On metric transformations with a $U(1)$ gauge field

TL;DR

This paper explores metric transformations involving a $U(1)$ gauge field's field strength and its dual, revealing simplified forms under parity evenness and mathematical identities, extending previous work without dual tensors.

Contribution

It introduces a unified form of metric transformations incorporating both the field strength tensor and its dual, simplifying the structure under specific symmetry conditions.

Findings

Transformation reduces to a simple form under parity evenness.

Utilizes Cayley-Hamilton theorem and identities for simplification.

Extends previous work by including the dual tensor in metric transformations.

Abstract

We study metric transformations including not just the field strength tensor of a $U(1)$ gauge field, but also its dual tensor. We first consider an arbitrary symmetric matrix built up with these two tensors in the metric transformation. It turns out the form of transformation reduces to a quite simple form on imposing the parity evenness of the transformed metric and by utilising the Cayley-Hamilton theorem as well as other useful identities. Interestingly, the same form for the transformation was recently argued in the process of seeking for a generic metric transformation but without the inclusion of the dual tensor.