## The Essentials of ECE Theory

The essentials of ECE theory are simple to understand in a simplified symbolism in which the complicated indices of Riemann geometry are omitted. The theory is based directly on this geometry, which is very well known. Attempts to refute this geometry are futile. The geometry was devised by Cartan in the nineteen twenties and can be summarized in four equations:

T = D ^ q
R = D ^ omega
D ^ T := q ^ R
D ^ T tilde := q ^ R tilde

The tilde denotes Hodge dual and the fourth equation is an example of the third. The first two are the Cartan Maurer structure equations. The third is the Cartan Bianchi identity, the fourth is the Cartan Evans identity. The Hodge duals of the two-forms R and T are two-forms. Here T denotes the Cartan torsion two-form, R denotes the Cartan curvature two-form, q is the Cartan tetrad one-form, omega is the Cartan spin connection. The operator D ^ is defined by

D ^ := d ^ + omega ^

where d ^ is Cartan’s exterior derivative. This is all well known differential geometry, taught in all reputable universities. This geometry becomes ECE theory with the simplest possible hypotheses. For electrodynamics:

A = A(0) q

where A is the one-form that represents the electromagnetic potential. So electromagnetism is geometry within A(0). In S.I. units cA(0) is a voltage that pervades the universe and which is observable in the radiative corrections. This may be amplified using spin connection resonance and used to produce new energy devices. These are already being manufactured and marketed.