Metrics obtained by integrating within the generalised invariant formalism are structured around their intrinsic coordinates, and this considerably simplifies their invariant classification and symmetry analysis. We illustrate this by presenting a simple and transparent complete invariant classification of the conformally flat pure radiation metrics (except plane waves) in such intrinsic coordinates; in particular we confirm that the three apparently non-redundant functions of one variable are genuinely non-redundant, and easily identify the subclasses which admit a Killing and/or a homothetic Killing vector. Most of our results agree with the earlier classification carried out by Skea in the different Koutras-McIntosh coordinates, which required much more involved calculations; but there are some subtle differences. Therefore, we also rework the classification in the Koutras-McIntosh coordinates, and by paying attention to some of the subtleties involving arbitrary functions, we obtain complete agreement with the results obtained in intrinsic coordinates. We have corrected and completed statements and results by Edgar and Vickers, and by Skea, about the orders of Cartan invariants at which particular information becomes available.

A complete and simple invariant classification of the conformally flat pure radiation metrics with a negative cosmological constant that were obtained by integration using the generalised invariant formalism is presented. We show equivalence between these metrics and the corresponding type O subclass of the more general spacetime studied by Siklos. The classification procedure indicates that the metrics possess a one degree of null isotropy freedom which has very interesting repercussions in the symmetry analysis. The Killing and homothetic vector analysis in GHP formalism is then generalised to this case were there is only one null direction defined geometrically. We determine the existing Killing vectors for the different subclasses that arise in the classification and compare these results to those obtained in the symmetry analysis performed by Siklos for a larger class of metrics with Ricci tensor representing a pure radiation field and a negative cosmological constant. It is also shown that there are no homothetic Killing vectors present.

Using the generalised invariant formalism we derive a special subclass of conformally flat spacetimes whose Ricci tensor has a pure radiation and a Ricci scalar component. The method used is a development of the methods used earlier for pure radiation spacetimes of Petrov types O and N, respectively. In this paper we demonstrate how to handle, in the generalised invariant formalism, spacetimes with isotropy freedom and rich Killing vector structure. Once the spacetimes have been constructed, it is straightforward to deduce their Karlhede classification: the Karlhede algorithm terminates at the fourth derivative order, and the spacetimes all have one degree of null isotropy and three, four or five Killing vectors.

We prove that a Lanczos potential L-abc for the Weyl candidate tensor W-abcd does not generally exist for dimensions higher than four. The technique is simply to assume the existence of such a potential in dimension n, and then check the integrability conditions for the assumed system of differential equations, if the integrability conditions yield another non-trivial differential system for L-abc and W-abcd, then this system's integrability conditions should be checked, and so on. When we find a non-trivial condition involving only W-abcd and its derivatives, then clearly Weyl candidate tensors failing to satisfy that condition cannot be written in terms of a Lanczos potential L-abc.

6.

Edgar, SB

et al.

Linkoping Univ, Dept Math, S-58183 Linkoping, Sweden Univ Alberta, Dept Math Sci, Edmonton, AB T6G 2G1, Canada.

Ludwig, G

Linkoping Univ, Dept Math, S-58183 Linkoping, Sweden Univ Alberta, Dept Math Sci, Edmonton, AB T6G 2G1, Canada.

In order to achieve efficient calculations and easy interpretations of symmetries, a strategy for investigations in tetrad formalisms is outlined: work in an intrinsic tetrad using intrinsic coordinates. The key result is that a vector field xi is a Killing vector field if and only if there exists a tetrad which is Lie derived with respect to xi, this result is translated into the GHP formalism using a new generalised Lie derivative operator L-xi with respect to a vector field xi. We identify a class of intrinsic GHP tetrads, which belongs to the class of GHP tetrads which is generalised Lie derived by this new generalised Lie derivative operator L-xi in the presence of a Killing vector field xi. This new operator L-xi also has the important property that, with respect to an intrinsic GHP tetrad, it commutes with the usual GHP operators if and only if xi is a Killing vector field. Practically, this means, for any spacetime obtained by integration in the GHP formalism using an intrinsic GHP tetrad, that the Killing vector properties can be deduced from the tetrad or metric using the Lie-GHP commutator equations, without a detailed additional analysis. Killing vectors are found in this manner for a number of special spaces.

7.

Herberthson, Magnus

Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.

It is well known that, for asymptotically flat spacetimes, one cannot in general have a smooth differentiable structure at spacelike infinity, i(0). Normally, one uses direction dependent structures, whose regularity has to match the regularity of the (rescaled) metric. The standard C->1-structure at i(0) ensures sufficient regularity in spacelike directions, but examples show very low regularity on I+ and I-. The alternative C1+-structure shows that both null and spacelike directions may be treated on an equal footing, at the expense of some manageable logarithmic singularities at i(0). In this paper, we show that the Kerr spacetime may be rescaled and given a structure which is C->1 in both null and spacelike directions from i(0).

8.

Ludwig, G

et al.

Univ Alberta, Dept Math Sci, Edmonton, AB T6G 1S6, Canada.

Edgar, SB

Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.

Finding (conformal) Killing vectors of a given metric can be a difficult task. This paper presents an efficient technique for finding Killing, homothetic, or even proper conformal Killing vectors in the Newman-Penrose (NP) formalism. Leaning on, and extending, results previously derived in the GHP formalism we show that the (conformal) Killing equations can be replaced by a set of equations involving the commutators of the Lie derivative with the four NP differential operators, applied to the four coordinates. It is crucial that these operators refer to a preferred tetrad relative to the (conformal) Killing vectors, a notion to be defined. The equations can then be readily solved for the Lie derivative of the coordinates, i.e. for the components of the (conformal) Killing vectors. Some of these equations become trivial if some coordinates are chosen intrinsically (where possible), i.e. if they are somehow tied to the Riemann tensor and its covariant derivatives. If part of the tetrad, i.e. part of null directions and gauge, can be defined intrinsically then that part is generally preferred relative to any Killing vector. This is also true relative to a homothetic vector or a proper conformal Killing vector provided we make a further restriction on that intrinsic part of the tetrad. If because of null isotropy or gauge isotropy, where part of the tetrad cannot even in principle be defined intrinsically, the tetrad is defined only up to (usually) one null rotation parameter and/or a gauge factor, then the NP-Lie equations become slightly more involved and must be solved for the Lie derivative of the null rotation parameter and/or of the gauge factor as well. However, the general method remains the same and is still much more efficient than conventional methods. Several explicit examples are given to illustrate the method.