The Inductance Gradient

Now consider a region where there is no static charge, but there is electric current flowing such that the magnetic vector potential is not zero. The magnetic flux is the integral of the flux density through the surface area bound by the closed curve. By Stokes theorem it is also the integral of the magnetic vector potential around the closed curve .

(16)

The magnetic flux is valued as the inductance times the current.

(17)

An inductor is defined as a tube of magnetic flux with no conductors required. The energy of the magnetic field is stored in the inductance. If for exampleis everywhere constant and parallel to an infinitesimal length around a small closed circle, or is everywhere orthogonal to the bound surface , then the inductance can be calculated as simple integrals of the loop,

(18)

The four-vector gradient of the inductance is,

(19)

Inductance then is also considered an effective potential that leads to a constant ratio of inductance per unit length, and a permeability vector . A function of the constant of permeability and the geometrical integral of the path.

Now consider that as constants the permeability and permittivity of free space are also Lorentz invariant. So what happens when the inductance and capacitance at the point change due to the presence of polarizable matter nearby? The answer is obvious. Inductance and capacitance are strictly functions of the geometry. So in the same manner that Lorentz transformations preserve the constancy of the speed of light, they must also preserve the constancy of permeability and permittivity. Therefore natural coordinate transformations on the space-time manifold are functions of the polarizability represented by the inductance and capacitance at each point. Variation of these variables will necessitate transformations of the coordinates in order to preserve the Lorentz invariant constants.

Gauge Transformation