Curvature and torsion of implicit hypersurfaces and the origin of charge

Abstract

A formal correspondence is established between the curvature theory of generalized implicit hypersurfaces, electromagnetism as expressed in terms of exterior differential systems, and thermodynamics. Starting with a generalized implicit surface whose normal field is represented by an exterior differential 1-form, it is possible to deduce the curvature invariants of the implicit surface and to construct a globally closed vector density in terms of the Jacobian properties of the normal field. When the closed vector density is assigned the role of an intrinsic charge current density, and the components of the normal field are assigned the roles of the electromagnetic potentials, the theory is formally equivalent to an exterior differential system that generates the PDE's of both the Maxwell Faraday equations and the Maxwell Ampere equations. The interaction energy density between the potentials and the induced closed charge current density is exactly the similarity curvature invariant of highest degree (N-1) for the implicit surface. Although developed without direct contact with M-brane theory, these ideas of generalized implicit surfaces should have application to the study of p-branes that can have multiple components and envelopes. The theory suggests that gravitational collapse of mass energy density should include terms that involve the interaction between charge-current densities and electromagnetic potentials.