Vacuum Kundt Waves

TL;DR

This paper classifies vacuum Kundt waves using the Cartan-Karlhede algorithm, establishing a sharp upper bound of 4 iterations for the classification process and identifying a unique subclass requiring the fourth covariant derivatives.

Contribution

It introduces a new upper bound of 4 iterations for classifying vacuum Kundt waves and identifies a unique subclass with maximal invariant count requiring fourth derivatives.

Findings

Established a sharp upper bound q ≤ 4 for the classification iterations.

Identified a unique subclass with invariant count (0,1,3,4,4) requiring fourth derivatives.

Provided a complete invariant classification of vacuum Kundt waves.

Abstract

We discuss the invariant classification of vacuum Kundt waves using the Cartan-Karlhede algorithm, and the upper bound on the number of iterations of the Karlhede algorithm to classify the vacuum Kundt waves. 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 $q \leq 4$ and show 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.