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.
13652721
Irreducible Killing Tensors from Conformal Killing Vectors
Koutras has proposed some methods to construct reducible proper conformal Killing tensorsand Killing tensors (which are, in general, irreducible) when a pair of orthogonal conformalKilling vectors exist in a given space. We give the completely general result demonstratingthat this severe restriction of orthogonality is unnecessary. In addition we correct andextend some results concerning Killing tensors constructed from a single conformal Killingvector. We give as an example a Kimura metric and demonstrate how it is possible toconstructalargerclassofreducibleproperconformalKillingtensorsandKillingtensorsthanpermitted by the Koutras algorithms. In addition, from our new result that all conformalKilling tensors are reducible in conformally ﬂat spaces, we have a method of constructingall conformal Killing tensors and hence all the Killing tensors (which will in general beirreducible) of conformally ﬂat spaces using their conformal Killing vectors.
120435397
Irrotational perfect fluids with a purely electric Weyl tensor
All perfect fluid spacetimes with a purely electric Weyl tensor are shown to have an alignment between the fluid 4-velocity and a canonical null tetrad determined by the Weyl tensor. If, in addition, it is assumed that the flow is irrotational, the eigenframes of the shear and Weyl tensors coincide. In all but two rather special cases, it is proved that the vectors of this eigenframe are hypersurface orthogonal and consequently that a coordinate system exists in which the metric, shear and Eab (the electric part of the Weyl tensor) are all diagonal. Geodesic Petrov type D spacetimes are shown to be either Bianchi type 1 or to belong to the class of solutions considered by Szekeres (1975) and Szafron (1977). The Allnutt solutions (1982) are shown to be the only purely electric type D fields in which the shear is non-degenerate and in which the acceleration vector lies in the plane spanned by the principal null vectors. The field equations are partially integrated in two classes where no solutions are yet known.
121708777
A class of homogeneous Einstein-Maxwell fields
The author considers solutions of the Einstein-Maxwell field equations satisfying the conditions Fij;k+lk=fFij+, Fij;k+nk=gFij+ where Fij+ is the self-dual Maxwell bivector and li and ni are the principal null vectors of Fij+. The fields are algebraically general and the most general solution is found in closed form when li and ni have non-zero twist. Several twist-free solutions are also obtained. Vacuum space-times admitting a test Maxwell field are also considered. In all cases the metric admits a three-parameter group of motions acting transitively on three-dimensional orbits. Fields satisfying the dual conditions with li and ni replaced by mi and mi are also investigated and similar results are obtained.
125608804
Note 65.39: Inverse and reciprocal functions
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.
123442940
The initial neutrino events from supernovae as evidence for matter versus antimatter