A metric with no symmetries or invariants

Abstract

A metric is given which has no scalar invariants formed from its Riemann tensor or derivatives of its Riemann tensor and which admits, in its general form, no local homotheties or isometries. It is conformally flat and describes pure radiation. Subcases of this metric are the plane wave metric and a metric of Wils.

All conformally flat pure radiation metrics

“…Using a coordinate transformation w = v/x, Koutras and McIntosh [3] (see also [4]) have recently written this metric in a slightly different coordinate system (u, w, x, y),…”

Complete K-Curvature Homogeneous Pseudo-Riemannian Manifolds 0-Modeled on an Indecomposible Symmetric Space

“…There is a related result concerning scalar invariants: This result fails in the pseudo-Riemannian setting; Koutras and McIntosh [20] gave examples of non-flat manifolds all of whose scalar Weyl invariants vanish; see also related examples by Pravda, Pravdová, Coley, and Milson [25].…”

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. 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.…”