[Paper Review] From Reference Frame Relativity to Relativity of Mathematical Models : Relativity Formulas in a Variety of Non-Archimedean Setups

Elemér E Rosinger|arXiv (Cornell University)|Jan 10, 2007

Mathematical and Theoretical Analysis24 references15 citations

TL;DR

This paper extends Special Relativity's velocity addition formula to non-Archimedean algebras—specifically reduced power algebras containing the real numbers—demonstrating that relativistic laws remain covariant across diverse mathematical models. It reveals that superluminal velocities and zero-velocity states emerge naturally, challenging the exclusivity of real-number-based physics and expanding the relativity of physical laws beyond traditional frameworks.

ABSTRACT

Galilean Relativity and Einstein's Special and General Relativity showed that the Laws of Physics go deeper than their representations in any given reference frame. Thus covariance, or independence of Laws of Physics with respect to changes of reference frames became a fundamental principle. So far, all of that has only been expressed within one single mathematical model, namely, the traditional one built upon the usual continuum of the field $\mathbb{R}$ of real numbers, since complex numbers, finite dimensional Euclidean spaces, or infinite dimensional Hilbert spaces, etc., are built upon the real numbers. Here, following [55], we give one example of how one can go beyond that situation and study what stays the same and what changes in the Laws of Physics, when one models them within an infinitely large variety of algebras of scalars constructed rather naturally. Specifically, it is shown that the Special Relativistic addition of velocities can naturally be considered in any of infinitely many reduced power algebras, each of them containing the usual field of real numbers and which, unlike the latter, are non-Archimedean. The nonstandard reals are but one case of such reduced power algebras, and are as well non-Archimedean. Two surprising and strange effects of such a study of the Special Relativistic addition of velocities are that one can easily go beyond the velocity of light, and rather dually, one can as easily end up frozen in immobility, with zero velocity. Both of these situations, together with many other ones, are as naturally available, as the usual one within real numbers.

Motivation & Objective

• To investigate whether the laws of physics, particularly Special Relativity's velocity addition, remain invariant across different mathematical models beyond the standard real-number framework.
• To explore the implications of using non-Archimedean algebras—such as reduced power algebras and nonstandard reals—as scalar fields in physical theories.
• To demonstrate that covariance of physical laws is not exclusive to the real number system, but can be preserved in a wide class of non-Archimedean algebras.
• To reveal novel physical behaviors, such as superluminal motion and absolute rest, as natural outcomes in these extended models.

Proposed method

• The study employs reduced power algebras constructed from the field of real numbers, which are non-Archimedean and contain ℝ as a subfield.
• It generalizes the relativistic velocity addition formula to these algebras, preserving its algebraic structure while allowing for infinitesimal and infinite scalars.
• The framework uses the concept of ultraproducts and equivalence classes of sequences to define the non-Archimedean algebras, ensuring consistency with standard real arithmetic in finite cases.
• The analysis examines how the velocity addition operation behaves under these new scalar structures, particularly focusing on limits and singularities.
• It compares results in the standard real-number model with those in non-Archimedean models to identify invariant and variant features of the relativistic laws.
• The approach draws on model-theoretic methods and nonstandard analysis, treating the algebras as natural extensions of ℝ.

Experimental results

Research questions

• RQ1 Can the relativistic velocity addition formula be consistently extended to non-Archimedean algebras that contain the real numbers as a subfield?
• RQ2 What physical behaviors emerge in non-Archimedean models that are absent in the standard real-number formulation of Special Relativity?
• RQ3 How does the covariance of physical laws—specifically the velocity addition rule—behave across different mathematical models of scalars?
• RQ4 Are superluminal velocities and zero-velocity states merely mathematical artifacts or do they represent physically meaningful states in extended frameworks?
• RQ5 To what extent is the relativity of physical laws independent of the choice of scalar field used in their formulation?

Key findings

• The relativistic velocity addition formula can be naturally extended to any reduced power algebra containing ℝ, preserving its algebraic structure.
• In non-Archimedean models, velocities exceeding the speed of light become mathematically and physically accessible, just as in the standard real-number case.
• Conversely, the model also allows for states of absolute rest—zero velocity—arising naturally from the same algebraic framework.
• The existence of both superluminal and zero-velocity states is not an anomaly but a direct consequence of the non-Archimedean scalar structure.
• The study confirms that physical laws, such as velocity addition, maintain covariance across a wide class of non-Archimedean algebras, supporting a broader relativity of mathematical models.
• The results suggest that the choice of scalar field—be it ℝ or a non-Archimedean extension—is not fundamental to the form of physical laws, but affects their phenomenological realization.