The type N Karlhede bound is sharp

Details The type N Karlhede bound is sharp The type N Karlhede bound is sharp

2007-10-03

arxiv.org/abs/0710.0688v3

Class.Quant.Grav.25:012001,2008

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

7 pages, typos corrected, added citation and acknowledgement

Scientific paper

10.1088/0264-9381/25/1/012001

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 $\nabla^{(6)}R$.

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