Axial symmetry and conformal Killing vectors

Section: Proposition 1 the Axial Killing Vector Is Spacelike On Umentioning

confidence: 94%

Section: Axially Symmetric Spacetimes Admitting Further Symmetriesmentioning

confidence: 99%

Some developments on axial symmetry

Carot

2000

“…Consequences of the definition are that G 2 group has to be Abelian [17], and that the set of fixed points must form a timelike two-surface [18], this is the axis. The axial Killing η is then intrinsically defined by normalising it demanding ∂ α η 2 ∂ α η 2 /4 η 2 → 1 at the axis.…”

Section: Remark 21mentioning

confidence: 99%

Axially symmetric equilibrium regions of Friedmann–Lemaître–Robertson–Walker universes

Nolan

Vera

2005

Abstract. The study of the matching of stationary and axisymmetric spacetimes with Friedmann-Lemaître-Robertson-Walker spacetimes preserving the axial symmetry is presented. We show, in particular, that any orthogonally transitive stationary and axisymmetric region in FLRW must be static, irrespective of the matter content. Therefore, previous results on static regions in FLRW cosmologies apply. As a result, the only stationary and axisymmetric vacuum region that can be matched to a (nonstatic) FLRW spacetime is a spherically symmetric region of Schwarzschild. This constitutes another uniqueness result for the Einstein-Straus model (as well as its Oppenheimer-Snyder counterpart), and hence another indication of its unsuitability as an answer to the influence of the cosmic expansion on local physics.

“…The amount of work that represents to consider the fourteen inequivalent three-dimensional Lie algebras arising when the orbits of one of the generators are closed is substantially restricted due to a recent result by the authors [7] which states that in an axially symmetric space-time (stationary or not) a conformal Killing vector must necessarily commute with the axial Killing vector whenever no more conformal symmetry exists in the space-time. This result is purely geometric and does not depend on orthogonal transitivity or any matter contents of the space-time.…”

Section: Introductionmentioning

confidence: 99%

Stationary and axisymmetric perfect fluid solutions with conformal motion

Mars

Senovilla

1994

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.