Local freedom in the gravitational field

In a cosmological context, the electric and magnetic parts of the Weyl tensor, 0264-9381/14/7/025/img5 and 0264-9381/14/7/025/img6, represent the locally free curvature, i.e. they are not pointwise determined by the matter fields. By performing a complete covariant decomposition of 0264-9381/14/7/025/img7 and 0264-9381/14/7/025/img8, 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 0264-9381/14/7/025/img5 and 0264-9381/14/7/025/img6 together with their spatial curls. These parts of the derivatives are shown to be crucial for the existence of gravitational waves.