The characterization conjecture for curvature-invariant spacetimes

A spacetime is characterised by its invariants if its scalar curvature invariants determine it in the weak or strong sense described in the paper. A spacetime is type D$^k$ when it is of type D to all orders, and it is \mathcal{I}-non-degenerate when its curvature invariants locally determine its metric up to the relevant equivalences.

Characterization conjecture. If a spacetime is characterised by its invariants, weakly or strongly, then it is either $\mathcal{I}$-non-degenerate or of type D to all orders; that is, of type D$^k$.

This conjecture proposes a classification of spacetimes determined by their scalar curvature invariants. It isolates the degenerate type-D-to-all-orders case as the only alternative to $\mathcal{I}$-non-degeneracy; the surrounding discussion contrasts this with Lorentzian spacetimes that are not characterised by their invariants and with the special VSI and CSI classes.