118984350
Comment on"Conformally flat stationary axisymmetric metrics"
Garcia and Campuzano claim to have found a previously overlooked family of stationary and axisymmetric conformally flat spacetimes, contradicting an old theorem of Collinson. In both these papers it is tacitly assumed that the isometry group is orthogonally transitive. Under the same assumption, we point out here that Collinson's result still holds if one demands the existence of an axis of symmetry on which the axial Killing vector vanishes. On the other hand if the assumption of orthogonal transitivity is dropped, a wider class of metrics is allowed and it is possible to find explicit counterexamples to Collinson's result.
118781109
Killing tensors from conformal Killing vectors
Some years ago Koutras presented a method of constructing a conformal Killing tensor from a pair of orthogonal conformal Killing vectors. When the vector associated with the conformal Killing tensor is a gradient, a Killing tensor (in general irreducible) can then be constructed. In this paper it is shown that the severe restriction of orthogonality is unnecessary and thus it is possible that many more Killing tensors can be constructed in this way. We also extend, and in one case correct, some results on Killing tensors constructed from a single conformal Killing vector. Weir's result that, for flat space, there are 84 independent conformal Killing tensors, all of which are reducible, is extended to conformally flat spacetimes. In conformally flat spacetimes it is thus possible to construct all the conformal Killing tensors and in particular all the Killing tensors (which in general will not be reducible) from conformal Killing vectors.
125608804
Note 65.39: Inverse and reciprocal functions
119366478
On the symmetries of the Edgar?Ludwig metric
The conformal Killing equations for the most general (non-plane wave) conformally flat pure radiation field are solved to find the conformal Killing vectors. As expected 15 independent conformal Killing vectors exist, but in general the metric admits no Killing or homothetic vectors. However for certain special cases a one-dimensional group of homotheties or motions may exist and in one very special case, overlooked by previous investigators, a two-dimensional homethety group exists. No higher dimensional groups of motions or homotheties are admitted by these metrics.
15659787
On Stephani's rotating dust solutions
Some years ago Stephani derived two classes of rotating dust solutions. In this paper Stephani's results are generalized to the case where the cosmological constant is non-zero. The solutions are of Petrov type D and the magnetic part of the Weyl tensor vanishes. The solutions involve seven arbitrary functions of a spacelike coordinate and, in general, they admit no Killing vectors. Nevertheless the spacetime geometry is relatively simple and admits a foliation of timelike hypersurfaces of constant intrinsic curvature and zero extrinsic curvature. The (geodesic) flow-lines of the fluid are tangent to these hypersurfaces and, in general, the flow is rotating, shearing and expanding. In the vacuum limit the solutions reduce to spacetimes of constant curvature.
10348027
Some restrictions on the symmetry groups of axially symmetric spacetimes
Lie transformation groups containing a one-dimensional subgroup acting cyclically on a manifold are considered. The structure of the group is found to be considerably restricted by the existence of a one-dimensional subgroup whose orbits are circles. The results proved do not depend on the dimension of the manifold nor on the existence of a metric, but merely on the fact that the Lie group acts globally on the manifold. Firstly, some results for the general case of an (m + 1)-dimensional Lie group are derived: those commutators of the associated Lie algebra involving the generator of the cyclic subgroup, X0 say, are severely restricted and, in a suitably chosen basis, take a simple form. The Jacobi identities involving X0 are then applied to show that there are further restrictions on the structure of the Lie algebra. All Lie algebras of dimensions two and three compatible with cyclic symmetry are obtained. In the two-dimensional case the group must be Abelian. For the three-dimensional case, the Bianchi type of the Lie algebra must be I, II, III, VII0, VIII or IX and furthermore in all cases the vector X0 forms part of a basis in which the algebra takes its canonical form. Finally, four-dimensional Lie algebras compatible with cyclic symmetry are considered and the results are related to the Petrov–Kruchkovich classification of all four-dimensional Lie algebras.
122426348
Pion Charge Exchange Scattering at High-Energies
We report on a study of pion charge-exchange scattering carried out at Fermilab in the energy range 20 to 200 GeV. The results can be described remarkably well by a simple Regge-pole model. The charge-exchange cross sections in the forward direction lead to a prediction for the difference in total cross sections of ..pi../sup -/p and ..pi../sup +/p which is in satisfactory agreement with direct measurements of this difference in another experiment at Fermilab. (AIP)