# Faraday's Law Does Not Follow from Either Weber's or Ampere's Law....and its Implications For GWs

It has recently been emphasized in the literature that tidal waves are gravitational waves. However, nothing could be further from the truth. We will demonstrate this using the fact that Weber's law conserves energy, and can therefore be derived from a scalar potential. The condition is that the vector potential be constant. This effectively removes the acceleration term from Weber's law, and with it the possibility of deriving Faraday's law. The absence of the inductive terms of Faraday's law and Maxwell's displacement current eliminate the possibility of forming a wave equation. This is entirely equivalent to the existence of a scalar potential, whose force is time-independent.

Ampere divided his currents into longitudinal and parallel components. He showed, experimentally that the force of the latter is twice that of the former. Their ratio is equal to the ratio of the electrodynamic to the electromagnetic units of measurements. Weber began his derivation by considering the longitudinal current proportional to the velocities of the electric particles. So that the force between two electric particles in longitudinal motion should be independent of the direction of their motion, he added a square term to the static Coulomb potential,

F=(ee'/r^2){1-a^2(dr/dr)^2},

where a^2 is the constant of proportionality. It is clear that if the force is repulsive, longitudinal motion will decrease its magnitude.

In the parallel case, it may happen that two charges approaching each other will momentarily have zero relative velocity, but have relative acceleration. To account for this Weber added an acceleration term to his equation

F= (ee'/r^2){1-a^2(dr/dt)^2+b d^2r/dt^2},

where b is another "constant" to be determined. Although all the books refer to it as a constant, it will turn out to be anything but a constant. Since the particles in the parallel current do not move along the same straight line, Weber (1846) found it necessary to introduce the condition

d^2r/dt^2=(dr/dt)^2/r. (A)

Thus his force reduced to

F=(ee'/r^2){1-(a^2-b/r)(dr/dt)^2},

thereby effectively eliminating the acceleration term.

The ratio of the "constants", b/r/a^2, was the force between parallel and longitudinal current elements that Ampere had determine two decades earlier. Their ratio was found by him to be 2, so Weber's equation could be written as

F=(ee'/r^2){1-a^2(dr/dt)^2+2a^2r d^2r/dt^2)}, (B)

which is also equivalent to

F=(ee'/r^2){1+a^2(dr/dt)^2}.

The constant a was measured by Weber and Kohlrausch to be 1/a=1.1 X 10^(11) mm/sec.

From the above we deduce the following:

Weber's condition (A) implies that the vector potential, A=dr/dt/r is constant so that the electric field, or the force is a gradient of a scalar potential.

The subsequent ratio that Ampere found gives a scalar potential U=(ee'/r){1-(a^2/2)(dr/dt)^2} (C) whose derivative is precisely the Weber force, (B).

As far as gravity is concerned, (C) is equivalent to a tidal potential

As a far as GR is concerned, tidal stresses are supposedly accounted for by the Weyl tensor, which roughly speaking is the Riemann tensor less the Ricci tensor. The Weyl tensor can be decomposed into electric and magnetic tensors, both of which are symmetric and trace-free. The condition that the static space-time is that the Weyl tensor be of the electric type [E N Glass, "Weyl tensor decomposition in stationary vacuum space-times"] Only in this case, will the tidal forces be derived from the electric component of the Weyl tensor.

If E_(ab) is the electric component, then

E_(ab)- E_(ba) (D)

would give Faraday's law for the time rate of change of the magnetic component were it not for the fact that being symmetric, the above vanishes. The tidal stresses follow from the electric component of the electric field, and this property would be destroyed if (D) did not vanish.

Hence, there is nothing comparable to induction in gravitational interactions, and without induction there can be no gravitational waves!