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.
The basis of the calculations in this section is the static 4-potential solution in spherical coordinates
Ar = -G m rr^2 exp(-r/rr)/r^2 - G m rr exp(-r/rr)/r
Ψ = q exp(-r/rr)/r
Ar is the radial vector potential; Ψ is the scalar potential. G is the gravitational constant, and m is the mass of the particle. rr is the range of the fields.
q is the charge of the particle, which is not considered in this section. The gravitational solutions also acquire a scalar potential after transformation, but it does not result in electrical fields for the first derivative solutions.
The mathematical basis for the equation was developed in section IV of the paper at http://arxiv.org/abs/1409.2101 The solution was obtained by restructuring the Proca equations, but it is not a solution to the Proca equations. The motivation for restructuring the Proca equations follows from the calculations at s-4.com/tensor. Additional relevant calculations are shown at s-4.com/som1
Some solutions for the undifferentiated vector potential are developed at s-4.com/retard. Those calculations are now mostly obsolete, but the section may contain some useful background material.
The solutions of this section are for the expansion factor, div(A)+1/c dΨ/dt. The acceleration is the gradient of the expansion factor, but it cannot be computed until the global solution for the expansion factor has been obtained. It would probably be better to directly retard the acceleration, and those calculations are in progress. The equivalence principle applies to the acceleration in the solutions, meaning that an inertial observer will not perceive it. The expansion factor, which is traditionally known as the Lorentz condition, is always zero in the solutions of the Liénard-Wiechert retardation equations for a charged particle. It is also zero in the solutions of the Maxwell potential equations in the Lorentz gauge.
A retardable equation for the first derivative
The total differential for the first derivative is derived. An equivalent and retardable vector equation for the 1/r terms of the total differential is obtained. There are also 1/r^3 terms that drop out of low order solutions. They can be arbitrarily larger than the terms that do make a contribution. The mathematical behavior is unusual, and it tends to frustrate attempts at applying the Taylor theorem to the above exponential vector equation.
These relationships are still being developed, and updates will be posted here at regular intervals. I will eventually have to use these equations to compute the precession of the periastron in order for them to be believable, and I do not know yet what the solution will be. A calculation of the Lense-Thirring precession would be easier (if the equations predict the effect).
The electrical fields of the mass solution for the first derivatives are zero. However, especially for the E field, the null result is a consequence of a cancellation of a vast number of seemingly unrelated terms, which is interesting. It seems unlikely that the cancellation will occur in the solution for the second derivative, but we will not know until we get there. If the cancellation does not occur then that will open the door to some fascinating laboratory experiments.
mass calculations v^3 a^1
mass calculations v^6 a^1
The solution for the first derivative is the same with or without the Thomas precession. That is because, unlike the solution for a charged particle, the acceleration of the particle drops out of the solution. It could be that the gravitational solution for the first derivative is of too low an order to be useful.
Computing the second derivative entails computing the first derivative at a displaced point. The first time derivative can be computed at a point displaced in either space or time at the field point, and the first space derivative can similarly be computed at a point that is displaced in either space or time. The three light cone solutions needed for computing the second derivatives for a retarded particle are shown here. There are no retardation equations in this section.
without HTML with HTML dr1 dt1 or dt1 dr1
without HTML with HTML dt1 dt2
cone.a program listing
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. On the other hand, the solution for the Lense-Thirring precession inside a rotating mass shell has some similarites to this problem, but the solution does not predict the existence of a linear acceleration term for a moving test particle.
There are electrical duals of these relationships, and those solutions could be investigated with the Liénard-Wiechert retardation equations -- in the frame of reference of a moving test particle. The fields perceived by a moving observer are not simply related to those perceived by a stationary observer. The magnetic field does not even exist as a separate entity. It is a transformed E field. Each particle that makes a contribution to the field has to be considered separately when obtaining the solution for an observer with a different velocity. The fields that exist in the laboratory frame of reference have no absolute significance. These relationship may be relevant to the calculation of the fields for the Aharonov-Bohm effect.
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Last update 7 Sept 2015 Revision History