Abelian Isometry Group Research Articles

Repères symétriques de solutions de type D des équations de Maxwell-Einstein avec constante cosmologique

Rigid symmetrical null tetrad already considered by McLenaghan, Taricq and Debever [4] are proven to exist in a family D1 of solutions of Maxwell-Einstein equations with cosmological constant. One deduces the general form of a metic with an abelian isometry group G2, orthogonally transitive. The case of null orbits is presented.

On stationary, axially symmetric solutions of Jordan’s five-dimensional, unified theory

It is shown that the association of a linear eigenvalue problem for solutions of Einstein’s equations admitting a two-parameter Abelian group of isometries can be extended to Jordan’s five-dimensional, unified theory admitting three commuting Killing vectors. The reduction to a two-dimensional problem, the derivation of infinitely many conservation laws and the generation of one-parameter families of solutions can thereby be transcribed almost literally.

Riemannian-Maxwellian invertible structures in general relativity

It is shown that a space-time admitting a nonsingular 2-form satisfying the source-free Maxwell equations and a Lorentzian involution under which the 2-form and the exterior derivative of a related 2-form are skew invariant while the trace-free Ricci tensor and the covariant derivative of the involution itself are invariant possesses locally an invertible 2-parameter Abelian isometry group with nonsingular orbits.

Ehlers-Harrison-type transformations for Jordan's extended theory of gravitation

It is shown that the “hidden” symmetry of the stationary Einstein equations under the groupSL(2,R) can be extended to a symmetry under the groupSL(3,R) in the case of Jordan's five-dimensional, unified theory. More generally one obtains an action of the groupSL(n+2,R) on the set of (n+4)-dimensional Einstein spaces admitting a (n+1)-parameter Abelian group of isometries.

Spatially homogeneous and inhomogeneous cosmologies with equation of statep=?

This paper gives an algorithm for generating solutions of the Einstein field equations which have an irrotational perfect fluid, with equation of statep=μ, as source, and which admit a two-parameter Abelian group of local isometries. The algorithm is used to derive a variety of new and known spatially homogeneous cosmological models, both tilted and nontilted. However, since the solutions in general only admit two Killing vectors, spatially inhomogeneous models are also obtained. Finally, it is pointed out that the solution generation technique used in this paper is closely related to solution generation techniques that have been used to generate solutions of the source-free Brans-Dicke field equations, and of the Einstein field equations with a massless scalar field as source.

Exact solutions of Yang-Mills equations on curved space-times

By use of the complex vectorial formalism we give, in a simple way, the general solution of SU(2) Yang-Mills equations which is invariant on those space-times which admit an invertible 2-parameter Abelian isometry group or which are spherically symmetric.

A characterization of carter's separable space-times

In a space-time admitting a two-parameter Abelian isometry group, and a quadratic Killing tensor with the eigenvalues (λ, λ, μ, μ) and vanishing Lie derivatives with respect to the Killing vectors, we construct a canonical coordinate system. The isometry group acts orthogonally transitively. The Hamilton-Jacobi equation is separable. We give a necessary and sufficient condition for the separability of the Klein-Gordon equation. We obtain Carter's space-times with completely separable Klein-Gordon equation.

On space–time Killing tensors with a Segre characteristic [(11)(11)

We consider the set of all space–times which admit a Killing tensor Kαβ whose Segre characteristic is [(11)(11)] and whose eigenvalues λ1 and λ2 satisfy the condition that dλ1 - dλ2 is not null. We prove that there locally exist scalar fields φ3, φ4 such that λ1, λ2, φ3, φ4 constitute a chart, and dλ1⋅dλ2 =dλi⋅dφr=0 for i=1, 2 and r=3, 4. Also, relative to this coordinate system, the metric has the same general form as Carter’s Hamilton–Jacobi separable metric except that his arbitrary functions of λi are replaced by functions of λi, φ3, φ4. If Rαβ(∇αλ1)(∇βλ2) =0, there exists a two parameter Abelian group of isometries which commute with Kαβ; also (φ3,φ4) can be chosen so that ∂/∂φ3 and ∂/∂φ4 are the Killing vectors. Rαβ(∇αλ1)(∇βλ2) =0 is necessary and sufficient for Schrödinger equation separability in case (a) of Carter. A Newtonian analog of our results is discussed.

Structures prémaxwelliennes involutives en relativité générale

We describe a situation called pre-maxwellian involutive structure which implies for Lorentzian metrics the existence of a two dimensional abelian invertible group of isometries. Some consequences are presented.

Killing vector fields and the Einstein-Maxwell field equations in general relativity

A number of theorems concerning non-null electrovac spacetimes, that is space-times whose metric satisfies the source-free Einstein-Maxwell equations for some non-null bivector Fij, are presented. Firstly, we suppose that the metric is invariant under a one-parameter group of isornetries with Killing vector field ξ. It is proved that the electromagnetic field tensor Fij is invariant under the group, in the sense that its Lie derivative with respect to ξ vanishes, if and only if the gradient αij of the complexion scalar is orthogonal to ξ. It is is also proved that if in addition ξ is hypersurface orthogonal, it is necessarily parallel to α,i. These results are used to generalize theorems of Perjes and Majumdar concerning static electrovac space-times. Secondly, we suppose that the metric is invariant under a two-parameter othogonally transitive Abelian group of isometries. It is proved that in this case Fij is necessarily invariant under the group. The above results can be used to simplify many derivations of exact solutions of the Einstein-Maxwell equations.

Hamilton-Jacobi and Schrodinger Separable Solutions of Einstein’s Equations

This paper contains an investigation of spaces with a two parameter Abelian isometry group in which the Hamilton-Jacobi equation for the geodesies is soluble by separation of variables in such a way that a certain natural canonical orthonormal tetrad is determined. The spaces satisfying the stronger condition that the corresponding Schrodinger equation is separable are isolated in a canonical form for which Einstein’s vacuum equations and the source-free Einstein-Maxwell equations (with or without a Λ term) can be solved explicitly. A fairly extensive family of new solutions is obtained including the previously known solutions of de Sitter, Kasner, Taub-NUT, and Kerr as special cases.