Aidan J Keane

The conformal group SO(4,2) and Robertson-Walker spacetimes

The Robertson-Walker spacetimes are conformally flat and so are conformally invariant under the action of the Lie group SO (4,2), the conformal group of Minkowski spacetime. We find a local coordinate transformation allowing the Robertson-Walker metric to be written in a manifestly conformally flat form for all values of the curvature parameter k continuously and use this to obtain the conformal Killing vectors of the Robertson-Walker spacetimes directly from those of the Minkowski spacetime. The map between the Minkowski and Robertson-Walker spacetimes preserves the structure of the Lie algebra so (4,2). Thus the conformal Killing vector basis obtained does not depend upon k , but has the disadvantage that it does not contain explicitly a basis for the Killing vector subalgebra. We present an alternative set of bases that depend (continuously) on k and contain the Killing vector basis as a sub-basis (these are compared with a previously published basis). In particular, bases are presented which include the Killing vectors for all Robertson-Walker spacetimes with additional symmetry, including the Einstein static spacetimes and the de Sitter family of spacetimes, where the basis depends on the Ricci scalar R

An extension of the Newman–Janis algorithm

The Newman–Janis algorithm is supplemented with a null rotation and applied to the tensors of the Reissner–Nordstrom spacetime to generate the metric, Maxwell, Ricci and Weyl tensors for the Kerr–Newman spacetime. This procedure also provides a mechanism whereby the Carter Killing tensor arises from the geodesic angular momentum tensor of the underlying Reissner–Nordstrom metric. The conformal Killing tensor in the Kerr–Newman spacetime is generated in a similar fashion. The extended algorithm is also applied to the Killing vectors of the Reissner–Nordstrom spacetime with interesting consequences. The Schwarzschild to Kerr transformation is a special case.

Conformal symmetry classes for pp-wave spacetimes

We determine conformal symmetry classes for the pp-wave spacetimes. This refines the isometry classification scheme given by Sippel and Goenner (1986 Gen. Rel. Grav. 18 1229). It is shown that every conformal Killing vector for the null fluid type N pp-wave spacetimes is a conformal Ricci collineation. The maximum number of proper non-special conformal Killing vectors in a type N pp-wave spacetime is shown to be three, and we determine the form of a particular set of type N pp-wave spacetimes admitting such conformal Killing vectors. We determine the conformal symmetries of each type N isometry class of Sippel and Goenner and present new isometry classes.

A classification of spherically symmetric spacetimes

A complete classification of locally spherically symmetric four-dimensional Lorentzian spacetimes is given in terms of their local conformal symmetries. The general solution is given in terms of canonical metric types and the associated conformal Lie algebras. The analysis is based upon the local conformal decomposition into 2+2 reducible spacetimes and the Petrov type. A variety of physically meaningful example spacetimes are discussed.

Conformal symmetry inheritance in null fluid spacetimes

We define inheriting conformal Killing vectors for null fluid spacetimes and find the maximum dimension of the associated inheriting Lie algebra. We show that for non-conformally flat null fluid spacetimes, the maximum dimension of the inheriting algebra is seven and for conformally flat null fluid spacetimes the maximum dimension is eight. In addition, it is shown that there are two distinct classes of non-conformally flat generalized plane wave spacetimes which possess the maximum dimension, and one class in the conformally flat case.

Killing tensors in pp-wave spacetimes

The formal solution of the second-order Killing tensor equations for the general pp-wave spacetime is given. The Killing tensor equations are integrated fully for some specific pp-wave spacetimes. In particular, the complete solution is given for the conformally flat plane wave spacetimes and we find that irreducible Killing tensors arise for specific classes. The maximum number of independent irreducible Killing tensors admitted by a conformally flat plane wave spacetime is shown to be six. It is shown that every pp-wave spacetime that admits an homothety will admit a Killing tensor of Koutras type and, with the exception of the singular scale-invariant plane wave spacetimes, this Killing tensor is irreducible.

The classical Kepler problem and geodesic motion on spaces of constant curvature

In this paper we clarify and generalize previous work by Moser and Belbruno concerning the link between the motions in the classical Kepler problem and geodesic motion on spaces of constant curvature. Both problems can be formulated as Hamiltonian systems and the phase flow in each system is characterized by the value of the corresponding Hamiltonian and one other parameter (the mass parameter in the Kepler problem and the curvature parameter in the geodesic motion problem). Using a canonical transformation the Hamiltonian vector field for the geodesic motion problem is transformed into one which is proportional to that for the Kepler problem. Within this framework the energy of the Kepler problem is equal to (minus) the curvature parameter of the constant curvature space and the mass parameter is given by the value of the Hamiltonian for the geodesic motion problem. We work with the corresponding family of evolution spaces and present a unified treatment which is valid for all values of energy continuous...

Conformally reducible 1+3 spacetimes

Spacetimes which are conformally related to reducible 1+3 spacetimes are considered. We classify these spacetimes according to the conformal algebra of the underlying reducible spacetime, giving in each case canonical expressions for the metric and conformal Killing vectors, and providing physically meaningful examples.

Spectrum generating algebras for the classical Kepler problem

It is shown that the so(4, 2) spectrum generating algebra for the classical Kepler problem for non-zero energies can be obtained from the generators of the spacetime conformal group SO(4, 2). This is achieved by exploiting the equivalence of Kepler motion and null geodesic motion in conformally flat Einstein static spacetimes. We show that it is the existence of a time-dependent representation of the so(4, 2) spectrum generating algebra for null geodesic motion in the Einstein static spacetimes (originating from the so(4, 2) algebra of first integrals) which determines the corresponding spectrum generating algebra structure in the classical Kepler problem. Further, for the zero energy state, it is shown that only the iso(3) invariance subalgebra has a direct physical significance.

Some remarks on the symmetries of the Kerr spacetime

It is remarked that the computer-generated results (and more) regarding symmetries of the Kerr spacetime recently reported by Jerie et al (1999 Class. Quantum Grav. 16 2885) are well known and that similar results for other standard spacetimes can be obtained by geometrical techniques.