Perfect Fluid Metrics Research Articles

Generalising certain cosmological solutions sourced by a stiff perfect fluid

In this article, we show that a number of well-known time-dependent, spherically symmetric solutions of the Einstein’s field equations sourced by a stiff, perfect fluid with a cosmological constant can be generalised into a solution with arbitrary metric functions. This metric can be applied to construct stiff, perfect fluid metrics with (or without) a cosmological constant. We also explore the possibility that this metric may allow us to generate a singularity-free, spherically symmetric cosmological model.

Exact solutions admitting isometry groups Gr ⫆ Abelian G3

Metrics admitting a minimal three dimensional Abelian isometry group, G3 are classified according to their Petrov types and metrics, giving all type O and D metrics explicitly, without imposing a source condition. The corresponding maximal Lie algebras for these metrics are obtained and identified as well. The type O metrics admit a maximal G3 with r = 4, 6, 7 and 10, whereas the classes of metrics of type D admit G3 with r = 3, 4, 5 and 6 as the maximal isometry groups. Type O metrics with a perfect fluid source are then found explicitly and are shown to admit a maximal Gr with r = 4, 7 and 10. Type D perfect fluid metrics are found explicitly which admit either a maximal G3 or G4. This classification also proves that the only non-null Einstein-Maxwell field admitting a maximal G3 is the type D metric () which is of Segre type [(1, 1) (1 1)] and is isometric to the McVittie solution.

Perfect fluid metrics conformal to the Schwarzschild exterior spacetime

We construct perfect fluid spacetimes by performing a conformal transformation on a non-conformally flat vacuum solution, namely the Schwarzschild exterior metric. It should be noted that conformally Ricci flat perfect fluid solutions, except those that are conformally flat, are rarely reported explicitly. In this article it is demonstrated that perfect fluid metrics conformal to the Schwarzschild exterior line element are necessarily static. The Einstein field equations for the static case reduce to a fully determined system of three differential equations in three unknowns and the conformal factor is uniquely determined in closed form. The solution is analysed for physical plausibility by establishing the positivity of the energy density and pressure profiles graphically. Additionally, the solution is observed to be causal in an appropriate limit and both the energy density and pressure is shown to be decreasing outwards towards the boundary. Finally, the weak, strong and dominant energy conditions are found to be satisfied in the region under investigation. Accordingly, the most common elementary physical conditions are met and the model is seen to be suitable for a core-envelope stellar model.

From 2-dimensional surfaces to cosmological solutions

We construct perfect fluid metrics with two symmetries by means of a recently developed geometrical method [1]. The Einstein equations are reduced to a single equation for a conformal factor. Under additional assumptions we obtain new cosmological solutions of Bianchi type II, VI 0 and VII 0 . The solutions depend on an arbitrary function of time, which can be specified in order to satisfy an equation of state.

Ideally embedded space–times

Due to the growing interest in embeddings of space–time in higher-dimensional spaces we consider a specific type of embedding. After proving an inequality between intrinsically defined curvature invariants and the squared mean curvature, we extend the notion of ideal embeddings from Riemannian geometry to the indefinite case. Ideal embeddings are such that the embedded manifold receives the least amount of tension from the surrounding space. Then it is shown that the de Sitter spaces, a Robertson–Walker space–time and some anisotropic perfect fluid metrics can be ideally embedded in a five-dimensional pseudo-Euclidean space.

Ellipsoidal shapes in general relativity: general definitions and an application

A generalization of the notion of ellipsoids to curved Riemannian spaces is given and the possibility of using it in describing the shapes of rotating bodies in general relativity is examined. As an illustrative example, stationary, axisymmetric perfect-fluid spacetimes with a so-called confocal inside ellipsoidal symmetry are investigated in detail under the assumption that the 4-velocity of the fluid is parallel to a timelike Killing vector field. A class of perfect-fluid metrics representing interior NUT-spacetimes is obtained along with a vacuum solution with a non-zero cosmological constant.

A self-similar inhomogeneous dust cosmology

A detailed study of an inhomogeneous dust cosmology contained in a -law family of perfect-fluid metrics recently presented by Mars and Senovilla is performed. The metric is shown to be the most general orthogonally transitive, Abelian, on solution admitting an additional homothety such that the self-similar group is of Bianchi type VI and the fluid flow is tangent to its orbits. The analogous cases with Bianchi types I, II, III, V, VIII and IX are shown to be impossible thus making this metric privileged from a mathematical viewpoint. The differential equations determining the metric are partially integrated and the line-element is given up to a first-order differential equation of Abel type of first kind and two quadratures. The solutions are qualitatively analysed by investigating the corresponding autonomous dynamical system. The spacetime is regular everywhere except for the big bang and the metric is complete both into the future and in all spatial directions. The energy density is positive, bounded from above at any instant of time and with an spatial profile (in the direction of inhomogeneity) which is oscillating with a rapidly decreasing amplitude. The generic asymptotic behaviour at spatial infinity is a homogeneous plane wave. Well known dynamical system results indicate that this metric is very likely to describe the asymptotic behaviour in time of a much more general class of inhomogeneous dust cosmologies.

Towards realistic singularity-free cosmological models.

We present an explicit general family of inhomogeneous cosmological models. The family contains an arbitrary function of comoving time (interpretable as the cosmological scale factor) and four arbitrary parameters. In general, it is a solution of Einstein's field equations for a fluid with anisotropic pressures, but it also includes a big subfamily of perfect-fluid metrics. The most interesting feature of this family is that it contains both all the diagonal separable singularity-free cosmological models recently found and all the Friedmann-Lema\^{\i}tre-Robertson-Walker standard models. This property allows one to speculate on the construction of some interesting models in which the Universe has been FLRW-like from some time on (for instance, since the nucleeosynthesis time), but it also went through primordial singularity-free inhomogeneous epochs (in fact, there are quite natural possibilities in which these primordial epochs are inflationary) without ever violating energy conditions or other physical properties. Nevertheless, the physical processes leading to the isotropization and homogenization of the Universe are not fixed nor indicated by the models themselves. The interesting properties of the general model are studied in some detail. \textcent{} 1996 The American Physical Society.

Stationary and axisymmetric perfect fluid solutions with conformal motion

Stationary and axisymmetric perfect fluid metrics are studied under the assumption of the existence of a conformal Killing vector field and in the general case of differential rotation. The possible Lie algebras for the conformal group and corresponding canonical line elements are explicitly given. It turns out that only four different cases appear, the Abelian and three others called I, II and III. We explicitly find all the solutions in the Abelian and I cases. For the Abelian case the general solution depends on an arbitrary function of a single variable and the perfect fluid satisfies the equation of state . This class of metrics is the one presented recently by one of us. The general solution for case I is a new Petrov type D metric, with the velocity vector outside the 2-space spanned by the two principal null directions and a barotropic equation of state . For the cases II and III, the general solution has been found only under the further assumption of a natural separation of variables ansatz. The conformal Killing vectors in the solutions that come out here are, in fact, homothetic. No barotropic equation of state exists in these metrics unless for a new Petrov type D solution belonging to case II and with which cannot be interpreted as an axially symmetric solution and such that the velocity vector points in the direction of one of the Killing vectors. This solution has the previously unknown curious property that both commuting Killing vectors are timelike everywhere.

New and inhomogeneous cosmological models from the generalized Kerr--Schild transformation

New perfect-fluid solutions of Einstein's field equations have been found by means of the generalized Kerr--Schild transformation for perfect-fluid metrics. All these solutions are Petrov type D with the velocity vector of the fluid out of the two-space spanned by the two null principal directions of the Weyl tensor. In general, the solutions have just one Killing vector, but some special cases admit two commuting Killing vectors. All the solutions can be interpreted as inhomogeneous cosmological models, and an interesting property is that they contain some Friedman--Robertson--Walker metrics as special cases. Thus, they can be seen as inhomogeneous generalizations of the standard FRW cosmologies.

Inhomogeneous perfect fluid cosmologies with a nonorthogonally transitive symmetry group

Inhomogeneous perfect fluid cosmologies are studied which have a nonorthogonally transitive two-dimensional Abelian symmetry group with one hypersurface-orthogonal Killing vector. The study is limited to nonrotating perfect fluids with an equation of state p=( gamma -1) mu and separable metric functions. Except when gamma =2, solutions generally turn out to be self-similar and have a big-bang singularity. Some particular solutions are given, among which are new Petrov type II and D stiff perfect fluid metrics.

Physical properties of Buchdahl's three-parameter static spherically symmetric perfect fluid metrics

The physical properties of a static spherically symmetric perfect fluid metric proposed by Buchdahl are investigated. The general solution can be written in terms of associated Legendre functions. If the adiabatic sound speed is to be less than the velocity of light at the boundary of the sphere, its radius will be larger than 1.238 times the Schwarzschild radius. Two particular solutions are studied in detail.

Petrov types D and II perfect-fluid solutions in generalized Kerr–Schild form

Petrov types D and II perfect-fluid solutions are obtained starting from conformally flat perfect-fluid metrics and by using a generalized Kerr–Schild ansatz. Most of the Petrov type D metrics obtained have the property that the velocity of the fluid does not lie in the two-space defined by the principal null directions of the Weyl tensor. The properties of the perfect-fluid sources are studied. Finally, a detailed analysis of a new class of spherically symmetric static perfect-fluid metrics is given.

New LRS perfect-fluid cosmological models

Starting from locally rotationally symmetric (LRS) perfect-fluid metrics and by using a Kerr-Schild method the author obtains new exact perfect-fluid metrics of the same type. The method is used to give a large class of new LRS cosmological models explicitly starting from the known ones.

Stationary axisymmetric perfect-fluid metrics with q+3 p=const

All Petrov type D stationary axisymmetric rigidly rotating perfect-fluid metrics with an equation of state q+3 p=const are explicity obtained. There are two classes of metrics, the general solution of class I is the Wahlquist solution. Class II contains Kramer's metric and any of the solutions in this class can be obtained as a limiting case of Wahlquist's solution. The limiting procedure leading to Kramer's metric is given explicitly.

A class of perfect fluid metrics with flat three-dimensional hypersurfaces

The class of perfect fluid and vacuum space-times with a family of flat three-slices and a tensor of exterior curvature covariantly constant within these slices is examined and the corresponding solutions are found. It is shown that this class contains the class of metrics with three commuting Killing vectors. Therefore, e.g., all known stationary metrics with cylindrical or plane symmetry are generalized. An instruction is given for constructing perfect fluid metrics with this symmetry and a connection to a vacuum across surfaces p=0. Thereby the equation of state of the interior rotating perfect fluid can be arbitrarily chosen and the positivity of density and pressure can be forced. A geometric criterion of the interior metric with rotating matter is found that decides whether the exterior solution is stationary or static. Besides solutions with three symmetries, inhomogeneous metrics also are found. Among them is a solution with one symmetry and rotating, expanding, shearing, and accelerating perfect fluid. All resulting vacuum solutions are already known.

Petrov type D perfect-fluid solutions in generalized Kerr–Schild form

Generalized Kerr–Schild space-times for a perfect-fluid source are investigated. New Petrov type D perfect fluid solutions are obtained starting from conformally flat perfect-fluid metrics.