Perfect Fluid Spacetimes Research Articles

Spacetimes with Semisymmetric Energy-Momentum Tensor

The object of the present paper is to introduce spacetimes with semisymmetric energy-momentum tensor. At first we consider the relation R(X,Y)⋅T=0, that is, the energy-momentum tensor T of type (0,2) is semisymmetric. It is shown that in a general relativistic spacetime if the energy-momentum tensor is semisymmetric, then the spacetime is also Ricci semisymmetric and the converse is also true. Next we characterize the perfect fluid spacetime with semisymmetric energy-momentum tensor. Then, we consider conformally flat spacetime with semisymmetric energy-momentum tensor. Finally, we cited some examples of spacetimes admitting semisymmetric energy-momentum tensor.

Spacetimes admitting W2-curvature tensor

The object of this paper is to study spacetimes admitting W2-curvature tensor. At first we prove that a W2-flat spacetime is conformally flat and hence it is of Petrov type O. Next, we prove that if the perfect fluid spacetime with vanishing W2-curvature tensor obeys Einstein's field equation without cosmological constant, then the spacetime has vanishing acceleration vector and expansion scalar and the perfect fluid always behaves as a cosmological constant. It is also shown that in a perfect fluid spacetime of constant scalar curvature with divergence-free W2-curvature tensor, the energy-momentum tensor is of Codazzi type and the possible local cosmological structure of such a spacetime is of type I, D or O.

Pseudo-Z symmetric space-times

In this paper, we investigate Pseudo-Z symmetric space-time manifolds. First, we deal with elementary properties showing that the associated form Ak is closed: in the case the Ricci tensor results to be Weyl compatible. This notion was recently introduced by one of the present authors. The consequences of the Weyl compatibility on the magnetic part of the Weyl tensor are pointed out. This determines the Petrov types of such space times. Finally, we investigate some interesting properties of (PZS)4 space-time; in particular, we take into consideration perfect fluid and scalar field space-time, and interesting properties are pointed out, including the Petrov classification. In the case of scalar field space-time, it is shown that the scalar field satisfies a generalized eikonal equation. Further, it is shown that the integral curves of the gradient field are geodesics. A classical method to find a general integral is presented.

ALGORITHMIC CONSTRUCTION OF EXACT SOLUTIONS FOR NEUTRAL STATIC PERFECT FLUID SPHERES

Although it ranks amongst the oldest of problems in classical general relativity, the challenge of finding new exact solutions for spherically symmetric perfect fluid spacetimes is still ongoing because of a paucity of solutions which exhibit the necessary qualitative features compatible with observational evidence. The problem amounts to solving a system of three partial differential equations in four variables, which means that any one of four geometric or dynamical quantities must be specified at the outset and the others should follow by integration. The condition of pressure isotropy yields a differential equation that may be interpreted as second-order in one of the space variables or also as first-order Ricatti type in the other space variable. This second option has been fruitful in allowing us to construct an algorithm to generate a complete solution to the Einstein field equations once a geometric variable is specified ab initio. We then demonstrate the construction of previously unreported solutions and examine these for physical plausibility as candidates to represent real matter. In particular we demand positive definiteness of pressure, density as well as a subluminal sound speed. Additionally, we require the existence of a hypersurface of vanishing pressure to identify a radius for the closed distribution of fluid. Finally, we examine the energy conditions. We exhibit models which display all of these elementary physical requirements.

PSEUDO-Q-SYMMETRIC RIEMANNIAN MANIFOLDS

In this paper, we introduce a new kind of tensor whose trace is the well-known Z tensor defined by the present authors. This is named Q tensor: the displayed properties of such tensor are investigated. A new kind of Riemannian manifold that embraces both pseudo-symmetric manifolds ( PS )n and pseudo-concircular symmetric manifolds [Formula: see text] is defined. This is named pseudo-Q-symmetric and denoted with ( PQS )n. Various properties of such an n-dimensional manifold are studied: the case in which the associated covector takes the concircular form is of particular importance resulting in a pseudo-symmetric manifold in the sense of Deszcz [On pseudo-symmetric spaces, Bull. Soc. Math. Belgian Ser. A44 (1992) 1–34]. It turns out that in this case the Ricci tensor is Weyl compatible, a concept enlarging the classical Derdzinski–Shen theorem about Codazzi tensors. Moreover, it is shown that a conformally flat ( PQS )n manifold admits a proper concircular vector and the local form of the metric tensor is given. The last section is devoted to the study of ( PQS )n space-time manifolds; in particular we take into consideration perfect fluid space-times and provide a state equation. The consequences of the Weyl compatibility on the electric and magnetic part of the Weyl tensor are pointed out. Finally a ( PQS )n scalar field space-time is considered, and interesting properties are pointed out.

Conformally 2 + 2-decomposable perfect fluid spacetimes with constant curvature factor manifolds

Conformally 2 + 2-decomposable non-conformally flat spacetimes, which are the solutions of the Einstein field equations with a perfect fluid source, are discussed. These spacetimes are necessarily of Petrov-type D and, when the fluid velocity is aligned with the plane of principal null directions, form a subset of the LRSII class, determined up to two first-order partial differential equations. When, on the contrary, the fluid is non-aligned, the general solution can be given in terms of elementary functions.

Parallel tensors and Ricci solitons in N (k)-quasi Einstein manifolds

The Eisenhart problem of finding parallel and symmetric tensors is considered in the framework ofN (k)-quasi Einstein manifolds and the result is connected with Ricci solitons. If the generator of the manifold provides a Ricci soliton then this is: i) shrinking on a class of conformally flat perfect fluid space-times and on quasi-umbilical hypersurfaces, in particular unit spheres; ii) expanding if the generator is of torse-forming type.

On Conformally Flat Almost Pseudo-Ricci Symmetric Spacetimes

We consider a conformally flat almost pseudo-Ricci symmetric spacetime. At first we show that a conformally flat almost pseudo-Ricci symmetric spacetime can be taken as a model of the perfect fluid spacetime in general relativity and cosmology. Next we show that if in a conformally flat almost pseudo-Ricci symmetric spacetime the matter distribution is perfect fluid whose velocity vector is the vector field corresponding to 1-form B of the spacetime, the energy density and the isotropic pressure are not constants. We also show that a conformally flat almost pseudo-Ricci symmetric spacetime is the Robertson-Walker spacetime. Finally we give an example of a conformally flat almost pseudo-Ricci symmetric spacetime with non-zero non-constant scalar curvature admitting a concircular vector field.

Group analysis of a conformal perfect fluid spacetime

We find new exact solutions of the Einstein field equations for a perfect fluid metric conformal to a spacetime of type D in the Petrov classification scheme. We analyse the complete system of equations using Lie group analysis. While previous work was confined to conformal factors of the form U = U(t, x), we investigate the complete situation U = U(t, x, y, z) as well as an auxiliary integrability condition. New classes of solutions are generated via the symmetry generators. The resulting solutions are examined for physical plausibility. Expressions for the energy density and pressure are obtained explicitly and empirical results suggest that these dynamical quantities are positive as expected.

Conformal Einstein Perfect Fluid Spacetimes

A review is given of results obtained in connection with perfect fluid spacetimes which are conformally related to Einstein spaces.

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.

ON LORENTZIAN QUASI-EINSTEIN MANIFOLDS

The notion of quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. The object of the present paper is to study Lorentzian quasi-Einstein manifolds. Some basic geometric properties of such a manifold are obtained. The applications of Lorentzian quasi-Einstein manifolds to the general relativity and cosmology are investigated. Theories of gravitational collapse and models of Supernova explosions [5] are based on a relativistic fluid model for the star. In the theories of galaxy formation, relativistic fluid models have been used in order to describe the evolution of perturbations of the baryon and radiation components of the cosmic medium [32]. Theories of the structure and stability of neutron stars assume that the medium can be treated as a relativistic perfectly conducting magneto fluid. Theories of relativistic stars (which would be models for supermassive stars) are also based on relativistic fluid models. The problem of accretion onto a neutron star or a black hole is usually set in the framework of relativistic fluid models. Among others it is shown that a quasi-Einstein spacetime represents perfect fluid spacetime model in cosmology and consequently such a spacetime determines the final phase in the evolution of the universe. Finally the existence of such manifolds is ensured by several examples constructed from various well known geometric structures.

Lanczos Potential and Perfect Fluid Spacetimes

Using method of general observers and the spin coefficient formalism of Newman and Penrose, the Lanczos potential has been studied for perfect fluid spacetimes.

Thermodynamics of magnetized binary compact objects

Binary systems of compact objects with electromagnetic field are modeled by helically symmetric Einstein-Maxwell spacetimes with charged and magnetized perfect fluids. Previously derived thermodynamic laws for helically symmetric perfect-fluid spacetimes are extended to include the electromagnetic fields, and electric currents and charges; the first law is written as a relation between the change in the asymptotic Noether charge $\ensuremath{\delta}Q$ and the changes in the area and electric charge of black holes, and in the vorticity, baryon rest mass, entropy, charge and magnetic flux of the magnetized fluid. Using the conservation laws of the circulation of magnetized flow found by Bekenstein and Oron for the ideal magnetohydrodynamic fluid, and also for the flow with zero conducting current, we show that, for nearby equilibria that conserve the quantities mentioned above, the relation $\ensuremath{\delta}Q=0$ is satisfied. We also discuss a formulation for computing numerical solutions of magnetized binary compact objects in equilibrium with emphasis on a first integral of the ideal magnetohydrodynamic-Euler equation.

Separating expansion from contraction in spherically symmetric models with a perfect fluid: Generalization of the Tolman-Oppenheimer-Volkoff condition and application to models with a cosmological constant

We investigate spherically symmetric perfect-fluid spacetimes and discuss the existence and stability of a dividing shell separating expanding and collapsing regions. We perform a 3+1 splitting and obtain gauge invariant conditions relating the intrinsic spatial curvature of the shells to the Misner-Sharp mass and to a function of the pressure that we introduce and that generalizes the Tolman-Oppenheimer-Volkoff equilibrium condition. We find that surfaces fulfilling those two conditions fit, locally, the requirements of a dividing shell and we argue that cosmological initial conditions should allow its global validity. We analyze the particular cases of the Lema\^itre-Tolman-Bondi dust models with a cosmological constant as an example of a cold dark matter model with a cosmological constant (\Lambda-CDM) and its generalization to contain a central perfect-fluid core. These models provide simple, but physically interesting illustrations of our results.

Maximally inhomogeneous Gödel-Farnsworth-Kerr generalizations

It is pointed out that physically meaningful aligned Petrov type D perfect fluid space-times with constant zero-order Riemann invariants are either the homogeneous solutions found by Gödel (isotropic case) and Farnsworth and Kerr (anisotropic case), or new inhomogeneous generalizations of these with non-constant rotation. We present the construction of the line element and the local geometric properties for the latter.

Perfect fluid space-times whose energy-momentum tensor is conformal Killing

We show that the energy-momentum tensor T of an expanding perfect fluid space-time (M,g) is a nontrivial conformal Killing tensor if and only if M is shear-free, vorticity-free, and satisfies certain energy and force equations. In particular, if T is conformal Killing, we show that the perfect fluid is stationary and its energy-density and pressure are constant.

On Ricci Riemannian Manifolds

The object of this paper is to study some properties of a type of Riemannian manifold called Ricci Riemannian manifold (M, g), n > 2. The perfect fluid space time (M, g) of general relativity is also studied. Mathematics Subject Classification 2000: 53C25.

Concircular Curvature Tensor and Fluid Spacetimes

In the differential geometry of certain F-structures, the importance of concircular curvature tensor is very well known. The relativistic significance of this tensor has been explored here. The spacetimes satisfying Einstein field equations and with vanishing concircular curvature tensor are considered and the existence of Killing and conformal Killing vectors have been established for such spacetimes. Perfect fluid spacetimes with vanishing concircular curvature tensor have also been considered. The divergence of concircular curvature tensor is studied in detail and it is seen, among other results, that if the divergence of the concircular tensor is zero and the Ricci tensor is of Codazzi type then the resulting spacetime is of constant curvature. For a perfect fluid spacetime to possess divergence-free concircular curvature tensor, a necessary and sufficient condition has been obtained in terms of Friedmann-Robertson-Walker model.

Matching stationary spacetimes

Using the quasi-Maxwell formalism, we derive the necessary and sufficient conditions for the matching of two stationary spacetimes along a stationary time-like hypersurface, expressed in terms of the gravitational and gravitomagnetic fields and the 2-dimensional matching surface on the space manifold. We prove existence and uniqueness results to the matching problem for stationary perfect fluid spacetimes with spherical, planar, hyperbolic and cylindrical symmetry. Finally, we find an explicit interior for the cylindrical analogue of the NUT spacetime.