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We prove that, in a space-time of dimension n>3 with a velocity field that isshear-free, vorticity-free and acceleration-free, the covariant divergence ofthe Weyl tensor is zero if the contraction of the Weyl tensor with the velocityis zero. The other way, if the covariant divergence of the Weyl tensor is zero,then the contraction of the Weyl tensor with the velocity has recurrentgeodesic derivative. This partly extends a property found in GeneralisedRobertson-Walker spacetimes, where the velocity is also eigenvector of theRicci tensor. Despite the simplicity of the statement, the proof is involved.As a product of the same calculation, we introduce a curvature tensor with aninteresting recurrence property.

A simple property of the Weyl tensor for a shear, vorticity and acceleration-free velocity field