According to the decomposition of the Riemann tensor, the distribution of stresses, other than the gravitational ones, and masses is described by the Ricci tensor, and the remaining part, the Weyl tensor, describes the gravitational field that, supposedly, is created by the local spacetime curvature. The Weyl tensor describes the long range gravitational field through tidal forces, but no gravitational waves.
The non-existence of gravitational waves is the result of fact that the traceless electric and magnetic components of the Weyl tensor are both symmetric. Hence, there is no Faraday induction, nor a transverse displacement current. Without induction there can be no wave equation. The electric, or Newtonian, component of the Weyl tensor accounts for tidal stresses. The magnetic, or non-Newtonian, component of the Weyl tensor does not vanish if there is a transverse current density. The remaining two Maxwell-like equations are given by the Bianchi identities.
This says that the gravitational field must be static in order to be described by tidal forces. In contrast to the Faraday tensor, which is completely anti-symmetric, the Weyl tensor is symmetric. This implies the absence of induction, which is uniquely related to the interaction of moving charges.
It has been claimed that the divergence of the electric component is related to tidal forces and the magnetic component to Ampere's law without the displacement current [Costa & Herdeiro, Phys Rev. D 78 (2008)], and, as such, coincide exactly with the time-time and time-space projections of the Einstein equations. This is a little difficult to swallow since the Ricci tensor is not trace-free while the Weyl tensor is. And the Weyl tensor is the complementary component of the Ricci tensor which accounts for all forms of energy-stresses other than gravitational. From the fact that the electric and magnetic components of the Weyl tensor together with the Ricci tensor fully determine the curvature of spacetime, for any given "threading" (like 3+1), it would be redundant, to say the least, that the two sets of Maxwell type equations be identical for static fields, i.e. those that do not contain inductances.
In fact, comparing the homogeneous model of the universe, which comprises the standard model, with the Weber force law, we see that the space-space component of the Ricci tensor does not coincide with it. It cannot even be rationalized physically. Moreover the time time component of the Ricci tensor contains the pressure as well as the density. It is the density, and not the pressure, which acts on the surface, that is related to the tidal force components. Hence, there should be no relation between the equations that the Weyl components obey and those of the Ricci tensor. They are, in fact, complementary to one another.
According to Bertschinger (9503235v1), there is no displacement current because the electric component of the Weyl tensor is non-radiative. That is, it is not needed to ensure conservation of mass which is already accounted for by the scalar potential. Rather, the vanishing of the displacement current ensures that the electric component of the Weyl tensor is tidal, or, as some would say, Newtonian. He, however, still holds to the claim that the Faraday like equation still holds in which there is a time-varying magnetic component of Weyl's tensor. But, this is ridiculous since the electric component of the Weyl tensor is symmetric so that the covariant analog of the curl of the electric field must vanish. Nevertheless, without both components of the Weyl tensor time-varying there can be no gravitational waves. Masses do not behave as electric charges which can repel as well as attract.