Local freedom in the gravitational field

In a cosmological context, the electric and magnetic parts of the Weyl tensor, $E_{ab}$ and $H_{ab}$, represent the locally free curvature -- i.e. they are not pointwise determined by the matter fields. By performing a complete covariant decomposition of the derivatives of $E_{ab}$ and $H_{ab}$, 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 spatial symmetrised trace--free parts and the curls.