The type N Karlhede bound is sharp

TL;DR

This paper constructs specific Lorentzian manifolds demonstrating that the Karlhede bound of seven covariant derivatives for invariant classification is sharp, correcting previous assumptions that six derivatives suffice.

Contribution

It provides explicit examples of spacetimes requiring the seventh covariant derivative, establishing the sharpness of Karlhede's bound for four-dimensional Lorentzian manifolds.

Findings

The constructed spacetimes are null radiation, type N solutions on anti-de Sitter background.

Invariant classification requires the seventh covariant derivative of the curvature tensor.

This work corrects the previous belief that six derivatives are sufficient.

Abstract

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 $CH_2$, i.e., curvature homogeneous of order 2 but non-homogeneous. This means that tetrad components of $R, \nabla R, \nabla^{(2)}R$ are constant, and that essential coordinates first appear as components of $\nabla^{(3)}R$. Covariant derivatives of orders 4,5,6 yield one additional invariant each, and $\nabla^{(7)}R$ 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 $\nabla^{(6)}R$.