Some restrictions on algebraically general vacuum metrics

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Abstract
Vacuum Einstein metrics of Petrov type I, general, are considered. It is shown that the only solution of this sort in which one of the Petrov scalars is zero is the trivial flat−space one. Further, it is shown that the point at which the four Petrov scalars vanish simultaneously (zero curvature tensor) cannot be included as a regular point of a neighborhood over which the scalars are functionally independent. In fact, for type I all derivatives of the Petrov scalars must vanish at a point at which the curvature tensor does so that this point cannot be a regular point of any nontrivial analytic solution.

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