Hawking-Ellis classification of stress-energy: test-fields versus back-reaction

TL;DR

This paper examines the Hawking-Ellis classification of stress-energy tensors, contrasting test-field scenarios with those including back-reaction, revealing that back-reaction often constrains stress-energy to type I in many physically relevant spacetimes.

Contribution

It demonstrates that back-reaction governed by Einstein equations restricts stress-energy types to type I in many important spacetime configurations, unlike test-field cases which can produce type IV.

Findings

Test fields can produce type IV stress-energy tensors.

Back-reaction constrains stress-energy to type I in static and stationary spacetimes.

Type I stress-energy is guaranteed in many cosmological models.

Abstract

We consider the Hawking-Ellis (Segre-Plebanski) classification of stress-energy tensors, both in the test-field limit, and in the presence of back-reaction governed by the usual Einstein equations. For test fields it is not too difficult to get a type~IV stress-energy via quantum vacuum polarization effects. (For example, consider the Unruh quantum vacuum state for a massless scalar field in the Schwarzschild background.) However, in the presence of back-reaction driven by the ordinary Einstein equations the situation is often much more constrained. For instance: (1) in any static spacetime the stress-energy is always type I in the domain of outer communication, and on any horizon that might be present; (2) in any stationary axisymmetric spacetime the stress-energy is always type I on any horizon that might be present; (3) on any Killing horizon that is extendable to a bifurcation 2-surface the stress-energy is always type I; (4) in any stationary axisymmetric spacetime the stress-energy is always type I on the axis of symmetry; (5) some of the homogeneous Bianchi cosmologies are guaranteed to be Hawking-Ellis type I (for example, all the Bianchi type I cosmologies, all the FLRW cosmologies, and all the "single mode" Bianchi cosmologies). That is, in very many physically interesting situations once one includes back-reaction the more unusual stress-energy types are automatically excluded.