Vacuum Kundt waves

We discuss the invariant classification of vacuum Kundt waves using the Cartan–Karlhede algorithm and determine the upper bound on the number of iterations of the Karlhede algorithm to classify the vacuum Kundt waves (Collins (1991 Class. Quantum Grav. 8 1859–69), Machado Ramos (1996 Class. Quantum Grav. 13 1589)). By choosing a particular coordinate system we partially construct the canonical coframe used in the classification to study the functional dependence of the invariants arising at each iteration of the algorithm. We provide a new upper bound, q ⩽ 4, and show that this bound is sharp by analyzing the subclass of Kundt waves with invariant count beginning with (0, 1,…) to show that the class with invariant count (0, 1, 3, 4, 4) exists. This class of vacuum Kundt waves is shown to be unique as the only set of metrics requiring the fourth covariant derivatives of the curvature. We conclude with an invariant classification of the vacuum Kundt waves using a suite of invariants. (paper)