The following explanation comes courtesy of another forum from a user named publius, who knows GR well and has given me permission to repost his words. I do this to provide some sort of formalism, which I am not able to provide myself:
Let me elaborate on why EM behavior is so relevant. You'll also note Carlip started out with EM analogy, too, before he delved into the much more complex GR calculations.
First of all, EM is simpler. Second, we're all familiar with it, and no one disputes that EM influences propagate at 'c'. I mean, that's what the speed of light is all about. Third, we know that EM forces involve more than just a basic inverse square Coulomb interaction. There's the magnetic side, and what I like to call the "Maxwellian dance", which is the coupling between the two through time variation (or just go to the 4-vector form where it's all one tensor field, not two vector fields).
Thus, it is no great surprise that when a source charge is in motion, there will components to the field beyond the simple inverse square Coulomb part, and these components are velocity dependent. Indeed, calculating the fields directly is a bit of chore, so we use the simple potentials, and we learn that E = -del(phi) - dA/dt and B = curl A.
When a point charge is moving, there is an A, and it is the dA/dt part that is responsible for the "correction" from your basic inverse square Coulomb part, which comes from the gradient of the scalar potential part.
That's what makes the force point ahead of the retarded, light image position of the source.
Now, that falls right out of the equations, and should surprise no one.
Now, enter SR. From the spirit of relativity, we know that we should be able to switch to the frame of an inertially moving source charge and get the same invariant results as in a frame where the source is moving. Going to that frame, the source isn't moving, and the field is static, and pure inverse square Coulomb. We know that the test particle force has to point toward the static source.
Now, whether the force "misses" or not is an invariant. It cannot point at the source in one frame and not point at it in another!
Thus, we deduce that if EM is going to be compatible with SR, then it must somehow "extrapolate" for inertial motion in frames where a source is moving. It has to.
So, relatively simple EM does this "extrapolation" and in a way that should surprise no one.
Thus, the objection to "extrapolation" should be dispensed with. Then we move on to gravity. Because of the additional complexity of gravity, it extrapolates on higher order than EM. It's the same basic behavior at work, just doing it a little better than EM.
As we learn from EM, because of Lorentz invariance (compatibility with SR), we see that there must be these velocity dependent components of the forces (in the 4-vector form, you see that is nothing but the components of a tensor changing with coordinate transforms -- it's the same tensor, it just has different components in different frames).
So, if gravity is to be compatible with relativity, then by George, it's going to have to have velocity dependent components as well, as further, it's going to "extrapolate" to at least velocity as EM must.
But gravity also must obey the Equivalence Principle, which makes it more complex, and also requires it to "extrapolate" to second order, to acceleration, as well as velocity.
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The best way to think of it is the GR gravitational field contains information of the position, velocity, *and acceleration* of the sources and the first order of radiation is quadrapole. By contrast, the EM field contains only velocity and position information and the first order is dipole.
Let r be the light retarded position of the source, and v its velocity, and a its acceleration. That is, that's the position of vector, in your coordinates, of the light delayed image of the source you would see. It's rather straightfoward to show that the E field of a moving source points in the direction:
r + v(r/c)
(r/c) is the light travel mean time. If the source is moving at constant velocity, that points exactly at the instantanous position of the source. Hence EM can be said to "extrapolate" the velocity of the source. One can, as shown by Carlip, get that very quickly by the elegant and succinct, but more advanced, 4-vector formulation of EM, but you can get it from the more familiar 3-vector form Maxwell as well, it just takes more groking.
Now, as Carlip shows, GR gravity (in the weak field, low velocity limit) points along:
r + v(r/c) + 1/2 a (r/c)^2
IOW, it further extrapolates for the acceleration of the source in addition to velocity, something not in EM. And the magnitude of that acceleration in that limit is just GM/R^2, where R is that "extrapolated" position magnitude. And that's just instantaneous Newton!
So, as long as the source doesn't change acceleration, gravity points at the instantaneous position of the source, following Newton exactly (in the limit).
