In a cosmological context, the electric and magnetic parts of the Weyl tensor, Eab and Habt, represent the locally free curvature, i.e. they are not pointwise determined by the matter fields. By performing a complete covariant decomposition of ∇cEab and ∇cHabt we show that the parts of the derivative of the curvature which are locally free (i.e. not pointwise determined by the matter via the Bianchi identities) are exactly the symmetrized trace-free spatial derivatives of Eab and Hab together with their spatial curls. These parts of the derivatives are shown to be crucial for the existence of gravitational waves.