Local freedom in the gravitational field

In a cosmological context, the electric and magnetic parts of the Weyl tensor, E_ab and H_abt, represent the locally free curvature, i.e. they are not pointwise determined by the matter fields. By performing a complete covariant decomposition of ∇_cE_ab and ∇_cHab_t 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 E_ab and H_ab together with their spatial curls. These parts of the derivatives are shown to be crucial for the existence of gravitational waves.