Super-energy tensors

TL;DR

This paper introduces a general algebraic framework for super-energy tensors applicable to various fields in any dimension, highlighting their mathematical properties and physical relevance, including conserved quantities and energy exchange mechanisms.

Contribution

It provides a unified, algebraic construction of super-energy tensors for arbitrary fields in any dimension, encompassing classical tensors like Bel-Robinson and defining new super-energy tensors with physical significance.

Findings

Super-energy tensors have positive definite timelike components and satisfy the dominant property.

In flat spacetime, they yield infinitely many conserved quantities.

Conserved super-energy currents exist in curved spacetime for scalar fields with symmetries.

Abstract

A purely algebraic construction of super-energy tensors for arbitrary fields is presented in any dimensions. These tensors have good mathematical and physical properties, and they can be used in any theory having as basic arena an n-dimensional manifold with a metric of Lorentzian signature. In general, the completely timelike component of these s-e tensors has the mathematical features of an energy density: they are positive definite and satisfy the dominant property. Similarly, the super-momentum vectors have mathematical properties of s-e flux vectors. The classical Bel-Robinson tensor is included in our general definition. The energy-momentum and super-energy tensors of physical fields are also obtained, and the procedure is illustrated by writing down these tensors explicitly for the cases of scalar, electromagnetic, and Proca fields. Moreover, `(super)$^k$-energy' tensors are defined and shown to be meaningful and in agreement for the different physical fields. In flat spacetimes, they provide infinitely many conserved quantities. In non-flat spacetimes conserved s-e currents are found for any minimally coupled scalar field whenever there is a Killing vector. Furthermore, the exchange of gravitational and electromagnetic super-energy is also shown by studying the propagation of discontinuities.