A restatement of the normal form theorem for area metrics

TL;DR

This paper revisits and proves a normal form theorem for area metrics on 4-manifolds, classifying them into 23 metaclasses with explicit coordinate forms, enhancing understanding of electromagnetic media representations.

Contribution

It restates and proves the normal form theorem for skewon-free (2,2)-tensors, clarifying coordinate representations for each metaclass, and extends the classification of area metrics.

Findings

23 metaclasses with explicit forms

Each metaclass has three coordinate representations

Metaclasses I-VII have a single coordinate form

Abstract

An area metric is a (0,4)-tensor with certain symmetries on a 4-manifold that represent a non-dissipative linear electromagnetic medium. A recent result by Schuller, Witte and Wohlfarth provides a pointwise normal form theorem for such area metrics. This result is similar to the Jordan normal form theorem for (1,1)-tensors, and the result shows that any area metric belongs to one of 23 metaclasses with explicit coordinate expressions for each metaclass. In this paper we restate and prove this result for skewon-free (2,2)-tensors and show that in general, each metaclasses has three different coordinate representations, and each of metaclasses I, II, ..., VI, VII need only one coordinate representation.