The gravitational field of a source that is moving with respect to the field point is not the same as that of a stationary source at the same retarded distance. The calculation can be extended to a swarm of moving particles. If the integral over all the sources is not zero then the swarm will tend to drag a test particle along with it. Conversely, when the test particle is moving and the swarm is stationary, the test particle will experience a retarding force, and a photon will be redshifted. A redshifted photon still propagates at c, but its energy is less. In effect, it would be uphill all the way when traversing the cosmos.
More specifically, the gravitational force on a test particle moving inside a stationary spherical mass shell is not necessarily zero when the effects of retardation are considered, even though it is zero in Newtonian gravity. Newtonian gravity does not contain the speed of light, so those equations cannot be retarded. Alternatively, the Newton equations can be viewed as being the retardation equations for the case where the speed of light is infinite.
From these considerations, it may be possible to use retardation equations to compute the mass density of the universe from the Hubble constant. The Lense-Thirring precession does not predict the existence of such an effect, so it is fairly definite that low order retardation solutions will not be adequate. The retardation equations used in the calculation would of course need validation in the laboratory. The measurement techniques required are not esoteric. It is simply that the gravitational field of a moving mass is not the same as that of a stationary mass. However, the equations are parameterized by v/c, so it will not be an easy experiment. Some ingenuity will be required in devising a practical experimental configuration. The possibility that a jerked mass produces electrical fields can also be considered [arXiv.org/abs/1409.2101 ], even though the solution for the first derivative is a null result [calculations]. (On the other hand, a constant velocity charge does produce a local magnetic field.)
The general theory of relativity is a field theory. The method of
retardation is so different from the methods of field equations that it
is often difficult to see why there should even be a connection, but
there is. There is no known way of deriving retardation equations from
field equations, which is not to say that a way of doing so could not be
discovered. Until then, we need both perspectives.
Comments and corrections to the material shown in these pages are always welcome.
Gary Osborn
Anaheim, California
Last update 6 Sept 2015 Revision History