Homogeneous and conformally Ricci flat pure radiation fields

Abstract

Some unsolved problems concerning pure radiation fields in general relativity are studied. The general solutions for homogeneous and for conformally Ricci flat metrics are given. They are all of Petrov type N or O, and contain a hitherto unidentified conformally flat metric. Non-aligned Petrov type III pure radiation metrics are shown not to exist.

All conformally flat pure radiation metrics

“…Wils [1] has recently given an example of a conformally flat metric representing pure radiation (i.e. energy-momentum tensor T ij = Φ 2 l i l j , where l i is a null vector), and he points out, since this metric is not a plane wave, it contradicts Theorem 32.17 in Kramer et al [2].…”

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

“…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. Koutras and McIntosh [24] have given a slightly more general metric in different coordinates; this form includes plane waves as well as the Wils metric.…”

“…and the Wils spacetime [42] given in the same coordinates but with g(u) = 0 = h(u), in order to determine whether the Edgar-Ludwig spacetime could actually be reduced to the Wils spacetime by a coordinate transformation. We shall consider the more general problem of the complete invariant classification of the Edgar-Ludwig spacetime, in particular checking on the redundancy of the three apparently non-redundant functions; we shall then specialise these results to the equivalence problem of the Edgar-Ludwig and Wils spacetimes.…”

All homogeneous pure radiation spacetimes satisfy the Einstein–Maxwell equations

“…[1] based upon results of Refs. [2] and [3]. Such solutions are all pp waves, and they are either plane waves or they are diffeomorphic to…”

“…However, the metric (2) does arise from an electromagnetic source. In fact, infinitely many electromagnetic fields can serve as the source for the spacetime defined by (2). For any function f (u) the null two form…”