[Paper Review] Geometries with intransitive equivalence relation

Yuri A. Rylov|ArXiv.org|Jul 13, 2008

Relativity and Gravitational Theory7 references3 citations

TL;DR

This paper proposes a physical geometry based on a deformed, intransitive equivalence relation, using the world function σ (half the squared distance) as the fundamental geometric element. Unlike traditional axiomatized geometries, this approach enables description of discrete, granular space-time with limited divisibility—key for explaining elementary particle properties—without relying on coordinate systems or linear vector spaces.

ABSTRACT

One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry cannot be axiomatized, in general. It is obtained as a result of deformation of the proper Euclidean geometry. Class of physical geometries is more powerful, than the class of axiomatized geometries. The physical geometry admits one to describe such geometric properties as discreteness, granularity and limited divisibility. These properties are important in application to the space-time. They admits one to explain the discrimination properties of the space-time, which generate discrete parameters of elementary particles. Mathematical formalism of a physical geometry is very simple. The physical geometry is formulated in geometrical terms (in terms of points and world function) without a use of means of description (coordinate system, space dimension, manifold, etc.).

Motivation & Objective

• To develop a geometric framework that describes space-time with discrete, granular properties incompatible with continuous, axiomatized geometries.
• To replace traditional axiomatic geometry with a deformation-based method rooted in the world function σ, enabling physical geometries not amenable to standard axiomatization.
• To explain the origin of discrete parameters in elementary particles through geometric granularity and multivariance arising from intransitive equivalence.
• To provide a geometric foundation for dynamics that avoids differential equations and relies solely on the world function, supporting finite-difference dynamics.
• To unify physical geometry with the principles of relativity and quantum structure by geometrizing mass and particle composition via world chains.

Proposed method

• Construct physical geometry via deformation of proper Euclidean geometry, using the world function σ(P,Q) = ½ρ²(P,Q) as the sole geometric primitive.
• Define geometric objects and relations (e.g., segments, angles, equivalence) entirely in terms of σ, avoiding coordinates, manifolds, or vector spaces.
• Introduce an intransitive equivalence relation on points, where P ~ Q and Q ~ R does not imply P ~ R, leading to multivariance and geometric granularity.
• Model composite particles as world chains—sequences of points connected by links of fixed σ-length—where mass is geometrically determined by link length.
• Derive dynamics from the geometry using finite-difference equations in σ, showing that free particle motion in curved geometry is equivalent to motion in a field in Minkowski space.
• Establish that physical geometry is not axiomatizable in general, but is fully determined by σ, distinguishing it from mathematical geometries like projective or affine geometry.

Experimental results

Research questions

• RQ1 Can a physical geometry be constructed without relying on axiomatic systems, using only the world function σ?
• RQ2 How does intransitive equivalence in a geometric relation lead to discrete, granular space-time properties?
• RQ3 What is the role of multivariance in generating limited divisibility and discrete parameters of elementary particles?
• RQ4 Can dynamics of composite particles be fully described by the geometry of the world function without invoking external fields or quantum formalism?
• RQ5 How does the deformation of Euclidean geometry via σ lead to a consistent physical geometry that includes general relativity and quantum-like features?

Key findings

• Physical geometry based on the world function σ is not axiomatizable in general, distinguishing it from traditional geometries and enabling description of discrete, granular space-time.
• The intransitive equivalence relation leads to multivariance, which is the geometric source of space-time granularity and limited divisibility.
• Elementary particle mass and structure can be geometrically defined as the length of links in a world chain, eliminating the need for wave functions or quantum operators.
• Dynamics of free composite particles is described by finite-difference equations in σ, which reduce to differential equations only under specific limiting conditions.
• The geometry of Minkowski space can be used to describe dynamics in curved space-time, but this corresponds to motion in an effective field, not free motion.
• The method of deformation via σ allows construction of physical geometries without testing consistency of axioms or proving theorems, bypassing limitations of the Euclidean method.