Dark Fields do Exist in Weyl Geometry

TL;DR

This paper introduces a generalized Weyl integrable geometry (GWIG) that incorporates interactions with arbitrary dark fields and anisotropic dilation, leading to new mathematical formulations and potential solutions to classical physics problems.

Contribution

It extends classical Weyl geometry to include dark field interactions and anisotropic effects, enabling new approaches to physics equations and singularity-free models.

Findings

Derived Maxwell's equations within GWIG framework

Established boundary conditions using conformal infinity analogy

Constructed a non-singular point charge model

Abstract

A generalized Weyl integrable geometry (GWIG) is obtained from simultaneous affine transformations of the tangent and cotangent bundles of a (pseudo)-Riemannian manifold. In comparison with the classical Weyl integrable geometry (CWIG), there are two generalizations here: interactions with an arbitrary dark field, and, anisotropic dilation. It means that CWIG already has interactions with a {\it null} dark field. Some classical mathematics and physics problems may be addressed in GWIG. For example, by derivation of Maxwell's equations and its sub-sets, the conservation, hyperbolic, and elliptic equations on GWIG; we imposed interactions with arbitrary dark fields. Moreover, by using a notion analogous to Penrose conformal infinity, one can impose boundary conditions canonically on these equations. As a prime example, we did it for the elliptic equation, where we obtained a singularity-free potential theory. Then we used this potential theory in the construction of a non-singular model for a point charged particle. It solves the difficulty of infinite energy of the classical vacuum state.