The invariant classification of conformally flat pure radiation spacetimes

Abstract

We study conformally flat pure radiation spacetimes by means of their invariant classifications and show that no such spacetime requires higher than the fourth covariant derivative of the Riemann tensor in its invariant classification. Additional side results that we obtain are as follows. The Edgar - Ludwig metric for conformally flat pure radiation is shown to be a true generalization of the Wils metric; the subclass of the Edgar - Ludwig spacetimes which admit exactly one Killing vector is identified, generalizing Koutras's G1 subclass of the Wils metric; it is shown that the Edgar - Ludwig spacetimes with no Killing vectors can further be split into three discrete subsets, depending on the Cartan invariant where the fourth essential coordinate is found.

Integration using invariant operators: conformally flat radiation metrics

“…Subsequently further examples which required the fourth derivative were found by Edgar and Ludwig [9]. Finally it was shown by Skea [19] (and confirmed by the above calculation) that for all conformally flat pure radiation solutions, all the information about the spacetime is contained in the Riemann tensor and its covariant derivatives to no higher than fourth order.…”

“…In this case it is necessary to go to 4th order to show that no further relations exist. That such cases can arise in practice was first shown by Koutras [18] who showed that a solution of Wils [19] required the fourth covariant derivative for its invariant classification. Subsequently further examples which required the fourth derivative were found by Edgar and Ludwig [9].…”

Invariant classification and the generalised invariant formalism: conformally flat pure radiation metrics

“…The subclass of CFPR (conformally flat pure radiation) spacetimes, which are not plane waves, has provided a very good illustration of the benefits of the GIF integration procedure [16]; furthermore, regarding their classification, Skea has also pointed out that "this application of the equivalence problem provides a non trivial didactic application of the use of the invariant classifications in practice" [40]. Integrating in the NP formalism, Wils [42] obtained a metric (containing one apparently non-redundant function of one coordinate) which was claimed to represent the whole class of CFPR spacetimes which were not plane waves; subsequently Koutras [23] showed that this was the first metric from which a new essential base coordinate was obtained at third order of the Cartan scalar invariants, which means that its invariant classification formally requires investigation of fourth order Cartan scalar invariants.…”

“…When Edgar and Ludwig [10], [11], proposed what appeared to be a more general metric (containing three apparently nonredundant functions of one coordinate) to represent all CFPR spacetimes, this provided an ideal opportunity for a non-trivial application of the equivalence problem. (The nontrivial nature of the invariant classification of this metric was emphasised when Skea's investigation of it [40] revealed a bug in the CLASSI computer programme which is used to handle the complicated calculations in the usual NP form; furthermore, when the Edgar-Ludwig metric was used to test the new Maple based invariant classification package in GRTensor, problems occurred and the results of the invariant classification published in [39] are clearly in error, as Barnes [2] has pointed out.) Skea [40] carried out a detailed investigation of the equivalence problem in Koutras-McIntosh coordinates for the Edgar-Ludwig and Wils metrics: this analysis confirmed that the Edgar-Ludwig spacetime is a genuine generalisation of the Wils spacetime, and all information about essential coordinates was obtained by third order, which means that the procedure will formally terminate with fourth order Cartan scalar invariants; Skea also found the particular subclass of the Edgar-Ludwig spacetime which coincided with the Wils spacetime, and as well he investigated the particular subclass of the Edgar-Ludwig spacetime which permitted one Killing vector.…”

Symmetry analysis of radiative spacetimes with a null isotropy using GHP formalism

“…On the other hand, if a more mathematical analysis of a spacetime is required the original coordinates may not be the most convenient. Recently, [6], we have investigated in detail the invariant classification and symmetry analysis of the conformally flat pure radiation spacetimes with zero cosmological constant [14], [13], [20]; we demonstrated how these procedures were much simpler and transparent using the version of the metric generated by the Generalised Invariant Formalism (GIF) integration procedure in [20], as compared to the version given in more familiar Kundt-type coordinates in [50]. A symmetry analysis of these metrics can also be found in [3].…”