Based on the analogy with Maxwell's field equations electrodynamics Einstein attempted a similar procedure for gravitation. Whereas Maxwell's equations are determined by their sources, i.e., charges and their currents, Einstein's equations are determined by the distribution of matter and their currents. However, whereas the sources of Maxwell's equations are independent of the field equations, in Einstein's theory, the motion of matter is determined by them. In other words, fields and the motion of matter are not independent.
Say we are completely ignorant of Einstein's theory and its relation to geodesic motion. A presumably good place to go to is the Wikipedia article "Einstein field equations". After a detailed expose of the field equations, the concluding sentence states:
"These equations [referring to Einstein's equations] together with the geodesic equation [ref: S. Weinberg, Dreams of a Final Theory, pp. 107, 233], which dictates how freely-falling matter moves through space-time, form the core of the mathematical formulation of general relativity."
After a heavy dose of mathematics, we are given a reference to a popularized and lackadaisical account of one scientists opinions of what we should expect on how physics will evolve. Be that as it may, there is neither a reference to a geodesic equation in the index, nor can any be found on the pages cited, or anywhere else for that matter. This is skulduggery at its worst, and highlights the deficiencies of the theory of general relativity. You don't pass off a mathematical expose with reference to a popularized account of the personal whims of a single scientist which doesn't even mention that expose.
In electrodynamics, we know that charged particles radiate. In general relativity accelerating masses (should) radiate---but the geodesic equation prevents this from happening. Geodesic equation describe uniform motion, whether it be free-falling or otherwise. While it is true that Einstein's equations can be linearized to give a wave equation, but the masses that supposedly give rise to such solution cannot accelerate. No acceleration, no radiation! And no acceleration, no aberration: the field is in direction of particle motion. We need a field that is oblique to the direction of the source to the observer. And this is afforded by the acceleration vector which makes an angle with the direction to the source: the so-called aberration angle.
If we turn to Weber's theory which is not a field theory but a theory of particle motion, the electric field is in the direction of the particle's present position. While it is a combination of the static Coulomb law and Ampere's law, it does contain a term due to acceleration arising in the parallel case where the relative velocity of the particles is zero but they have relative acceleration. Weber combined this with Ampere's law of longitudinal elements describing the motion of positive and negative charges in relative motion. It is this acceleration term that justified his conclusion regarding the constant in his equation that represents the maximum velocity which "electricatical masses e and e' have and must retain, if they are not to act on each other at all." No such term exists in the geodesic equation, where the constant c appears in the static coefficients in the geodesic equation that does not represent any limiting speed for the quadratic velocity term in the geodesic equation.
There has been much ado about demonstrating why gravitational waves do not aberrate and yet travel at a finite speed---that of light. An electromagnetic example is used of an electric field created by a uniformly moving charge. The electric field is written in terms of past positions of a moving source s',
VE=K[(Vs')-s'(Vv)/c]/(s'-Vs' * Vv/c)^3,
where the factor K affects the magnitude but not the direction of the field, (Vs') stands for vector of the past position of the particle towards the direction of the observer and Vv is its velocity at the past position. The dot product of the two appears in the denominator Taken separately, the first term is oriented to the past position of the source, and the second term is the aberrational term; the sine of the angle between the direction of the velocity Vv and the direction Vs' is proportional to the ratio v/c for small velocities, where c is the limiting velocity. But, taken together,
Vs'-s'Vv/c=Vs,
is the vector toward the observer of the present position of the source, Vs. Hence, the addition of the two terms conspire to cancel all aberrational effects. This conspiracy should be carried over to the geodesic equation of general relativity considers mass in uniform motion, the same should apply there [Ibison, et al, "The speed of gravity revisited"]. Wrong!
This miraculous cancellation has also been promulgated by Steven Carlip. It is both misleading and wrong. This should have been apparent from the observation that whereas the electric field for uniform motion varies as 1/s^2, that for radiation involving acceleration varies as 1/s. It is that term that shows how electromagnetic waves aberrate, and determines the flow of energy.
There is nothing analogous to this in the geodesic equation of general relativity nor the field equations of Einstein. By a mere quirk of fate, the Ricci tensor, which is what the Einstein tensor is composed of, can be linearized to yield the wave equation. The balancing terms on the other side of the equation represent sources so that it appears as a bona fide equation for wave motion. But the motion of matter must give rise to such a field, and it can't because it can't accelerate. So what was linked together by a common root, namely the field equations and the geodesic equation, are not compatible with radiation and aberration. This problem does not subsist in electrodynamics since Maxwell's field equations are independent of the force equation that relates the fields to particle motion, namely the Lorentz force.
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