Non-Newtonian mathematics instead of non-Newtonian physics: Dark matter and dark energy from a mismatch of arithmetics

Abstract

Newtonian physics is based on Newtonian calculus applied to Newtonian dynamics. New paradigms such as MOND change the dynamics, but do not alter the calculus. Calculus is dependent on arithmetic, e.g. in special relativity we add and subtract velocities by means of addition $β_1\oplus β_2=\tanh\big(\tanh^{-1}(β_1)+\tanh^{-1}(β_2)\big)$, although multiplication $β_1\odot β_2=\tanh\big(\tanh^{-1}(β_1)\cdot\tanh^{-1}(β_2)\big)$ does not seem to appear in the literature. The map $f_\mathbb{X}(β)=\tanh^{-1}(β)$ defines an isomorphism of the arithmetic in $\mathbb{X}=(-1,1)$ with the standard one in $\mathbb{R}$. The new arithmetic is non-Diophantine in the sense of Burgin. Velocity of light plays a role of non-Diophantine infinity. The new arithmetic allows us to define the corresponding derivative and integral, and thus a new calculus which is non-Newtonian in the sense of Grossman and Katz. Treating he above example as a paradigm, we ask what can be said about the set $\mathbb{X}$ and the isomorphism $f_{\mathbb{X}}:\mathbb{X}\to \mathbb{R}$, if we assume the standard form of Newtonian mechanics and general relativity (formulated by means of the new calculus) but demand agreement with astrophysical observations. It turns out that for $f_\mathbb{X}(t/t_H)\approx 0.8\sinh (t-t_1)/(0.8\, t_H)$ the resulting non-Newtonian Friedman equation with $Ω_Λ=0$ is exactly quivalent to the standard Newtonian one with $Ω_Λ=0.7$, $Ω_M=0.3$. Asymptotically flat rotation curves are obtained if `zero', the neutral element of addition, is nonzero from the point of view of the standard arithmetic of $\mathbb{R}$. We do not yet know if the proposed generalization ultimately removes any need of dark matter, but it will certainly change estimates of its parameters.