Algebraic Rainich theory and antisymmetrisation in higher dimensions

TL;DR

This paper extends Rainich theory to higher dimensions, using antisymmetrisation to find new identities for superenergy tensors of general p-forms, and provides a complete five-dimensional generalization.

Contribution

It introduces algebraic Rainich conditions for general p-forms in higher dimensions and establishes new antisymmetrisation identities, including a full five-dimensional extension.

Findings

Derived new identities for superenergy tensors of non-simple forms.

Proved sufficiency of identities to determine the form in some cases.

Generalized Rainich theory to five dimensions.

Abstract

The classical Rainich(-Misner-Wheeler) theory gives necessary and sufficient conditions on an energy-momentum tensor $T$ to be that of a Maxwell field (a 2-form) in four dimensions. Via Einstein's equations these conditions can be expressed in terms of the Ricci tensor, thus providing conditions on a spacetime geometry for it to be an Einstein-Maxwell spacetime. One of the conditions is that $T^2$ is proportional to the metric, and it has previously been shown in arbitrary dimension that any tensor satisfying this condition is a superenergy tensor of a simple $p$-form. Here we examine algebraic Rainich conditions for general $p$-forms in higher dimensions and their relations to identities by antisymmetrisation. Using antisymmetrisation techniques we find new identities for superenergy tensors of these general (non-simple) forms, and we also prove in some cases the converse; that the identities are sufficient to determine the form. As an example we obtain the complete generalisation of the classical Rainich theory to five dimensions.