In electrodynamics, Faraday's law of induction and Maxwell's displacement current are necessary for the propagation of electromagnetic waves. No such forms of induction exist in gravitation, and, consequently, there is no such thing as gravitational waves.
The derivation of Ampere's law also eliminates induction so Faraday's law does not follow from Ampere's law. Ampere's equation reads
F_A=e/r^2{v^2-(3/2)(r^.v)^2}, (1)
where r" is the radial unit vector and r^v=dr/dt is the radial component of the velocity, v. To make it compatible with Weber's law, we must add an acceleration term, e/r(dv/dt), which supposedly accounts for Faraday induction. Ampere's equation is basically an "angle" equation, so the acceleration is extraneous to it. But if we want it to be equivalent (not identical) to the corresponding component of Weber's equation,
F_W=e/r^2{-(1/2)(dr/dt)^2+r(d^2r/dt^2)}, (2)
then we must add it to Eqn (1).
Now, equating (1) with (2) there results
d^2r/dt^2-r(d theta/dt)^2 = dv/dt,
which is simply Newton's second law, where we used
v^2=(v^xv)^2+(r^ x v)^2=(dr/dt)^2+r^2(d theta/dt)^2,
So the equivalence of Weber's and Ampere's extended equation yields Newton's second law for an arbitrary acceleration. Yet, this acceleration is not so arbitrary.
Weber used the condition
d^2r/dt^2=(dr/dt)^2/r, (3)
to determine the coefficients in his expression. Having introduced this into his equation with two unknown, made the comparable: Their ratio is nothing other than the ratio of the force between parallel current elements to the force between longitudinal elements. This ratio Ampere determined to be 1/2, which is that appearing in Eq. (2). Once condition (3) has been imposed, the acceleration term in (2) has essentially been "tamed" so that it does not produce any inductive effects. This is confirmed by the fact that Weber's force can be derived from a scalar potential, as we shall now show.
It allows Ampere's force, with the acceleration term to be derived from a scalar potential. Furthermore, if we extremize the Lagrangian
L=m{1/2[(dr/dt)^2+r^2(d theta/dt)^2]+cGM/r[1-(1/2)(dr/dt)^2]}
we get
(1-cGM/r)(d^2r/dt^2-r(dtheta/dt)^2=-cGM/r^2{1-(1/2(dr/dt)^2}.
This yields the exact perihelion advance for Mercury when we set c=6. This was done by Schrodinger in 1925.
The condition (3) Weber used to determine the coefficients a^2 and b in his
equation,
F=e/r^2{a^2(dr/dt)^2+b(d^2r/dt^2},
implies that the vector potential,
A=(dr/dt)/r (4)
is constant. If we didn't know the coefficient in Ampere's law
F_A=e/r^2{v^2-k(r^.v)^2 +rdv/dt}, (5)
we would impose the condition of constant vector potential,
A' =v/r,
and equating it with Weber's equation on the condition (4) is constant, we get
k=3/2 for v=(r^.v)=dr/dt, entirely radial.
The fact that Weber's equation can be derived from a potential implies conservation of energy and the absence of all inductive effects. Inductive effects convert the conservation of energy into an energy balance equation where the rate of decrease of energy in time is balanced exactly by the negative divergence of the Poynting vector. Obviously Einstein's condition of energy conservation does not apply to such situations. Moreover, Poynting's vector does not exist since there is nothing corresponding to a magnetic field in gravitation.
To ensure the absence of inductive effects, the force cannot contain a term dA/dt that explicitly depends on time.
The fact that it is only the radial component which subsist in v, is guaranteed by Hamilton's equations, derived from the hamiltonian
H=*1/2v^2+V(r),
for any arbitrary scalar potential V. Hamilton's equations read
dr/dt=dH/dv=v, and dv/dt=-dH/r=-dV/dr.
The first term identifies v=r^.v with the radial velocity.
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