MOND vs. dark matter in light of historical parallels

Milgrom, Mordehai

doi:10.1016/j.shpsb.2020.02.004

“…Based on the idea of ramifications from initial conditions given approximately (RICA), this approach was developed in a series of earlier papers, offering also a new generalized framework for control problems [105][106][107][108][109]. Another attempt along this direction has been labeled as MOND, the Modified Newtonian Dynamics approach [110][111][112], which is also time-nonlocal.…”

Section: Nonlocality In Timementioning

confidence: 99%

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Sytnyk

¹

,

Melnik

²

2021

MCA

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Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture effects and phenomena that traditional models cannot. While spatial nonlocalities have received considerable attention in the research community, the same cannot be said about nonlocality in time, in particular when nonlocal initial conditions are present. This paper aims at filling this gap, providing an overview of the current status of nonlocal models and focusing on the mathematical treatment of such models when nonlocal initial conditions are at the heart of the problem. Specifically, our representative example is given for a nonlocal-in-time problem for the abstract Schrödinger equation. By exploiting the linear nature of nonlocal conditions, we derive an exact representation of the solution operator under assumptions that the spectrum of Hamiltonian is contained in the horizontal strip of the complex plane. The derived representation permits us to establish the necessary and sufficient conditions for the problem’s well-posedness and the existence of its solution under different regularities. Furthermore, we present new sufficient conditions for the existence of the solution that extend the existing results in this field to the case when some nonlocal parameters are unbounded. Two further examples demonstrate the developed methodology and highlight the importance of its computer algebra component in the reduction procedures and parameter estimations for nonlocal models. Finally, a connection of the considered models and developed analysis is discussed in the context of other reduction techniques, concentrating on the most promising from the viewpoint of data-driven modelling environments, and providing directions for further generalizations.

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“…The term 'non-Newtonian' refers here to the level of calculus, and not to the laws of physics. It should not be confused with Milgrom's 'modified Newtonian dynamics' (MOND) (Milgrom 1983(Milgrom , 2019Bekenstein 2004), Moffat's 'modified gravity' (MOG) (Moffat and Toth 2009;Green and Moffat 2019), or similar theories. A non-Newtonian calculus is based on a non-Diophantine arithmetic.…”

Section: Dark Universe and Its Dark Arithmeticmentioning

confidence: 99%

Non-Newtonian Mathematics Instead of Non-Newtonian Physics: Dark Matter and Dark Energy from a Mismatch of Arithmetics

Czachor¹

2020

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Newtonian physics is based on Newtonian calculus applied to Newtonian dynamics. New paradigms such as ‘modified Newtonian dynamics’ (MOND) change the dynamics, but do not alter the calculus. However, calculus is dependent on arithmetic, that is the ways we add and multiply numbers. For example, in special relativity we add and subtract velocities by means of addition $$\beta _1\oplus \beta _2=\tanh \big (\tanh ^{-1}(\beta _1)+\tanh ^{-1}(\beta _2)\big )$$ β 1 ⊕ β 2 = tanh ( tanh - 1 ( β 1 ) + tanh - 1 ( β 2 ) ) , although multiplication $$\beta _1\odot \beta _2=\tanh \big (\tanh ^{-1}(\beta _1)\cdot \tanh ^{-1}(\beta _2)\big )$$ β 1 ⊙ β 2 = tanh ( tanh - 1 ( β 1 ) · tanh - 1 ( β 2 ) ) , and division $$\beta _1\oslash \beta _2=\tanh \big (\tanh ^{-1}(\beta _1)/\tanh ^{-1}(\beta _2)\big )$$ β 1 ⊘ β 2 = tanh ( tanh - 1 ( β 1 ) / tanh - 1 ( β 2 ) ) do not seem to appear in the literature. The map $$f_{\mathbb{X}}(\beta )=\tanh ^{-1}(\beta )$$ f X ( β ) = tanh - 1 ( β ) defines an isomorphism of the arithmetic in $${\mathbb{X}}=(-1,1)$$ X = ( - 1 , 1 ) with the standard one in $${\mathbb{R}}$$ R . The new arithmetic is projective and non-Diophantine in the sense of Burgin (Uspekhi Matematicheskich Nauk 32:209–210 (in Russian), 1977), while ultrarelativistic velocities are super-large in the sense of Kolmogorov (Technika Molodezhi 10:16–19 (11:30–33 in Russian), 1961). 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 (Non-Newtonian calculus, Lee Press, Pigeon Cove, 1972). Treating the above example as a paradigm, we ask what can be said about the set $${\mathbb{X}}$$ X of ‘real numbers’, and the isomorphism $$f_{{\mathbb{X}}}:{\mathbb{X}}\rightarrow {\mathbb{R}}$$ f X : X → 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 the observable accelerated expansion of the Universe can be reconstructed with zero cosmological constant if $$f_{\mathbb{X}}(t/t_H)\approx 0.8\sinh (t-t_1)/(0.8\, t_H)$$ f X ( t / t H ) ≈ 0.8 sinh ( t - t 1 ) / ( 0.8 t H ) . The resulting non-Newtonian model is exactly equivalent to the standard Newtonian one with $$\Omega _\Lambda =0.7$$ Ω Λ = 0.7 , $$\Omega _M=0.3$$ Ω M = 0.3 . Asymptotically flat rotation curves are obtained if ‘zero’, the neutral element $$0_{{\mathbb{X}}}$$ 0 X of addition, is nonzero from the point of view of the standard arithmetic of $${\mathbb{R}}$$ R . This implies $$f^{-1}_{{\mathbb{X}}}(0)=0_{{\mathbb{X}}}>0$$ f X - 1 ( 0 ) = 0 X > 0 . The opposition Diophantine versus non-Diophantine, or Newtonian versus non-Newtonian, is an arithmetic analogue of Euclidean versus non-Euclidean in geometry. 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. Physics of the dark universe seems to be both geometry and arithmetic.

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“…This is why the physics of gravitational interaction lies in the heart of cosmology and, in fact, it determines the structure and all main predictions for observations. A comparison of initial principles and main predictions for some alternative cosmological models, based on alternative gravitation theories, is presented in Milgrom 2020 [240], Debono & Smoot 2016 [5], Baryshev & Teerikorpi 2012 [8], Clifton et al, 2012 [7], and Pavsic 1975 [241].…”

Section: Gravitation Theory As the Basis Of Cosmological Modelsmentioning

confidence: 99%

Einstein’s Geometrical versus Feynman’s Quantum-Field Approaches to Gravity Physics: Testing by Modern Multimessenger Astronomy

Baryshev¹

2020

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Modern multimessenger astronomy delivers unique opportunity for performing crucial observations that allow for testing the physics of the gravitational interaction. These tests include detection of gravitational waves by advanced LIGO-Virgo antennas, Event Horizon Telescope observations of central relativistic compact objects (RCO) in active galactic nuclei (AGN), X-ray spectroscopic observations of Fe Kα line in AGN, Galactic X-ray sources measurement of masses and radiuses of neutron stars, quark stars, and other RCO. A very important task of observational cosmology is to perform large surveys of galactic distances independent on cosmological redshifts for testing the nature of the Hubble law and peculiar velocities. Forthcoming multimessenger astronomy, while using such facilities as advanced LIGO-Virgo, Event Horizon Telescope (EHT), ALMA, WALLABY, JWST, EUCLID, and THESEUS, can elucidate the relation between Einstein’s geometrical and Feynman’s quantum-field approaches to gravity physics and deliver a new possibilities for unification of gravitation with other fundamental quantum physical interactions.

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MOND vs. dark matter in light of historical parallels

Cited by 60 publications

References 149 publications

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Non-Newtonian Mathematics Instead of Non-Newtonian Physics: Dark Matter and Dark Energy from a Mismatch of Arithmetics

Einstein’s Geometrical versus Feynman’s Quantum-Field Approaches to Gravity Physics: Testing by Modern Multimessenger Astronomy

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