The publication of Weber's force was a watershed in physics for it separated a physical force which does not depend on the frame of reference, nor upon an observer. The force contains only relative quantities, difference in position, velocity, and acceleration of two charges so they are the same no matter what frame is chosen.

Weber's force is considered as equivalent to the earlier Ampere law that describes the motion of Amperian current elements, which cannot be made infinitesimally small. These depend on the choice of the origin from where they are viewed, and so can have different values for different observers.

Whereas Weber's force considers pairs of point charges, Ampere's force is given in terms of the angular orientation of two current elements. Both forces are said to account for mutual induction. But, mutual induction will change when one of the circuits is turned about an arbitrary axis with respect to the other. Such an effect can only be brought about by a mechanical torque, and is certainly not something that can be handled by Weber's force.

Maxwell does a brilliant job of mixing the two types of variables in Vol. 2 of his Treatise. As a rule of thumb, r, dr/dt, and d^2r/dt^2 are all relative and the same in whatever frame is chosen. The variables, v and a, are the velocity of the circuit elements, and their accelerations, a=dv/dt. Their values depend on the origin of the frame chosen and will be different for different observers. The relation to the relative variables are

dr/dt=i_r.v,

where i_r is the unit radial vector and

r d^2r/dt^2=v^2-(dr/dt)^2+ r.a

In reference to current, I, and current element, ds, we have

I ds=e v.

And since a=dv/dt

d/dt( I ds)=dI/dt ds= e a,

if the current element doesn't rotate. So Induction is related to the acceleration of the current elements, ds, and not to the acceleration of the charges, d^2r/dt^2.

The difference is underlined by the Grassmann force. It is

F_G= v X B =I ds X mu (I'ds' X i_r)/4pi r^2,

where mu is the vacuum permeability, and the prime denotes the second current element. Expanding and taking the inner product with respect to the unit radial vector, we get

F_G.i_r=-mu(II'/4 pi r^2){(ds.ds')-(i_r.ds)(i_r.ds')}

=-mu(II'/4pi r^2){v^2-(dr/dt)^2}

=-mu(II'/4\pi r{d^2r/dt^2)-a},

which would vanish if the two accelerations were equal. The current elements ds and ds' are not in relative motion like the charges e and e'. The relative velocity of the latter is dr/dt which, in general, is not v. Inductive effects are related to a and not to d^2r/dt^2.

Inductive effects depend on the observer. If Weber had prevailed, we would have no need of observers, and no need of relativity, only relational quantities would hold. The fact that Weber's force conserves energy means that there is no energy flux into or out of the system that is determined by the divergence of Poynting's vector in the energy balance equation. If you consider Einstein's equation as a static conservation of energy, there is no energy flux into or out of the system so that all inductive terms would be zero. This is consonant with the fact that Einstein's equations always yield geodesics that are derived from a variational principle.

Many, if not all, the paradoxes that relativity theory possesses would disappear if the force relations were relational in that they would be valid in any frame of reference, inertial or not. It is a real pity that Weber's formalism did not prevail.

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