Stationary Perfect Fluid Solutions Research Articles

Static and stationary dark fluid universes: a gravitoelectromagnetic perspective

The usual characterization of exact solutions of Einstein field equations, including cosmological solutions, is based on the symmetry properties of their corresponding metrics which is obviously mathematically involved. Here we present a physical characterization of the static and stationary perfect fluid solutions of the Einstein field equations by employing the 1+3 formulation of spacetime decomposition which introduces the so-called quasi-Maxwell form of the Einstein field equations in the broader context of gravitoelectromagnetism. These solutions have a single or 2-component perfect fluid sources, and are characterized according to their gravitoelectric and gravitomagnetic fields which are the gravitational analogs of the electromagnetic fields. It is shown that the absence or presence of either or both of these fields could restrict the equations of state of the contributing perfect fluid sources. As the representative of each family of solutions, we consider those spaces that include the cosmological term as a dark fluid source with the equation of state p=-rho = constant.

Comment on ‘Stationary perfect fluid solutions with differential rotation’

We vindicate our results in Mars and Senovilla (Phys Rev D 54, 6166–6180, 1996), which have been recently put in doubt or misunderstood in Garcia and Ulloa (Gen Rel Grav 39, 1639–1650, 2007). In particular, we maintain that there indeed exist axially symmetric differentially rotating perfect-fluid solutions satisfying all energy conditions within the family of solutions presented in Senovilla (Class Quant Grav 9, L167–L169, 1992). We remark that the existence of an axis of symmetry can never be “re-interpreted” as the existence of a Killing horizon, and vice versa. We prove that such horizons are simply impossible for the perfect fluids within the family, and that regular axes of symmetry (or curvature singularities) are the only possibilities. Other inaccuracies or misunderstandings appeared in Garcia and Ulloa (Gen Rel Grav 39, 1639–1650, 2007) are clarified.

Stationary perfect fluid solutions with differential rotation

In the framework of a class of metrics allowing for a timelike and a spacelike symmetries, usually referred to as stationary and axisymmetric gravitational fields, coupled to differentially rotating perfect fluids, the existing exact solutions to the corresponding Einstein equations are analyzed from the point of view of the fulfillment of the energy conditions and the existence of an axis of rotation. The main conclusion is that none of the reported exact solutions fulfills all these physical requirements, and at most, they can be thought of as stationary cyclic symmetric cosmological spacetimes.

Stationary generalized Kerr–Schild spacetimes

In this paper we have applied the generalized Kerr–Schild transformation finding a new family of stationary perfect-fluid solutions of the Einstein field equations. The procedure used combines some well-known techniques of null and timelike vector fields, from which some properties of the solutions are studied in a coordinate-free way. These spacetimes are algebraically special being their Petrov types II and D. This family includes all the classical vacuum Kerr–Schild spacetimes, excepting the plane-fronted gravitational waves, and some other interesting solutions as, for instance, the Kerr metric in the background of the Einstein Universe. However, the family is much more general and depends on an arbitrary function of one variable.

Generating new perfect-fluid solutions from known ones: II

The properties of a transformation previously considered for generating new perfect-fluid solutions from known ones are further investigated. It is assumed that the 4-velocity of the fluid is parallel to the stationary Killing field, and also that the norm and the twist potential of the stationary Killing field are functionally related. This case is complementary to the case studied in our previous paper. The transformation can be applied to generate possibly new perfect-fluid solutions from known ones only for the case of the barotropic equation of state or, alternatively, for the case of a static spacetime. For static spacetimes our method recovers the Buchdahl transformation. It is demonstrated, moreover, that Herlt's technique for constructing stationary perfect-fluid solutions from static ones is, actually, a special case of the method considered in this paper.