Enrico Barausse, Thomas P. Sotiriou, John C. Miller
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.
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After briefly reviewing the results about polytropic stars in Palatini f(R)-theories, we first show how these results rely on the assumption of a regular function f(R). In particular, singular models allow to extend the parameter interval in which no singularity is formed. Furthermore, we present how the conformal metric can be matched smoothly in the cases where the original metric generates a singularity. In fact, the singularity comes from a singular conformal factor which is continuous though not differentiable at the stellar surface. This suggests that the correct metric to be considered …
We will briefly discuss the necessary conditions for stability of polytropies in $f(\hat R)$ Palatini gravity and the differences with the General Relativity ones.
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 have derived a modified Lane-Emden equation for the Starobinsky model in Palatini gravity which is numerically solvable. Comparing the results to the ones provided by General Relativity we observe a significant difference depending on the theory parameter for the $M-R$ relations.
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.
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 work out the junction conditions for $f(R)$ gravity formulated in metric-affine (Palatini) spaces using a tensor distributional approach. These conditions are needed for building consistent models of gravitating bodies with an interior and exterior regions matched at some hypersurface. Some of these conditions depart from the standard Darmois-Israel ones of General Relativity and from their metric $f(R)$ counterparts. In particular, we find that the trace of the stress-energy momentum tensor in the bulk must be continuous across the matching hypersurface, though its normal derivative need n…
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…
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.
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 rati…
In a previous paper we showed that static spherically symmetric objects which, in the vicinity of their surface, are well-described by a polytropic equation of state with 3/2<Gamma<2 exhibit a curvature singularity in Palatini f(R) gravity. We argued that this casts serious doubt on the validity of Palatini f(R) gravity as a viable alternative to General Relativity. In the present paper we further investigate this characteristic of Palatini f(R) gravity in order to clarify its physical interpretation and consequences.
Anisotropic Generalized Polytropic Spheres: Regular 3D Black Holes
该研究针对各向异性流体构成的相对论三维球体，通过求解广义TOV方程得到三类带正则核心的渐近AdS三维黑洞解，验证其正则性并解释了成因，还分析了其在能量流经柯西视界时的稳定性。
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 …
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.
Static Charged Polytropic Spheres with a Cosmological Constant: Physical Acceptability and Trapped Orbits
研究带电多项式流体球体在不同n和Γ值下的物理和几何性质，分析圆轨道被俘获区域，并探讨中性光子与带电粒子的俘获机制。
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 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…
Revisiting relativistic electrically charged polytropic spheres
本文研究相对论性带电多方球的解族、稳定性与因果性，发现电荷密度与静止质量密度之比的微观界限与极端Reissner-Nordström黑洞的宏观界限一致，并指出黑洞模仿者只能通过电平衡尘埃解的精细极限获得。
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, …
We find that if general relativity is modified at the Planck scale by a Ricci-squared term, electrically charged black holes may be nonsingular. These objects concentrate their mass in a microscopic sphere of radius $r_{core}\approx N_q^{1/2}l_P/3$, where $l_P$ is the Planck length and $N_q$ is the number of electric charges. The singularity is avoided if the mass of the object satisfies the condition $M_0^2\approx m_P^2 α_{em}^{3/2} N_q^3/2$, where $m_P$ is the Planck mass and $α_{em}$ is the fine-structure constant. For astrophysical black holes this amount of charge is so small that their e…
The aim of this paper is to explore the complexity factor (CF) for those self-gravitating relativistic spheres whose evolution proceeds non-dynamically. We are adopting the definition of CF mentioned in \cite{PhysRevD.97.044010}, modifying it to the static spherically symmetric case, within the framework of a modified gravity theory (the Palatini $f(R)$ theory). In this respect, we have considered radial dependent anisotropic matter content coupled with spherical geometry and determined the complexity factor involved in the patterns of radial evolution. We shall explore the field and a well-kn…
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…
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…
Dark matter halos modeled by polytropic spheres influenced by the relict cosmological constant and trapping polytropes forming supermassive black holes
本文用含遗迹宇宙学常数的广义相对论多方球建模暗物质晕，通过测地线与稳定性分析给出匹配星系和星系团尺度的参数区间，并界定可形成超大质量黑洞的俘获多方球条件。
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 …
We study the structure and stability of traversable wormholes built as (spherically symmetric) thin shells in the context of Palatini $f(\mathcal{R})$ gravity. Using a suitable junction formalism for these theories we find that the effective number of degrees of freedom on the shell is reduced to a single one, which fixes the equation of state to be that of massless stress-energy fields, contrary to the general relativistic and metric $f(R)$ cases. Another major difference is that the surface energy density threading the thin-shell, needed in order to sustain the wormhole, can take any sign, a…
Analysis of the Fractional Relativistic Polytropic Gas Sphere
本文在mRL框架下建立分数阶TOV方程并结合Euler-Abel变换与Pade近似求解，揭示了分数阶与相对论参数对不同多方指数恒星模型的影响，发现降低分数阶参数可使白矮星质量极限升至1.63348 M。
Propagation and lensing of gravitational waves in Palatini $f(\hat R)$ gravity
本文系统研究 Palatini $f(\hat R)$ 引力中引力波的传播与透镜效应，推导光度距离修正和偏转角变化，并分析其对偏振面旋转的影响，为区分该理论与广义相对论提供依据。
We study both analytically and numerically the gravitational fields of stars in f(R) gravity theories. We derive the generalized Tolman-Oppenheimer-Volkov equations for these theories and show that in metric f(R) models the Parameterized Post-Newtonian parameter $γ_{\rm PPN} = 1/2$ is a robust outcome for a large class of boundary conditions set at the center of the star. This result is also unchanged by introduction of dark matter in the Solar System. We find also a class of solutions with $γ_{\rm PPN} \approx 1$ in the metric $f(R)=R-μ^4/R$ model, but these solutions turn out to be unstable …
We study electrically charged compact stars in the framework of extended theory of gravity (ETG). We assume that the charge density is proportional to the energy density. The polytropic equation of state is chosen to describe the state of the charged perfect fluid. We aim to find the Oppenheimer Volkoff (OV) mass limit for charged compact stars. A detailed numerical study is performed. We show the dependence of the mass-radius diagram of the spheres on the values of the perturbatif parameter $β$, the polytropic exponent $γ$ and the charge fraction $α$. Our results are compared with those found…
Stability Analysis Of Fractional Relativistic Polytropes
本研究利用分数阶微积分分析多方指数1-3的相对论多方球稳定性，通过临界点方法确定临界参数，发现临界相对论值随分数阶参数减小而增大，明确了不同多方指数下的稳定/不稳定阈值。
Thermodynamics and orbital structure of anti-de Sitter black holes in Palatini-inspired nonlinear electrodynamics
本文研究了基于帕拉蒂尼启发的非线性电动力学模型的反德西特黑洞的热力学和轨道结构，分析了其相变行为及几何特性。
This paper investigates the complexity of a charged static sphere filled with anisotropic matter in the background of energy-momentum squared gravity. For this purpose, we evaluate the modified field and conservation equations to determine the structure of celestial system. The mass function is calculated through Misner-Sharp as well as Tolman mass definitions. The complexity of a self-gravitating system depends on different factors such as anisotropic pressure, electromagnetic field, energy density inhomogeneity, etc. We formulate the structure scalars by the orthogonal decomposition of the R…
The scalar-tensor theory of gravity with the Higgs field as scalar field is presented. For central symmetry it reproduces the empirically measured flat rotation curves of galaxies. We approximate the galaxy by a polytropic gas sphere with the polytropic index $γ=2$ and a massive core.
Equilibrium tori orbiting Reissner-Nordström naked singularities
本研究在广义相对论框架下，针对Reissner-Nordström裸奇点时空，构建了比角动量均匀的多方平衡环面，补充了极端带电致密天体周围的轨道结构研究。
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…
Optical and orbital characterization of spherically symmetric static black holes of self-gravitating new nonlinear electrodynamics model
本文研究了基于新非线性电磁动力学模型的球对称静态黑洞的光学和轨道特性，通过测地线分析确定关键半径，并探讨了光子球、阴影大小及质量探测器轨道特性，为未来引力透镜观测提供理论支持。
Relativistic Signatures of Dark Matter Equations of State in Static Spherically Symmetric Spacetimes
该研究分析PFDM、常数ω暗物质、BEC暗物质三种模型对静态球对称黑洞时空几何的影响，计算强引力可观测量并结合EHT约束，证实强引力可观测量可用于区分暗物质模型。
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