The type N Karlhede bound is sharp

We present a family of four-dimensional Lorentzian manifolds whose invariant classification requires the seventh covariant derivative of the curvature tensor. The spacetimes in questions are null radiation, type N solutions on an anti-de Sitter background. The large order of the bound is due to the fact that these spacetimes are properly CH2, i.e., curvature homogeneous of order 2 but non-homogeneous. This means that tetrad components of R, ∇R, ∇(2)R are constant, and that essential coordinates first appear as components of ∇(3)R. Covariant derivatives of orders 4, 5, 6 yield one additional invariant each, and ∇(7)R is needed for invariant classification. Thus, our class proves that the bound of 7 on the order of the covariant derivative, first established by Karlhede, is sharp. Our finding corrects an outstanding assertion that invariant classification of four-dimensional Lorentzian manifolds requires at most ∇(6)R. (fast track communication)