Homogeneous and hypersurface-homogeneous shear-free perfect fluids in general relativity

Shear-free, general-relativistic perfect fluids are investigated in the case where they are either homogeneous or hypersurface-homogeneous (and, in particular, spatially homogeneous). It is assumed that the energy density and the presurep of the fluid are related by a barotropic equation of statep = p(), where +p 0. Under such circumstances, it follows that either the fluid's volume expansion rate or the fluid's vorticity (i.e., rotation) must vanish. In the homogeneous case, this leads to only two possibilities: either = = 0 (the Einstein static solution), or 0, = 0 (the Gödel solution). In the hypersurface-homogeneous case, the situation is more complicated: either = 0, 0 (as exemplified,inter alia, by the Friedmann-Robertson-Walker models), or 0, = 0 (which pertains, for example, in general stationary cylindrically symmetric fluids with rigid rotation, or = = 0 (as occurs for static spherically symmetric solutions). Each possibility is further subdivided in an invariant way, and related to the studies of other authors, thereby unifying and extending these earlier works.     

Collapsing shear-free perfect fluid spheres with heat flow

A global view is given upon the study of collapsing shear-free perfect fluid spheres with heat flow. We apply a compact formalism, which simplifies the isotropy condition and the condition for conformal flatness. The formulas for the characteristics of the model are straight and tractable. This formalism also presents the simplest possible version of the main junction condition, demonstrated explicitly for conformally flat and geodesic solutions. It gives the right functions to disentangle this condition into well known differential equations like those of Abel, Riccati, Bernoulli and the linear one. It yields an alternative derivation of the general solution with functionally dependent metric components. We bring together the results for static and time-dependent models to describe six generating functions of the general solution to the isotropy equation. Their common features and relations between them are elucidated. A general formula for separable solutions is given, incorporating collapse to a black hole or to a naked singularity.     

Spherically symmetric charged perfect fluid distributions executing shear-free motion

A method to obtain exact solutions characterizing spherically symmetric charged perfect fluid distributions undergoing shear-free motion has been discussed. This method makes use of the criterion that the solution be free from movable critical points as has been employed earlier by Shah and Vaidya. Two solutions have been obtained, one of which is new and the other is the recent solution due to Sussman.     

Comment: On charged analogue of shear-free fluids

Recently, Gupta et al. [2] claim that solutions with NDG have no uncharged analogue. This claim seems to be partially correct as it is seen in present article that neutral shear-free fluid (f =α₁ x ^−5/2α₁ being a constant) given by Stephani [3] can be charged.     

Nonrotating and nonexpanding perfect fluids

It is fairly well-known that nonstationary spacetimes exist which are filled with a nonrotating and nonexpanding perfect fluid, provided the fluid does not admit an equation of state. It is less well-known that the same is true when an equation of statedoes exist. Even whenall possible invariants of a perfect fluid solution depend on a single spatial coordinate only, the corresponding solutions need not be stationary.     

Irrotational and conformally Ricci-flat perfect fluids

When a space-time, containing an irrotational perfect fluid withw + p 0, is conformally Ricci-flat, three possibilities arise: (a) When the gradient of the conformal scalar field is aligned with the fluid velocity, the solution is conformally flat; (b) when the gradient is orthogonal to the fluid velocity, solutions are either shearfree, nonexpanding and (pseudo-) spherically or plane-symmetric, or they are conformally related to a particular new vacuum solution admitting a three-dimensional group of motions of Bianchi type VI_o on a timelike hypersurface; (c) in the general case solutions are (pseudo) spherically or plane-symmetric and have nonvanishing expansion.     

Shearfree and conformally Ricci-flat perfect fluids

A proof is given that shearfree perfect fluids, the metric of which is conformally related to a vacuum solution, are either of Petrov type D, as well as locally rotationally symmetric, or are conformally flat. In both cases the general solutions are well known.     

Thermodynamics of relativistic rotating perfect fluids

A new formulation of thermodynamics for special and general relativistic rotating perfect fluids is developed. Both isolated systems and portions of isolated systems electrically uncharged or charged are treated. Exploiting the symmetry of motion of stationary axisymmetric fluids, the global thermodynamic functions, including total energy and spin, are defined as free scalars, represented by hypersurface integrals of conserved vectors. Local equilibrium parameters such as local temperature and chemical potential are scalar functions. There also exist global equilibrium parameters, global temperature and global chemical potential, which are free scalars. The connection between local and global conditions of thermodynamic equilibrium is made clear and explicit. Thermodynamic potentials are introduced in the context of treating open systems in a relativistically invariant way.     

Singularity-free cosmological solutions with non-rotating perfect fluids

A conjecture stated by Raychaudhuri which claims that the only physical perfect fluid non-rotating non-singular cosmological models are comprised in the Ruiz–Senovilla and Fernández–Jambrina families is shown to be incorrect. An explicit counterexample is provided and the failure of the argument leading to the result is explicitly pointed out.     

Higher-dimensional perfect fluids and empty singular boundaries

In order to find out whether empty singular boundaries can arise in higher dimensional Gravity, we study the solution of Einstein’s equations consisting in a (N + 2)-dimensional static and hyperplane symmetric perfect fluid satisfying the equation of state ρ = ηp, being η an arbitrary constant and N ≥ 2. We show that this spacetime has some weird properties. In particular, in the case η > −1, it has an empty (without matter) repulsive singular boundary. We also study the behavior of geodesics and the Cauchy problem for the propagation of massless scalar field in this spacetime. For η > 1, we find that only vertical null geodesics touch the boundary and bounce, and all of them start and finish at z = ∞; whereas non-vertical null as well as all time-like ones are bounded between two planes determined by initial conditions. We obtain that the Cauchy problem for the propagation of a massless scalar field is well-posed and waves are completely reflected at the singularity, if we only demand the waves to have finite energy, although no boundary condition is required.     

High-speed cylindrical collapse of two perfect fluids

In this paper, the study of the gravitational collapse of cylindrically distributed two perfect fluid system has been carried out. It is assumed that the collapsing speeds of the two fluids are very large. We explore this condition by using the high-speed approximation scheme. There arise two cases, i.e., bounded and vanishing of the ratios of the pressures with densities of two fluids given by c _s , d _s . It is shown that the high-speed approximation scheme breaks down by non-zero pressures p ₁, p ₂ when c _s , d _s are bounded below by some positive constants. The failure of the high-speed approximation scheme at some particular time of the gravitational collapse suggests the uncertainty on the evolution at and after this time. In the bounded case, the naked singularity formation seems to be impossible for the cylindrical two perfect fluids. For the vanishing case, if a linear equation of state is used, the high-speed collapse does not break down by the effects of the pressures and consequently a naked singularity forms. This work provides the generalisation of the results already given by Nakao and Morisawa (Prog Theor Phys 113:73, 2005) for the perfect fluid.     

Projection tensor hydrodynamics: Generalized perfect fluids

The geometry of a projection tensor field in curved space-time is expressed in a way that does not restrict the dimensionality of the projection. The thermohydrodynamics of a perfect fluid then takes a compact and symmetrical form which also describes such exotic media as classical string fluids.Work supported in part by the National Science Foundation under grant No. MPS 74-18386-A01 and in part by the Alfred P. Sloan Foundation.     

Relativistic perfect fluids in local thermal equilibrium

Every evolution of a fluid is uniquely described by an energy tensor. But the converse is not true: an energy tensor may describe the evolution of different fluids. The problem of determining them is called here the inverse problem. This problem may admit unphysical or non-deterministic solutions. This paper is devoted to solve the inverse problem for perfect energy tensors in the class of perfect fluids evolving in local thermal equilibrium (l.t.e.). The starting point is a previous result (Coll and Ferrando in J Math Phys 30:2918–2922, 1989) showing that thermodynamic fluids evolving in l.t.e. admit a purely hydrodynamic characterization. This characterization allows solving this inverse problem in a very compact form. The paradigmatic case of perfect energy tensors representing the evolution of ideal gases is studied in detail and some applications and examples are outlined.     

On shear-free motion of charged perfect fluid obeying an equation of state in general relativity

In this note, a spherically symmetric charged perfect fluid executing shear-free motion has been investigated under the assumption of validity of an equation of state. Mashoon and Partovi [4] have studied this problem and have established several theorems giving the properties of such fluid distributions. We find that there is one more solution which is new. It is characterized by three parameters, one of which is the charge parameter. In the limit of vanishing charge the metric goes over to the Wyman metric [2]. It has been shown that the solution does not match with the Reissner-Nordström solution at a boundary and hence it is not suitable to represent a bounded system. We have also discussed the possibility of this solution representing the physical universe. We have found that the solution after a proper choice of constants may satisfy the physical requirements3p,d/dp 1 but will violate the condition /1 wherep, , and represent, respectively, the pressure, matter density, and charge density of the fluid. Therefore, the charged Wyman solution is unsuitable to represent the physical universe. Thus we conclude that for a charged perfect fluid distribution executing shear-free motion the field equations do not admit any physically meaningful solution if we assume the validity of an equation of state.This paper was partially presented at the 12th meeting of Indian Association for General Relativity and Gravitation and the symposium on Applications of General Relativity to Astrophysics and Cosmology held at Puna, India, November 9–12, 1983.     

Nonlocal shear stress for homogeneous fluids

It has been suggested that for fluids in which the rate of strain varies appreciably over length scales of the order of the intermolecular interaction range, the viscosity must be treated as a nonlocal property of the fluid. The shear stress can then be postulated to be a convolution of this nonlocal viscosity kernel with the strain rate over all space. In this Letter, we confirm that this postulate is correct by a combination of analytical and numerical methods for an atomic fluid out of equilibrium. Furthermore, we show that a gradient expansion of the nonlocal constitutive equation gives a reasonable approximation to the shear stress in the small wave vector limit.     

Dynamo-optical effect in homogeneous newtonian fluids

It is experimentally found that homogeneous polar fluids (water, glycerol, ethanol, etc.) become optically anisotropic under shear. As a result, the permittivity, which is a macroscopic characteristic of a fluid, becomes anisotropic as well. Results obtained indicate that quiescent polar fluids have signs of an ordered structure, which readily disappears under weak shear stresses and then relaxes to the initial state for several hours (depending on the fluid temperature).     

Generalized shear-free singularities

The possibility of matter singularities in a class of cosmological models is considered. The results are applied to shear-free models and suggest that in these the fluid cannot simultaneously expand and rotate.     

The Hamiltonian structure of general relativistic perfect fluids

We show that the evolution equations for a perfect fluid coupled to general relativity in a general lapse and shift, are Hamiltonian relative to a certain Poisson structure. For the fluid variables, a Lie-Poisson structure associated to the dual of a semi-direct product Lie algebra is used, while the bracket for the gravitational variables has the usual canonical symplectic structure. The evolution is governed by a Hamiltonian which is equivalent to that obtained from a canonical analysis. The relationship of our Hamiltonian structure with other approaches in the literature, such as Clebsch potentials, Lagrangian to Eulerian transformations, and its use in clarifying linearization stability, are discussed.Research supported in part by NSF grant MCS 81-08814(A02)Research supported in part by NSF grant MCS 81-07086