Stationary and axisymmetric perfect fluid solutions with conformal motion

Abstract

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 other three 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 ρ = p+const. 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 ρ + 3p = 0. 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 ρ + 3p = 0 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.

An approximate global solution of Einstein’s equations for a rotating finite body

“…Only a few exact solutions of the Einstein equations can describe the gravitational field inside a rotating star, i. e., the gravitational field of an axisymmetric rotating perfect fluid (see [6] for a brief review and [7,8]). So far it has not been possible to match any of them to an asymptotically flat vacuum solution, the gravitational field outside the star, to obtain a global star model similar to those provided by Newtonian theory (some results point out that matching may be meaningless or doubtful for such metrics [9,10]).…”

Stationary and axisymmetric perfect fluids with one conformal Killing vector

“…Let us end this paper by adding a corrigendum to our previous paper [1]. The general solution for the Case I was explicitly found there.…”

“…which implies that the rotation Ω of the fluid is constant. Thus the general perfect fluid solution in Case I is rigidly rotating contrarily to what was stated in [1]. As a consequence, we have that this solution must be contained in the general Wahlquist family [6] which is the general solution (together with its limit cases [8] [9][5]) for Petrov type D stationary and axisymmetric rigidly rotating perfect fluids with equation of state ρ + 3p =const.…”

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We study the stationary and axisymmetric non-convective differentially rotating perfect-fluid solutions of Einstein's field equations admitting one conformal symmetry. We analyse the two inequivalent Lie algebras not exhaustively considered in [1] and show that the general solution for each Lie algebra depends on one arbitrary function of one of the coordinates while a set of three ordinary differential equations for four unknowns remains to be solved. The conformal Killing vector of these solutions is necessarily homothetic.

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“…The conformal Killing vector of these solutions is necessarily homothetic. We summarize in a table all the possible solutions for all the allowed Lie algebras and also add a corrigendum to an erroneous statement in [1] concerning the differentially rotating character of one of the solutions presented.In a recent paper [1] we considered the study of stationary (non-static) and axisymmetric non-convective differentially rotating perfect-fluid solutions admitting one proper conformal Killing vector. A purely geometric result [2] for stationary and axisymmetric spacetimes (with a well-defined axis of symmetry) admitting one conformal motion implies that both the stationary and the axial Killing vectors necessarily commute with the conformal Killing.…”