Perfect Fluid Models Research Articles

Gravitational-Wave Driven Instability of Rotating Relativistic Stars

A series of recent surprises appear dramatically to have improved the likelihood that the spin of rapidly rotating, newly formed neutron stars is limited by a nonaxisymmetric instability driven by gravitational waves — and that the emitted waves may be detectable. The first of these was the discovery that the r-modes, rotationally restored modes that have axial parity for spherical models, are unstable in perfect fluid models with arbitrarily slow rotation. First indicated in numerical work by Andersson, 1) the instability is implied in a nearly newtonian context by the newtonian expression for the r-mode frequency (3.1), and a computation by Friedman and Morsink 19) of the canonical energy of initial data showed (independent of assumptions on the existence of discrete modes) that the instability is a generic feature of axial-parity perturbations of relativistic stars. Studies of the viscous and radiative timescales associated with the rmodes, 38), 43), 2), 29), 37) have revealed a second surprising result: The growth time of r-modes driven by current-multipole gravitational radiation is significantly shorter than had been expected, so short, in fact, that the instability to gravitational radiation reaction easily dominates viscous damping in hot, newly formed neutron stars (see Fig. 1 below). As a result, a neutron star that is rapidly rotating at birth now appears likely to spin down by radiating most of its angular momentum in gravitational waves. (See, however, the caveats below.) Nearly simultaneous with these theoretical surprises was the discovery by Marshall et al. 41) of a fast (16ms) pulsar in a supernova remnant (N157B) in the Large Magellanic Cloud. Estimates of the initial period put it in the 6-9ms range, implying the existence of a class of neutron stars that are rapidly rotating at birth. Hence,

Kinematic self-similar locally rotationally symmetric models

A brief summary of results on kinematic self-similarities in general relativity is given. Attention is focused on locally rotationally symmetric models admitting kinematic self-similar vectors. Coordinate expressions for the metric and the kinematic self-similar vector are provided. Einstein's field equations for perfect fluid models are investigated and all the homothetic perfect fluid solutions admitting a maximal four-parameter group of isometries are given.

Timelike self-similar spherically symmetric perfect-fluid models

Einstein's field equations for timelike self-similar spherically symmetric perfect-fluid models are investigated. The field equations are rewritten as a first-order system of autonomous differential equations. Dimensionless variables are chosen in such a way that the number of equations in the coupled system is reduced as far as possible and so that the reduced phase space becomes compact and regular. The system is subsequently analysed qualitatively using the theory of dynamical systems. Using this approach, we obtain a clear picture of the full phase space and the full space of solutions. Solutions of physical interest, e.g. the solution associated with criticality in black hole formation, are easily singled out. We also discuss the various `band structures' that are associated with certain one-parameter sets of solutions.

Asymptotic Analysis Relating Spectral Models in Fluid--Solid Vibrations

An asymptotic study of two spectral models which appear in fluid--solid vibrations is presented in this paper. These two models are derived under the assumption that the fluid is slightly compressible or viscous, respectively. In the first case, min-max estimations and a limit process in the variational formulation of the corresponding model are used to show that the spectrum of the compressible case tends to be a continuous set as the fluid becomes incompressible. In the second case, we use a suitable family of unbounded non-self-adjoint operators to prove that the spectrum of the viscous model tends to be continuous as the fluid becomes inviscid. At the limit, in both cases, the spectrum of a perfect incompressible fluid model is found. We also prove that the set of generalized eigenfunctions associated with the viscous model is dense for the L2-norm in the space of divergence-free vector functions. Finally, a numerical example to illustrate the convergence of the viscous model is presented.

Static perfect fluid cylinders

The field equations for cylindrically symmetric perfect fluid models with a non-scale invariant equation of state where the energy density is linearly related to the pressure are rewritten as a set of first order coupled ordinary differential equations. Variables are chosen such that the resulting phase space is compact and everywhere regular. The dynamical systems approach, used in the past mainly for scale invariant matter sources, is subsequently used to gain qualitative information about the space of solutions. It leads to a simple way of demonstrating the existence and properties of interior sources for cylindrically symmetric vacuum solutions. The models are shown to exhibit asymptotic scale invariance.

Spatially self-similar spherically symmetric perfect-fluid models

Einstein's field equations for spatially self-similar spherically symmetric perfect-fluid models are investigated. The field equations are rewritten as a first-order system of autonomous differential equations. Dimensionless variables are chosen in such a way that the number of equations in the coupled system is reduced as far as possible and so that the reduced phase space becomes compact and regular. The system is subsequently analysed qualitatively with the theory of dynamical systems.

Rigidly rotating stationary cylindrically symmetric perfect fluid models

Einstein's field equations for rigidly rotating stationary Bianchi type I models are investigated, focusing on the cylindrically symmetric cases. The equations are rewritten as a first-order system of autonomous ordinary differential equations and dimensionless variables are introduced such that the reduced phase space is compact. The system is then studied qualitatively using the theory of dynamical systems. The cylindrically symmetric submanifold is studied in detail.

Inhomogeneous cosmological models with heat flux

We present a general class of inhomogeneous cosmological models filled with non-thermalized perfect fluid by assuming that the background spacetime admits two space-like commuting Killing vectors and has separable metric coefficients. The singularity structure of these models depends on the choice of the parameters and the metric functions. A number of previously known perfect fluid models follow as particular cases of this general class. Physical and geometrical features of these models are studied and the general expression for temperature distribution is given.

Hypersurface homogeneous and hypersurface self-similar models

Dynamical systems for hypersurface homogeneous and hypersurface self-similar models with non-null symmetry surfaces are derived. The equations are cast in a geometric form based on properties of the symmetry surfaces that emphasize the close connection between the various models. Perfect fluid models are discussed in particular. It is shown how the models form a hierarchical structure where simpler models act as building blocks for more complicated ones. Expressions for the fluid's kinematic properties are given and discussed in the context of various special cases.

Stationary Bianchi type II perfect fluid models

Einstein’s field equations for stationary Bianchi type II models with a perfect fluid source are investigated. The field equations are rewritten as a system of autonomous first-order differential equations. Dimensionless variables are subsequently introduced for which the reduced phase space is compact. The system is then studied qualitatively using the theory of dynamical systems. It is shown that the locally rotationally symmetric models are not asymptotically self-similar for small values of the independent variable. A new exact solution is also given.

Relativistic Diskoseismology. I. Analytical Results for “Gravity Modes”

We generalize previous calculations to a fully relativistic treatment of adiabatic oscillations which are trapped in the inner regions of accretion disks by non-Newtonian gravitational effects of a black hole. We employ the Kerr geometry within the scalar potential formalism of Ipser and Lindblom, neglecting the gravitational field of the disk. This approach treats perturbations of arbitrary stationary, axisymmetric, perfect fluid models. It is applied here to thin accretion disks. Approximate analytic eigenfunctions and eigenfrequencies are obtained for the most robust and observable class of modes, which corresponds roughly to the gravity (internal) oscillations of stars. The dependence of the oscillation frequencies on the mass and angular momentum of the black hole is exhibited. These trapped modes do not exist in Newtonian gravity, and thus provide a signature and probe of the strong-field structure of black holes. Our predictions are relevant to observations which could detect modulation of the X-ray luminosity from stellar mass black holes in our galaxy and the UV and optical luminosity from supermassive black holes in active galactic nuclei.

The Role of Shear in Expanding G2 Orthogonal Perfect Fluid Models

For G2 orthogonal expanding perfect fluid spacetimes we prove that vanishing of shear implies vanishing of acceleration or spacetime is plane-symmetric. That means inhomogeneous spacetimes must always be shearing. Non-singular G2 perfect fluid models will thus be both inhomogeneous and anisotropic.

A study of variable G and Lambda in Bianchi type II cosmological models

In this work, Bianchi type II cosmological models have been discussed treating G and Λ as non-constant scalars. Some solutions are obtained assuming an equation of state for the perfect fluid model and a power law form between the scale factors. An inflationary scenario has been observed.

A conformal mapping and isothermal perfect fluid model

Instead of the metric conformal to flat spacetime, we take the metric conformal to a spacetime which can be thought of as “minimally” curved in the sense that free particles experience no gravitational force yet it has non-zero curvature. The base spacetime can be written in the Kerr-Schild form in spherical polar coordinates. The conformal metric then admits the unique three-parameter family of perfect fluid solutions which are static and inhomogeneous. The density and pressure fall off in the curvature radial coordinates asR −2, for unbounded cosmological model with a barotropic equation of state. This is the characteristic of an isothermal fluid. We thus have an ansatz for an isothermal perfect fluid model. The solution can also represent bounded fluid spheres.

Inhomogeneous Kasner-type cylindrically symmetric models in Kaluza–Klein space–time

We consider a cylindrically symmetric metric separable in space and time variables having Kasnerian time dependence in the Kaluza–Klein space–time. We obtain perfect fluid models that admit the dimensional reduction yielding the usual four-dimensional space–time as the universe expands. The models are inhomogeneous and they have the Kasnerian vacuum solution as their matter-free limit. The five-dimensional Kasnerian vacuum solution goes over on dimensional reduction to the radiation Friedman–Robertson–Walker flat model.

Spatially self-similar locally rotationally symmetric perfect fluid models

Einstein's field equations for spatially self-similar locally rotationally symmetric perfect fluid models are investigated. The field equations are rewritten as a first-order system of autonomous ordinary differential equations. Dimensionless variables are chosen in such a way that the number of equations in the coupled system of differential equations is reduced as far as possible. The system is subsequently analysed qualitatively for some of the models. The nature of the singularities occurring in the models is discussed.

A study of Brans-Dicke theory in FRW-model

In FRW space time Brans-Dicke theory is developed for two cases: (i) the vacuum and (ii) the perfect fluid model. The field equations are transformed into a much simpler form under a change of time co-ordinates and then the solutions are determined for the above cases. An equation of statep =ρ/3 (radiation) is assumed in the case of perfect fluid.

Perfect and viscous fluid cosmological models in presence of scalar fields for anisotropic metric

An exact solution of Einstein equations representing a perfect fluid model in the presence of a zero-mass scalar field is obtained for an anisotropic metric. The effect of dissipative processes on the model is also discussed. Different physically valid forms of the scale factors in the three spatial directions give rise to some distinct models. The physical behaviour of these models is discussed in detail. Some of the perfect fluid models exhibit the ‘cigar’ singularity.

Axionic vorticity variational formulation for relativistic perfect fluids

For an arbitrary equation of state giving the pressure P as a function of the energy density , it is shown that the perfect fluid model defined thereby (and also the corresponding Ginzburg--Landau type generalization) can be represented by a variational theory in which the independent fields are an antisymmetric Kalb--Ramond gauge form, , a pair of scalar vorticity gauge potentials, , and a dilatonic scalar amplitude . The associated physical fields consist of the dilatonic ampitude itself, together with an axionic field, , and a vorticity field, . The Lagrangian for the perfect fluid case will be given by for a dilatonic self-coupling potential V whose form as a function of depends on the equation of state in such a way that its on-shell value will be given by .

Viscous fluid cosmology

In this paper we shall examine the effects of both bulk and shear viscosities upon a variety of cosmological models. We assume that the viscous terms can be modelled by dimensionless equations of state, and this allows us to write the Einstein equations as a system of autonomous differential equations. After briefly discussing the Friedmann--Robertson--Walker and Bianchi I models, we discuss in some detail the Bianchi V and Kantowski--Sachs models. In all cases we find the critical points of the flow and elucidate their nature, making use of the energy conditions. Because of our choice of dimensionless variables, (almost) all critial points represent self-similar solutions to the field equations. We also study the case of a Bianchi V perfect fluid model with a non-linear equation of state. We find that the models are structurally stable under the addition of shear viscosity, whereas they are structurally unstable under the introduction of bulk viscosity. Almost all of the Bianchi V models examined have initial singularities where the matter is dynamically unimportant; the Kantowski--Sachs models have final singularities where the matter is dynamically important.