Gravitational waves are based on the assumption of the existence of a gravito-magnetic field, just like the magnetic field in electrodynamics. The fact that Faraday's assumption that a moving conductor in a fixed magnetic field is the same as a fixed circuit in a moving in a varying magnetic field is not borne out in gravitation.

This would equate Ampere's law of the action of moving current elements and Faraday's law of induction of a moving magnetic field in the presence of stationary currents. Ampere's law would be analogous to the action of a Coriolis force the electric field, while Faraday's law would correspond to the Euler force which is completely tangential. The curl extracts the rotation component of the electric field in Faraday's law. Without this symmetry there can be no wave propagation, like in Maxwell's laws.

Ampere's law has defied a mechanical explanation thus far. Moreover, its relation to Grassmann law is not understood except to say that both give the same results when the circuits are closes. We will show that Ampere's circuit law is a combination of the Coriolis force and the relativistic contraction of the electric field in motion, while the Grassmann force is the combination of the centrifugal and Coriolis forces. Moreover, the Biot-Savart law is, in essence, the definition of the angular speed.

It is to be attributed to the genius of Ampere that these phenomena can be interpreted as the attraction between two elements and the oblique forces that depend only on the angles these elements make with the line joining them. The phenomenon of induction can only be attributed to the motion of the gravitational field and the fictitious forces that create a magnetic field. There is no counterpart of a moving magnetic field in a stationary electric field implying the non-existence of a gravito-magnetic field. The breaking of the symmetry that Faraday envisioned for the electromagnetic field precludes the existence of gravitational waves.

There is no symmetry between the Lorentz force and the Coriolis force. It is the latter that determines the motion of the gravitational field. For the Lorentz force would subsist in an inertial frame while the Coriolis force not.

And because the there is a transformation from stationary to moving frames of the electric field does not mean that the electric field in the moving frame acquires an explicit dependency on time, i.e., there is no vector potential. Thus, there is no displacement current in the generalization of Ampere's law. It would be the same as if a Doppler shift in the frequency would make it a time-dependent quantity.

In analogy with the Lorentz force

F =e'(E+ dr/dt X B/c^2) (1)

where the electric field is E=e/r^2, mechanics tells us the the fictitous forces are

E' = e^2/r^2 r' + wXwXr - 2(vXw) (2)

where the centrifugal and Coriolis forces have been appended onto the static gravitational force. Introducing the definition of the angular speed,

w= e^2/c^2 (v X r')/r^2 (3)

where r' is a unit vector pointing the radial direction, (2) boils down to

E'= E{ r'+ 1/c^2(v^2 r' - (dr/dt) v)}. (4)

Whereas (3) is the Biot-Savart law, (4) is none other than force that Grassmann proposed. Its projection in the radial direction is

r'.E'= E{1+ (wr)^2/c^2} (5)

We could have simply written (2) as

E'= E - v Xw (6)

in analogy with the Lorentz force, (1), The two differ in that the former subsists in an inertial frame while the latter not. Hence, there is no real analogy between the two.

Ampere's law, which is more complicated can be considered as a synthesis of the Coriolis force and the relativistic contraction of the

E'= E {(1--(dr/dt)^2/c^2)r' -2 v X w} (7)

whose projection in the radial direction gives

r'.E' = E { 1+ (2v^2-3(dr/dt)^2)c^2} (8)

which is Ampere's law. So actually Ampere's law can be considered relativistic.

Ampere's law consists of an attraction of two conducting elements s and s' that are connected along a line, r. The angles that the elements make with respect to the line are cos theta = dr/ds and cos theta'= dr/ds'. Both of these will merge in the velocity dr/dt. The angle between the two elements, phi is related to the velocity v by e e'v^2=j j'cos phi where j and j' are the currents i ds and i'ds', respectively. Thus, Ampere's law takes the more familiar form

r'.E'= j j'/r^2 (2 cos phi - 3 dr/ds dr/ds'). (8')