Essential core of the Hawking--Ellis types

TL;DR

This paper simplifies the Hawking-Ellis classification of stress-energy tensors by isolating their core structures, revealing algebraic properties and stability characteristics of each type.

Contribution

It introduces simplified core types for the Hawking-Ellis classification, providing clearer algebraic properties and insights into their classical and semi-classical interpretations.

Findings

Types I and IV are stable under perturbations.

Types II and III are definitively unstable.

Simplified cores have elegant algebraic properties.

Abstract

The Hawking-Ellis (Segre-Plebanski) classification of possible stress-energy tensors is an essential tool in analyzing the implications of the Einstein field equations in a more-or-less model-independent manner. In the current article the basic idea is to simplify the Hawking-Ellis type I, II, III, and IV classification by isolating the "essential core" of the type II, type III, and type IV stress-energy tensors; this being done by subtracting (special cases of) type I to simplify the (Lorentz invariant) eigenvalue structure as much as possible without disturbing the eigenvector structure. We will denote these "simplified cores" type II$_0$, type III$_0$, and type IV$_0$. These "simplified cores" have very nice and simple algebraic properties. Furthermore, types I and II$_0$ have very simple classical interpretations, while type IV$_0$ is known to arise semi-classically (in renormalized expectation values of standard stress-energy tensors). In contrast type III$_0$ stands out in that it has neither a simple classical interpretation, nor even a simple semi-classical interpretation. We will also consider the robustness of this classification considering the stability of the different Hawking-Ellis types under perturbations. We argue that types II and III are definitively unstable, whereas types I and IV are stable.