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 ${\beta }_1\oplus {\beta }_2=\tanh ({\tanh }^{-1}({\beta }_1)+{\tanh }^{-1}({\beta }_2))$, although multiplication ${\beta }_1\odot {\beta }_2=\tanh ({\tanh }^{-1}({\beta }_1)\cdot {\tanh }^{-1}({\beta }_2))$ does not seem to appear in the literature. The map $f_{\mathbb{X}}(\beta )={\tanh }^{-1}(\beta )$ 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 $f_{\mathbb{X}}(t/t_H)\approx 0.8\sinh (t-t_1)/(0.8t_H)$ the resulting non-Newtonian Friedman equation with ${\Omega }_{\Lambda }=0$ is exactly quivalent to the standard Newtonian one with ${\Omega }_{\Lambda }=0.7$, ${\Omega }_M=0.3$. 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.

arXiv commentin v2 an arithmetic that exactly reconstructs the observable acceleration of the Universe is derived

References in corpus (14)

• Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant
• Measurements of Omega and Lambda from 42 High-Redshift Supernovae
• Relativistic gravitation theory for the MOND paradigm
• Fundamental parameter-free solutions in Modified Gravity
• MOND vs. dark matter in light of historical parallels
• Modified Gravity (MOG) fits to observed radial acceleration of SPARC galaxies
• Waves along fractal coastlines: From fractal arithmetic to wave equations
• Relativity of arithmetic as a fundamental symmetry of physics
• Towards a Coherent Theory of Physics and Mathematics: The Theory-Experiment Connection
• Fourier transforms on Cantor sets: A study in non-Diophantine arithmetic and calculus
• Crystallization of space: Space-time fractals from fractal arithmetic
• Simple fractal calculus from fractal arithmetic
• Dark energy as a manifestation of nontrivial arithmetic
• The Theory Experiment Connection: Rn Space and Inflationary Cosmology

Cited by in corpus (4)

• Unifying Aspects of Generalized Calculus
• A loophole of all `loophole-free' Bell-type theorems
• Imitating quantum probabilities: Beyond Bell's theorem and Tsirelson bounds
• On the Relativity of Quantumness as Implied by Relativity of Arithmetic and Probability