The universal 'energy' operator

TL;DR

This paper introduces a universal, dimension-independent definition of the positive square of tensors in Lorentzian manifolds, unifying energy concepts and revealing new structures of principal null directions.

Contribution

It provides a universal formulation of the positive square of tensors, applicable across arbitrary dimensions and tensor symmetries, unifying energy-momentum and super-energy tensors.

Findings

Standard energy-momentum tensors are special cases of the general square.

A richer structure of principal null directions is identified.

The formulation is valid in Lorentzian manifolds of any dimension.

Abstract

The "positive square" of any tensor is presented in a universal and unified manner, valid in Lorentzian manifolds of arbitrary dimension, and independently of any (anti)-symmetry properties of the tensor. For rank-m tensors, the positive square has rank 2m. Positive here means future, that is to say, satisfying the dominant property. The standard energy-momentum and super-energy tensors are recovered as appropriate parts of the general square. A richer structure of principal null directions arises.