Hawking-Ellis type III spacetime geometry

TL;DR

This paper investigates the elusive type III stress-energy tensors in the Hawking-Ellis classification by exploring specific spacetime geometries and models that could support such tensors, which are not known to arise from classical sources.

Contribution

The authors explicitly construct spacetime geometries and a Lagrangian model that produce type III stress-energy tensors, advancing understanding of their possible physical origins.

Findings

Identified spacetime geometries with type III Einstein tensors.

Developed a Lagrangian model leading to type III stress-energy in Minkowski space.

Provided insights into the physical plausibility of type III stress-energy tensors.

Abstract

The type III (and the "essential core" type III$_0$) stress-energy tensors in the Hawking-Ellis (Segre-Plebanski) classification stand out in that there is to date no known source (either classical or semi-classical) leading to type III stress-energy. (In contrast the Hawking-Ells types I and II occur classically, and type IV is known to occur semi-classically). We instead start by asking the obverse question: What sort of spacetime (assuming the Einstein equations) needs a type III stress-energy to support it? One key observation is that type III is incompatible with either planar or spherical symmetry, so one should be looking at spacetimes of low symmetry (or no symmetry). Finding such a type III spacetime is a matter of somehow finding an appropriate ansatz for the metric, calculating the Einstein tensor, and analyzing the pattern of (Lorentz invariant) eigenvalues and eigenvectors. Herein we report some (partial) success along these lines - we explicitly exhibit several (somewhat unnatural) spacetime geometries with a type III Einstein tensor. We then build an explicit but somewhat odd Lagrangian model leading (in Minkowski space) to type III stress-energy. While we still have no fully acceptable general physical model for type III stress-energy, we can at least say something about what such a stress-energy tensor would entail.