The type III stress-energy tensor: Ugly Duckling of the Hawking-Ellis classification

We present some advances in the understanding of type III stress-energy tensors as per the Hawking-Ellis classification. Type I and type II naturally appear in classical situations, and can also describe semiclassical effects. Type IV often shows up in semiclassical gravity. Type III is much more subtle. We focus our attention on type III$_0$ stress-energy tensors, which capture the essence ("essential core") of type III. Reflecting on known purely phenomenological examples, ("gyratons"), we are able to generalize the geometry generated by those type III$_0$ stress-energy tensors. Moreover, we also succeed in extending work by Griffiths based on massless Weyl spinors by finding a fundamental classical bosonic Lagrangian description of these type III$_0$ stress-energy tensors. To the best of our knowledge this is the first time in the literature that a consistent classical bosonic Lagrangian formulation for type III$_0$ stress-energy has been found.