Family of Rotating Anisotropic Fluid Solutions Which Match to Kerr's Solution

Abstract

We present a family of exact rotating anisotropic fluid solutions, which satisfy all energy conditions for certain values of their parameters. The components of the Ricci tensor R µν the eigenvalues of the tensor R ν µ and the energy-momentum tensor T µν of the solutions are given explicitly. All members of the family have the ring singularity of Kerr's solution and most of them one or two more singularities. The solutions can be matched to the solution of Kerr on three closed surfaces, which for proper values of the parameters of the solutions approximate oblate spheroids. All matching surfaces are thin shells.For some values of a constant the surface density in one of them is positive everywhere and in this surface and in its interior all energy conditions are satisfied. PACS number(s):

Interior solution for the Kerr metric

“…where (15,16) have been used, and the subscript 0 indicates that the quantity is evaluated at the center of the distribution. Using the conditions above in (26) we may write for Λ, Ξ and Π…”

“…Furthermore, the good behaviour of the function Ω on the symmetry axis is fulfilled since w Σ and w ′ Σ vanish when y = ±1. Finally, let us note that the energy-momentum tensor components (14), (15) do not diverge on the symmetry axis because the first derivative with respect to the angular variable θ of both Ω ′ and Ω ′′ vanishes on the symmetry axis since not only w Σ and w ′ Σ vanish there, but their first derivatives with respect to θ vanish as well.…”

“…Since the discovery of the Kerr metric [1] there have been many attempts to find a physically meaningful matter distribution that could serve as its source (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and references therein). However, no satisfactory solution has yet been found to this problem.…”

Physical properties of a source of the Kerr metric: Bound on the surface gravitational potential and conditions for the fragmentation

“…In [1] we provided a general procedure to choose the interior metric functionsâ,ĝ and Ω producing physically meaningful models. With this aim, besides the fulfilment of the junction conditions (16), we required that all physical variables be regular within the fluid distribution and the energy density be positive. Following this procedure, we have for the interior of the Kerr metric:â…”

Stability analysis of a class of anisotropic spacetimes

“…Spacetimes sourced by rotating anisotropic fluids have been used to model the interior of a rotating star such that the metric matches to the Kerr metric outside the star [5]. Although the spacetimes in [5] have ring singularities, this approach is interesting in view of the fact that to date, there is no example of a stellar interior sourced by a perfect fluid, that matches (with standard matching conditions) to the Kerr metric outside the star. There are other related compact objects that have been proposed with anisotropic interiors, such as gravastars [6].…”