If Gravity is Geometry, is Dark Energy just Arithmetic? - Publication - Bridge of Knowledge


If Gravity is Geometry, is Dark Energy just Arithmetic?


Arithmetic operations (addition, subtraction, multiplication, division), as well as the calculus they imply, are non-unique. The examples of four-dimensional spaces, 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.


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artykuł w czasopiśmie wyróżnionym w JCR
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INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS no. 56, edition 4, pages 1364 - 1381,
ISSN: 0020-7748
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Bibliographic description:
Czachor M.: If Gravity is Geometry, is Dark Energy just Arithmetic?// INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS. -Vol. 56, iss. 4 (2017), s.1364-1381
Digital Object Identifier (open in new tab) 10.1007/s10773-017-3278-x
Bibliography: test
  1. Czachor, M.: Relativity of arithmetic as a fundamental symmetry of physics. Quantum Stud.: Math. Found. 3, 123-133 (2016). arXiv:1412.8583 [math-ph] open in new tab
  2. Aerts, D., Czachor, M., Kuna, M.: Crystallization of space: Space-time fractals from fractal arithmetic. Chaos, Solitons and Fractals 83, 201-211 (2016). arXiv:1506.00487 [gr-qc] open in new tab
  3. Aerts, D., Czachor, M., Kuna, M.: Fourier transforms on Cantor sets: A study in non-Diophantine arithmetic and calculus. Chaos, Solitons and Fractals 91, 461-468 (2016). arXiv:1603.05471 [math-ph] open in new tab
  4. Aerts, D., Czachor, M., Kuna, M.: Fractal arithmetic and calculus on Sierpiński sets. arXiv:1606.01337 [math.GN] open in new tab
  5. Baird, J.C., Noma, E.: Fundamentals of Scaling and Psychophysics. Wiley, New York (1978)
  6. Norwich, K.H.: Information, Sensation, and Perception. Academic Press, San Diego (1993) open in new tab
  7. Czachor, M.: Information processing and Fechner's problem as a choice of arithmetic. In: Burgin, M., Hofkirchner, W. (eds.) Information Studies and the Quest for Transdisciplinarity: Unity in Diversity. World Scientific, Singapore (2016). arXiv:1602.00587 [q-bio.NC] open in new tab
  8. Fechner, G.T.: Elemente der Psychophysik. Breitkopf und Hartel, Leipzig (1860)
  9. Burgin, M.: Non-Diophantine Arithmetics, Ukrainian Academy of Information Sciences Kiev. (in Russian) (1997) open in new tab
  10. Burgin, M.: Introduction to projective arithmetics. arXiv:1010.3287 [math.GM] (2010) open in new tab
  11. Benioff, P.: New gauge field from extension of space time parallel transport of vector spaces to the underlying number systems. Int. J. Theor. Phys. 50, 1887 (2011) open in new tab
  12. Benioff, P.: Fiber bundle description of number scaling in gauge theory and geometry. Quantum Stud. Math. Found. 2, 289 (2015). arXiv:1412.1493 open in new tab
  13. Benioff, P.: Space and time dependent scaling of numbers in mathematical structures: Effects on physical and geometric quantities. Quantum Inf. Proc. 15, 1081 (2016). arXiv:1508.01732 open in new tab
  14. Jackson, J.D.: Classical Electrodynamics. Wiley, New York (1962)
  15. Hartle, J.B.: Gravity. An Introduction to Einstein's General Relativity. San Francisco, Benjamin Cummings (2003)
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