General Electric-Magnetic decomposition of fields, positivity and Rainich-like conditions

TL;DR

This paper generalizes the electric-magnetic decomposition of fields to arbitrary tensors in any dimension, introduces super-energy tensors with positivity properties, and extends Rainich conditions beyond four dimensions.

Contribution

It provides a unified framework for tensor decomposition, positivity, and Rainich-like conditions applicable in higher-dimensional spacetimes.

Findings

Super-energy tensors include energy-momentum, Bel, and Bel-Robinson tensors.

Any tensor with the dominant property decomposes into super-energy tensors.

Results enable generalized Rainich conditions in arbitrary dimensions.

Abstract

We show how to generalize the classical electric-magnetic decomposition of the Maxwell or the Weyl tensors to arbitrary fields described by tensors of any rank in general $n$-dimensional spacetimes of Lorentzian signature. The properties and applications of this decomposition are reviewed. In particular, the definition of tensors quadratic in the original fields and with important positivity properties is given. These tensors are usually called "super-energy" (s-e) tensors, they include the traditional energy-momentum, Bel and Bel-Robinson tensors, and satisfy the so-called Dominant Property, which is a straightforward generalization of the classical dominant energy condition satisfied by well-behaved energy-momentum tensors. We prove that, in fact, any tensor satisfying the dominant property can be decomposed as a finite sum of the s-e tensors. Some remarks about the conservation laws derivable from s-e tensors, with some explicit examples, are presented. Finally, we will show how our results can be used to provide adequate generalizations of the Rainich conditions in general dimension and for any physical field.