Self-gravitating Fluid Research Articles

Evolution of expansion-free spherically symmetric self-gravitating non-dissipative fluids and some analytical solutions

We consider the distribution of spherically symmetric self-gravitating non-dissipative (but anisotropic) fluids under the expansion-free condition which requires the existence of vacuum cavity within the fluid distribution. The Darmois junction condition is investigated for matching the spherically symmetric metric to an internal vacuum cavity (Minkowski space-time). We have studied some analytical models, total of three family of solutions out of which two satisfy the junction conditions over both the hypersurfaces. The models are investigated under some known dynamical assumptions which further provide analytical solution in each family.

Energy transfer in compressible magnetohydrodynamic turbulence for isothermal self-gravitating fluids.

Three-dimensional, compressible, magnetohydrodynamic turbulence of an isothermal, self-gravitating fluid is analyzed using two-point statistics in the asymptotic limit of large Reynolds numbers (both kinetic and magnetic). Following an alternative formulation proposed by Banerjee and Galtier [Phys. Rev. E 93, 033120 (2016)2470-004510.1103/PhysRevE.93.033120; J. Phys. A: Math. Theor. 50, 015501 (2017)1751-811310.1088/1751-8113/50/1/015501], an exact relation has been derived for the total energy transfer. This approach results in a simpler relation expressed entirely in terms of mixed second-order structure functions. The kinetic, thermodynamic, magnetic, and gravitational contributions to the energy transfer rate can be easily separated in the present form. By construction, the new formalism includes such additional effects as global rotation, the Hall term in the induction equation, etc. The analysis shows that solid-body rotation cannot alter the energy flux rate of compressible turbulence. However, the contribution of a uniform background magnetic field to the flux is shown to be nontrivial unlike in the incompressible case. Finally, the compressible, turbulent energy flux rate does not vanish completely due to simple alignments, which leads to a zero turbulent energy flux rate in the incompressible case.

New definition of complexity for self-gravitating fluid distributions: The spherically symmetric, static case

We put forward a new definition of complexity, for static and spherically symmetric self--gravitating systems, based on a quantity, hereafter referred to as complexity factor, that appears in the orthogonal splitting of the Riemann tensor, in the context of general relativity. We start by assuming that the homogeneous (in the energy density) fluid, with isotropic pressure is endowed with minimal complexity. For this kind of fluid distribution, the value of complexity factor is zero. So, the rationale behind our proposal for the definition of complexity factor stems from the fact that it measures the departure, in the value of the active gravitational mass (Tolman mass), with respect to its value for a zero complexity system. Such departure is produced by a specific combination of energy density inhomogeneity and pressure anisotropy. Thus, zero complexity factor may also be found in self--gravitating systems with inhomogeneous energy density and anisotropic pressure, provided the effects of these two factors, on the complexity factor, cancel each other. Some exact interior solutions to the Einstein equations satisfying the zero complexity criterium are found, and prospective applications of this newly defined concept, to the study of the structure and evolution of compact objects, are discussed.

Anisotropic generalization of well-known solutions describing relativistic self-gravitating fluid systems: an algorithm

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.

The Hall-induced stability of gravitating fluids

We analyze the stability behavior of low-density partially ionized self-gravitating magnetized unbounded dusty plasma fluid in the presence of the Hall diffusion effects (HDEs) in the non-ideal magnetohydrodynamic (MHD) equilibrium framework. The effects of inhomogeneous self-gravity are methodically included in the basic model tapestry. Application of the Fourier plane-wave perturbative treatment decouples the structuration representative parameters into a linear generalized dispersion relation (sextic) in a judicious mean-fluid approximation. The dispersion analysis shows that the normal mode, termed as the gravito-magneto-acoustic (GMA) mode, is drastically modified due to the HDEs. This mode is highly dispersive, and driven unstable by the Hall current resulting from the symmetry-breaking of electrons and ions relative to the magnetic field. The mode feature, which is derived from a modified induction with the positive Hall, is against the ideal MHD. It is further demonstrated that the HDEs play stabilizing roles by supporting the cloud against gravitational collapse. Provided that the HDEs are concurrently switched off, the collapse occurs on the global spatial scale due to enhanced inward accretion of the gravitating dust constituents. It is seen explicitly that the enhanced dust-charge leads to stabilizing effects. Besides, the Hall-induced fluctuations, as propagatory wave modes, exhibit both normal and anomalous dispersions. The reliability checkup of the entailed results as diverse corollaries and special cases are illustratively discussed in the panoptic light of the earlier paradigmatic predictions available in the literature.

Density distribution function of a self-gravitating isothermal compressible turbulent fluid in the context of molecular clouds ensembles

We have set ourselves the task of obtaining the probability distribution function of the mass density of a self-gravitating isothermal compressible turbulent fluid from its physics. We have done this in the context of a new notion: the molecular clouds ensemble. We have applied a new approach that takes into account the fractal nature of the fluid. Using the medium equations, under the assumption of steady state, we show that the total energy per unit mass is an invariant with respect to the fractal scales. As a next step we obtain a nonlinear integral equation for the dimensionless scale Q which is the third root of the integral of the probability distribution function. It is solved approximately up to the leading-order term in the series expansion. We obtain two solutions. They are power-law distributions with different slopes: the first one is -1.5 at low densities, corresponding to a equilibrium between all energies at a given scale, and the second one is -2 at high densities, corresponding to a free fall at small scales.

Thermodynamical stability for a perfect fluid

According to the maximum entropy principle, it has been proved that the gravitational field equations could be derived by the extrema of the total entropy for a perfect fluid, which implies that thermodynamic relations contain information as regards gravity. In this manuscript, we obtain a criterion for the thermodynamical stability of an adiabatic, self-gravitating perfect fluid system by the second variation of the total entropy. We show, for Einstein’s gravity with spherical symmetry spacetime, that the criterion is consistent with that for the dynamical stability derived by Chandrasekhar and Wald. We also find that the criterion could be applied to cases without spherical symmetry, or under general perturbations. The result further establishes the connection between thermodynamics and gravity.

An approximate global solution of Einstein’s equations for a differentially rotating compact body

We obtain an approximate global stationary and axisymmetric solution of Einstein's equations which can be thought as a simple star model: a self-gravitating perfect fluid ball with a differential rotation motion pattern. Using the post--Minkowskian formalism (weak field approximation) and considering rotation as a perturbation (slow rotation approximation), we find approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic coordinates. Interior and exterior solutions are matched, in the sense of Lichnerowicz, on the surface of zero pressure to get a global solution. The resulting metric depends on four arbitrary constants: mass density, rotational velocity at $r=0$, a parameter which takes into account of the change in the rotational velocity through the star and the star radius in the non-rotation limit. The mass, angular momentum, quadrupole moment and other constants of the exterior metric are determined in terms of these four parameters.

Cosmological Model with Ultrarelativistic Ideal Gas using Gravitational and Thermomechanical Hamiltonian Formulation

A brief review of the Hamiltonian theory of self-gravitating perfect fluid, which has been established by Kijowski et al. (Phys Rev D 41:1875, 1990), has been discussed in this work. The formulation of the spherically symmetric cosmological problem has been derived. The most general three-dimensional metric in the case of spherically symmetric space–time has been considered. In addition, the parameters that govern the dynamics have been fixed. The problem of homogeneous universe has been considered for the ultrarelativistic ideal gas. The dynamical equations have been derived. Three analytical solutions have been obtained.

Self-gravitating fluid systems and galactic dark matter

We study gravitational collapse with anisotropic pressures, whose end stage can mimic space-times that are seeded by galactic dark matter. To this end, we identify a class of space-times (with conical defects) that can arise out of such a collapse process, and admit stable circular orbits at all radial distances. These have a naked singularity at the origin. An example of such a space-time is seen to be the Bertrand space-time discovered by Perlick, that admits closed, stable orbits at all radii. Using relativistic two- fluid models, we show that our galactic space-times might indicate exotic matter, i.e one of the component fluids may have negative pressure for a certain asymptotic fall off of the associated mass density, in the Newtonian limit. We complement this analysis by studying some simple examples of Newtonian two-fluid systems, and compare this with the Newtonian limit of the relativistic systems considered.

Self–gravitating fluid flows with Gowdy symmetry near cosmological singularities

ABSTRACTWe consider self-gravitating fluids in cosmological spacetimes with Gowdy symmetry on the torus T3 and, in this class, we solve the singular initial value problem for the Einstein–Euler system of general relativity, when an initial data set is prescribed on the hypersurface of singularity. We specify initial conditions for the geometric and matter variables and identify the asymptotic behavior of these variables near the cosmological singularity. Our analysis of this class of nonlinear and singular partial differential equations exhibits a condition on the sound speed, which leads us to the notion of sub-critical, critical, and super-critical regimes. Solutions to the Einstein–Euler systems when the fluid is governed by a linear equation of state are constructed in the first two regimes, while additional difficulties arise in the latter one. All previous studies on inhomogeneous spacetimes concerned vacuum cosmological spacetimes only.

On the Motion of a Self-Gravitating Incompressible Fluid with Free Boundary

We consider the motion of the interface separating a vacuum from an inviscid, incompressible, and irrotational fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid motion is described by the Euler-Poission system in moving bounded simply connected domains. A family of equilibrium solutions of the system are the perfect balls moving at constant velocity. We show that for smooth data which are small perturbations of size $\epsilon$ of these static states, measured in appropriate Sobolev spaces, the solution exists and remains of size $\epsilon$ on a time interval of length at least $c\epsilon^{-2},$ where $c$ is a constant independent of $\epsilon.$ This should be compared with the lifespan $O(\epsilon^{-1})$ provided by local well-posdness. The key ingredient of our proof is finding a nonlinear transformation which removes quadratic terms from the nonlinearity. An important difference with the related gravity water waves problem is that unlike the constant gravity for water waves, the self-gravity in the Euler-Poisson system is nonlinear. As a first step in our analysis we also show that the Taylor sign condition always holds and establish local well-posedness for this system.

On the admissible range of the radial temperature gradient and Brünt-Väisäla frequency in the mantle and core: I. Main relations

The question of ambiguity in the solution of the inverse problem for determining the Brunt-Vaisala frequency in the Earth’s mantle from the entire set of the up-to-date data on seismicity, free oscillations, and forced nutations of the Earth, as well as the data on the Earth’s total mass and total moment of inertia, is considered. Based on the results of a series of numerical experiments, the band of admissible distributions of the Brunt-Vaisala frequency and mantle density with depth is calculated. This estimate is used for investigating the convective and gravitational stability of the different regions of the mantle against relatively small adiabatic and nonadiabatic perturbations. The generalization of the known Rayleigh criterion of convective stability of homogeneous and a nonself-gravitating incompressible viscous fluid for the case of a compressible self-gravitating fluid is given. A system of the ordinary eight-order differential equations with complex coefficients and homogeneous boundary conditions, whose eigenvalues determine the transition from the stable state to instability, is obtained. Examples of the numerical determination of these eignevalues are presented. For interpreting the data about the band of the admissible distributions of the Brunt-Vaisala frequency with depth, the notion of the effective bulk modulus of the medium at different depths is introduced. This quantity governs the depth changes in temperature in a convecting mantle and allows us to make a conclusion about the role of heat conduction and the radial heterogeneity of the mantle composition without imposing any constraints on the convection mechanism. It is shown that within the present-day observation errors in the frequencies of the Earth’s free oscillations, the simplest reasonable model is that in which the ratio of the effective bulk modulus to its adiabatic value in the lower and middle mantle is 1.043 ± 0.05. The closeness of this value to unity indicates that convection in the lower and middle mantle is fairly close to adiabatic. At the same time, when the analysis only relies on seismic data and on the information about the periods of the free oscillations of the Earth, there is a significant uncertainty in the models of the effective bulk modulus distribution in the upper mantle and crust. This uncertainty precludes us from making purely empirically derived conclusions that reliably and unambiguously describe the role of the effects of heat conduction and radially heterogeneous material composition in the convection in the upper mantle.

Torsional Alfvén resonances as an efficient damping mechanism for non-radial oscillations in red giant stars

Stars are self-gravitating fluids in which pressure, buoyancy, rotation and magnetic fields provide the restoring forces for global modes of oscillation. Pressure and buoyancy energetically dominate, while rotation and magnetism are generally assumed to be weak perturbations and often ignored. However, observations of anomalously weak dipole mode amplitudes in red giant stars suggest that a substantial fraction of these are subject to an additional source of damping localised to their core region, with indirect evidence pointing to the role of a deeply buried magnetic field. It is also known that in many instances the gravity-mode character of affected modes is preserved, but so far no effective damping mechanism has been proposed that accommodates this aspect. Here we present such a mechanism, which damps the oscillations of stars harbouring magnetised cores via resonant interactions with standing Alfv\'en modes of high harmonic index. The damping rates produced by this mechanism are quantitatively on par with those associated with turbulent convection, and in the range required to explain observations, for realistic stellar models and magnetic field strengths. Our results suggest that magnetic fields can provide an efficient means of damping stellar oscillations without needing to disrupt the internal structure of the modes, and lay the groundwork for an extension of the theory of global stellar oscillations that incorporates these effects.

On the oscillation spectrum of a magnetized core in a giant star

The spectrum of gravito-acoustic modes is depleted in dipolar modes for a significant fraction of the giant stars observed by the Kepler mission, a feature that has been explained by the presence of magnetic fields in the core of these stars (Fuller et al. 2015, Cantiello et al. 2016). We further investigate this possible scenario by considering first the oscillation spectrum of the core of a giant star modeled by a stably stratified, self-gravitating fluid of uniform density in a sphere pervaded by a uniform magnetic field. Our results show that the first effect of a magnetic field on the g-modes is to reduce their wavenumber and therefore reduce their damping. The magnetic effect, on this model, is therefore opposite Fuller’s et al scenario. Moreover, the model shows that it is not possible to change the damping rate without changing the frequency of the modes and this latter change is not observed. Because of the simplicity of our model, the magnetized core scenario cannot be dismissed but further investigations are needed, and other ways of explaining the presence of depressed modes should also be considered.

Nonlinear wave breaking in self-gravitating viscoelastic quantum fluid

The stability of a viscoelastic self-gravitating quantum fluid has been studied. Symmetry breaking instability of solitary wave has been observed through ‘viscosity modified Ostrovsky equation’ in weak gravity limit. In presence of strong gravitational field, the solitary wave breaks into shock waves. Response to a Gaussian perturbation, the system produces quasi-periodic short waves, which in terns predicts the existence of gravito-acoustic quasi-periodic short waves in lower solar corona region. Stability analysis of this dynamical system predicts gravity has the most prominent effect on the phase portraits, therefore, on the stability of the system. The non-existence of chaotic solution has also been observed at long wavelength perturbation through index value theorem.

Structure scalars and evolution equations in f(G) cosmology

In this paper, we study the dynamics of self-gravitating fluid using structure scalars for spherical geometry in the context of f(G) cosmology. We construct structure scalars through orthogonal splitting of the Riemann tensor and deduce a complete set of equations governing the evolution of dissipative anisotropic fluid in terms of these scalars. We explore different causes of density inhomogeneity which turns out to be a necessary condition for viable models. It is explicitly shown that anisotropic inhomogeneous static spherically symmetric solutions can be expressed in terms of these scalar functions.

Collapse in self-gravitating turbulent fluids

Motivated by the nonlinear star formation efficiency found in recent numerical simulations by a number of workers, we perform high-resolution adaptive mesh refinement simulations of star formation in self-gravitating turbulently driven gas. As we follow the collapse of this gas, we find that the character of the flow changes at two radii, the disk radius $r_d$, and the radius $r_*$ where the enclosed gas mass exceeds the stellar mass. Accretion starts at large scales and works inwards. In line with recent analytical work, we find that the density evolves to a fixed attractor, $\rho(r,t ) \rightarrow \rho(r)$, for $r_d<r<r_*$; mass flows through this structure onto a sporadically gravitationally unstable disk, and from thence onto the star. In the bulk of the simulation box we find that the random motions $v_T \sim r^p$ with $p \sim 0.5$, in agreement with Larson's size-linewidth relation. In the vicinity of massive star forming regions we find $ p \sim 0.2-0.3$, as seen in observations. For $r<r_*$, $v_T$ increases inward, with $p=-1/2$. Finally, we find that the total stellar mass $M_*(t)\sim t^2$ in line with previous numerical and analytic work that suggests a nonlinear rate of star formation.

Regularity of weak solutions of compressible isentropic self-gravitating fluid

In this paper, the regularity of weak solutions to the three-dimensional compressible isentropic self-gravitating viscous fluids is considered in periodic domain. Inspired by the idea of Desjardins (1997) [3,4], we get the higher integrability of density ρ, and we remove the additional assumption of ρ∈L∞(0,T;Lq0) in the work of Choe and Jin (2003). Finally, we prove the supremum of the density is finite and the infimum of the density is positive in the domain (0,T)×T3 for a positive time T. Moreover, Moser type iteration scheme is used to obtain the L∞ norm estimate of the velocity.

Cosmological self-gravitating fluid solutions of shape dynamics

Shape Dynamics is a 3D conformally invariant theory of gravity which possesses a large set of solutions in common with General Relativity. When looked closely, these solutions are found to behave in surprising ways, so in order to probe the fitness of Shape Dynamics as a viable alternative to General Relativity one must find and understand increasingly more complex, less symmetrical exact solutions, on which to base perturbative studies and numerical analyses in order to compare them with data. Spherically symmetric exact solutions have been studied, but only in a static vacuum setup. In this work we construct a class of time-dependent exact solutions of Shape Dynamics from first principles, representing a central inhomogeneity in an evolving cosmological environment. By assuming only a perfect fluid source in a spherically symmetric geometry we show that this fully dynamic non-vacuum solution satisfies in all generality the Hamiltonian structure of Shape Dynamics. The simplest choice of solutions is shown to be a member of the McVittie family.