Most of the material in this section has been superseded by the calculations at www.s-4.com/mass
The following calculations are exploratory and speculative, and they contain inconsistencies. Readers interested in learning more about gravity should refer to the general theory of relativity.
A set of retardation equations for the gravitational field are derived. The equations are duals of the Liénard-Wiechert retardation equations for charge. The derivation consists of applying the Lorentz transform to a 4-potential representation of Newtonian gravity. The solutions would be expected to give the retarded gravitational stress arising from the relativistic motion of an accelerated mass.
In the case of two equal masses in a mutual circular orbit these solutions predict a radiated power that is 1/3 of the value given by the general theory. The gravitational wave has not yet been directly detected, but the observations of the pulsar binaries are now accurate enough that the orbital decay due to gravitational radiation can be measured . These measurements show that the GR equation is the correct one.
The power calculation is of an assumed form. In being obtained from the Lagrangian, another way of computing the power would be to utilize the energy-momentum tensor, but I have not yet carried through these calculations.
It is a reasonable expectation that a set of weak-field retardation equations exist that will give the correct value for the radiated power, so these investigations will continue. This material is presented only for those who wish to pursue independent investigations. A relevant consideration is that, in being direct duals of the Liénard Wiechert retardation equations for charge, any shortcomings of the gravitational solution are probably present in the electrical solutions also.
The gravitational solutions predict that the gravitational wave is an acceleration wave. The fully developed acceleration wave cannot exist in an infinite universe. The equations are singular at infinity. The acceleration wave can be projected into an infinite universe by taking a limit and then gauge transforming the potentials, which has the effect of removing the singularity. If the solution is transformed to the frame of reference of one of the accelerated particles then quadrupole terms appear, but the solutions are still not the same as those of the general theory.
Now it seems that the radiative solutions of the general theory of relativity might have the same difficulty at infinity. For example, if the acceleration vector is constant over an extended region then the curvature is zero and the GR equations are singular. But the existing derivations of the plane wave GR solutions suppose from the beginning that the solutions can exist in an infinite universe , and it is conceivable that solutions obtained with the simplifying assumption misrepresent some aspects of the gravitational wave. In particular, in the weak field of the Schwarzschild solution the acceleration is the gradient of the expansion factor, yet this equation plays no role in the radiative field, with the consequence being that the gravitational wave is not an acceleration wave. It would be highly desirable to obtain the full weak field spherical radiative solutions of the general theory.
Suppose that a swarm of stationary particles exists in some isolated region of space. A weak acceleration wave passes through the swarm. If there is to be any observable effect, when viewed from afar, then each of the particles must be independently accelerated by the wave. For a weak wave, the motion of each particle must be quite nearly Newtonian, for all observationally established gravitational and kinematic relationships reduce to the Newtonian in the asymptotic limits.
The wave is due to gravitational field of a distant mass. The mass is in motion, so its field must be retarded. In order to do that, first transform to a frame of reference that is co-moving with the source at the retarded time. In that frame of reference, and in the weak field limit, the gravitational 4-potential along the future light cone is equivalent to the simple inverse square law relationship of the Newton theory if the particle is moving at a constant velocity. It might be otherwise if the particle is accelerated. In the case of charge, there are no experimental results showing that the acceleration makes a difference, and it will be assumed that the gravitational field behaves similarly. In other words, even though the acceleration makes a vast difference in the final global solution, the retardation equations for charge do not contain any acceleration terms. The potentials are the integral of the fields, causing them to be of lower order. The acceleration terms do not appear until the potential solution is differentiated.
In the second frame of reference, compute the gravitational 4-potential at the location of the distant field point, and then transform the 4-potential back to the first frame of reference. Compute the acceleration of a test particle from it.
The calculation is a direct dual of the derivation of the LW equations for an accelerated charge. The assumptions are essentially the same as those of Liénard and Wiechert, but they evidently do not work for the gravitational field. We can consider the possibility that they do not actually work for the electrical fields either, but that experiments of a appropriate kind have not yet been performed to show the discrepancy. The gravitational radiated power varies as ω6. A small electrical antenna varies as ω4. A gravitational solution therefore imposes more stringent tests of the high order terms of the retardation equations. A satisfactory low-order electrical computational technique can fail altogether for a gravitational solution.
It may be of interest to look for these indicated shortcomings of the
Liénard Wiechert solutions in the laboratory utilizing rotating
electrical machinery. To the best of my knowledge experiments of an
appropriate configuration and sensitivity have never been performed.
The velocity of conduction electrons is so small that no discrepancy
would be observable in stationary conductors. In the continued study of
electricity, magnetism, and gravity we need a keener understanding of
the method of retardation.
Other online material: A
An Oscillatory Pulsar Model
If you find technical or conceptual error in any of the material at this site then please email me the details of it.
Last update 2 Jan 2000