Whose Wave Equation Is It, Anyway?
Gravitational waves result from the linearization of the Ricci tensor and applying what is analogous to a Lorentz gauge on the perturbed metric. What is supposedly perturbed is the flat, Minkoswski metric of special relativity, and this results in the wave speed equal to that of light. But, is there any difference between the wave equation of electrodynamics and the wave equation for a weak gravitational field? Sadly, we will come to the conclusion that there is no difference.
The Lorentz gauge says that the time derivative of the gravitational (electrostatic) field plus the divergence of the "vector" potential (called the gravitomagnetic potential) is equal to zero. If the scalar potential is associated with the mass density and the "vector" potential with the current, the density times the velocity, then the above condition will coincide with the continuity equation which is Einstein's condition that the 4-divergence of the energy stress tensor be zero.
What one does is to write down the geodesic equation with the analog of the Lorentz force producing the acceleration. One then introduces the vanishing of the divergence of the perturbed metric in the divergence of the Lorentz force, and out pops the wave equation with a source term, if the divergence doesn't vanish. If the motion is sufficently slow, the flux term in the perturbed metric can be neglected and what you get is Poisson's equation.
Now the flux term in the perturbed metric means that there will appear a cross term in the metric, i.e. a term involving the product of the time and space increments. Generally, a coordinate transition can be applied to obtain a time axis that is orthogonal to the spatial curves. Even if we forget about such a transformation, we would have to explain why the flux is time dependent. Recall that it is usually assumed that the metric coefficients are slowly varying functions of the radial coordinate that are independent of time. Such is the case in the Schwarzschild metric. And it is even more surprising that when the metric is prolonged beyond its boundary (i.e. the Schwarzschild radius) which is indicative of a black hole, it is said that time and space "swap" roles, but never the two together. So it is indeed surprising that the Schwarzschild solution does not predict gravitational waves, even when it is prolonged beyond its natural boundary.
Having the wave equation equation in hand with a source density proportional to the density of matter (the gravitational constant playing the role of the inverse permettivity), one claims that when the motion is slow enough, the second derivative in time can be neglected and their results the Poisson equation. But, what if the motion is not slow enough, what then will the wave equation describe? Certainly, not an electromagnetic wave since we are considering the slow motion of a particle, or its density. When do gravitational waves kick in? at what speed of the particles?
There error in this is easily seen when instead of linearizing the Ricci tensor, one begins with the geodesic equation. It is then concluded that whereas the gradient term is fist order in the relative velocity, the time dependent terms are "at least second order in the relativity velocity." (M. Ibison, et al. "The speed of gravity revisited"). Hence the space and time derivatives are not of the same order of magnitude so that a linear wave equation will never exist: Space and time derivatives are not of the same order in the relativity velocity.
There is nothing wrong with first writing down the nonlinear geodesic equation and then linearizing. Both path should lead to the same wave equation at the same order in the relativity velocity. That they do not means there is a "fly in the ointment".
From a physical perspective thing are even worse. Moving charges create magnetic fields. Do moving masses create gravitomagnetic fields? What are they anyway? Accelerating charges create electromagnetic radiation. Accelerating masses are said to create gravitational waves. But where are the acceleration terms? In the electromagnetic case they have been neglected. Is this legit in the gravitational case? Is masses do not accelerate there will be no source of gravitational waves. In the electromagnetic case, the electric field acquires terms that depend on acceleration directly. In the gravitational case there should be no waves at all without accelerating masses. However, one can argue that the wave equation is a good approximation far from the sources. But do the sources have to be so far that it is legit to consider a small perturbation on a flat metric?
The condition that Einstein introduced: All paths follow geodesics led to the greatest triumphs of GR, like the advance of the perihelion, and the bending of light by a gravitational field, but it also led to its greatest defeat: The inability of considering the dynamics of gravitational waves that result from the cataclysmic collision of black holes or binary neutron stars. The same formalism will definitely not work, even with the greatest imagination possible!