Most retardation solutions are radiative. In a radiative solution the fields decay with distance as 1/r in the far field. The energy density of the field is proportional to the square of the field intensity, so the energy density decays by the inverse square law. All antenna solutions also have a near-field region, where some of the terms decay more rapidly than by the inverse square law, even though they propagate at c.
The retardation solutions are always non-trivial solutions to the Maxwell potential equations. The E and B fields are zero, so they are trivial solutions to the Maxwell equations proper. Conversely, the acceleration field of the solutions to the Liénard-Wiechert retardation equations is always zero. The two sets of retardation equations are perfectly orthogonal. This relationship is shown in each of the calculations. The far-field acceleration vector is always parallel to the direction of propagation.
Since mechanical stresses propagate at the speed of sound, while the acceleration wave propagates at the speed of light, any realizable mechanical structure behaves much like a fluid if the frequency of the wave is high. A chunk of metal and a wet sponge respond in quite nearly the same way. In principle, the equations represent the gravitational stress measured by an anchored accelerometer, but this method of measurement is obviously not realizable in practice.
The total radiated power is computed in each solution. The calculation assumes that the energy density of the radiative acceleration field is the same as that of the static field. The acceleration at the outer surface of a thin spherical mass shell is G m/r02. The acceleration at the inner surface is zero, so the average acceleration within the thickness of the shell is 1/2 G m/r02. The total radial force on the shell is computed from the equation f=m a, then the work done in shrinking the shell from infinity is computed. This energy is equated to the energy obtained by integrating the equation 1/2 η0 a dot a over all space, and the result is solved for η0. In computing the Lagrangian of a system the sum of the positive and negative energies is zero, so it seems justifiable to interpret the field energy density as a positive quantity, even though the gravitational potential energy is negative. The solution for η0 is 1/(4 π G). In the case of two equal masses in a mutual circular orbit the radiated power is 1/3 of the value predicted by the general theory.
All the solutions in this section were obtained with the same
computer program with different control settings.
The calculations shown are mostly mostly raw computer output, and are not intended to be read from start to finish.
Each mass orbits about the barycenter, with the barycenter being the point that conserves the momentum of the system. It is obvious from the symmetry of the problem that the radiative terms for two equal masses must be at the second harmonic of the orbital frequency. It is unobvious that it should be that way when the masses are unequal, but it does work out that way for the radiative terms in this solution. However, if the accuracy of the calculation is extended to order r03 then a weak radiative term at the fundamental frequency does occur if the masses are different.
A side-calculation in this file shows that the solution for a rotating mass ring is the same as for a stationary ring.
This solution is radiative. The mechanism is that the shell radiates the expansion factor, and the acceleration is the gradient of the expansion factor. A spherically symmetric solution cannot radiate a transverse wave, from which it follows that the general theory predicts that a pulsating (or collapsing) mass does not radiate.
The dominant radiation term is at the fundamental frequency in this solution. Regardless of the mass ratio, both masses contribute equally to the radiation.
These calculations have been carried through to high order (typically
r05), and all the essential characteristics of
the solutions remain the same.
The solutions remain solutions to the Maxwell equations, and there are
no electrical fields. The far field terms acquire minor relativistic
corrections when the orbital velocities are high, but no transverse