I solutions. These solutions are well-behaved in all respects, satisfy all energy conditions and the resulting compactness parameter is also within Buchdahl limit. The well-behaved nature of the solutions for a particular star solely depends on index n. We have discussed the solutions in detail for the neutron star XTE J1739-285 (M = 1.51M$\odot$, R = 10.9 km). For this particular star, the solution is well behaved in all respects for $8 \le n \le 20$. However, the solutions with n < 8 possess increasing trend of sound speed and the solutions belong to n > 20 disobey causality condition. Further, the well-behaved nature of the solutions for PSR J0348+0432 (2.01M$\odot$, 11 km), EXO 1785-248 (1.3M$\odot$, 8.85 km) and Her X-1 (0.85M$\odot$, 8.1 km) are specified by the index n with limits $24 \le n \le 54$, $1.5 \le n \le 4$ and $0.8 \le n \le 2.7$ respectively.

Buchdahl’s inequality in five dimensional Gauss–Bonnet gravity

The Buchdahl limit for static spherically symmetric isotropic stars is generalised to the case of five dimensional Gauss-Bonnet gravity. Our result depends on the sign of the Gauss-Bonnet coupling constant $\alpha$. When $\alpha>0$, we find, unlike in general relativity, that the bound is dependent on the stellar structure, in particular the central energy density. We find that stable stellar structures can exist arbitrarily close to the event horizon. Thus stable stars can exist with extra mass in this theory compared to five dimensional general relativity. For $\alpha<0$ it is found that the Buchdahl bound is more restrictive than the general relativistic case.

Polytropic spheres with electric charge: Compact stars, the Oppenheimer-Volkoff and Buchdahl limits, and quasiblack holes

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 speed and their dependence on the polytropic exponent. We find that for the polytropic exponent \gamma=4/3 the Chandrasekhar mass limit coincides with the Oppenheimer-Volkoff mass limit. We test the Oppenheimer-Volkoff limit for such compact objects. We also analyze the Buchdahl limit for these charged polytropic spheres, which happens in the limit of very high polytropic exponents, i.e., for a stiff equation of state. It is found that this limit is extremal and it is a quasiblack hole.

The Character of an Anisotropic and Spherically Symmetric Star in the Presence of a Cosmological Constant

From the gravitational structure equations of an anisotropic and spherically symmetric star in the presence of a cosmological constant, we derive the inequality which limits the mass-radius ratio, imposing energy conditions on the matter. From this inequality, we display how the cosmological constant modifies the Buchdahl limit, and express the bounds on both the mass-radius ratio and surface redshift in terms of the ratio between the cosmological constant and mean energy density of the star.

Buchdahl limit for d-dimensional spherical solutions with a cosmological constant

We investigate the conditions for which a d-dimensional perfect fluid solution is in hydrostatic equilibrium with a cosmological constant. We find a generalization of Buchdahl inequality and obtain an upper bound for the degree of compactification. Using the Tolman–Oppenheimer–Volkoff equation to get a lower bound for the degree of compactification we analyse the regions where the solution is in hydrostatic equilibrium. We obtain the inner metric solution and the pressure for the constant fluid density model.

Radiative falloff in neutron star spacetimes

We systematically study late-time tails of scalar waves propagating in neutron star spacetimes. We consider uniform density neutron stars, for which the background spacetime is analytic and the compaction of the star can be varied continously between the Newtonian limit 2M/R << 1 and the relativistic Buchdahl limit 2M/R = 8/9. We study the reflection of a finite wave packet off neutron stars of different compactions 2M/R and find that a Newtonian, an intermediate, and a highly relativistic regime can be clearly distinguished. In the highly relativistic regime, the reflected signal is dominated by quasi-periodic peaks, which originate from the wave packet bouncing back and forth between the center of the star and the maximum of the background curvature potential at R ~ 3 M. Between these peaks, the field decays according to a power-law. In the Buchdahl limit 2M/R -> 8/9 the light travel time between the center and the maximum or the curvature potential grows without bound, so that the first peak arrives only at infinitely late time. The modes of neutron stars can therefore no longer be excited in the ultra-relativistic limit, and it is in this sense that the late-time radiative decay from neutron stars looses all its features and gives rise to power-law tails reminiscent of Schwarzschild black holes.

A Maximum Mass-to-Size Ratio in Scalar-Tensor Theories of Gravity

We derive a modified Buchdahl inequality for scalar-tensor theories of gravity. In general relativity, Buchdahl has shown that the maximum value of the mass-to-size ratio, $2M/R$, is 8/9 for static and spherically symmetric stars under some physically reasonable assumptions. We formally apply Buchdahl's method to scalar-tensor theories and obtain theory-independent inequalities. After discussing the mass definition in scalar-tensor theories, these inequalities are related to a theory-dependent maximum mass-to-size ratio. We show that its value can exceed not only Buchdahl's limit, 8/9, but also unity, which we call {\it the black hole limit}, in contrast to general relativity. Next, we numerically examine the validity of the assumptions made in deriving the inequalities and the applicability of our analytic results. We find that the assumptions are mostly satisfied and that the mass-to-size ratio exceeds both Buchdahl's limit and the black hole limit. However, we also find that this ratio never exceeds Buchdahl's limit when we impose the further condition, $\rho-3p\ge0$, on the density, $\rho$, and pressure, $p$, of the matter.