Dark energy as a manifestation of nontrivial arithmetic

TL;DR

This paper explores how different choices of arithmetic and calculus frameworks can alter the geometric and physical interpretation of spacetime, suggesting dark energy effects may stem from mismatched mathematical assumptions.

Contribution

It introduces non-Diophantine arithmetic as a means to reconcile Minkowskian and Lorentzian geometries, offering a novel perspective on dark energy phenomena.

Findings

Non-Diophantine arithmetic makes space globally Minkowskian.

Switching to natural arithmetic yields Lorentzian manifolds.

Dark energy effects may result from arithmetic mismatches in models.

Abstract

Arithmetic operations (addition, subtraction, multiplication, division), as well as the calculus they imply, are non-unique. The examples of four-dimensional spaces, $\mathbb{R}_+^4$ and $(-L/2,L/2)^4$, are considered where different types of arithmetic and calculus coexist simultaneously. In all the examples there exists a non-Diophantine arithmetic that makes the space globally Minkowskian, and thus the laws of physics are formulated in terms of the corresponding calculus. However, when one switches to the `natural' Diophantine arithmetic and calculus, the Minkowskian character of the space is lost and what one effectively obtains is a Lorentzian manifold. I discuss in more detail the problem of electromagnetic fields produced by a pointlike charge. The solution has the standard form when expressed in terms of the non-Diophantine formalism. When the `natural' formalsm is used, the same solution looks as if the fields were created by a charge located in an expanding universe, with nontrivially accelerating expansion. The effect is clearly visible also in solutions of the Friedman equation with vanishing cosmological constant. All of this suggests that phenomena attributed to dark energy may be a manifestation of a miss-match between the arithmetic employed in mathematical modeling, and the one occurring at the level of natural laws. Arithmetic is as physical as geometry.