In vector analysis, a vector field is irrotational if its curl vanishes. On the other hand, a source-free vector field is determined by the magnitude of its divergence. Complete freedom is attained if and only if its divergence vanishes. A vector field satisfying both conditionals is said to possess double irrotational freedom. That is both curl V=0 and div V=0 for an arbitrary vector V.
When applied to the electromagnetic field, four conditionals arise such that div E=0, div B=0, curl E=0, and curl B=0 where E is the electric field and B is the magnetic field. Moreover, these are all time independent and exist throughout the complete space-time evolution of the universe. Since both the curl and the divergence are differential space operators they imply the existence of spatial frequency, deriving a scalar potential and a vector potential. However, the energy density (W) of the field is still given by the partial time derivatives as W=(E+B)/8p. This energy density implies the existence of temporal frequency and for vanishing mean curvature of space-time evolution the ratio of spatial over temporal is equivalent to the speed of light (c). Nevertheless, total freedom from all conservative force fields is attainable if and only if the rest mass is zero and the charge density vanishes such that the linear momentum (p) is given by p=W/c and implying vanishing volume and constant double Hopf rings.


LinkBack URL
About LinkBacks
Reply With Quote

