[Paper Review] The Euclidean geometry deformations and capacities of their application to microcosm space-time geometry

Yuri A. Rylov|ArXiv.org|Jun 5, 2002

Biofield Effects and Biophysics11 references3 citations

TL;DR

This paper proposes T-geometry, a generalized space-time geometry derived from deformed Euclidean geometry via a world function, enabling intrinsic geometric description of quantum effects and particle mass without relying on quantum theory. By relaxing constraints of Riemannian geometry—especially continuity, fixed dimension, and coordinate dependence—T-geometry achieves absolute parallelism, geometrizes mass, and explains non-relativistic quantum effects as geometric consequences of nondegenerate, asymmetric world functions.

ABSTRACT

Usually a Riemannian geometry is considered to be the most general geometry, which could be used as a space-time geometry. In fact, any Riemannian geometry is a result of some deformation of the Euclidean geometry. Class of these Riemannian deformations is restricted by a series of unfounded constraints. Eliminating these constraints, one obtains a more wide class of possible space-time geometries (T-geometries). Any T-geometry is described by the world function completely. T-geometry is a powerful tool for the microcosm investigations due to three its characteristic features: (1) Any geometric object is defined in all T-geometries at once, because its definition does not depend on the form of world function. (2) Language of T-geometry does not use external means of description such as coordinates and curves; it uses only primordially geometrical concepts: subspaces and world function. (3) There is no necessity to construct the complete axiomatics of T-geometry, because it uses deformed Euclidean axiomatics, and one can investigate only interesting geometric relations. Capacities of T-geometries for the microcosm description are discussed in the paper. When the world function is symmetric and T-geometry is nondegenerate, the particle mass is geometrized, and nonrelativistic quantum effects are described as geometric ones, i.e. without a reference to principles of quantum theory. When world function is asymmetric, the future is not geometrically equivalent to the past, and capacities of T-geometry increase multiply. Antisymmetric component of the world function generates some metric fields, whose influence on geometry properties is especially strong in the microcosm.

Motivation & Objective

• To develop a more general space-time geometry than Riemannian geometry by removing arbitrary constraints.
• To overcome limitations of Riemannian geometry, such as dependence on coordinates, curves, and continuity.
• To enable intrinsic geometric description of microcosmic phenomena, including particle mass and quantum effects.
• To demonstrate that quantum-like behavior can emerge from geometric nondegeneracy and asymmetry in the world function.
• To establish a foundation for geometrizing physical properties like mass and explaining quantum phenomena purely through geometric structure.

Proposed method

• Defining geometry via the world function Σ(P,Q) = ½ρ²(P,Q), which fully encodes geometric structure without coordinates or curves.
• Constructing T-geometry as a deformation of Euclidean geometry by modifying the world function, preserving all geometric objects universally across geometries.
• Using only intrinsic geometric concepts—subspaces and world function—eliminating reliance on external tools like coordinates or parametric curves.
• Introducing nondegenerate and asymmetric world functions to model stochastic particle motion and geometric mass.
• Deriving geometric relations directly from deformed Euclidean axioms without constructing a full independent axiomatic system.
• Analyzing the influence of antisymmetric components in the world function on metric fields and geometric stochasticity in microcosmic regimes.

Experimental results

Research questions

• RQ1 Can space-time geometry be generalized beyond Riemannian geometry by removing unjustified constraints?
• RQ2 How can quantum effects be explained geometrically without invoking quantum postulates?
• RQ3 Can particle mass be derived from geometric structure alone via nondegeneracy of the world function?
• RQ4 What role does asymmetry in the world function play in breaking future-past geometric equivalence and generating new geometric fields?
• RQ5 How does the absence of coordinate dependence and curves in T-geometry enable absolute parallelism and intrinsic geometric description?

Key findings

• T-geometry is fully defined by the world function Σ(P,Q), which contains complete geometric information and allows universal definition of geometric objects across all geometries.
• Nondegenerate T-geometry with symmetric world function leads to geometric mass and explains non-relativistic quantum effects as intrinsic geometric phenomena.
• Asymmetric world functions break future-past geometric equivalence and generate additional metric fields that strongly influence microcosmic geometry.
• The antisymmetric component of the world function increases the thickness of geometric tubes, inducing intrinsic stochasticity in particle motion without stochastic geometry.
• T-geometry overcomes key limitations of Riemannian geometry—such as coordinate dependence, lack of absolute parallelism, and inability to geometrize mass—by using a coordinate-free, intrinsic language based on subspaces and world functions.
• Solutions to geometric equations show that at timelike infinity, antisymmetric perturbations lead to finite, non-zero tube radii, indicating persistent geometric effects in the microcosm.