Curvature Invariants in Algebraically Special Spacetimes

Let us define a curvature invariant of the order k as a scalar polynomial constructed from gαβ, the Riemann tensor Rαβγδ, and covariant derivatives of the Riemann tensor up to the order k. According to this definition, the Ricci curvature scalar R or the Kretschmann curvature scalar RαβγδR αβγδ are curvature invariants of the order zero and Rαβγδ;εR αβγδ;ε is a curvature invariant of the order 1. We consider only vacuum spacetimes so that the Riemann tensor is equal to the Weyl tensor. An arbitrary curvature invariant can thus be expressed in terms of the Weyl spinor ΨABCD and its covariant derivatives. We can use a standard basis oA, ιA, which satisfies