Curvature of Implicit Surfaces, and the Origin of Internal Energy and Charge

The origin of charge and internal energy remains one of the mysteries of physics. In that which follows, techniques of differential geometry yield a clue that charge density and internal energy density, like inertia and mass, may have its roots in the curvature properties of implicit surfaces.

The differential geometry of implicit surfaces can be studied in terms of a map from a variety xk into a normal vector field, Ak which is homogeneous of degree zero in its component functions. The Jacobian matrix of the map is singular and will induce a metric on the variety which is also singular. The similarity invariants of the singular Jacobian matrix yield expressions in terms of the curvatures of the implicit surface. In the simplest of cases, the normal vector field of the implicit surface may be used to construct a 1-form, A=Akdxk which is integrable in the sense of Frobenius. That is, the one-form may be reduced to the product of a single function times the total differential (gradient field) of a second function, A=a db. On the otherhand it is possible to start from an arbitrary 1-form of Action potentials, A0 which need not satisfy the Frobenius unique integrability theorem, and thereby describe more complex implicit surfaces.

For if an arbitrary 1-form is divided by a suitable Holder norm l the component functions, A=A0/l can be made homogeneous of degree zero, and most of the results for the simple implicit surfaces also follow. Recall that A0 may be used to formulate the exterior differential system F0 - dA0 = 0, which, when dA0 is not zero, leads to the Maxwell-Faraday equations as a system of constraints on the variety. These observations imply a possible connection between electromagnetism and curvature of implicit surfaces. The conjecture is further strengthened by noting that the Adjoint matrix to the singular Jacobian matix can be used to construct a well defined N-1 form density, Js. This N-1 form has global zero divergence, d Js = 0. Hence, Js - dGs = 0, which defines an exterior differential system equivalent to the Maxwell-Ampere equations, and leads to concept that Js represents an electromagnetic current density.

For those Holder norms which are equivalent to the Gauss map, the interaction between these closed currents Js and the potentials, A, generated by the N-form A^Js has remarkable properties. For 2D surfaces in 3 dimensions, it has been observed that the interaction coefficient of A^Js is exactly equal to the Gauss curvature of the implicit surface defined by the homogeneous 1-form, A. In any dimension, the N-form of interaction A^Js has a coefficient equal to the curvature similarity invariant of degree N-1. For N = 4, this result implies that the charge current density is cubic in the curvatures of the implicit surface. As gravity and mass energy density are currently considered to be artifacts of quadratic (Gauss) curvature, the conjecture follows that the energy density of interaction between potentials and currents, as well as charge, may be artifacts of cubic (Adjoint) curvature, and not Gauss sectional curvature. By using the known relationship between A^Js and the first Poincare invariant, it would appear that Adjoint curvature is a specific gauge addition, d (A^Gs) to the Lagrangian of the field, F^Gs This result raises the credence level in the physical significance of what has been called topological spin, A^Gs and suggest a possible role for studying the electromagnetic interaction and modification or manipulation of the gravitational field.

By studying the evolution of the physical system described by the Action 1-form, with respect to a vector field generated by the charge current density via Cartan's Magic formula , it also becomes apparent that the Adjoint curvature and the "internal Energy" are formally the same. Hence internal energy density, interaction energy density, and Adjoint curvature of the implicit hypersurface generated by the 1 form of Action Potentials are equivalent concepts.