The type N Karlhede bound is sharp

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 $C{H}_2$, 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 ${\nabla }^{(6)}R$.

7 pages, typos corrected, added citation and acknowledgement

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