Weber's law of electrodynamics, like Coulomb's and Ampere's laws, which it comprises are action-at-a-distance laws. It, therefore, should have raised some eyebrows that Weber's law predicts a finite speed of propagation of electromagnetic waves. How was this possible?
Weber's law, like Lorentz's, is a union of static and motional laws. In Lorentz's case, it is the sum of Coulomb's law and Grassmann's law. O'Rahilly took exception to such a union saying that they were not compatible with one another. Yet, Kohlrausch and Weber were able to determine the constant that allowed Ampere's law to be written alongside of Coulomb's law to be c times the square-root of 2, which is the square root of two greater than the present value of the speed of light.
In another development, the electric field of a uniformly moving charge will always point in the present direction of the charge, and not in the retarded position of the source. Since it was assumed that the gravitational field also propagates at the same speed, it was taken to imply that the gravitational force cannot preferentially point toward a retarded--or an advanced--position, and therefore it too should not exhibit aberration. Although Ibison et al thought it was necessary to write an entire article, "The speed of gravity revisited", on the subject in which the electric field of a uniform propagating charge is compared to the gravitational field of a mass following the geodesic equation of general relativity, their conclusion is completely superfluous because it is contained in their footnote . For in that footnote they admit "[w]hen the velocity is constant, there is no radiation, no radiation reaction, and time-symmetry is then present both in the Maxwell equations for the fields, and the Lorentz-Dirac equation for the sources. In that case, the motion must be time reversible. It follows immediately that the force cannot preferentially point toward an historical as opposed to a future position." This is not entirely true since the Lorentz force contains a motional term which violates Newton's third law. But, the force always points in the direction of the electric field, E, since the motion term is vxH. The velocity v is normal to the plane containing the orthogonal vectors E and H. This is also the direction of the Poynting vector.
Ampere's law describes the angular dependencies of two conducting elements s and s' separated by a distance r. The remarkable thing is that if s and s' stand for the present positions of the charge instead, and the distance of separation replaced by the uniform speed of their motion, the electric field is determined by the present position of the charge alone independent of its past history. Instead of the angles being formed by the orientation of the conducting elements and the line joining them, the angles are formed by the direction of the position of the charges with regard to the observer and the direction of the uniform velocity.
However, this value of the electric field does not correspond to Weber's expression. Weber's development of Ampere's law consisted of a combination of longitudinal elements and parallel ones. The longitudinal elements depended on the uniform velocity with which the charges moved, based on Gustav Fechner's incorrect hypothesis that a current consists of the opposite flow of positive and negative electrical particles, and the parallel case in which the relative velocity of the particles was zero, but they had a relative acceleration. In the longitudinal case, the force is diminished by the relative motion of the two current elements, whereas in the parallel case, the electric field is increased by the relative acceleration of the particles. There is also no guarantee that the particles will move in the same direction as in the longitudinal case.
As far back as the end of the eighteenth-century, Laplace knew that aberrational effects due to time retardation do not act in the direction of the separation of two bodies. Due to the finite speed of propagation of gravity, Laplace reasoned, the force exerted on the moon by the earth at any given time was produced at an earlier time. And due to this time retardation, the action of the force will make an angle to the line connecting the moon and the earth proportion to the ratio of the orbital velocity of the earth about the sun to the unknown speed of gravity, which he hoped to determine by supposing that the moon will "fall" to earth after about 30 billion years. Although the supposition was wrong, Laplace's conclusion that "the gravitating fluid has a velocity which is at least a hundred times greater than that of light" held sway for almost two centuries. Although LIGO claims the contrary, no one has succeeded in measuring the speed of gravity. Inferences are not measurements.
So we must look for a force that is oblique to the direction connecting the two bodies. This comes from the radiative term of the electric field. This radiative term has the appearance of Grassmann's force which is a triple vector product of the two conducting elements and a unit vector in the direction which connects them. If we replace the conducting elements by the position of the charges at earlier and later times and the unit vector by their relative acceleration, then the radiative force in the past direction will contain two terms: one along the present direction of the charge and the other in the direction of their acceleration. It is the former term that when added to the field in uniform motion in the present direction of the charge that will give Weber's force. The angles in the former are determined with respect to the direction of the charges and their uniform velocity of propagation whereas the acceleration term, the angle is between the acceleration and the past position of the charge. The coefficient of the acceleration time is between the direction of the charges at the two different times and this does not depend upon their separation, whether it be their relative velocity or relative acceleration.
The calculation of the time retarded fields goes under the name of Lienard-Wiechert potentials. It was this method that allowed Paul Gerber to deduce the correct anomalous advance of the perihelion of Mercury, although it was difficult to rationalize by the square of the retarded distance was needed in the denominator of the gravitational potential, rather than merely the linear term. It is rather surprising that the present present position of the charge, expressed in terms of its past position, and the uniform speed of propagation, can be expressed entirely in terms of the present position in terms of the Beltrami metric. The relation is
where s' and s are past and present positions. Dividing both sides by [1-(v/c)^2]^2, gives the Beltrami metric exactly and its inverse is proportional to the electric field in the direction of s. What is needed to complete Weber's force is the term (a.s')s, where a is the relative acceleration, and the angle is between the direction of relative acceleration and the past position of the charge, s'.
If there is any deviation from rectilinear motion, it must come from the radiative component of the field, not the one that depends on the uniform motion of the charges. Now Weber found that in order to determine the coefficients in his expression of the force, he had to reduce the acceleration term to one quadratic in the speed. This he did by imposing the condition that A=v/r=constant, and it is none other than the requirement that the vector potential be constant. Then the ratio of the coefficients was found using none other than Ampere's relation between the force between longitudinal current elements to the force between parallel current elements. This was the same value that Ampere determined to have an absolute value equal to one-half. It was supposed that this relation still held when the condition of a constant vector potential was not imposed.
We have seen that the Beltrami metric gives the increment in distance above. It is tantalizing to generalize this to any quantity, like the velocity or acceleration. The former is referred to as the Lobachevsky-Einstein metric in velocity space by Fock in his book, The Theory of Space, Time, and Gravitation, while the latter is the increment in acceleration that appears in the expression for the total radiated energy, which is proportional to: