Essential core of the Hawking--Ellis types

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.

V1: 18 pages; V2: reformatted, now 14 pages; some clarifications added; no significant physics changes. This version accepted for publication in Classical and Quantum Gravity

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