Perfect Fluid Approximation Research Articles

Theoretical foundations of the reduced relativistic gas in the cosmological perturbed context

The Reduced Relativistic Gas (RRG) is a simplified version of the ideal relativistic gas, which assumes that all particles have the same momentum magnitude. Although this is a very idealized situation, the resulting model preserves the phenomenology of Maxwell-Boltzmann distribution and, in some situations, can be described as a perfect fluid, without introducing large errors in both cosmological background and first-order perturbations. The perfect fluid description of RRG model was already used to study the warmness of dark matter, massive neutrinos and interaction of baryons and photons before recombination, showing very good agreement with previous works based on the full Einstein-Boltzmann system of equations. In order to understand these results and construct a more general and formal framework for RRG, we develop a theoretical description of first-order cosmological perturbations of RRG, based on a distribution function which encodes the simplifying assumption that all particles have the same momentum magnitude. The full set of Einstein-Boltzmann equations for RRG distribution are derived and quantities beyond the perfect fluid approximation are studied. Using RRG to describe warm dark matter, we show that, for particles with m ∼ keV, the perfect fluid approximation is valid on scales k < 10 h/Mpc, for most of the universe evolution. We also determine initial conditions for RRG in the early universe and study the evolution of potential in a toy model of universe composed only by RRG.

Evolution of dark matter velocity dispersion

Cosmological perturbation theory for the late Universe dominated by dark matter is extended beyond the perfect pressureless fluid approximation by taking the dark matter velocity dispersion tensor as an additional field into account. A proper tensor decomposition of the latter leads to two additional scalar fields, as well as a vector and a tensor field. Most importantly, the trace of the velocity dispersion tensor can have a spatially homogeneous expectation value. While it decays at early times, we show that a back-reaction effect quadratic in perturbations makes it grow strongly at late times. We compare sterile neutrinos as a candidate for comparatively warm dark matter to weakly interacting massive particles as a rather cold dark matter candidate and show that the late time growth of velocity dispersion is stronger for the latter. Another feature of a non-vanishing velocity dispersion tensor is that it can account for multiple streams within the fluid description and thereby allows to treat times and scales beyond shell-crossing.

Dynamical wormhole definitions confronted

Crude comparison between four alternative proposals for the very definition of a wormhole is provided, all of which were intended to apply to the dynamical cases. An interesting dynamical solution, based upon large scale magnetic fields, is used for the comparisons. Such a solution goes beyond the perfect fluid approximation due to an anisotropic pressure component, bringing to the fore some unsuspected features of those definitions. Certain notions as reversible traversability are claimed as a way to select among those definitions the best suited one to represent our intuition of what a wormhole solution is expected to be.

Black brane viscosity and the Gregory-Laflamme instability

We study long wavelength perturbations of neutral black p-branes in asymptotically flat space and show that, as anticipated in the blackfold approach, solutions of the relativistic hydrodynamic equations for an effective p + 1-dimensional fluid yield solutions to the vacuum Einstein equations in a derivative expansion. Going beyond the perfect fluid approximation, we compute the effective shear and bulk viscosities of the black brane. The values we obtain saturate generic bounds. Sound waves in the effective fluid are unstable, and have been previously related to the Gregory-Laflamme instability of black p-branes. By including the damping effect of the viscosity in the unstable sound waves, we obtain a remarkably good and simple approximation to the dispersion relation of the Gregory-Laflamme modes, whose accuracy increases with the number of transverse dimensions. We propose an exact limiting form as the number of dimensions tends to infinity.

Gravity currents in rotating channels. Part 1. Steady-state theory

A theory is developed for the speed and structure of steady-state non-dissipative gravity currents in rotating channels. The theory is an extension of that of Benjamin (1968) for non-rotating gravity currents, and in a similar way makes use of the steady-state and perfect-fluid (incompressible, inviscid and immiscible) approximations, and supposes the existence of a hydrostatic ‘control point’ in the current some distance away from the nose. The model allows for fully non-hydrostatic and ageostrophic motion in a control volume V ahead of the control point, with the solution being determined by the requirements, consistent with the perfect-fluid approximation, of energy and momentum conservation in V, as expressed by Bernoulli's theorem and a generalized flow-force balance. The governing parameter in the problem, which expresses the strength of the background rotation, is the ratio W = B/R, where B is the channel width and R = (g′H)1/2/f is the internal Rossby radius of deformation based on the total depth of the ambient fluid H. Analytic solutions are determined for the particular case of zero front-relative flow within the gravity current. For each value of W there is a unique non-dissipative two-layer solution, and a non-dissipative one-layer solution which is specified by the value of the wall-depth h0. In the two-layer case, the non-dimensional propagation speed c = cf(g′H)−1/2 increases smoothly from the non-rotating value of 0.5 as W increases, asymptoting to unity for W → ∞. The gravity current separates from the left-hand wall of the channel at W = 0.67 and thereafter has decreasing width. The depth of the current at the right-hand wall, h0, increases, reaching the full depth at W = 1.90, after which point the interface outcrops on both the upper and lower boundaries, with the distance over which the interface slopes being 0.881R. In the one-layer case, the wall-depth based propagation speed Froude number c0 = cf(g′h0)−1/2 = 21/2, as in the non-rotating one-layer case. The current separates from the left-hand wall of the channel at W0 ≡ B/R0 = 2−1/2, and thereafter has width 2−1/2R0, where R0 = (g′h0)1/2/f is the wall-depth based deformation radius.

Numerical simulation of thick disc accretion on to a rotating black hole

We study two-dimensional axisymmetric accretion on to a rotating black hole by time-dependent fully relativistic hydro dynamical calculations. We investigate the innermost part of geometrically thick discs, in which most of the gravitationally released energy is advected into the black hole. Relativistic effects are important close to the cusp-like inner edge of the disc, where accretion proceeds with approximately constant angular momentum and the viscous time-scale exceeds the dynamical time-scale of accretion. We use the perfect fluid approximation and calculate isentropic flows with constant angular momentum. We confirm the previous analytical result that the structure of the innermost disc region strongly depends on the black hole spin. We find the mass accretion rate M to be in a good agreement with the analytical dependenceM oc (dW)rJ(r-I), where dWis the energy gap and r is the adiabatic index.

Relativistic fireworks: Not a viable model for cosmology

view Abstract Citations References (10) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Relativistic Fireworks: Not a Viable Model for Cosmology Widlund, Lennart Abstract The dynamics of the "fireworks model" for the universe is investigated, taking gravity fully into account. The fireworks model is a metagalactic model, which means that all matter observed is assumed to be a part of the metagalaxy: a huge, but finite aggregate of matter. In this way, the need for an initial singularity is avoided, and the Hubble expansion may be a result from some sort of explosion. However, it has proved to be surprisingly difficult to explain the highest redshifts observed in this way. The fireworks model is a possible remedy to that problem, where high redshifts are obtained in successive steps of explosion. In this paper, general relativistic Boltzmann-like equations are derived to describe the dynamics of such a model. The model is applied to cosmology in two ways. First, the nonrelativistic limit without gravity is investigated. In this case there are exact solutions to the equations. From these it is shown that a model with reasonable values of the density parameter and the random component of the peculiar velocity field cannot explain the highest redshifts observed. Then, a general relativistic perfect fluid approximation is treated numerically. Solutions are obtained where the redshifts have acceptable values. The velocity dispersion, in agreement with the nonrelativistic result, is by far too high to be compatible with observations. Publication: The Astrophysical Journal Pub Date: December 1994 DOI: 10.1086/174972 Bibcode: 1994ApJ...437...27W Keywords: Astronomical Models; Cosmology; Galactic Clusters; Relativity; Universe; Gravitational Effects; Mathematical Models; Relativistic Effects; Astrophysics; COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE; COSMOLOGY: THEORY; GALAXIES: CLUSTERING; RELATIVITY full text sources ADS |

Numerical treatment of the spherically symmetric general-relativistic Boltzmann equation for massless and massive particles.

The Arnowitt-Deser-Misner formalism is used to write the Einstein-Boltzmann coupled system of equations. The sources of gravitational field are represented by ordinary matter described by a perfect-fluid approximation together with a particle gas described by a phase-space distribution function obeying the general-relativistic Boltzmann transport equation. Through the use of the Liouville operator in phase space, we obtain a form of the Boltzmann equation that makes it very amenable for numerical treatment. The resulting system of equations can be used for the numerical study of either massless or massive particles interacting with ordinary matter.

O( d, d)-covariant string cosmology

The recently discovered O( d, d) symmetry of the space of (slowly-varying) cosmological string vacua in d+1 dimensions is shown to be preserved in the presence of bulk string matter. The existence of O( d, d) conserved currents allows all the equations of string cosmology to be reduced to first-order differential equations. The perfect-fluid approximation is not invariant under O( d, d) transformations, implying that stringly fluids possess in general a non-vanishing viscosity.

Numerical solution of the general relativistic Boltzmann equation for massive and massless particles

The ADM formalism of general relativity is applied to write the Einstein and matter equations for a spherically symmetric spacetime whose gravitational sources are taken to be represented by ordinary matter, described by a perfect fluid approximation, and either massless or massive weakly interacting particles, for which a kinetic-theory description is employed. The Boltzmann equation, which is solved directly, is written in a form which is particularly amenable to numerical treatment, and for which massive particles can be accommodated as easily as massless ones. Some of the numerical methods used to solve these equations are described and the results of several code tests are discussed. The behavior of massive and massless particles is compared, and results are presented from a code for the cooling of a crude model of a 'neutron star' by emission of massive particles. 33 refs.

The cosmological effects of generalized field equations in relativity

An autonomous system of equations, describing uniform cosmological models, is formulated by using the perfect fluid approximation of Einstein’s equations. These equations contain an arbitrary function related to the matter content of the universe, which may include negative energy fields. This function, designated α , is assumed to depend on the density and expansion rate of the universe only. Geometrical methods of analysis are used to study the behaviour of all models described by this system. The analysis shows that there are only three possible modes of behaviour that can be exhibited by a uniform universe. Examples of the first two classes are well known in the ‘big-bang’ and ‘steady-state’ theories. However, it is shown that the familiar theories are not unique, but an infinite number of both such types of model exist for various α . It is also shown that all steady-state models in an expanding universe are stable. The third class of model, associated with periodic behaviour, is of two types. The first is demonstrated by a universe which oscillates between expansion and contraction but never achieves infinite density. The second consists of ever-expanding (or contracting) models in which the density and expansion rate oscillate between finite values. These latter models possess evolutionary characteristics on ‘short’ time-scales, while satisfying the ‘perfect cosmological principle’ in the large, and only arise in the presence of gross non-linearities introduced by the function α . Both the periodic and steady-state classes occur only in the case of negative energy fields.

Systems of Self-Gravitating Particles in General Relativity and the Concept of an Equation of State

A method of self-consistent fields is used to study the equilibrium configurations of a system of self-gravitating scalar bosons or spin-\textonehalf{} fermions in the ground state without using the traditional perfect-fluid approximation or equation of state. The many-particle system is described by a second-quantized free field, which in the boson case satisfies the Klein-Gordon equation in general relativity, ${\ensuremath{\nabla}}_{\ensuremath{\alpha}}{\ensuremath{\nabla}}^{\ensuremath{\alpha}}\ensuremath{\varphi}={\ensuremath{\mu}}^{2}\ensuremath{\varphi}$, and in the fermion case the Dirac equation in general relativity ${\ensuremath{\gamma}}^{\ensuremath{\alpha}}{\ensuremath{\nabla}}_{\ensuremath{\alpha}}\ensuremath{\psi}=\ensuremath{\mu}\ensuremath{\psi}$ (where $\ensuremath{\mu}=\frac{\mathrm{mc}}{\ensuremath{\hbar}}$). The coefficients of the metric ${g}_{\ensuremath{\alpha}\ensuremath{\beta}}$ are determined by the Einstein equations with a source term given by the mean value $〈\ensuremath{\varphi}|{T}_{\ensuremath{\mu}\ensuremath{\nu}}|\ensuremath{\varphi}〉$ of the energy-momentum tensor operator constructed from the scalar or the spinor field. The state vector $〈\ensuremath{\varphi}|$ corresponds to the ground state of the system of many particles. In both cases, for completeness, a nonrelativistic Newtonian approximation is developed, and the corrections due to special and general relativity explicitly are pointed out. For $N$ bosons, both in the region of validity of the Newtonian treatment (density from ${10}^{\ensuremath{-}80}$ to ${10}^{54}$ g ${\mathrm{cm}}^{\ensuremath{-}3}$, and number of particles from 10 to ${10}^{40}$) as well as in the relativistic region (density \ensuremath{\sim}${10}^{54}$ g ${\mathrm{cm}}^{\ensuremath{-}3}$, number of particles \ensuremath{\sim}${10}^{40}$), we obtain results completely different from those of a traditional fluid analysis. The energy-momentum tensor is anisotropic. A critical mass is found for a system of $N\ensuremath{\sim}{[\frac{(\mathrm{Planck}\mathrm{mass})}{m}]}^{2}\ensuremath{\sim}{10}^{40}$ (for $m\ensuremath{\sim}{10}^{\ensuremath{-}25}$ g) self-gravitating bosons in the ground state, above which mass gravitational collapse occurs. For $N$ fermions, the binding energy of typical particles is ${G}^{2}{m}^{5}{N}^{\frac{4}{3}}{\ensuremath{\hbar}}^{\ensuremath{-}2}$ and reaches a value $\ensuremath{\sim}{\mathrm{mc}}^{2}$ for $N\ensuremath{\sim}{N}_{\mathrm{crit}}\ensuremath{\sim}{[\frac{(\mathrm{Planck}\mathrm{mass})}{m}]}^{3}\ensuremath{\sim}{10}^{57}$ (for $m\ensuremath{\sim}{10}^{\ensuremath{-}24}$ g, implying mass \ensuremath{\sim}${10}^{33}$ g, radius \ensuremath{\sim}${10}^{6}$ cm, density \ensuremath{\sim}${10}^{15}$ g/${\mathrm{cm}}^{3}$). For densities of this order of magnitude and greater, we have given the full self-consistent relativistic treatment. It shows that the concept of an equation of state makes sense only up to ${10}^{42}$ g/${\mathrm{cm}}^{3}$, and it confirms the Oppenheimer-Volkoff treatment in extremely good approximation. There exists a gravitational spin-orbit coupling, but its magnitude is generally negligible. The problem of an elementary scalar particle held together only by its gravitational field is meaningless in this context.

Buoyant vortex rings

It is pointed out that there is a fundamental difference in the behaviour of vortex rings projected upwards, according as they do or do not contain fluid which is lighter than the surroundings. A theory based on the perfect fluid approximation is developed to describe the motion of buoyant rings in a uniform fluid. The essential assumption is that the circulation remains constant with time while the buoyancy force acts to increase the impulse of a ring. This leads to the prediction that increasing the buoyancy will give a greater rate of expansion and a lower velocity of rise. The theory is extended to the case of a ring rising through a stably stratified fluid having a constant density gradient; in this case increasing the buoyancy should lead to a lower final height. The predictions of the theory have been verified by carrying out experiments in the laboratory with small vortex rings formed in water, using methylated spirits and salt to produce the density differences. The observations suggest that substantially the same analysis may be applicable to phenomena on a larger scale in the atmosphere; the velocity and final height of an explosion cloud should be determined by the buoyancy and circulation generated near the ground and the stability conditions of the atmosphere.