Hawking-Ellis classification of stress-energy tensors: Test fields versus backreaction

We consider the Hawking-Ellis (Segré-Plebański) classification of stress-energy tensors, both in the test-field limit and in the presence of backreaction 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 backreaction 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 Friedmann–Lemaître–Robertson–Walker cosmologies, and all the “single mode” Bianchi cosmologies). That is, in very many physically interesting situations once one includes backreaction the more unusual stress-energy types are automatically excluded.