Self-gravitating Fluid Research Articles

General-relativistic Rotation Laws in Rotating Fluid Bodies: Constant Linear Velocity

New rotation laws have been recently found for general-relativistic self-gravitating stationary fluids. It was not clear whether they apply to systems rotating with a constant linear velocity. In this paper we fill this gap. The answer is positive. That means, in particular, that these systems should exhibit the recently discovered general-relativistic weak-field effects within rotating tori: the dynamic anti-dragging and the deviation from the Keplerian motion induced by the fluid selfgravity.

General relativistic, nonstandard model for the dark sector of the Universe

We present a general relativistic version of the self-gravitating fluid model for the dark sector of the Universe (darkon fluid) introduced in Stichel and Zakrzewski (Phys Rev D 80:083513, 2009) and extended and reviewed in Stichel and Zakrzewski (Entropy 15:559, 2013). This model contains no free parameters in its Lagrangian. The resulting energy-momentum tensor is dustlike with a nontrivial energy flow. In an approximation valid at sub-Hubble scales we find that the present-day cosmic acceleration is not attributed to any kind of negative pressure but it is due to a dynamically determined negative energy density. This property turns out to be equivalent to a time-dependent spatial curvature. The obtained cosmological equations, at sub-Hubble scales, agree with those of the nonrelativistic model but they are given a new physical interpretation. Furthermore, we have derived the self-consistent equation to be satisfied by the nonrelativistic gravitational potential produced by a galactic halo in our model from a weak-field limit of a generalized Tolman–Oppenheimer–Volkoff equation.

Jeans instability of a rotating self-gravitating viscoelastic fluid

The effect of rotation on the Jeans instability of a self-gravitating viscoelastic fluid has been investigated using generalized hydrodynamic fluid equations. A dispersion relation is obtained using the linearized perturbation equations and normal mode analysis which is discussed for directions of rotation parallel and perpendicular to the wave propagation in classical (hydrodynamic) as well as kinetic limits. The Jeans criterion of instability is also obtained and it is found that it is unaffected by the presence of rotation in both the classical hydrodynamic and kinetic limits. The effects of Mach number, shear viscosity, sound velocity and rotation on the growth rate of the Jeans instability are also discussed numerically and we found that the shear viscosity and rotation have stabilizing influences on the growth rates of instability in both perpendicular and parallel propagations with finite angular rotation frequency. The stability of the rotating viscoelastic fluid is discussed using the Routh-Hurwitz criterion.

The formation of trapped surfaces in spherically-symmetric Einstein–Euler spacetimes with bounded variation

We study the evolution of a self-gravitating compressible fluid in spherical symmetry and we prove the existence of weak solutions with bounded variation for the Einstein–Euler equations of general relativity. We formulate the initial value problem in Eddington–Finkelstein coordinates and prescribe spherically symmetric data on a characteristic initial hypersurface. We introduce here a broad class of initial data which contain no trapped surfaces, and we then prove that their Cauchy development contains trapped surfaces. We therefore establish the formation of trapped surfaces in weak solutions to the Einstein equations. This result generalizes a theorem by Christodoulou for regular vacuum spacetimes (but without symmetry restriction). Our method of proof relies on a generalization of the “random choice” method for nonlinear hyperbolic systems and on a detailed analysis of the nonlinear coupling between the Einstein equations and the relativistic Euler equations in spherical symmetry.

General proof of the entropy principle for self-gravitating fluid in static spacetimes

We show that for any perfect fluid in a static spacetime, if the Einstein constraint equation is satisfied and the temperature of the fluid obeys Tolman's law, then the other components of Einstein's equation are implied by the assumption that the total entropy of the fluid achieves an extremum for fixed total particle number and for all variations of metric with certain boundary conditions. Conversely, one can show that the extrema of the total entropy of the fluid are implied by Einstein's equation. Compared to previous works on this issue, we do not require spherical symmetry for the spacetime. Our results suggest a general and solid connection between thermodynamics and general relativity.

Jeans analysis in modified gravity

MOdified Gravity (MOG) is a covariant modification of Einstein's general relativity. This theory is one of the current alternatives to dark matter models. We describe dynamics of collisionless self-gravitating systems in the context of MOG. By studying the weak field approximation of this theory, we derive the equations governing the dynamics of the self-gravitating systems. More specifically, we consider the Jeans instability for self-gravitating fluid and stellar systems, and derive new Jeans mass limit $\tilde{M}_J$ and wave-number $\tilde{k}_J$. Furthermore, considering the gravitational instability in star forming regions, we show that MOG has not a significant difference with general relativity on this astrophysical scale. However, at larger scales such as intergalactic space MOG may lead to different galaxy and structure formation processes.

Effects of some physical factors on the inhomogeneity in planar symmetry

This paper is devoted to identify some physical causes of energy density inhomogeneity and stability of self-gravitating relativistic fluids in plane symmetry such as Weyl tensor, local anisotropy, dissipative terms and their specific combination. We first develop a relationship between matter variables and the Weyl tensor and then formulate dynamical equations using Bianchi identities. For the non-dissipative dust fluid, we conclude that the system will remain homogeneous if and only if it is conformally flat which implies the shear-free condition. However, the converse is not true for the non-dissipative isotropic fluid. For non-dissipative anisotropic fluid, the inhomogeneity factor is identified to be one of the structure scalars. A particular case of geodesic with dissipation is also discussed.

The interplay of group and dynamical systems analysis: The case of spherically symmetric charged fluids in general relativity

We investigate the relationship between the Dynamical Systems analysis and the Lie Symmetry analysis of ordinary differential equations. We undertake this investigation by looking at a relativistic model of self-gravitating charged fluids. Specifically we look at the impact of specific parameters obtained from Lie Symmetries analysis on the qualitative behaviour of the model. Steady states, stability and possible bifurcations are explored. We show that, in some cases, the Lie analysis can help to simply the dynamical systems analysis.

The effective gravitational decoupling between dark matter and the CMB

e present a detailed and self-contained analytical derivation of the evolution of sub-horizon cosmological perturbations before decoupling, based on previous work by S. Weinberg. These solutions are valid in the minimal ΛCDM scenario, to first order in perturbation theory, in the tight-coupling limit and neglecting neutrino shear stress. We compare them to exact numerical solutions computed by a Boltzmann code, and we find the two to be in very good agreement. The analytic solutions show explicitly that CDM and the baryon-photon fluid effectively behave as separate self-gravitating fluids until the epoch of baryon drag. This in turn leads to the surprising conclusion that the CMB is much less sensitive to the clustering properties of minimally coupled Dark Matter models than what would be naively expected.

Effect of quantum corrections on the Jeans instability of self-gravitating viscoelastic dusty fluid

In this paper we investigate the effects of quantum correction on the Jeans instability of self-gravitating viscoelastic dusty electron-ion quantum fluids. The massive self-gravitating dust grains are assumed to be strongly coupled and non-degenerate having both viscous and elastic behavior while the inertialess electrons and ions are considered as weakly coupled and Fermi degenerate. The hydrodynamic model is modified and a linear dispersion relation is derived employing the plane wave solutions on the linearized perturbation equations for the considered system. It is observed that the dispersion properties are affected due to the presence of viscoelastic effects and quantum statistical corrections. The modified condition of Jeans instability and expression of critical Jeans wavenumber are obtained. Numerically it is shown that viscoelastic effects, dust plasma frequency and quantum statistical effects all have stabilizing influence on the growth rate of gravitationally Jeans mode. The growth rates are also compared in kinetic and hydrodynamic limits and it is found that decay in the growth of unstable Jeans mode is larger under the kinetic limits than the hydrodynamic limits. The results are discussed for the understanding of formation of dense degenerate dwarf star through gravitational collapsing which is assumed to be strongly coupled dusty quantum fluid where the strongly coupled dust provides inertia and Fermi degenerate electron and ions provide quantum statistical effects.

Фигуры равновесия неоднородной самогравитирующей жидкости

This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with a stratified density and a steady-state velocity field. As in the classical formulation of the problem, it is assumed that the figures, or their layers, uniformly rotate about an axis fixed in space. It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with a constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the corresponding Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification. We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.

Cosmology in a petri dish? Simulation of collective of colloids at fluid interfaces

Interfacially trapped, micrometer-sized colloidal particles interact via ranged capillary attraction which is analogous to two-dimensional screened Newtonian gravity with the capillary length λ as the tuneable screening length. Using Brownian namics simulations and density functional theory, we study the dynamics of an initially prepared distribution of colloids, either a random homogeneous distribution, or a finitely- sized patch of colloids. Whereas the limit λ → ∞ corresponds to the global collapse self-gravitating fluid, for smaller λ the dynamics crosses over to spinodal decomposition showing a coarsening of regions of enhanced density which emerge from initial fluctuations. For the finite patch of colloids and intermediate λ we predict theoretically and serve in simulations a ringlike density peak at the outer rim of the disclike patch, moving as an inbound shock wave. Experimental realizations of this crossover scenario appear be well possible for colloids trapped at water interfaces and having a radius of around micrometer. Finally, the influence of hydrodynamic interactions on this capillary collapse will be discussed briefly.

Colloidal particles at fluid interfaces: Effective interactions, dynamics and a gravitation–like instability

Colloidal particles of micrometer size usually become irreversibly trapped at fluid interfaces if they are partially wetted by one phase. This opens the chance to create two–dimensional model systems where the effective interactions between the particles are possibly influenced by the presence of the interface to a great extent. We will review recent developments in the quantitive understanding of these effective interactions with a special emphasis on electrostatics and capillarity. Charged colloids of micrometer size at an interface form effective dipoles whose strength sensitively depends on the double layer structure. We discuss the success of modified Poisson–Boltzmann equations with regard to measured colloidal dipole moments. On the other hand, for somewhat larger particles capillary interactions arise which are long–ranged and analogous to two–dimensional screened Newtonian gravity with the capillary length λ as the screening length. For colloidal diameters of around 10 micrometer, the collective effect of these long–ranged capillary interactions will dominate thermal motion and residual, short–ranged repulsions, and results in an instability towards a collapsed state for a finite patch of particles. Such long–ranged interactions with the associated instability are also of interest in other branches of physics, such as self-gravitating fluids in cosmology, two–dimensional vortex flow in hydrodynamics, and bacterial chemotaxis in biology. Starting from the colloidal case we develop and discuss a dynamical “phase diagram” in the temperature and interaction range variables which appears to be of more general scope and applicable also to other systems.

Central density dependent anisotropic compact stars

Stars can be treated as a self-gravitating fluid. In this connection, we propose a model for an anisotropic star under the relativistic framework of Krori and Barua (J. Phys. A, Math. Gen. 8:508, 1975) spacetime. It is shown that the solutions are regular and singularity free. The uniqueness of the model is that interior physical properties of the star solely depend on the central density of the matter distribution.

Gravothermal catastrophe: The dynamical stability of a fluid model

A re-investigation of the gravothermal catastrophe is presented. By means of a linear perturbation analysis, we study the dynamical stability of a spherical self-gravitating isothermal fluid of finite volume and find that the conditions for the onset of the gravothermal catastrophe, under different external conditions, coincide with those obtained from thermodynamical arguments. This suggests that the gravothermal catastrophe may reduce to Jeans instability, rediscovered in an inhomogeneous framework. We find normal modes and frequencies for the fluid system and show that instability develops on the dynamical time scale. We then discuss several related issues. In particular: (1) For perturbations at constant total energy and constant volume, we introduce a simple heuristic term in the energy budget to mimic the role of binaries. (2) We outline the analysis of the two-component case and show how linear perturbation analysis can be carried out also in this more complex context in a relatively straightforward way. (3) We compare the behavior of the fluid model with that of the collisionless sphere. In the collisionless case the instability seems to disappear, which is at variance with the linear Jeans stability analysis in the homogeneous case; we argue that a key ingredient to understand the difference (a spherical stellar system is expected to undergo the gravothermal catastrophe only in the presence of some collisionality, which suggests that the instability is dissipative and not dynamical) lies in the role of the detailed angular momentum in a collisionless system. Finally, we briefly comment on the meaning of the Boltzmann entropy and its applicability to the study of the dynamics of self- gravitating inhomogeneous gaseous systems.

Magnetogravitodynamic stability of streaming fluid cylinder under the effect of capillary force

The magnetohydrodynamic stability criterion of self-gravitating streaming fluid cylinder under the combined effect of self-gravitating, magnetic, and capillary forces has been derived. The results are discussed analytically and some data are verified numerically for different parameters of the problem. The magnetic and capillary forces are stabilizing, but the streaming is destabilizing while the self-gravitating is stabilizing or destabilizing according to restrictions. The stable and unstable domains are identified and, moreover, the influences of the magnetic and capillary forces on the self-gravitating instability of the model have been examined. Including the magnetic force together with self-gravitating force improves the instability of the model. However, the self-gravitating instability will never be suppressed whatever the effects of the MHD force stabilizing effects are.

Ultrarelativistic Black Hole Formation

We study the head-on collision of fluid particles well within the kinetic energy dominated regime (γ = 8 to 12) by numerically solving the Einstein-hydrodynamic equations. We find that the threshold for black hole formation is lower (by a factor of a few) than simple hoop conjecture estimates, and, moreover, near this threshold two distinct apparent horizons first form postcollision and then merge. We argue that this can be understood in terms of a gravitational focusing effect. The gravitational radiation reaches luminosities of 0.014 c(5)/G, carrying 16 ± 2% of the total energy.

Maximum entropy principle for self-gravitating perfect fluid in Lovelock gravity

We consider a static self-gravitating system consisting of perfect fluid with isometries of an $(n-2)$-dimensional maximally symmetric space in Lovelock gravity theory. A straightforward analysis of the time-time component of the equations of motion suggests a generalized mass function. Tolman-Oppenheimer-Volkoff like equation is obtained by using this mass function and gravitational equations. We investigate the maximum entropy principle in Lovelock gravity, and find that this Tolman-Oppenheimer-Volkoff equation can also be deduced from the so called "maximum entropy principle" which is originally customized for Einstein gravity theory. This investigation manifests a deep connection between gravity and thermodynamics in this generalized gravity theory.

Nonstandard Approach to Gravity for the Dark Sector of the Universe

We summarize the present state of research on the darkon fluid as a model for the dark sector of the Universe. Nonrelativistic massless particles are introduced as a realization of the Galilei group in an enlarged phase space. The additional degrees of freedom allow for a nonstandard, minimal coupling to gravity respecting Einstein’s equivalence principle. Extended to a self-gravitating fluid the Poisson equation for the gravitational potential contains a dynamically generated effective gravitational mass density of either sign. The equations of motion (EOMs) contain no free parameters and are invariant w.r.t. Milne gauge transformations. Fixing the gauge eliminates the unphysical degrees of freedom. The resulting Lagrangian possesses no free particle limit. The particles it describes, darkons, exist only as fluid particles of a self-gravitating fluid. This darkon fluid realizes the zero-mass Galilean algebra extended by dilations with dynamical exponent z = 5/3 . We reduce the EOMs to Friedmann-like equations and derive conserved quantities and a unique Hamiltonian dynamics by implementing dilation symmetry. By the Casimir of the Poisson-bracket (PB)-algebra we foliate the phase space and construct a Lagrangian in reduced phase space. We solve the Friedmann-like equations with the transition redshift and the value of the Casimir as integration constants. We obtain a deceleration phase for the early Universe and an acceleration phase for the late Universe in agreement with observations. Steady state equations in the spherically symmetric case may model a galactic halo. Numerical solutions of a nonlinear differential equation for the gravitational potential lead to predictions for the dark matter (DM) part of the rotation curves (RCs) of galaxies in qualitative agreement with observational data. We also present a general covariant generalization of the model.

Spherically Symmetric Equilibria for Self-Gravitating Kinetic or Fluid Models in the Nonrelativistic and Relativistic Case---A Simple Proof for Finite Extension

We consider a self-gravitating collisionless gas as described by the Vlasov-Poisson or Einstein-Vlasov system or a self-gravitating fluid ball as described by the Euler-Poisson or Einstein-Euler system. We give a simple proof for the finite extension of spherically symmetric equilibria, which covers all these models simultaneously. In the Vlasov case the equilibria are characterized by a local growth condition on the microscopic equation of state, i.e., on the dependence of the particle distribution on the particle energy, at the cut-off energy E_0, and in the Euler case by the corresponding growth condition on the equation of state p=P(\rho) at \rho=0. These purely local conditions are slight generalizations to known such conditions.