Riemannian geometry without the hypotheses of homogeneity and symmetry

TL;DR

This paper generalizes Riemannian geometry by removing homogeneity and symmetry assumptions, revealing a natural emergence of the Pythagorean distance formula and proposing a unified geometric framework for electromagnetic and gravitational fields.

Contribution

It introduces a minimal assumption-based generalization of Riemannian geometry, leading to a unified geometric model for fundamental interactions without relying on traditional hypotheses.

Findings

Reemergence of the Pythagorean distance formula as a first approximation.

Identification of a one-parameter family of Riemann-Randers line elements.

Framework accounts for electromagnetic and gravitational interactions, including charge and CPT symmetry.

Abstract

A generalisation of Riemannian geometry is considered, based exclusively on the minimal assumptions that the line element $ds$ is a regular function of position and direction and that the distance of every point from itself is equal to zero. Besides the Riemannian line element, also Riemann's residual hypotheses of homogeneity and symmetry are dropped. Surprisingly, the infinitesimal Pythagorean distance formula reemerges, without the need of being postulated, as a first approximation to $almost$ $every$ geometry that is invariant with respect to direction reversal. More in general, the first approximation to $almost$ $every$ geometry is a one-parameter family of homogeneous Riemann-Randers line elements, naturally providing the geometrical framework for a unified theory of the classical electromagnetic and gravitational fileds. Geometry naturally accounts for the hierarchy between electromagnetic and gravitational interactions, their different attractive/repulsive nature, electric charge, CPT symmetry, Maxwell and Einstein equations. Within this framework higher order terms could describe new spacetime degrees of freedom of possible large-scale relevance.